DONE: Objectives of ORCHIDEE

[1]PeylinPhilippe

[1]AbadieCamille

[1]AllonJulien

[2]ArdanehKazem

[1]BarichivichJonathan

[1]BastrikovVladislav

[2]CadulePatricia

[14]CalixLouis-Gabriel

[16]ChasteEmeline

[3]CheruyFrederique

[1]CuynetAmélie

[4]DucharneAgnes

[5]DumasChristophe

[6]GaillardRémi

[2]GhattasJosefine

[1]GollDaniel

[6]GuenetBertrand

[8]GuimberteauMatthieu

[9]JeongJina

[7]KiałkaFilip

[10]KirchnerAnna

[9]LanetMarine

[10]LansøAnne Sofie

[1]LathireJuliette

[14]LippmannTanya J.R.

[11,12]MacBeanNatasha

[13]McgrathMatthew

[1]MaignanFabienne

[1]MaugisPascal

[1]MarieGuillaume

[14]NaudtsKim

[1]OttlCatherine

[15]PolcherJan

[1]SalmonElodie

[3,4]TiengouPierre

[15]ValadeAude

[1]VuichardNicolas

[1]ViovyNicolas

[1]WangXiaoni

[17]YueChao

[9]LuyssaertSebastiaan

Description of the land surface model ORCHIDEE Peylin et al. Philippe Peylin (peylin@lsce.ipsl.fr)

2CO$_2$ 2oH$_2$O 2oN$_2$O NO$_{y}$ NH$_{x}$ NH$^+_4$ NO$^-_3$

W m$^{-2}$ gC m$^{-2}$ gC m$^{-2}$ s$^{-1}$ gN m$^{-2}$ gN m$^{-2}$ s$^{-1}$ m s$^{-1}$ mm s$^{-1}$ g g$^{-1}$ GtC yr$^{-1}$ 2m$^2$

rxxxx rxxxx r8520 v2.0 v3.0 v4.2

CMIP-5 CMIP-6 CMIP-7

TEXT

NOTES FOR AUTHORS:

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The model description is a collage of new text with substantial contributions from model descriptions from previous and ongoing manuscripts. This will be mentioned in the cover letter.

WRITING GUIDELINES FOR AUTHORS:

The LaTeX document of this manuscript can be found at https://www.overleaf.com/project/668a343bbd918ed0abd9362f
VOCABULARY: use GRID CELL instead of pixel
VOCABULARY: biogeographical submodel instead of DGVM. The term DGVM has been used for more or less everything. I witnessed people claiming their LSM is a DGVM because the LAI is dynamic…
use vvvv for the version number. These symbols are linked to the correct numbers such that they can be easily changed. Note the title needs to be manually adjusted because the symbols are defined after the title.
use \left( and \right) to better format brackets
use \frac{}{} to better show fractions
use {}/{} when using a fraction in a sentence
use \exp instead of e^{} to enhance the readability of the exponential
use \cdot instead of *
use \, to separate different elements in the units (don’t use the French .). Thus m\,s$^{-2}$ instead of m.s$^{-2}$

3. DONE: Objectives of ORCHIDEE#

As a common tool shared by a large interdisciplinary scientific community, the ORCHIDEE land surface model is aiming to integrate a comprehensive and process-based description of land biophysical, biogeochemical and demographic processes. The purpose of ORCHIDEE is to predict how the land surface responds to changing environmental conditions including unprecedented climate and atmospheric conditions. ORCHIDEE has been developed to respond to changes in: (1) atmospheric 2, (2) weather, i.e., air temperature, radiation, precipitation, air specific humidity, wind, and surface pressure, (3) nitrogen inputs (available from ORCHIDEE onwards), and (4) land cover and land use ?Others environmental changes that should be listed? ADD: (5) land management (i.e., irrigation, fertilisation,…). The simulated response variables can be grouped as: (1) energy (among others, surface temperature and albedo) (2) water (among others, evapotranspiration and soil water content) (3) carbon (among others, net biome production and soil carbon), (4) nitrogen (among others, leaching and n2o emissions), and (5) yield (among others, river discharge and wood, grass and crop production) responses.

Ongoing developments aim to extent the applicability of the ORCHIDEE model towards assessing: (1) the impact of the so-called Nature-Based Solutions [] on the energy budget, the water cycle, and the carbon and nitrogen cycles at the land surface as well as on the Earth’s climate system, and (2) the resilience, i.e., the capacity to return to the reference state or dynamic after a temporary disturbance ES: is this include extreme weather events? or should it be added after temporary extreme weather events [], of terrestrial ecosystems to ongoing and future climate change, and (3) the impact of land management on the energy budget, the water cycle, and the carbon and nitrogen cycles at the land surface [] as well as on the Earth’s climate system [].

4. DONE: Structural assumptions#

Every process accounted for in ORCHIDEE comes with at least one, if not several, assumptions. As such, there are 100’s of assumptions underlying the ORCHIDEE model. The complexity of this kind of model makes it no longer feasible to state all assumptions []. The focus is therefore on listing the six structural assumptions of the ORCHIDEE model, in other words, the assumptions that determine the applicability of ORCHIDEE. Unless mentioned otherwise, these assumptions are common to all ORCHIDEE versions.

Assumption on vegetation demography. Although ecosystem attributes such as demography, growth rates, mortality, extinction rates, and community structure are all emergent properties of individual-based systems [], the ORCHIDEE model, assumes that interactions between individuals can be represented through statistical-averaging. For the purpose of the ORCHIDEE land surface model, simulating ecosystem dynamics in forests, grasslands and croplands does not require the inclusion of every individual within the community. From an ecological point of view, the simulation unit of ORCHIDEE is thus the community level which sets ORCHIDEE apart from individual-based models, as is the case for most land surface models except for SEIB-DGVM [] which is an individual-based land surface model. Following a broadening of the purpose of ORCHIDEE to study the resilience of ecosystems, a hybrid approach [] was needed in which the community is represented by a few individual model trees. Such an approach was introduced in ORCHIDEE [] and developed further in ORCHIDEE []. The introduction of this hybrid approach did not change the classification of the ORCHIDEE model as a community-level model, partly due to the following two assumptions.
Assumption on vegetation diversity. Although different tree, grass and crop species may respond differently to similar environmental conditions, it is assumed that for the purpose of ORCHIDEE, the daunting diversity, i.e., already more than 60,000 tree species [], can be represented by making use of a limited number of plant functional types [], which in turns relies on the assumption that all species within a single functional type show sufficiently similar land–atmosphere interactions irrespective of their geographical location []. In ORCHIDEE, as in many other land surface models [], the assumption on vegetation diversity hinders the model to address the response of the land surface to changes in species diversity. The assumption on vegetation diversity strengthens the classification of the ORCHIDEE land surface model as a community-level model (see Assumption on vegetation demography).
Assumption on interactions. Different vegetal communities compete for light, water and nutrients, affect each other’s demography, and affect the atmosphere in different ways resulting in micro-climates. Minus few exceptions, these interactions between communities are not accounted for as it is assumed that for the purpose of the ORCHIDEE model, landscape level interactions within and between grid cells can be ignored. Landscape level interactions within a grid cell, e.g., roughness as a function of the lay-out of the distribution of different vegetation types, are simulated assuming that statistical-averaging of the communities’ properties can be used. Landscape level interactions between grid cells, e.g., river routing, are explicitly simulated. The competition for soil water within an ORCHIDEE grid cell is the result of the discretisation of the model (section ??) rather than a deliberate scientific representation of a land surface process. The assumption on landscape interactions sets ORCHIDEE, as most other land surface models, apart from the landscape models and strengthens the classification of the ORCHIDEE land surface model as a community-level model (see Assumption on vegetation demography and diversity).
Assumption on vegetation evolution. Although natural selection operates in every generation and can often not be ignored when studying ecological phenomena [] at least for species with generation times substantially less than the time frame of the model application [], it is assumed that for the purpose of ORCHIDEE, micro-evolution and adaptation can be ignored. This is reflected by mostly using spatially and temporally constant parameters within a plant functional type. The diversity and evolutionary assumptions have been challenged and trait-based solutions [], informed by manipulation experiments [], global datasets [], and optimality theory [] begin to provide the land surface community with the insights required to overcome several of the limitations resulting from the assumptions on vegetation diversity and evolution. Stomatal closure under soil moisture stress, for example, could be refined by incorporating vegetation acclimation to long‐term vapour pressure deficit conditions [] and the maximum tree height was estimated from long-term precipitation []
Assumption on the canopy structure. Although vegetation canopies are three-dimensional heterogeneous media, the ORCHIDEE model, as well as many other land surface models [] assume that a bulk canopy approach adequately represents canopy processes, such as transpiration, and the exchange of sensible heat. The bulk canopy approach assumes that the canopy is an infinitesimal thin layer between the soil and the atmosphere. From a physical point of view, the simulation unit of ORCHIDEE is thus the bulk canopy. Broadening the purpose of ORCHIDEE to study the resilience of ecosystems required representing three-dimensional heterogeneous canopies in ORCHIDEE []. This new representation is used in the calculation of the radiative transfer in ORCHIDEE and efforts to use it in the calculations of the energy budget [] are ongoing.
Assumption on spatial heterogeneity. With a range of a few to hundreds of metres, the heterogeneity of the land surface is 10$^{3}$ to 10$^{5}$ higher than the current grid resolution of typical land surface models. Hence, land surface models, including ORCHIDEE, rely on simplified, statistical sub-grid tiling schemes that treat complex, interconnected landscapes as disconnected sets of patches. In ORCHIDEE only one energy budget is calculated for the entire grid-cell. For the water budget, three soil columns are considered, respectively, for short vegetation, tall vegetation, and bare soil, which prevent soil water competition between these three groups of plant functional types. For carbon and nitrogen, the budgets are calculated for each plant functional type separately.

5. DONE: Symbolic notation#

The symbolic notation in this document is structured as $X^{descriptor}_{discretisation}$. Central to this symbolic notation is the symbol of the variable group ($X$ in the example above) which helps the reader to understand whether the symbol represents a flux, a mass, or a dimension, to mention a few of the common variable groups used in this document (Table 1). When appropriate, the symbol is complemented with a subscript that denotes the spatial and temporal discretisation. Where multiple subscripts are needed, they are separated by a comma to maintain readability. As c is a constant and there is no spatial and temporal discretisation, its name is in the subscript.

:::{list-table} Symbols of the main variable groups used in the symbolic notation of this document :header-rows: 0 :name: tab:variable group

- Symbol
- Variable group
- $b$
- Temporary variable
- $c$
- Prescribed parameter
- $d$
- Vegetation dimension such as diameter, height, and basal area
- $f$
- Fraction
- $F$
- Water, energy, carbon, or nitrogen flux
- $g$
- A function
- $i$
- Indicator of process
- $k$
- Calculated parameter
- $m$
- Modulator of a pool or flux
- $M$
- Water, carbon, or nitrogen pool
- $p$
- Pressure
- $P$
- Liquid and solid precipitation
- $q$
- Atmospheric humidity
- $R$
- Resistance in an electric circuit analogy
- $T$
- Soil, water, biomass, and atmospheric temperature
- $z$
- Height above ground level or depth below ground level
- $\alpha$
- albedo
- $\psi$
- Water potential :::

The majority of the calculations is performed within a single discrete time step $t$. When this is the case, the time dimension is omitted from the symbolic notation to enhance readability. Where variables from another time step than the current one are used (e.g., $t-1$ or $t+1$), this is indicated in the subscript of the symbolic notation.

Several processes have been discretised for different vertical atmospheric, canopy, and soil layers. Different layers are represented by the subscripts $i$, $k$, and $l$. The subscript $l$ strictly refers to the different circumference classes of forests. Note that subscripts $i$ and $k$ may represent different discretisation schemes between different sections in the document. When different spatial discretisations apply at the same time, the subscripts are separated by a comma.

The superscript contains an explanatory descriptor of the variable. Most often this descriptor is explicit and specifies the process underlying the flux, i.e., $F^{gpp}$, or the litter pool for which the biomass is given, i.e., $M^{structural}$. When the same calculations are applied to different biomass components such as plant organs or litter qualities, a general descriptor, $o$, is used in the symbolic notation, i.e., $M^{o}$. When several keywords are used for the explanatory descriptor, they are separated by a comma.

The numbering of the parameters is relative to each section: the parameters $c_1$, $c_2$, … from different sections have different meaning and values. Where the same parameters are used across sections, this is explicitly mentioned in the text.

The majority of the calculations in ORCHIDEE are performed for each PFT present in a grid cell. Calculations thus have to be repeated for the different PFTs within a grid cell and for the different grid cells within the spatial domain of the simulation. Where the calculations follow this nesting approach, the PFT and grid cell dimensions are omitted from the symbolic notation to enhance readability.

6. DONE: Spatial and temporal discretisation#

The real-world space and time continua are discretized in the ORCHIDEE model. Discretisation serves one or several of the following purposes: it allows for a rational use of the computational resources, it helps to better account for heterogeneity, and it enhances the simulation of highly non-linear processes. The spatial and temporal discretisation of the model has far-reaching consequences for its scope, its numerical approaches, and its computational costs.

Typical resolutions for global scale simulations range between 0.1 ° x 0.1 ° and 2.0 ° x 2.0 ° which is a trade-off between data availability of the boundary conditions (section ??) and the exponential increase in computation cost for higher resolution grids. At such resolutions each individual grid cell contains in reality different: (1) vegetation types, (2) soil types, (3) hydrological catchments, (4) elevations, (5) hillslope aspects, and (6) groundwater depths. This heterogeneity is addressed through the discretisation of the model. When an increase in model resolution is supported by high resolution boundary conditions (section ??), it is likely to better account for spatial and temporal heterogeneity. However, at higher resolution the validity of several of the structural model assumptions (section ??) is at risk.

6.1. DONE: Land surface#

The domain of an ORCHIDEE simulation is represented by an equidistant or unequal grid of cells where the number of cells will depend on the resolution of the grid. The domain of an ORCHIDEE simulation can range from a single grid cell representing a few 100 m² of vegetation to thousands of grid cells describing a specific region or the global landmass. The grid and its resolution are implicitly set through the choice of the climatic forcing. If ORCHIDEE is coupled to an atmospheric model, it uses the same grid as the atmospheric model. If ORCHIDEE is run as a land-only model, its grid is identical to the grid of the meteorological forcing. A land-sea mask is applied to extract all terrestrial grid cells within the domain. The land-sea mask gives the continental fraction ($f_{cont}$; unitless) of each grid cell and only grid cells with $f_{cont}$ > 0 will be considered by ORCHIDEE.

6.2. DONE: Vegetation classes#

ORCHIDEE accounts for the heterogeneity of the vegetation in each grid cell by combining fractions of plant functional types (PFTs) as proposed by . Land cover products that are prepared to be used in ORCHIDEE prescribe the fraction of each PFT within that grid cell ($f^{veg,max}$; unitless). The distribution of PFTs comes from historical land cover reconstructions, contemporary remote sensing products, future land cover maps, or a combination of these sources (see ??). The product used to prescribe the PFT distribution should have the same number of PFTs as the model configuration, unless age classes are used (section ??).

At the highest hierarchical level, vegetation is classified in 13 meta-classes (MTCs). Each MTC can be split in a user-defined number of PFTs. Different PFTs that offspring from the same MTC will differ by at least one parameter value. For all other parameters, the PFT inherits the values of the MTC it originates from. This hierarchical approach has been exploited to introduce new PFTs in several applications, including regional ones [], to account for age classes in forest MTCs [] or to assess gross land cover changes []. ORCHIDEE currently uses its 13 MTCs to create 15 PFTs (including one PFT for bare soil, eight PFTs for various combinations of leaf-type and climate zones of forests, four PFTs for various climate zones and photosynthetic pathways for grasslands, and two PFTs for different photosynthetic pathways for croplands). Although PFTs differ in their parameter values, they mostly share the same equations with exceptions for the calculation of photosynthesis for C3 and C4 plants, leaf phenology, and leaf senescence.

The sum of all PFT fractions where individual fractions could be zero, should be less or equal to one:

\[f^{veg,max} = \sum_{i=1}^{npft} f^{veg,max}_{i} \le 1\]

If the sum of fraction is less than 1, the remaining fraction of the grid cell is considered as a non-biological fraction ($f_n$) and is treated as glacier.

\[f^{glacier}=1-\sum_{i=1}^{npft} f^{veg,max}_{i}\]

Following finalisation of the integration of lakes, future version of ORCHIDEE, will read a surface functional type map (SFT) instead of the current PFT map. One of the SFTs will then be lakes. For the time being, the lake fraction in each gridcell is read through a separate map (See ??). When ORCHIDEE pre-processes its PFT maps, lakes are placed on bare soil. If the the bare soil fraction cannot satisfy the areal demand of lakes in that grid cell, lakes are placed on grassland PFTs. The addition of lakes thus reduces the share of bare soil and grasslands in the grid cells where lakes are present.

6.3. DONE: Age classes#

At the beginning of a simulation, the number of age classes, the number of circumference classes, and the MTCs for which more than one age class will be used have to be set by the user. Although not all MTCs need to be run with several age classes, all MTCs that are run with age classes have the same number of age classes. The number of circumference classes is the same for all forest PFTs and is fixed to one for grassland and cropland PFTs.

If age classes are used in the simulation, all PFTs representing different age classes of the same vegetation should be matched by a single PFT on the land cover map (See ??). This implies that the age class distribution is an emerging property of the simulation.

For each forest MTC, several age class can be defined. Within an MTC, each age class is simulated as a separate PFT. Contrary to what its name suggests, the age class boundaries are determined by the tree diameter rather than the age of the trees. Different age classes are distinguished to better account for forest succession and its effects on the carbon, nitrogen, energy and water fluxes. For example, when several age classes are used, forest regrowth following land-cover change, forest management, or a natural disturbance will end up in a separate age class. However, when a single class is used, the regrowth is mixed with the mature vegetation typically diluting the effect of regrowth on forest structure and thus albedo, roughness length, and evapotranspiration to mention a few.

6.4. DONE: Diameter classes#

Each forest PFT in ORCHIDEE contains a mono-specific forest stand that is structured by a user-defined but fixed number of circumference classes ($ncirc$, three by default). Throughout the simulation, the boundaries of the circumference classes are adjusted to accommodate changes in the stand structure, while the number of classes remains constant. Flexible class boundaries provide a computationally efficient approach to simulate different forest structures. For instance, an even-aged forest is simulated by using a small diameter range between the smallest and largest trees, resulting in all trees belonging to the same stratum. In contrast, an uneven-aged forest is simulated by applying a wide range between diameter classes such that different classes represent different canopy strata. Circumference classes are taken into account in ORCHIDEE to better simulate canopy structure. Since the canopy acts as the interface between the land and the atmosphere, this feature has implications that extend beyond forest management. The structure of the stand has been shown to influence albedo, transpiration, photosynthesis, soil temperature, roughness length, and recruitment [].

PFTs are simulated independently from each other with the sole exception that all forest PFTs share the same water column. Consequently, the water consumption of one forest PFT affects the water availability of all other forest PFTs. Because age classes are simulated as PFTs in ORCHIDEE this implies that also the age classes of the same MTC are competing for soil water (See ??).

6.5. DONE: Vertical canopy layers#

The canopy space of each PFT is discretised in a user-defined number ($nlev$, ten by default) of equidistant vertical canopy layers that start at the top of the crown of the tallest individual in the PFT and extend to the bottom of the crown of the smallest individual in the PFT or 0.001 m in case of grassland and cropland PFTs. The calculation of the crown and canopy dimensions as well as the leaf area contained in each vertical canopy layer is detailed in section ??.

6.6. DONE: Soil water columns#

Each grid cell distinguishes three independent soil water columns also referred to as soil tiles. A soil water column is linked to the vegetation discretization such that each of the 13 vegetation meta-classes (MTC) is associated to a single soil water column, and each of the three columns gather distinct types of vegetation; one soil water column is reserved for the bare soil MTC, one for tree-based MTCs, and one for grass and crop MTCs. The soil water content and all processes controlling its dynamics (i.e., infiltration, root uptake, diffusion, etc.) are calculated independently for each soil water columns, which do not communicate horizontally with each other. This enables limiting competition between bare soils, tree-based MTCs and graas and crop based MTCs for water resources in the soil [], as well as allowing for differences in root profiles between these soil water colums []. Note that the different MTCs within a single soil water column, compete for water among each other.

6.7. DONE: Soil texture classes#

:::{figure} Figures/USDA_Triangle_with_CP_percent.pdf :name: fig:textural_triangle

Correspondence between the soil granulometric composition and the USDA textural classes. “Z” denotes the three classes of the simplified classification (coarse, medium, and fine, corresponding to sandy loam, loam, and clay loam, respectively). :::

Two soil texture classifications are available in ORCHIDEE: (1) the default scheme consisting of US Department of Agriculture (USDA) textural classes. For this, the map of is adapted for use in ORCHIDEE by splitting the clay texture class into regular (swelling) clay and clay oxisols according to the FAO Soil Order Map []. After this split the map distinguishes 12 textural classes plus oxisols: sand, silt, clay loam, loamy sand, loam, sandy clay, sandy loam, sandy clay loam, silty clay, silt loam, silty clay loam, clay, and oxisols. (2) A simplified scheme based on consists of only three textural classes, i.e., coarse, medium, and fine, corresponding to USDA sandy loam, loam, and clay loam, respectively . In both cases, the textural class is determined by the relative proportions of three granulometric fractions: sand (particle diameter between 0.05 mm and 2 mm), silt (diameter between 0.002 mm and 0.05 mm), and clay (diameter below 0.002 mm). The correspondence between the granulometric composition and the USDA textural classes is shown in Fig. 1.

6.8. DONE: Vertical soil layers#

:::{figure} Figures/soil_layers-general.pdf :name: fig:soil_layers :align: center :::

ORCHIDEE solves the equations of heat and water transport in the soil in one dimension using a finite difference method. For this, it uses a staggered grid consisting of nodes — at which water content and temperature are calculated — and of layer boundaries — at which heat and water fluxes are calculated.

The general layering scheme for hydrology is shown in Fig. 2a. It extends down to $z^\text{maxh}$ and consists of layers whose thickness increases geometrically with a ratio of 2, starting from a prescribed thickness $c^\text{topthick}$ of the first layer. Optionally, the geometric increase in thickness can be stopped below a depth of $z^\text{cstthick}$, producing layers with constant thickness. These can optionally be followed by layers with decreasing thickness (the decrease is geometric with a ratio of 1/2). The optional layers with constant thickness and the refinement at the bottom are intended for configurations with an impermeable bottom, in which a water table and thus a strong moisture gradient can form anywhere in the soil column. The hydrological layers of the default ORCHIDEE configuration are shown in Fig. 2b. The default configuration uses a free-drainage boundary condition and thus only layers with a geometrically increasing thickness. The thickness of the first layer $c^\text{topthick} \approx 1\:\unit{mm}$, and $z^\text{maxh} = z^\text{cstthick} = 2\:\unit{m}$, resulting in 11 layers.

Since ORCHIDEE , the thermodynamic soil layering is identical to the hydrological one in the region where the two overlap, but extends further to a depth of $z^\text{maxt} > z^\text{maxh}$ []. As shown in Fig. 2a, this extension consists of layers that increase in thickness mirroring the refinement at the bottom of the hydrological soil column (if the refinement is used), followed optionally by layers with constant depth down to $z^\text{geom}$, below which the geometric increase in layer thickness resumes. In the default configuration, shown in Fig. 2b, $z^\text{geom} = z^\text{maxh}$ and the thermodynamic layers continue the geometric progression of the thickness of hydrological layers, reaching a depth of $z^\text{maxt} = 90\:\unit{m}$ with a total of 18 layers. ES: should a sentence here be added to explain the soil C and N discretization or bucket? or it has been decided that it should be explained elsewhere?

6.9. DONE: River basins#

For the lateral transport of water over the continents and down to the oceans, the resolution of the climatic forcing is typically too coarse to allow a good representation of the topography, which is the main driving factor for the water surface flows. ORCHIDEE features two different approaches for the spatial discretisation of the river basins.

The first approach, which is often used when ORCHIDEE is used in an ESM-configuration, is to disentangle the grid of the climate forcing or climate simulations, on which the one-dimensional vertical water fluxes are computed, and the grid on which the lateral river routing is done. In the latter grid, the river basin discretisation follows the cells of the input hydrological digital elevation model (HDEM; See ??) or an aggregation of these cells. The water fluxes and pools are exchanged between the two grids through interpolations, ensuring mass conservation as proposed by . This approach is referred to as the “interpolated” routing method in ORCHIDEE. Not clear how many river basins we have, which is the actual discretisation. “In a typical global configuration this approach distinguishes XX rivers.”

The second approach is a hybrid approach that uses a sub-grid tiling of the climatic forcing or the climate simulations to reach the resolution required for the hydrological lateral transfers []. Within the supermesh of the HDEM grid cells overlapping with the coarser ORCHIDEE grid, a multistep algorithm is used to build, sub-divide, and merge elements that eventually constitute a sub-basin dimension [].

In fact, each ORCHIDEE grid cell is thus discretised into at most $n_{bas}^{\max}$ hydrologically consistent and connected tiles, yielding graphs of hydrological transfer units (HTUs) along which the water flows within and from grid cell to grid cell. This is referred to as the “subgrid” routing methods in ORCHIDEE. Not clear how many river basins we have, which is the actual discretisation. “In a typical global configuration this approach distinguishes XX rivers.”

describe here or in the spatial discretisation that there is a slow flow from the groundwater and a fast flow from the runoff that all enters the rivers.

6.10. DONE: Lakes#

Given the standard spatial resolution of several tens of kilometres in ORCHIDEE, the model distinguishes different lakes classes. The number of lake classes us a user-defined setting but by default ORCHIDEE distinguishes three classes according to the depth if the lakes. This choice was driven by the results of a prior sensitivity analysis with the FLake model REF showing that lake depth was the main parameter that affected the surface energy budget and fluxes, followed by some radiative properties such as the surface albedo or the extinction coefficient []. The three classes are representative of shallow (< 5 m), medium, and deep lakes (> 25 m).

6.11. DONE: Elevation, aspect and groundwater depth#

Although ORCHIDEE reads a digital elevation model, elevation differences within a grid cell are not taken into account except for the pre-calculation of the river basins (See ??). When using a stand-alone configuration, the differences in elevation between grid cells are implicitly accounted for through the climate reconstructions and explicitly through the topography index for the river routing calculations. The coupled land-atmosphere configuration considers differences in elevation between grid cells in its climate calculations and river routing.

The aspect of hill slope and groundwater depth are not taken into account in ORCHIDEE, neither the heterogeneity between grid cells nor the heterogeneity within a grid cell.

6.12. DONE: time steps#

ORCHIDEE combines three different time steps (Table 7): (1) few minutes (30 minutes for stand-alone simulations and less than 20 minutes when coupled to a GCM), (2) daily, and (3) annual. The minutes-scale processes include the soil water budget (for each soil column) and the exchanges of energy (for each grid cell), 2o and 2 through photosynthesis (for each PFT in each grid cell) between the atmosphere and the biosphere. They also account for litter decomposition and soil carbon dynamics. The daily processes include river routing for each watershed but mostly calculate the carbon dynamics of the terrestrial biosphere and essentially represent processes such as growth respiration, carbon allocation and phenology (all calculated for each PFT in each grid cell). The annual processes simulate changes in the land cover, wood harvest, mortality of PFTs and establishment of new PFTs as part of global vegetation dynamics.

SL: table needs to be updated to contain all (sub)titles from the model description

PP: thinks that there is maybe some sub-stepping for the hydrology. Can someone confirm?

7. Boundary conditions#

7.1. DONE: Soil properties#

ORCHIDEE reads soil property maps prescribing soil texture, pH, and bulk density. The soil texture is used primarily to determine the hydraulic and thermal properties of the soil, and the soil pH is used in the soil nitrogen calculations. Bulk density is currently not used, but is required by upcoming developments.

The default soil texture map is based on the 5-arc-min (1/12°) map by , while the simplified three-class scheme uses the 1° map of . Both of these maps in turn are based on the 1:5,000,000 FAO/UNESCO Soil Map of the World. In addition to these, any soil texture map using the USDA classification, such as SoilGrids [], can be used directly by setting a runtime parameter.

The soil texture map is regridded onto the model mesh by assigning to each grid cell the soil texture class that covers the largest fraction of the grid cell. In addition to this, granulometric sand, silt, and clay fractions are calculated for each grid cell as the area-weighted means of the values typical for the textural classes present in the grid cell. In both cases, soil horizons are not distinguished and all tiles present in a grid cell share the same textural class and granulometric composition.

Soil pH and bulk density are both read from 1° maps. The pH map was generated using the SoilData program from the IGBP-DIS CD-ROM []. The bulk density map is derived from the reference topsoil bulk density of the Harmonized World Soil Database version 1.1 [] or 1.2 [] [which?]; The soil PH and BulkDens description should be checked !.

7.2. DONE: Vegetation distribution#

Various historical reconstructions of the vegetation distribution ($f^{veg,max}$) can be used with ORCHIDEE. We describe below the main one that is used for global applications. It combines information from the land use harmonisation database (LUH v3.1.1; Update ref LUH3 ?) [] with land cover information derived from satellite observations, the Medium Resolution (300 m) Land Cover product (MRLC 2.8) from the Climate Change Initiative (CCI) of the European Space Agency (ESA) (referred as CCI-MRLC, ). Note that in the latest CCI-MRLC product that we use (described in ), the original 38 land cover classes [] are directly re-mapped onto a set of 16 generic PFTs at 300 m resolution (see https://orchidas.lsce.ipsl.fr/dev/lccci/generic_pfts.php), covering the period 1992 to 2020. In addition, in this product the grassland PFTs are divided into separate PFTs for grassland with the C3 or C4 photosynthetic pathway. Combining LUH v3.1.1 with satellite-derived generic PFTs consists of several processing steps that are described briefly below and in details in Olivera et al., (in preparation):

The bioclimatic zones (Köppen–Geiger climate classification map) were used to split wide-spread generic PFTs into separate generic PFTs for the tropical, temperate and boreal zones.
The generic PFTs are mapped into the ORCHIDEE specific PFTs. In the default PFTs classification (Table 6) shrubs were classified as 60% tree PFT and 40% grass PFT (80% tree and 20% grass for boreal zone) and lakes were classified as bare soil, unless the Flake model is activated (see section ??).
LUH v3.1.1 contains a land cover reconstruction from 850 to 2024 and several land cover scenarios from present-day to 2100 [] (update ref). These maps had to be merged with the ORCHIDEE PFTs defined in the previous steps. Adjustments were as follow for each year of the CCI-MRLC period (1992 to 2020): a) for each grid cell the crop fraction was taken from LUH v3.1.1 using the proposed C3/C4 split of that product. The pasture fraction from LUH v3.1.1 was pre-assigned to ORCHIDEE grass PFTs (with further split into C3/C4 pathway) b) the remaining fraction of the grid cell ($f^{remain}$) corresponds to natural vegetation. Present day fractions of natural PFTs (ORCHIDEE PFTs from step 2) were rescaled to match $f^{remain}$. For the historical reconstruction (years 850-1991) we use the average over 1992-1996 (first five years of the satellite derived PFTs) of the present day fraction of natural PFTs. For the recent present-day maps (years 2021-2024) and future projection (years 2025-2100) we used the last available five years of satellite derived data (2016-2020). More details and illustrations are provided in the website https://orchidas.lsce.ipsl.fr/dev/lccci/tools.php.

For model simulations that only require vegetation reconstruction over the satellite-era period (i.e., 1992 - onwards) we directly use the CCI-MRLC product, without step-3 described above (except for the crops fractioning into C3/C4 pathway that is always derived from LUH v3.1.1). Note that the ORCHIDEE team follows the regular update of the CCI-MRLC product; the latest one being described in .

7.3. DONE: 2 forcing#

ORCHIDEE uses the atmospheric 2 concentration (ppm) to compute the stomatal conductance and 2 assimilation (see section ??). By default, the annual time-series from the TRENDY intercomparison protocol is used . It is derived from ice core 2 data (starting in 1700) merged with NOAA observations from 1958 onwards. The construction of the dataset, updated each year, is detailed in .

7.4. DONE: Climate forcing#

ORCHIDEE requires eight meteorological variables at a default, but configurable, reference height of 2 m or 10 m above the plane of zero displacement when the stand-alone configuration is used or the lowest atmospheric layer when the land-atmosphere configuration is used. The eight meteorological variables are: air temperature (K), incoming direct shortwave and diffuse longwave radiations ($\text{W\,m}^{-2}$), liquid and solid precipitation (mm), air specific humidity ($\text{g\,g}^{-1}$), surface wind ($\text{m\,s}^{-1}$, with the possibility to provide latitudinal and longitudinal components), and surface pressure (Pa).

The geographical domain of the forcing data has to be provided on a standard latitude and longitude coordinate system. It can be any spatial domain ranging from a single point up to the entire globe. In order to reduce the disk space used by the forcing data and as ORCHIDEE only runs over land points, the two-dimensional grid can be reduced to a subset of land points. This means that only the forcing data for land points are stored in the forcing files, together with an indexing table that allows scattering the grid points back onto the regular latitude longitude grid when needed.

Although the stand-alone configuration, which makes use of a climate forcing, runs at the half-hourly time step, different climate forcings could, depending on their source, come at a half-hourly to a daily time step. If the frequency of the forcing is less than 6-hourly, ORCHIDEE interpolates the climate forcing between two data time steps. Instantaneous fields and long-wave radiation are linearly interpolated for each model time step $t$, typically half-hour, based on the values available at $t_0$ and $t_0+dt$ where $t_0$ < $t$ < $t_0+dt$. Precipitation fields are spread over the interval defined by a user defined parameter, so that this interval is the one over which the precipitation lasts when the forcing interval has rain or snow. Shortwave radiation is interpolated using a function distribution that corresponds to the solar angle distribution over a forcing time period while conserving the average flux for the interval [$t_0$, $t_0+dt$].

If the frequency of the climate forcing is greater than 6-hours, ORCHIDEE needs to reconstruct the diurnal cycle. The solar angle is calculated at each model time step—typically $t$ along with the times of sunrise, sunset, and solar noon. Based on these solar parameters and the concurrent time information, solar radiation is interpolated from its daily mean, and the corresponding air temperature is estimated using the daily max and min values. In oRCHIDEE the diurnal variation of precipitation is reconstructed using a numerical weather generator. On any rainy or snowy day, the duration of precipitation is initially set to 2 hours and then adjusted based on air temperature. If the temperature is below 20 ° C, the duration increases up to a maximum of 8 hours. when the temperature exceeds 20 ° C, the duration remains fixed at 2 hours. Subsequently, the timing of precipitation within the day is random where the randomisation uses air temperature as the seed to ensure reproducibility. Due to the randomisation, the frequency and intensity of precipitation do not necessarily reproduce the local precipitation characteristics, e.g., drizzle at the global west coasts. Subsequently, wind speed, air pressure, and humidity are linearly interpolated between the days.

ORCHIDEE uses CRU-JRA as its default climate forcing. It is based on the Japanese Reanalysis data [] aligned with the CRU TS data from meteorological stations []. The realignment preserves the monthly means of the CRU TS dataset and concerns temperature (Tmin, Tmax, Tmean), vapor pressure and precipitation, as detailed in . The forcing is defined on a 0.5 ° regular grid at 6-hourly time steps that cover the time period from 1901 to 2024. Regridded CRU-JRA forcing to 2 ° resolution is used for systematic ORCHIDEE reference simulations.

Other forcing datasets prepared to be used with ORCHIDEE include:

GSWP has global coverage with a spatial resolution of 0.5 ° on a regular grid, and a temporal resolution of 3 hours from 1901 to 2010 [];
WFDE5-CRU-GPCC has global coverage with a spatial resolution of 0.5 ° on a regular grid and a temporal resolution of 1 hour from 1980 to 2018 [];
ERA5 has global coverage with a spatial resolution of 0.25 ° on a regular grid and a temporal resolution of 1 hour from 19XX to 20XX [];
ERA5Land has global coverage with a spatial resolution of 0.1 ° on a regular grid and a temporal resolution of 1 hour from 19XX to 20XX [];
SAFRAN covers the metropolitan area of France with a spatial resolution of 8 km on a Lambert II extended grid with a temporal resolution of 1 hour from 1959 to 2023 [];
CRU-JRA55 has a global coverage with a spatial resolution of 0.5 degree on a regular grid and a temporal resolution of 6 hours from 1901 to 2022 [].

The atmospheric forcing is required only when running the ORCHIDEE model in the so-called offline mode. When coupled to the atmospheric model (mainly LMDz in case of ORCHIDEE), the atmospheric forcing is not needed.

7.5. DONE: Lake properties#

The lake fraction per grid cell is based on the HydroLakes database []. ORCHIDEE reads in an additional file containing spatialized values of effective lake depth, wind fetch, water albedo and extinction coefficients for shallow (< 5 m), medium (between 5 and 25 m), and deep lakes (> 25 m). Both files are available at a 0.25 ° and a 0.5 ° resolution for ORCHIDEE .

The HydroLakes database maps 1.4 million lakes of size larger than 0.1 km$^2$ on the global scale and documents their main properties, such as surface area and average depth. To generate the ORCHIDEE lake parameters files, all lakes available in the HydrolLake database were clustered into the three depth classes, i.e., shallow, medium and deep. For each grid cell and each depth-class, the total surface area was calculated as well as a weighted average mean of their respective depth and wind fetch which was then used to calculate the effective lake area, depth and fetch of the each depth-class separately. The albedo and extinction coefficient of freshwater are currently prescribed as 0.07 and 1 $m^{-1}$, respectively.

7.6. DONE: Routing graphs#

For the river routing scheme, ORCHIDEE reads information from a hydrological digital elevation model (HDEM). The minimal information required is elevation, flow direction, flow accumulation and distance to the ocean for each pixel. Ideally, the elevation in the input file is hydrologically consistent, in the sense that no flow direction should lead water to gain elevation. Several datasets that fulfil these criteria are commonly used with ORCHIDEE:

STN-30p at 0.5° resolution, from . It has been enhanced by flow directions in Antarctica in order to carry meltwater to the ocean and close the global water cycle. It is thus commonly used for global simulations.
MERIT at 60 arcsec resolution, from . It is a global dataset, excluding Antarctica, but its high resolution makes it more suitable for regional simulations.
HydroSHEDS at 30 arcsec resolution, from . It only covers regions below 60° N and without Antarctica, and is also used for regional simulations.

These HDEM maps are then processed to provide the two minimal elements needed at each of the ORCHIDEE routing cells to create a routing graph:

A single water flow direction among 11 possibilities: 8 directions towards another routing cell (N, NE, E, SE, S, SW, W, NW), one direction towards the endorheic lakes - local inland convergences of the routing graph (also called lake inflow in ORCHIDEE), one direction towards the ocean from the main stream (riverflow) and one direction accounting for the small disperse flows into the ocean (coastalflow).
A value of the topographical water retention index $k^{Wat-TopoIndex}$ (in km), computed as :

\[k^{Wat-TopoIndex}= % \frac{d}{\sqrt{\tan\beta}}\cdot 10^{-3}= \sqrt{\frac{d^3}{\Delta z}}\cdot 10^{-3}\]

with stream length $d$ (in m) assumed as the distance to the downstream routing cell, and $\Delta z$ (in m) the elevation drop between the two cells.

Beside this minimal information for river routing, the routing graph files can be completed by information characterizing natural hydrological elements (flood plains, swamps, ponds) or anthropogenic infrastructures (dams, reservoirs, adduction channels, gauge stations) as described below [].

7.7. Reservoirs and irrigation#

for irrigation, floodplains, and ponds (we heave a flag doponds) refer to https://forge.ipsl.fr/orchidee/wiki/Documentation/Ancillary as well as Pedro and Patricia’s articles

For the irrigation scheme introduced in , two input files are required, which the irrigation scheme can update the map for every simulated year.

The first one is a map of irrigated fractions, which can be obtained from the Historical Irrigation Dataset [], as in . HID provides a map every 10 years before 1980 and every 5 years after, at 5 arcmin resolution. To run future climate simulations, used irrigated fractions from the Land Use Harmonization 2 dataset []. The dataset was used in the CMIP6 framework with historical and SSPs scenarios, and provides data at 0.25° resolution.

The second input map describes the available equipments for irrigation withdrawals, to preferentially withdraw water from the surface reservoirs (overland and rivers) or groundwater reservoir of the routing scheme. This map is derived from the inventory of areas equipped for irrigation of .

7.8. Slope#

Slope is required as input to the hydrology module to constrain the re-infiltration of surface runoff. A map at 15 arc-min resolution (1/4°) from the US Geological Survey is used. It contains … COMPLETE.

7.9. DONE: Nitrogen inputs#

With the implementation of the nitrogen cycle in ORCHIDEE, the nitrogen deposition rate of mineral nitrogen and the application of fertiliser onto the land surface are required as input to the model. Nitrogen fertiliser input datasets are taken from the NMIP2 project []. In the standard configuration we take the most recent update of gridded N application rates from the TRENDY project [].

It includes inorganic nitrogen fertiliser application, which only started after the Haber-Bosch process was developed in the early 20th century, and manure application. Since manure is organic nitrogen that comes from waste from animals that ate vegetation, it should in principle be taken from other organic nitrogen sources in the model, which is not done yet in ORCHIDEE . For nitrogen deposition, the historical deposition rates are supplied as a time-varying spatially-varying deposition rate provided by the TRENDY project []. The time series starts in 1850 and we thus use the 1850 gridded values for any year prior to 1850. The nitrogen deposition maps combines deposition of and species.

BNF is still missing

7.10. DONE: Forest management#

Historical simulations use a spatially explicit global reconstruction of dominant forest management strategies. The reconstruction is performed on a regular 0.25 ° x 0.25 ° grid between the years 1700 and 2022. The reconstruction distinguishes two forest management strategies: (1) unmanaged, and (2) managed. Although scientific forest management was rare to none existent before 1750 [], management was thought to be an acceptable proxy for historical forest use where stand density was also reduced.

As the forest management maps have to be consistent with the vegetation distribution in ORCHIDEE, the annual vegetation maps are the basis of the management maps. All forests located in Greenland, Canada, the United States, and Russia were assigned to the unmanaged class unless the LUH2v2 reconstruction considered the pixel a secondary forest []. All forests outside of Greenland, Canada, the United States, and Russia were assigned to the managed class unless the LUH2v2 reconstruction considered the pixel to be primary forest []. All forest PFTs within a grid cell are assigned to the same forest management strategy.

7.11. DONE: Albedo background#

A background albedo map for the visible (VIS) and near infrared (NIR) parts of the solar spectrum is used to compute the overall grid-cell albedo (See ??). The background maps are derived from an optimization process (See ??) that uses MODIS VIS and NIR surface albedos and the background albedo maps of the Joint Research Centre Two-stream Inversion Package (JRC-TIP; ) as prior information.

7.12. DONE: Litter raking#

European maps of litter demand were based on historical livestock estimates [], taken to be equal to 0.6, 0.5, and 0.3 head of livestock person$^{-1}$ for northern, central, and southern Europe, respectively. The dividing parallels between northern, central and southern Europe were taken to be respectively 55 and 45° N latitude. It has been reported that 200 to 480 of dry litter were collected per livestock unit per year []. It was assumed that 480 litter per livestock unit per year corresponded to the peak demand in the mid 1800s and faded out afterwards.

Litter maps were generated from the livestock density maps using peak demand for all years from 1600 to 2010. Next, these initial litter maps were multiplied by a correction factor to account for the temporal evolution in litter demand []. The correction factor was tuned to give the desired behaviour, based on historical information from Switzerland [].

8. The energy budget#

8.1. OK: The surface energy budget: principle and main equation#

ORCHIDEE’s energy budget follows a “big-leaf” approach, in which the surface (of temperature $T^{surf}$ (K) and saturated humidity $q^{surf} = g^{q_{sat}}(T^{surf})$ ($kg.kg^{-1}$) ) is considered as a layer of infinitesimal thickness that exchanges with the atmosphere. The energy budget of the surface is based on two main equations. The first corresponds to the radiative budget at the surface:

\[F^{R_n} = F^{LW\downarrow}+F^{SW\downarrow}-F^{LW\uparrow}-F^{SW\uparrow}\]

Where, $F^{R_n}$ (W/m$^2$) is the net radiation at the surface, $F^{LW\downarrow}$ and $F^{SW\downarrow}$ (W/m$^2$) are the downwelling long- and short-wave radiations received respectively from the atmosphere and the sun, and $F^{LW\uparrow}$ and $F^{SW\uparrow}$ (W/m$^2$) are the upwelling long- and short-wave radiations emitted and reflected by the surface.

The second equation represents how the energy available at the surface ($F^{R_n}$) is used. By heating the surface, the absorbed energy is either re-emitted to the atmosphere under the form of convective fluxes, latent and sensible heat fluxes ($F^{LE}$ and $F^{SH}$ respectively - $W/m^2$) or transmitted to the ground through the ground heat flux ($F^G$ - $W/m^2$). The resulting equation enables us to calculate the evolution of the temperature of the surface $T^{surf}$ ($c^{cp}$ being the surface heat capacity $\mathrm{m}^2.\mathrm{s}^{-2}.\mathrm{K}^{-1}$).

\[c^{cp}\frac{dT^{surf}}{dt}=F^{R_n}-F^{LE}-F^{SH}-F^G\]

At equilibrium, the surface energy budget equation corresponds to:

(8.1)#\[F^{R_n}=F^{LE}+F^{SH}+F^G\]

8.2. OK: Radiative transfers#

As presented in section ??, the radiative budget equation allows us to calculate the amount of energy absorbed by and available at the surface. It is composed of net short- and long-wave radiation budgets.

8.2.1. OK: Short-wave radiations budget#

The short-wave radiation budget calculates the amount of solar radiation absorbed by the surface. Incoming short-wave radiation ($F^{SW\downarrow}$) is an input variable of ORCHIDEE. The upwelling short-wave radiation ($F^{SW\uparrow}$) represents the amount of solar radiation that is reflected to the atmosphere, as a function of the albedo ($\alpha$) of the entire grid-cell. Section ?? details the computation of the albedo for all sub-grid tiles.

\[F^{SW\uparrow} = \alpha.F^{SW\downarrow}\]

8.2.2. OK: Long-wave radiations budget#

The incoming long-wave radiation emitted by the atmosphere and received by the surface ($F^{LW\downarrow}$) is, as for the incoming solar radiations, an input of ORCHIDEE.

The upwelling long-wave radiation ($F^{LW\uparrow}$) represents the emissions of the Earth surface, which is assumed to behave as a grey-body of emissivity $c^\epsilon = 1 ??$:

\[F^{LW\uparrow} = c^\epsilon \cdot c^\sigma \cdot {T^{{surf}}}^{4}\]

Where $c^\sigma = 5.67.10^{-8}$ W.m$^{-2}$.K$^{-4}$ is the Stefan-Boltzmann constant.

8.3. Surface albedo calculation#

8.3.1. Grid cell albedo#

The albedo scheme divides the solar spectrum into two wavelength bands: visible (VIS) and near-infrared (NIR), with a separation at XX nm. For each domain, the overall grid cell albedo result from the combination of the albedo of specific grid cell components. Overall, the surface is divided into three compartments: bare soil with fractional cover $f^{bs}$, vegetated surfaces with fractional cover $f^{veg}$, non-biological surfaces with fraction cover $f^{nobio}$.

For the vegetated cover, the maximum surface occupied by each PFT is divided into two parts for the radiative transfer scheme (see section below): a fraction covered by leaves and a fraction corresponding to bare soil which represents gaps in the canopy or the absence of leaves. These fractions thus evolve over time. For each grid cell the different bare soil fractions are grouped together with the same albedo, i.e. the so-called background albedo. Overall the surface is thus divided into three compartments: bare soil (bs), vegetation cover (veg), non-biological cover (nobio).

In addition, each of these land surface types is further divided between snow covered and snow-free fractions. For the snow covered fractions, the calculation of the albedo follows a more complex scheme that depends on snow age with a different formulation for the non-biological surface (see details below).

The overall albedo is thus calculated as a surface area-weighted mean of the albedo of each compartment:

\[{\alpha} = f^{veg} \left[ (1-f^{veg,snow}) \cdot \alpha^{veg} + f^{veg,snow} \cdot \alpha^{veg,snow} \right] + f^{nobio} \left[ (1-f^{nobio,snow}) \cdot \alpha^{nobio} + f^{nobio,snow} \cdot \alpha^{nobio,snow} ) \right]\]

Where vegetation albedo ($\alpha^{veg}$) is defined as the combination of bare soil albedo ($alb_\mathsf{bs}$) and leaf albedo for each vegetation type ($alb_\mathsf{leaf}$), weighted by their fractions:

\[alb_\mathsf{veg} = frac_\mathsf{bs} \cdot alb_\mathsf{bs} + \sum_{\mathsf{pft}=2}^{13} frac_\mathsf{pft} \cdot alb_\mathsf{leaf,pft}\]

Note that as such the scheme overlooks the effect of vegetation shading the bare soil for sparse canopies, in addition to giving the background for all PFTs the same albedo properties as bare soil fraction of the grid. Snow is allowed to cover both vegetated and non-vegetated areas based on the amount of snow present (see section xxxx). Note also that there is no dependence of the background albedo on soil humidity (i.e, lower albedo when the soil is humid), which may be a limitation in regions with large bare soil fractions.

8.3.2. Vegetation albedo#

:::{figure} Figures/multilayer.jpeg :name: schem_multilayer :width: 50% :align: center

Schematic representation of RT through a multilayer medium. The incoming solar flux $F_{\odot}$ is incident at an angle $\theta_0$ with the surface normal. The medium is split into $N$ layers, with an optical depth of $\Delta \tau_l$ for each layer. The total optical depth above the $l$th layer is $\tau_c$, while $\tau$ gives the optical depth within a layer measured from the top of that layer. For each layer $l$, the downward diffuse flux $F_l^{\downarrow}$ is partially scattered and absorbed, giving an upward diffuse flux $F_l^{\uparrow}$. :::

We developed a two-stream multilayer matrix-based radiation transfer (RT) solver for the vegetation canopy based on . We formulated the RT process as a system of linear equations that considers the interactions of radiative fluxes within and between multiple layers. This approach is particularly well suited for complex vegetation canopies for which their vertical heterogeneity can significantly impact RT. In the two-stream approximation, the basic equations for the upward ($F^\uparrow$) and downward ($F^\downarrow$) diffuse fluxes are given by []:

(8.2)#\[\begin{split}\begin{array}{lcl} \displaystyle{ \frac{{\rm d} F^\uparrow}{{\rm d} \tau}} &=&\gamma_1\, F^\uparrow - \gamma_2\, F^\downarrow - \gamma_3 \, \omega \, \pi \, F_{\odot} \, \exp \left(-\tau / \mu_{0}\right) \\ \\ \displaystyle{\frac{{\rm d} F^\downarrow}{{\rm d} \tau}} &=&\gamma_2 \, F^\uparrow - \gamma_1 \, F^\downarrow +\gamma_4 \, \omega \, \pi \, F_{\odot} \, \exp \left(-\tau / \mu_{0}\right) \end{array}\end{split}\]

The gamma coefficients ($\gamma_1$, $\gamma_2$, $\gamma_3$, and $\gamma_4=1-\gamma_3$) are key parameters in the two-stream approximation that prescribe the scattering and absorption properties of the medium and are thus essential in determining the radiative fluxes within each layer. In particular, $\gamma_1$, and $\gamma_2$ relate to the interactions of upward and downward fluxes, while $\gamma_3$ and $\gamma_4$ represent the source terms associated with scattering and absorption processes. The values of these coefficients can be derived from various approximation methods such as the Eddington approximation, quadrature, and $\delta$-methods []. In Eqs. ((8.2)), $\tau$ is the optical depth, $\omega$ is the single-scattering albedo, $\pi F_{\odot}$ is the incident flux at the top of the canopy, and $\mu_0$ is the cosine of the solar zenith angle.

When in Eqs. ((8.2)) $\gamma_1$ is substituted with $[{1 - \omega \, (1 - \beta)}]/{\bar{\mu}}$, $\gamma_2$ is replaced by ${\omega \, \beta}/{\bar{\mu}}$, $\gamma_3$ is taken as $\beta_0$, $\mu_0$ is redefined as ${1}/{K}$, $\pi F_{\odot}$ is set equal to $K$, and $\tau$ is replaced with $L$ (leaf area index) the two-stream RT equations for the vegetation canopies are obtained []:

\[\begin{split}\begin{array}{lcl} {\displaystyle \frac{{\rm d} F^\uparrow}{{\rm d} L}} & = & \displaystyle{\frac{[1-\omega \, (1-\beta) ]}{\bar{\mu}}} \, F^\uparrow- \displaystyle{ \frac{\omega \, \beta}{\bar{\mu}}} \, F^\downarrow - \beta_0 \, \omega \, K \, \exp (-K \, L) \\ \\ {\displaystyle \frac{{\rm d} F^\downarrow}{{\rm d} L}} & = & \displaystyle{\frac{\omega\, \beta}{\bar{\mu}}} \, F^\uparrow - \displaystyle{ \frac{ [1-\omega \, (1-\beta ) ]}{\bar{\mu}}} \, F^\downarrow + (1-\beta_0) \, \omega \, K \, \exp (-K \, L) \end{array}\end{split}\]

Extending the single-layer two-stream model [Eqs. ((8.2))] to multiple layers involves calculating the fluxes within each layer, considering the cumulative optical depths $\tau_c$. In a given layer $l$, the fluxes are expressed as follows:

(8.3)#\[\begin{split}\begin{array}{lcl} F_l^{\downarrow}(\tau) &=& \Gamma_{l} \, B_{l2} \, \exp (\lambda_{l} \, \tau) + B_{l1} \, \exp (-\lambda_{l} \, \tau) + C_l^{\downarrow} \, \exp [-(\tau + \tau_c) / \mu_{0}] \\ \\ F_l^{\uparrow}(\tau) &=& B_{l2} \, \exp (\lambda_{l} \, \tau) + \Gamma_{l} \, B_{l1} \, \exp (-\lambda_{l} \, \tau) + C_l^{\uparrow} \, \exp [-(\tau + \tau_c) / \mu_{0}] \end{array}\end{split}\]

where

\[\begin{split}\begin{array}{lcl} \lambda_l &=& \left(\gamma_{l1}^{2} - \gamma_{l2}^{2}\right)^{1 / 2} \\ \\ \Gamma_l &=& \displaystyle{\frac{\gamma_{l1} - \lambda_l}{\gamma_{l2}}} \\ \\ C_l^{\uparrow} &=& \displaystyle{ \frac{\omega_{0} \, \pi \, F_{\odot} [ (\gamma_{l1} - 1 / \mu_{0}) \, \gamma_{l3} + \gamma_{l4} \, \gamma_{l2} ]}{(\lambda_l^{2} - 1 / \mu_{0}^{2})}} \\ \\ C_l^{\downarrow} &=& \displaystyle{ \frac{\omega_{0} \, \pi \, F_{\odot} \, [ (\gamma_{l1} + 1 / \mu_{0} ) \, \gamma_{l4} + \gamma_{l2} \, \gamma_{l3}]}{ (\lambda_l^{2} - 1 / \mu_{0}^{2})}} \end{array}\end{split}\]

Equations ((8.3)) give the analytical solution for Eqs. ((8.2)), where $\tau$ is the optical depth within each layer, $\tau_c$ is the cumulative optical depth at the top of each layer (Fig. 3). The coefficients $B_{l1}$ and $B_{l2}$ are derived from the boundary conditions and the continuity at the interfaces between the layers. For numerical stability and to keep all arguments to the exponential terms negative, the variables $A_l$ and $B_l$ as in Eqs. ((8.4)) are introduced. This mitigates the risk of numerical overflow or underflow, improving the stability and accuracy of the solver.

(8.4)#\[\begin{split}\begin{array}{lcl} A_{l}= \displaystyle{ \frac{B_{l1}\exp (\lambda_{l} \, \Delta \tau_{l})+B_{l2}}{2}} \\ \\ B_{l}=\displaystyle{ \frac{B_{l1}\exp (\lambda_{l} \, \Delta \tau_{l})-B_{l2}}{2}} \end{array}\end{split}\]

In these equations, $\Delta\tau$ represents the total optical depth of the layer (Fig. 3). The downward and upward fluxes then read:

(8.5)#\[\begin{split}\begin{array}{lcl} F_{l}^{\uparrow}(\tau)= & A_{l} \, \{\exp [-\lambda_{l} (\Delta\tau_{l}-\tau ) ]+\Gamma_{l} \exp (-\lambda_{l} \tau ) \} + B_{l} \, \{\exp [-\lambda_{l} (\Delta\tau_{l}-\tau ) ]-\Gamma_{l}\exp (-\lambda_{l} \tau ) \} \\ & + C_{l}^{\uparrow} \, \exp [-(\tau+\tau_c) / \mu_{0} ] \\ F_{l}^{\downarrow}(\tau)= & A_{l} \, \{\Gamma_{l} \exp [-\lambda_{l} (\Delta\tau_{l}-\tau ) ]+\exp (-\lambda_{l} \tau ) \} + B_{l} \, \{\Gamma_{l} \exp [-\lambda_{l} (\Delta\tau_{l}-\tau ) ]-\exp (-\lambda_{l} \tau ) \} \\ & + C_{l}^{\downarrow} \, \exp [-(\tau+\tau_c) / \mu_{0} ] \end{array}\end{split}\]

In multilayer matrix-based RT, flux continuity and boundary conditions are maintained using a matrix of coefficients. The continuity equations for upward and downward fluxes at the interfaces of layers $l$ and $l+1$ are given by:

(8.6)#\[\begin{split}\begin{array}{lcl} F_{l}^{\uparrow}(\tau=\Delta\tau_{l})=F_{l+1}^\uparrow(\tau=0), & \hbox{for } l=1,2, \cdots,(N-1) \\ \\ F_{l}^{\downarrow}(\tau=\Delta\tau_{l})=F_{l+1}^\downarrow(\tau=0), & \hbox{for } l=1,2, \cdots,(N-1) \end{array}\end{split}\]

Using Eqs. ((8.5)), Eqs. ((8.6)) can be expressed as a system of linear equations:

(8.7)#\[\begin{split}\begin{array}{lcl} e_l \, A_l + d_l \, B_l -f_{l+1} \, A_{l+1} + g_{l+1} \, B_{l+1} = & C_{l+1}^{\uparrow}(0)-C_{l}^{\uparrow}(\Delta\tau_l) \\ \\ f_l \, A_l + g_l \, B_l - e_{l+1} \, A_{l+1} + d_{l+1} \, B_{l+1} = & C_{l+1}^{\downarrow}(0)-C_{l}^{\downarrow}(\Delta\tau_l) \end{array}\end{split}\]

where the coefficients $e_l$, $d_l$, $f_l$, and $g_l$ are defined as:

\[\begin{split}\begin{array}{lcl} e_{l} = & 1+\Gamma_{l} \, \exp (-\lambda_{l}\Delta\tau_{l}) \\ d_{l} = & 1-\Gamma_{l} \, \exp (-\lambda_{l}\Delta\tau_{l}) \\ f_{l} = & \Gamma_{l} + \exp (-\lambda_{l}\Delta\tau_{l}) \\ g_{l} = & \Gamma_{l} - \exp (-\lambda_{l}\Delta\tau_{l}) \end{array}\end{split}\]

We derived a tridiagonal system of equations to reduce computational efforts. The following steps outline the derivation process. Multiply the first equation in Eqs. ((8.7)) by $d_{l+1}$ and the second by $g_{l+1}$, subtract the resulting equations to eliminate $B_{l+1}$, leading to a new equation relating $A_l$, $B_l$, and $A_{l+1}$. Multiply the second equation in Eqs. ((8.7)) by $d_{l+1} e_l - g_{l+1} f_l$ and the first by $f_l$, subtract these results to eliminate $A_{l+1}$, leading to another new equation that relates $A_l$, $B_l$, and $B_{l+1}$. The final system of equations reads:

(8.8)#\[\begin{split}\begin{array}{lll} \overbrace{[d_{l+1} \, e_{l} - f_{l} \, g_{l+1}]}^{\alpha_{l1}} \, A_l &+ \overbrace{[d_{l+1} \, d_{l} - g_{l} \, g_{l+1}]}^{\beta_{l1}} \, B_l &+ \overbrace{[e_{l+1} \, g_{l+1} - f_{l+1}d_{l+1}]}^{\gamma_{l1}} \, A_{l+1} = \\ & & \underbrace{[C_{l+1}^{\uparrow}(0) - C_{l}^{\uparrow}(\Delta\tau_l)] \, d_{l+1} + [C_{l}^{\downarrow}(\Delta\tau_l) - C_{l+1}^{\downarrow}(0)] \, g_{l+1}}_{\chi_{l1}} \\ \overbrace{[d_{l} \, f_{l} - e_{l} \, g_{l}]}^{\alpha_{l2}} \, B_l &+ \overbrace{[e_{l+1} \, e_{l} - f_{l} \, f_{l+1}]}^{\beta_{l2}} \, A_{l+1} &+ \overbrace{[f_{l} \, g_{l+1} - e_{l} \, d_{l+1}]}^{\gamma_{l2}} \, B_{l+1} = \\ & & \underbrace{[C_{l+1}^{\uparrow}(0) - C_{l}^{\uparrow}(\Delta\tau_l)] \, f_{l} + [C_{l}^{\downarrow}(\Delta\tau_l) - C_{l+1}^{\downarrow}(0)] \, e_{l}}_{\chi_{l2}} \end{array}\end{split}\]

where the coefficients $\alpha_{l1}$, $\beta_{l1}$, $\gamma_{l1}$ and $\alpha_{l2}$, $\beta_{l2}$, $\gamma_{l2}$ are derived from the original coefficients $e_l$, $d_l$, $f_l$, $g_l$ and the source terms $C_l^{\uparrow}$, $C_l^{\downarrow}$.

The downward diffuse flux at the top of the multilayer structure is equal to any incident diffuse flux. In addition, the upward flux at the bottom of the multilayer structure is equal to the product of the downward flux and the reflectance of the surface ($R_{S}$), i.e. the incident flux reflected at the surface is diffuse. Hence, boundary conditions at the top and bottom boundaries are as follows:

\[\begin{split}\begin{array}{lcl} F_{1}^{\downarrow}(\tau=0) &=& A_{1} \{\Gamma_{1} \exp [-\lambda_{1} (\Delta\tau_{1} ) ]+1 \} + B_{1} \{\Gamma_{1} \exp [-\lambda_{1} (\Delta\tau_{1} ) ]-1 \} + C_{1}^{\downarrow}(0)\\ &=& \rm {Downward\ diffuse\ flux}\\ \\ F_{N}^{\uparrow}(\tau=\Delta\tau_{N}) &=& A_{N} \{1+\Gamma_{N} \exp (-\lambda_{N} \Delta\tau_{N} ) \} + B_{N} \{1-\Gamma_{N} \exp (-\lambda_{N} \Delta\tau_{N} ) \} + C_{N}^{\uparrow}(\Delta\tau_{N})\\ &=& R_{S}F_{N}^{\downarrow}(\tau=\Delta\tau_{N}) + R_{S}F_{\odot}(\tau=\Delta\tau_{N}) \end{array}\end{split}\]

We need to solve a tridiagonal system ($2N-2$ equations) to determine $A_l$, $B_l$, $A_{l+1}$, $B_{l+1}$ [Eqs. ((8.8))] and the related variables such as $F_l^{\downarrow}$ and $F_l^{\uparrow}$. We use the standard method of tridiagonal solving, known as the Thomas algorithm. This algorithm is efficient for solving tridiagonal systems, as it reduces computational complexity compared with general matrix solvers.

Several key parameters need to be precalculated to model RT in the vegetation canopy. These are $G(\mu)$, $K(\mu)$, $\bar{\mu}$, $\omega \, \beta$, and $\omega \, \beta_0$. The function $G(\mu)$, known as the asymmetry factor of the phase function, is given by []:

\[G(\mu)= \frac{1}{2\pi} \int_{\Omega^{\prime}} \mathrm{d}\hat{\mathbf{\Omega}}^{\prime} \, g(\mathbf{\theta}^{\prime}) h(\mathbf{\phi}^{\prime})\left| \hat{\mathbf{\Omega}} \cdot \hat{\mathbf{\Omega}}^{\prime} \right|,\]

where $g(\theta)$ and $h(\phi)$ are the probability density functions of leaf normal inclination and azimuth, respectively. Several common functions exist for the probability distribution $g(\theta)\sin \theta$. For instance, the planophile function is given by ${2}/{\pi} \left(1 + \cos2\theta\right)$, while the erectophile function is defined as ${2}/{\pi} \left(1 - \cos2\theta\right)$. The plagiophile function is expressed as ${2}/{\pi} \left(1 - \cos4\theta\right)$, and the extremophile function is represented by ${2}/{\pi} \left(1 + \cos4\theta\right)$. The function is $\sin\theta$ in the spherical case ($G=1/2$) while the uniform function is ${2}/{\pi}$. Once the $G$ function has been determined, the other parameters can be directly derived. These are:

(8.9)#\[\begin{split}\begin{array}{lcl} K(\mu) &=& \frac{G(\mu)}{\mu} \\ \bar{\mu} &=& \int_0^1 \mathrm{d}\mu^{\prime} \frac{\mu^{\prime}}{G\left(\mu^{\prime}\right)}\\ \omega &=& \left(r + t\right) \\ \delta &=& \left(r - t\right) \\ \omega \, \beta &=& \frac{1}{2} \left(\omega + \delta \int_0^{\pi / 2} \mathrm{d} \theta \sin \theta \cos^2 \theta \, g\left(\theta\right)\, \right) \\ \omega \, \beta_0 &=& \frac{1}{2} \left(\omega + \frac{\mu_0}{G\left(\mu_0\right)} \delta \int_0^{\pi / 2} \mathrm{d} \theta \sin \theta\cos^2 \theta \, g\left(\theta\right)\, \right) \end{array}\end{split}\]

where $r$ is the reflectivity and $t$ is the transmissivity of vegetation (leaf). The last two equations in Eqs. ((8.9)) represent the back-scattering parameters ($\beta$, $\beta_0$) for the diffuse and direct beams, respectively, as defined by . However, these two parameters need to be adjusted for other cases. In , the back-scattering parameter for the diffuse beam is defined as

\[\omega \, \beta=\frac{1}{2}\left(\omega+\delta \cos ^2 \Theta\right)\]

where $\Theta$ is the mean leaf inclination angle relative to the horizontal surface. The back-scattering parameter for direct beam, $\beta_0$, reads []:

\[\omega \, \beta_0=\frac{1+\bar{\mu}\, K}{\bar{\mu} \, K} \alpha_{ss}(\mu),\]

where $\alpha_{ss}(\mu)$, the single scattering albedo of the canopy, is given by

\[\alpha_{ss}(\mu)=\omega \int_0^1 \mathrm{d} \mu^{\prime} \, \frac{\mu^{\prime} \, G(\mu)}{\mu \, G(\mu^{\prime})+\mu^{\prime} \, G(\mu) } .\]

8.3.3. OK: Background and bare soil albedo#

We combine into a single variable referred as soil albedo i) the background albedo that corresponds to the albedo of the soil under a vegetation cover and that is used as a boundary condition of the vegetation radiative transfer scheme to compute the overall vegetation albedo (see section above) and ii) the bare soil albedo from non-vegetated surfaces (i.e. Bare soil PFT).

Currently, there are two options to define the albedo of the soil. The first option (the default) is a spatially and temporally variable soil albedo, derived from MODIS satellite observations (Ref) using an extensive optimization procedure (see https://orchidas.lsce.ipsl.fr/dev/albedo/optimization.php). A 3-D field (longitude, latitude, time) of soil albedos was derived at a 1° spatial resolution and for a climatology of 12 months. The computation arises from an inverse procedure (similar to ), where for each 1° grid-cell the monthly soil albedo results from an optimization that accounts for vegetation albedo (see above) and snow albedo (see below). The second option corresponds to the use of a mean value (fixed in time) for each plant functional type that the user can easily prescribe.

8.3.4. Snow albedo#

The snow albedo is computed following the formulation of for each PFT, following the equation:

\[\alpha_{snow,i} = A_{aged,i} + B_{dec,i} \exp \left( - \frac{age_{snow}}{age_{dec}} \right)\]

$A_{aged}$ represent albedo of old snow and the sum of $A_{aged}$ and $B_{dec}$ is the albedo of fresh snow. $age_{dec}$ is the time constant of the albedo decay of snow in days and $age_{snow}$ the age of snow parameterized as follows:

\[age_{snow}(t+dt) = \left[ age_{snow}(t) + \left( 1 - \frac{age_{snow}(t)} {age_{max}} \right) dt \right] \exp \left( -\frac{ P_{snow}}{\delta_c} \right)\]

where $age_{max}$ is the maximum snow age, $P_{snow}$ is the amount of snowfall during the time interval $dt$ and $\delta_c$ is the solid precipitation constant of snow age. Grid point albedo is weighted according to the fraction of each PFT, distinguishing between forested and grasses and crops PFTs.

\[\begin{split}\alpha_{snow} = \sum_{i=1}^{N_{PFT}} fracsnow_{i}. \alpha_{snow,i} \begin{cases} fracsnow_{i} = \frac{f^{veg,max}_{i}} {f_{veg,tot}}, & forested\;PFTs\\ fracsnow_{i} = \frac{f^{veg}_{i}} {f_{veg,tot}}, & grasses\;and\;crops\;PFTs\\ fracsnow_{i} = \frac{f_{totbaresoilnotree}} {f_{veg,tot}}, & i=1 \end{cases}\end{split}\]

For forest PFTs, the albedo is weighted with $f^{veg,max}_{i}$, the maximum cover fraction of a PFT, assuming that even snow-covered vegetation remains visible. For grasses and crops PFTs, the albedo is weighted with $f^{veg}_{i}$, the fraction of vegetation that covers the soil. When $f^{veg}_{i}$ is lower than $f^{veg,max}_{i}$, the albedo of the snow on the baresoil is used. Baresoil snow albedo is weighted by $f_{totbaresoilnotree}$ with

\[f_{totbaresoilnotree} = \sum_{i=1}^{N_{PFT}} f^{veg,max}_{i} - f^{veg}_{i} \; if\; i = no\; tree\; PFT\]

8.3.5. Ice sheet and glacier albedo#

The non-biological compartment currently represents area covered by ice sheet or glaciers. Low surface air temperatures found in this regions slow down the snow metamorphism which is why albedo evolution is specific to these regions. This effect is accounted for with the function $f_{age,nobio}$:

\[\begin{split}\begin{align} &age_{snow}(t+dt) = \left[ age_{snow}(t) + \left( 1 - \frac{age_{snow}(t)} {age_{max}} \right) dt \right] \exp \left( -\frac{P_{snow}}{\delta_c} \right) + f_{age,nobio}\\ &f_{age,nobio} = \frac{ \left[ age_{snow}(t) + \left( 1 - \frac{age_{snow}(t)} {age_{max}} \right) dt \right] \exp \left( -\frac{P_{snow}}{\delta_c} \right) - age_{snow}(t)} {1 + g_{temp}}\\ &g_{temp} = \left(\frac{max(T_0 - T^{surf},0)}{\omega1}\right)^{\omega2} \end{align}\end{split}\]

where $\omega1$ and $\omega2$ are tuning constant.

8.4. Turbulent transfers#

8.4.1. OK: Main equation for sensible heat, latent heat and momentum fluxes#

As shown in the surface energy budget in Eq. (8.1), part of the energy received at the surface is transferred to the atmosphere through turbulent fluxes. These turbulent exchanges are typically represented by three main fluxes: the sensible and latent heat fluxes, which are driven by heat and moisture exchanges between the surface and the atmosphere, and the momentum flux, which represents the drag exerted by the atmosphere on the surface.

Over the grid cell, ORCHIDEE explicitly computes the turbulent fluxes of sensible $F^{H}$ and latent $F^{LE}$ heat (W m$^{-2}$) between the surface and the atmosphere. These fluxes are represented using the diffusive formulation of and can be expressed as:

(8.10)#\[F^{H} = \rho \cdot c^{cp} \frac{T^{surf} - T^{atm}}{R^{aero}}\]

(8.11)#\[F^{LE} = \lambda \cdot \rho \cdot \beta \cdot \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

Here, $\rho$ is the air density (kg m$^{-3}$), $c^{cp}$ is the specific heat capacity of air (m$^2$ s$^{-2}$ K$^{-1}$), $\lambda$ is the latent heat of vaporization (or sublimation in the case of a snow-covered surface) (J kg$^{-1}$), $T^{surf}$ is the surface temperature (K) and $q^{sat}(T^{surf})$ is the saturated specific humidity at $T^{surf}$ (kg kg$^{-1}$), $T^{atm}$ and $q^{atm}$ are the atmospheric temperature and specific humidity (K and kg kg$^{-1}$), $\beta$ represents the combined effect of physiological and physical resistances that reduce the actual evapotranspiration relative to the potential evapotranspiration ($F^{ETP}$) (unitless) (see Section ??), and $R^{aero}$ is the aerodynamic resistance (s m$^{-1}$). The aerodynamic transport model used to compute the aerodynamic resistance is described in section ??, while the resistances for sensible and latent heat fluxes are detailed in sections ?? and ??, respectively.

Although the momentum flux itself is not computed directly in ORCHIDEE, it represents the drag force that generates turbulence and determines the efficiency of heat and moisture transfers through its control on aerodynamic resistances. The zonal ($\tau_x$) and meridional ($\tau_y$) components of the momentum flux can be expressed as:

\[\tau_x = \rho \frac{u^{surf} - u^{atm}}{R^{aero}}\]

\[\tau_y = \rho \frac{v^{surf} - v^{atm}}{R^{aero}}\]

Where $u^{surf}$ and $v^{surf}$ are the zonal and meridional wind components at the surface (assumed to be zero at the roughness height), $u^{atm}$ and $v^{atm}$ are the corresponding wind components in the atmosphere (m s$^{-1}$), and $R^{aero}$ is the aerodynamic resistance (s m$^{-1}$).

8.4.2. OK: Aerodynamic transfer: roughness length and drag coefficients#

Does this section describe the different options in ORCHIDEE? For example, rough_dyn, use_ratio_z0m_z0h, RATIO_Z0M_Z0H, and use_height_dom. The parameter names should not neceassarily be mentioned but the different options should as they are referred in section ??.

The efficiency of turbulent fluxes introduced in the previous section depends on how the surface interacts with the atmosphere, which is captured through drag coefficients and roughness lengths. When ORCHIDEE is coupled with LMDZ, the drag coefficients are provided by LMDZ, while ORCHIDEE computes the roughness lengths. In offline mode, ORCHIDEE calculates both the drag coefficients and the roughness lengths directly. Although the energy budget is resolved at the grid-cell scale, the drag coefficients are calculated for each PFT, as they depend on vegetation structure, and then aggregated to obtain a grid cell average.

In offline mode, the roughness length is first computed for bare soil $z^{m,bare}_{0}$ (m), as well as the drag coefficients for heat $C^{h,bare}$ and momentum $C^{m,bare}$ (unitless) using the von Karman constant:

\[z^{m,bare}_{0}= \left( 1 - f^{snow,veg}\right) \cdot z^{bare}_{0} + f^{snow,veg} \cdot \frac{z^{bare}_{0}}{10}\]

Where $f^{snow,veg}$ is the fraction of snow on vegetated area, and $z^{bare}_{0}$ is the bare soil roughness length set to 0.01m.

\[C^{m,bare}= f^{bare} \cdot \left(\frac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{z^{m,bare}_{0}}\right)} \right)^2\]

\[C^{h,bare}= f^{bare} \cdot \left(\frac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{z^{m,bare}_{0}/c^{z^{m}_{0},z^{h}_{0}}}\right)} \right)\left(\dfrac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{z^{m,bare}_{0}}\right)} \right)\]

Where $f^{bare}$ is the total evaporating bare soil fraction, $\kappa=0.41$ is the von Karman constant (unitless), $z^{atm}_{1,max}$ is the maximum height of the first atmospheric layer (set to at least 10m), and $c^{z^{m}_{0},z^{h}_{0}}$ a constant representing the ratio between $z^{m}_{0}$ and $z^{h}_{0}$, set by default to 1 for all PFTs.

Then for each vegetated PFT, several options exist to compute each of these two roughness lengths, which can be combined for a given simulation.

A dynamical computation uses the formulation proposed by based on the initial work of for $z^{m,pft}_{0}$, which is calculated as:

\[z^{m,pft}_{0}=h_{c,pft} \left( 1 - \frac{d_{0,pft}}{h_{c,pft}} \right) \exp \left(- \frac{k}{\eta} \right)\]

with ${h_{c,pft}}$ the canopy height (m), ${d_{0,pft}}$ the displacement height, assumed equal to $\frac{2}{3}{h_{c,pft}}$, ${k}$ the von Karman’s constant and $\mathsf{\eta}$, the ratio of friction velocity to the wind speed at the top of canopy.

The roughness length for heat is calculated, assuming an exponential relationship between $z^{m,pft}_{0}$ and $z^{h,pft}_{0}$, with the following equation:

\[z^{h,pft}_{0} = \frac{z^{m,pft}_{0}}{\exp\left(\frac{k}{B}\right)}.\]

with $B$ the Stanton number (see Appendix E of for a full description).

It is also possible to use fixed ratios to define the roughness height for momentum as a fraction of the canopy height, and the roughness height for heat as a fraction of the roughness height for momentum.

\[z^{m,pft}_{0}=\frac{h_{c,pft}}{15}\]

\[z^{h,pft}_{0}=\frac{z^{m,pft}_{0}}{10}\]

Using these values of roughness lengths, the drag coefficients for heat $C^{h,veg}$ and momentum $C^{m,veg}$ (unitless) are computed following:

\[C^{m,veg}= f^{veg} \cdot \left(\frac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{\max(d^{h}\cdot c^{z,h},z^{m,bare}_{0})}\right)} \right)^2\]

\[C^{h,veg}= f^{veg} \cdot \left(\frac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{\max(d^{h}\cdot c^{z,h},z^{m,bare}_{0})/c^{z^{m}_{0},z^{h}_{0}}}\right)} \right)\left(\dfrac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{\max(d^{h}\cdot c^{z,h},z^{m,bare}_{0})}\right)} \right)\]

Where $f^{veg}$ is the fraction of vegetated PFT. Note that in the case of forest PFTs, $f^{veg}$ is set to $f^{veg,max}$ because trees trunks influence the roughness even without leaves, while for grassland PFT, they influence the roughness only during the growing season. $d^{h}$ is the PFT vegetation height, and $c^{z,h}$ is a roughness scaling factor set to 0.0625 for all vegetated PFTs (unitless).

Then, for each none vegetated surface, the drag coefficients for heat $C^{h,nobio}$ and momentum $C^{m,nobio}$ are computed in a similar way:

\[C^{m,nobio}= f^{nobio} \cdot \left(\frac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{z^{m,nobio}_{0}}\right)} \right)^2\]

\[C^{h,nobio}= f^{nobio} \cdot \left(\frac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{z^{m,nobio}_{0}/c^{z^{m}_{0},z^{h}_{0}}}\right)} \right)\left(\dfrac{\kappa}{\ln\left(\dfrac{z^{atm}_{1,max}}{z^{m,nobio}_{0}}\right)} \right)\]

Where $f^{nobio}$ is the fraction of the non vegetated surface, and $z^{m,nobio}_{0}$ is the roughness height of the non vegetated surface set to 0.001m.

Finally, the grid-cell drag coefficients are obtained by summing the fraction-weighted contributions from bare soil, vegetated PFTs, and non-vegetated surfaces. For vegetated areas, the total is normalized by the effective vegetated fraction to ensure consistent weighting across surface types.

Then, at the grid cell level, the roughness length for heat $z^{h}_{0}$ and for momentum $z^{m}_{0}$ are calculated based on the inversion of the calculation of the drag coefficients aggregated over the grid cell ($C^{h}$ and $C^{m}$):

\[z^{m}_{0}=\frac{z^{atm}_{1,max}}{\exp\left(\dfrac{\kappa}{\sqrt{C^{m}}}\right)}\]

\[z^{h}_{0}=\frac{z^{atm}_{1,max}}{\exp\left(\dfrac{\kappa^2}{C^{h}\cdot\ln\left(\dfrac{z^{atm}_{1,max}}{z^{m}_{0}}\right)}\right)}\]

The zero-plane displacement height $z^{d}$ is also computed as a fraction of average vegetation height:

\[z^{d}=d^{h,ave} \cdot c^{z^{d}}\]

Where $d^{h,ave}$ is the average vegetation height (m), and $c^{z^{d}}=0.66$ is a factor to calculate the zero-plane displacement height (unitless).

The grid cell effective roughness height $z^{rough}$ is computed as the difference between the average vegetation height and displacement height:

\[z^{rough}=d^{h,ave} - z^{d}\]

Note that the vertical reference for heights differs between the land and atmospheric components: in ORCHIDEE, heights are expressed relative to the canopy top or to $z^{d}$, while in LMDZ they are referenced from the displacement height. Therefore, $z^{rough}$ provides a consistent means to convert between these reference levels.

The aerodynamic drag coefficient determines the strength of turbulent exchanges between the land surface and the atmosphere. It depends on both surface roughness and atmospheric stability, which is expressed through the Richardson number $Ri$. $Ri$ is defined as the ratio of the two source terms for turbulent kinetic energy: buoyancy and wind shear, which depend on the gradient of temperature near the surface and on the surface wind speed:

\[Ri= z^{atm}_{1}g \cdot \frac{\left(T^{v,atm}-T^{v,surf}\right)}{u^2T^{v,atm}}\]

Where $z^{atm}_{1}$ is the height of the first atmospheric layer (m), $g$ is the gravitational constant, $u$ is the wind speed ($m\,s^{-1}$), and $T^{v,atm}$ and $T^{v,surf}$ are the virtual temperature of the atmosphere and the surface (K).

Positive $Ri$ values correspond to stable conditions (suppressed turbulence), negative values to unstable conditions (enhanced turbulence), and $Ri=0$ corresponds to neutral conditions.

The neutral drag coefficient $C^{neut}$ (assuming neutral atmospheric stability) is computed as:

\[C^{neut}= \frac{\kappa^2}{\ln\left(\dfrac{z^{atm}_{1,max}+z^{rough}}{z^{m}_{0}}\right)\ln\left(\dfrac{z^{atm}_{1,max}+z^{rough}}{z^{h}_{0}}\right)}\]

The final stability-corrected drag coefficient $C$ is derived from $C^{neut}$ and a stability function $f(Ri)$ from , which depends on the sign of $Ri$:

\[C = C^{neut} f(Ri)\]

\[\begin{split}f(Ri) = \begin{cases} 1 - \dfrac{3 c^b Ri}{1 + 3 c^b c^c C^{neut} \sqrt{|Ri| \dfrac{z^{atm}_{1,max}+z^{rough}}{z^{m}_{0}}}} & \text{if } Ri < 0 \\ \\ 1 & \text{if } Ri=0\\ \dfrac{1}{1 + 3 c^b Ri \sqrt{1 + c^d |Ri|}} & \text{if } Ri > 0 \end{cases}\end{split}\]

Where $c^b=c^c=c^d=5$ (unitless).

It must be noted that when ORCHIDEE is coupled with LMDZ, ORCHIDEE does not fully compute the drag coefficients. Instead, it computes only the neutral coefficients and provides LMDZ with grid-cell effective roughness lengths for the grid cell:

\[z^{m}_{0}=z^{atm}_{1} \exp{\left(\frac{-\kappa}{\sqrt{C^{m,neut}}}\right)}\]

\[z^{h}_{0}=z^{atm}_{1} \exp{\left(\frac{-\kappa}{\sqrt{C^{h,neut}}}\right)}\]

LMDZ then computes the drag coefficients using the values of the surface layer for $Ri$ and its own stability function $f(Ri)$.

\[\begin{split}\begin{align} &C^{m} = C^{m,neut} f(Ri)\\ &C^{h} = C^{h,neut} f(Ri) \end{align}\end{split}\]

The formulation for $f(Ri)$ is similar for unstable cases, but for stable cases, functions from are used:

\[\begin{split}f(Ri) = \begin{cases} 1 - \dfrac{3 c^b Ri}{1 + 3 c^b c^c C^{h,neut} \sqrt{|Ri| \dfrac{z^{atm}_{1}}{z^{m}_{0}}}} & \text{if } Ri < 0 \\ 1 & \text{if } Ri=0\\ \left(1-\dfrac{Ri}{c^e}\right)^2 & \text{if } 0 < Ri < c^e/2\\ c^f \left(\dfrac{c^e}{Ri}\right)^2 & \text{if } Ri \geq c^e/2\\ \end{cases}\end{split}\]

With $c^e = 0.25$, $c^f = 0.0625$ (unitless).

8.5. Sensible heat flux - associated resistance#

As presented in section ??, the sensible heat flux $F^{H}$ is expressed as:

\[F^{H} = \rho \cdot c_p \frac{T^{surf} - T^{atm}}{R^{aero}}\]

The exchange is thus controlled by the aerodynamic resistance $R^{aero}$ ($s\,m^{-1}$), which represents the efficiency of turbulent transfer of heat between the surface and the atmosphere:

\[R^{aero} = \frac{1}{C \cdot {u}}\]

Where $u$ is the horizontal wind speed ($m\,s^{-1}$) and $C$ is the drag coefficient (unitless) described in section ??.

8.6. Latent heat flux components - associated resistances#

As seen in section ??, at the grid cell scale, the latent heat flux $F^{LE}$ (W m$^{-2}$) is defined by Eq. (8.11) and, when expressed in water units, corresponds to the actual evapotranspiration $F^{ET}$ (kg m$^{-2}$ s$^{-1}$). It can also be expressed in terms of potential evapotranspiration ($F^{ETP}$ in kg m$^{-2}$ s$^{-1}$) as:

\[F^{LE} = \lambda F^{ET} = \beta \lambda F^{ETP}\]

Where $\beta$ is the dimensionless stress factor that accounts for physiological and physical resistances to water transport from the surface to the atmosphere (as presented in Eq. (8.11)), and $\lambda$ is the latent heat of vaporization (or sublimation in the case of a snow-covered surface) (J kg$^{-1}$).

In the approach of , $F^{ETP}$ represents the amount of water that would evaporate if the surface was completely wet, and is calculated using the temperature of a hypothetically wet surface $T^{wet}$ (K):

\[F^{ETP} = \rho \frac{q^{sat}(T^{wet}) - q^{atm}}{R^{aero}}\]

Where $\rho$ is the air density (kg m$^{-3}$), $q^{sat}(T^{wet})$ is the saturation specific humidity at $T^{wet}$ (kg kg$^{-1}$), $q^{atm}$ is the atmospheric specific humidity (kg kg$^{-1}$), and $R^{aero}$ is the aerodynamic resistance (s m$^{-1}$).

proposed a simplification widely used in land surface models: computing $F^{ETP}$ using the actual surface temperature $T^{surf}$ instead of $T^{wet}$. While this avoids the need to calculate a separate energy balance for a wet surface, it leads to an overestimation of $F^{ETP}$, because $T^{surf}$ > $T^{wet}$ and $q^{sat}$ is a function that increases with temperature.

To avoid this overestimation, introduced a correction factor that reduces $F^{ETP}$ based on the $\beta$ stress factor as the difference between $T^{surf}$ and $T^{wet}$ increases. The corrected potential evapotranspiration $F^{ETP,corr}$ (kg m$^{-2}$ s$^{-1}$) is therefore expressed as:

\[F^{ETP,corr} = \rho \frac{q^{sat}(T^{surf}) - T^{atm}}{R^{aero}} \frac{1+\xi}{1+\beta \xi}\]

Where $\xi$ is the corrective factor of (unitless) calculated as:

(8.12)#\[\xi = \frac{1}{R^{aero}}\frac{\lambda \cdot \rho \cdot \frac{\partial q^{sat}}{\partial T}(T^{atm})}{4 \cdot \epsilon\cdot c^{\sigma} \cdot T^{{atm}^3} + \frac{\rho \cdot c^{cp}}{R^{aero}}}\]

Here, $c^{cp}$ is the specific heat capacity of air (m$^2$ s$^{-2}$ K$^{-1}$), $\epsilon$ is the surface emissivity (unitless), and $c^{\sigma}$ is the Stefan-Boltzmann constant (W m$^{-2}$ K$^{-4}$).

While the actual evapotranspiration flux $F^{ET}$ is computed at the grid-cell scale, it can be partitioned into its main components, representing the different pathways of water transfer to the atmosphere:

Floodplain evaporation
Snow sublimation
Canopy interception loss
Canopy transpiration
Bare soil evaporation

A $\beta$-resistance scheme is used to partition the total evapotranspiration flux $F^{ET}$ into its individual components. For each process $i$, a coefficient $\beta^i \in [0,1]$ is computed, representing the combined effect of the aerodynamic resistance $R^{aero}$ and any process-specific resistance, scaled by the fraction of the grid cell over which the process occurs.

In this framework, $\beta^i$ quantifies the relative contribution of process $i$ to the total resistance-limited evapotranspiration.

Each component of the evapotranspiration flux $F^{ET,i}$ (kg m$^{-2}$ s$^{-1}$) can then be computed as:

\[F^{ET,i} = \beta^i \cdot F^{ETP} = \beta^i \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

By construction, the sum of the flux components recovers the total resistance-controlled evapotranspiration: $\sum_i F^{ET,i} = F^{ET}$.

Table 2 describes the different components of $F^{ET}$, their corresponding grid-cell fraction, the resistances applied to the flux and the evapotranspiration used to calculate the flux.

:::{list-table} Components of the latent heat flux and their corresponding fractions, resistances and evapotranspiration used for their calculation. :header-rows: 0 :name: tab:ETP_sub-fluxes :align: center

- Flux
- Fraction
- Resistances
- ETP
- Floodplains evaporation
- $f^{flood}$
- $R^{aero}$
- $F^{ETP,corr}$
- Snow sublimation
- $f^{snow}$
- $R^{aero}$
- $F^{ETP}$
- Canopy interception loss
- $f^{inter}$
- $R^{aero},R^{struc}$
- $F^{ETP}$
- Canopy transpiration
- $f^{veg}$
- $R^{aero},R^{stom}$
- $F^{ETP}$
- Bare soil evaporation
- $f^{bare}$
- $R^{aero}, R^{soil}$
- $F^{ETP,corr}$ :::

8.6.1. Floodplain evaporation#

The floodplain evaporation flux $F^{ET,flood}$ is calculated over the portion of the grid-cell covered by floodplains $f^{flood}$ and is primarly controlled by the aerodynamic resistance $R^{aero}$.

The calculation uses a prediction-correction approach. A first estimate of the resistance factor for floodplain evaporation $\beta^{flood}_{pred}$ is based on $f^{flood}$ and the ratio between the corrected and uncorrected potential evaporation fluxes:

\[\beta^{flood}_{pred} = f^{flood} \frac{F^{ETP,corr}}{F^{ETP}}\]

The predicted floodplain evaporation is:

\[F^{ET,flood}_{pred} = \beta^{flood}_{pred} \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

If this predicted evaporation exceeds the water available in the floodplain reservoir $M^{flood}$ over the model timestep $\Delta t$, $\beta^{flood}_{pred}$ is reduced proportionally to conserve water:

\[\beta^{flood}_{corr} = \beta^{flood}_{pred} \cdot \min\left(1,\frac{M^{flood}}{F^{ET,flood}_{pred} \, \Delta t}\right)\]

The final floodplain evaporation flux is then computed as:

\[F^{ET,flood} = \beta^{flood}_{corr} \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

8.6.2. Snow sublimation#

The snow sublimation flux $F^{ET,snow}$ is calculated over the fraction of the grid-cell covered by snow $f^{snow}$ (see Section ??), including contributions from vegetated and non-vegetated surfaces (ice, lakes, cities, etc). This flux is controlled solely by the aerodynamic resistance $R^{aero}$.

The calculation follows a prediction–correction approach. First, a resistance coefficient for snow sublimation is estimated for each of the contributions from vegetated and non-vegetated surfaces:

For the vegetated surface:

\[\beta^{snow,veg}_{pred}=(1-f^{nobio})f^{snow,veg}\]

Here, $f^{nobio}$ is the fraction of the grid cell occupied by non-vegetated surfaces, and $f^{snow,veg}$ is the fraction of vegetation covered by snow.

The predicted sublimation flux on the vegetation is calculated as:

\[F^{ET,snow,veg}_{pred} = \beta^{snow,veg}_{pred} \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

If the predicted sublimation over the model timestep $\Delta t$ exceeds the available snow mass on the vegetation $M^{snow,veg}$, $\beta^{snow,veg}_{pred}$ is reduced proportionally to conserve snow:

\[\beta^{snow,veg}_{corr} = \beta^{snow,veg}_{pred} \cdot \min\left(1,\frac{M^{snow,veg}}{F^{ET,snow,veg}_{pred} \, \Delta t}\right)\]

Similarly, for each non-vegetated surface type $j$:

\[\beta^{snow,nobio}_j=f^{nobio}_jf^{snow,nobio}_j\]

Here, $f^{nobio}_j$ is the fraction of the grid cell of non-vegetated type $j$, and $f^{snow,nobio}_j$ is the fraction of that surface type covered by snow.

The predicted sublimation flux on the non-vegetation surface type $j$ is computed as:

\[F^{ET,snow,nobio}_{pred,j} = \beta^{snow,nobio}_{pred,j} \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

If the predicted sublimation exceeds the available snow mass on this surface type $M^{snow,nobio}_j$, $\beta^{snow,nobio}_{pred,j}$ is reduced proportionally to conserve snow mass:

\[\beta^{snow,nobio}_{corr,j} = \beta^{snow,nobio}_{pred,j} \cdot \min\left(1,\frac{M^{snow,nobio}_j}{F^{ET,snow,nobio}_{pred_j} \, \Delta t}\right)\]

Finally, the total resistance coefficient for snow sublimation is computed as:

\[\beta^{snow}_{corr} = \beta^{snow,veg}_{corr} + \sum_j\beta^{snow,nobio}_{corr,j}\]

This corrected resistance coefficient is used to compute the actual snow sublimation flux (reduced by flooded surfaces):

\[F^{ET,snow} = (1-\beta^{flood}_{corr})\beta^{snow}_{corr} \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

8.6.3. OK: Interception loss#

The canopy interception loss flux $F^{ET,inter}$ is calculated for each PFT over the fraction of vegetation that holds intercepted water $f^{inter}$. This flux is controlled by the aerodynamic resistance $R^{aero}$ and a structural resistance $R^{struc}$ (s,m$^{-1}$), which accounts for reduced air mixing within the canopy. $R^{struc}$ is fixed for tree and herbaceous PFTs.

The calculation of $F^{ET,inter}$ also follows a prediction-correction scheme. First, the resistance coefficient for interception is estimated based on $f^{inter}$ (see Section ?? and (9.1)), the vegetated fraction of the PFT $f^{veg}$, $R^{aero}$ and $R^{struc}$:

\[\beta^{inter}_{pred} = f^{veg} f^{inter} \frac{R^{aero}}{R^{aero} + R^{struc}}\]

The predicted interception flux is then:

\[F^{ET,inter}_{pred} = \beta^{inter}_{pred} \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

To ensure mass conservation, if the predicted flux over the model timestep $\Delta t$ exceeds the total intercepted water storage $M^{inter}$, $\beta^{inter}_{pred}$ is reduced to ensure that the evaporation does not exceed the available water:

\[\beta^{inter}_{corr} = \beta^{inter}_{pred} \cdot \min\left(1, \frac{M^{inter}}{F^{ET,inter}_{pred} \, \Delta t}\right)\]

In case of dew formation, the same framework applies but the wet fraction originates from dew deposition rather than rainfall interception. The resistance factor for dew depends on the fraction of dew on the leaves $g^{dew}$ ((9.3)):

\[\beta^{inter,dew} = g^{dew} f^{veg}\]

The total interception coefficient then includes both rainfall and dew contributions:

\[\beta^{inter} = \beta^{inter}_{corr} + \beta^{inter,dew}\]

The final canopy interception loss flux, reduced by flooded and snow-covered surfaces, is then expressed as:

\[F^{ET,inter} = (1-\beta^{flood}_{corr})(1-\beta^{snow}_{corr})\beta^{inter}_{corr} \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

8.6.4. Canopy transpiration#

The canopy transpiration flux $F^{ET,transp}$ is calculated for each PFT over the fraction of the PFT covered by vegetation $f^{veg}$. The flux is controlled by the aerodynamic resistance $R^{aero}$ and a stomatal and boundary layer resistance $R^{leaf}$ (s m$^{-1}$).

The leaf resistance $R^{leaf}$ is expressed as:

\[R^{leaf} = \frac{f^{veg}}{f^{veg,max}} \cdot R^{stom}_{tot}\]

Where $f^{veg,max}$ is the maximum cover fraction of the PFT, and $R^{stom}_{tot}$ is the stomatal resistance integrated over the whole canopy.

The total stomatal resistance is computed from the stomatal conductance calculated in the photosynthesis model (Section ??) for each LAI layer $k$, as follows:

\[R^{stom}_{tot} = \frac{1}{\sum_k g^{stom}_{k} LAI_k}\]

where $g^{stom}_{k}$ is the stomatal conductance of layer $k$ (m s$^{-1}$), and $LAI_k$ is the leaf area index of layer $k$ (m$^{2}$ m$^{-2}$).

The transpiration resistance coefficient $\beta^{transp}$ is then computed as the reduction of the aerodynamic flux by the leaf resistance, considering the fraction of vegetation covered by the PFT $f^{veg}$:

\[\beta^{transp} = f^{veg}\frac{R^{aero}}{R^{aero} + R^{leaf}}\]

Finally, the actual canopy transpiration flux for the PFT, accounting for reductions due to flooded surfaces and snow cover, is given by:

\[F^{ET,transp} = (1-\beta^{flood}_{corr})(1-\beta^{snow}_{corr})\beta^{transp} \cdot \rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

8.6.5. OK: Bare soil evaporation#

Does this section describes the do_rsoil option? It should not necessarily mention the parameter name but the option should be mentioned as it is referred to in the description of the configuration in Section ??.

The bare soil evaporation flux component $F^{ET,bare}$ follows a two-steps prediction-correction scheme, as detailed in section ?? and summarized below.

First, a predicted bare soil resistance coefficient $\beta^{bare}_{pred,i}$ is calculated for each soil column $i$. It is computed based on an initial estimate of the bare soil evaporation flux $F^{bare}_{lim,i}$, derived from the soil water budget. This flux represents the amount of water that the soil column can supply over the timestep $\Delta t$, given its current soil moisture profile, and is weighted by the soil tile fraction $f^{bare}_i$. In particular, the current water balance depends on both the bottom and top fluxes of each soil column. The top flux is determined by the corrected potential evaporation $F^{ETP,corr}$, which is limited by a soil resistance for each soil tile $R^{soil}_i$, scaled by the aerodynamic resistance $R^{aero}$ (see Eq.(9.27)). The soil resistance, presented in Eq.(9.26), constrains soil evaporation as soil moisture decreases, following .

The bare soil evaporation estimate is then normalized by the potential evaporation $F^{ETP}$ to compute the resistance factor for a given soil column $i$:

\[\beta^{bare}_{pred,i} = f^{bare}_i \frac{F^{bare}_{lim,i} \Delta t}{F^{ETP}}\]

This resistance factor is then averaged over the grid cell ($\beta^{bare}_{pred}$):

\[\beta^{bare}_{pred} = \sum_i \beta^{bare}_{pred,i}\]

Next, a correction is applied to ensure mass conservation, since the sum of canopy interception loss, canopy transpiration, and bare soil evaporation cannot exceed the total available water flux for these processes. In terms of $\beta$ stress factors, this means that the sum of $\beta^{bare}_{corr}$, $\beta^{inter}_{corr}$, and $\beta^{transp}_{corr}$ cannot exceed 1. If this limit is exceeded, bare soil evaporation is reduced so that intercepted water evaporation and transpiration can proceed at their calculated rates. In that case, $\beta^{bare}_{pred}$ is reduced to $1 - \beta^{inter}_{corr} - \beta^{transp}_{corr}$:

\[\beta^{bare}_{corr} = \min\left(\beta^{bare}_{pred},1 - \beta^{inter}_{corr} - \beta^{transp}_{corr}\right)\]

Finally, the corrected stress factor is used to compute the actual bare soil evaporation flux (reduced by flooded and snow-covered surfaces):

\[F^{ET,bare} = (1-\beta^{flood}_{corr})(1-\beta^{snow}_{corr})\beta^{bare}_{corr} \cdot\rho \frac{q^{sat}(T^{surf}) - q^{atm}}{R^{aero}}\]

8.7. Soil thermodynamics#

8.7.1. Soil thermodynamics model#

The soil thermodynamic model in ORCHIDEE was developed by coupling heat conduction with the energy transferred by liquid water transport in the soil vertical direction, as well as latent heat due to soil freezing/melting (; ; ):

(8.13)#\[C_P(\theta,st)\frac{\partial T}{\partial t}=\frac{\partial}{\partial z}\left[\lambda(\theta,st)\frac{\partial T}{\partial z}\right]-C_W \frac {\partial q_{L} T}{\partial z} + \rho_{ice}L\frac{\partial \theta_{ice}}{\partial t}\]

where $C_P$ and $C_W$ are volumetric heat capacities (J m$^{-3}$ K$^{-1}$) of soil and liquid water, respectively; $\theta$ is the volumetric soil moisture (m${^3}$ m$^{-3}$); st stands for the soil texture; $T$ is the soil temperature (K); $t$ is the time (s); $z$ is the soil depth (m); $\lambda$ is the soil thermal conductivity (J m$^{-1}$ s$^{-1}$ K$^{-1}$); $q_L$ is the flux density of liquid water (m s$^{-1}$); $\rho_{ice}$ is the ice density (kg m$^{-3}$); $L$ is the latent heat of fusion (J kg$^{-1}$); $\theta_{ice}$ is the volumetric ice content (m$^3$ m$^{-3}$). The last term $\rho_{ice}L~\partial \theta_{ice}/{\partial t}$ is optional and represents the latent heat source/sink term due to the freezing and melting of the soil water (). During freezing, latent heat release delays the progression of the freezing front. In contrast, latent heat consumption counteracts warming as a frozen soil layer reaches the freezing point.

For numerical computation, equation (8.13) can be rewritten as:

(8.14)#\[C_{app}\frac{\partial T}{\partial t}=\frac{\partial}{\partial z}\left[\lambda(\theta,st)\frac{\partial T}{\partial z}\right]-C_W \frac {\partial q_{L} T}{\partial z}\]

where $C_{app}$ includes the volumetric heat capacity of the soil but also a term representing the latent heat impact (.

\[C_{app} = C_P(\theta,st) - \rho_{ice}L\frac{\Delta \theta_{ice}}{\Delta T}\]

At the surface, the energy budget equation is:

\[Cs\frac{\partial T_s}{\partial t}=F_{rad}+F_1^h+LF_1^q+G_1\]

where $T_S$ is the soil surface temperature (K); $G_1$ is the soil heat flux due to heat conduction process; $F_{rad}$, $F_1^h$, and LF$_1^q$ are the net radiation, sensible heat and latent heat flux respectively (W m$^{-2}$); $C_S$ is the ‘layer’ heat capacity per unit area (J m$^{-2}$ K$^{-1}$) and is related to the thickness of the first soil layer.

The Equation (8.13) is solved by using an implicit finite difference method. The soil temperature and the soil heat flux are calculated at the node and at the interface of each layer, respectively. The evolution of the temperature in the middle of the layers is given by ($1<k<7$):

\[\frac{T_{k+1/2}^t-T_{k+1/2}^{t-\Delta{t}}}{\Delta{t}}=\frac{\lambda}{C}\cdot\frac{1}{z_{k+1}-z_k}\cdot\left(\frac{T_{k+3/2}^t-T_{k+1/2}^t}{z_{k+3/2}-z_{k+1/2}}-\frac{T_{k+1/2}^t-T_{k-1/2}^t}{z_{k+1/2}-z_{k-1/2}}\right)\]

The temporal evolution of the temperature in the first layer $T_{1/2}$ is calculated as the difference between the total fluxes $\sum F^\downarrow (T_s^t)$ at the surface (atmospheric radiative fluxes, soil radiation and turbulent fluxes) and the conductive flux at the bottom of the first layer:

\[\frac{T^{1/2}_{t}-T_{1/2}^{t-\Delta{t}}}{\Delta{t}}=\frac{\lambda}{C}\cdot\frac{1}{z_1-z_0}\cdot\left(\frac{T_{3/2}^t-T^{1/2}_{t}}{z_{3/2}-z_{1/2}}+\frac{\sum F^\downarrow(T_s^t)-\epsilon\sigma(T_s^t)^4}{\lambda}\right)\]

The soil surface temperature ($T_s$) is calculated from the surface energy balance equation (see section ??), and it is used as boundary forcing for calculating the subsurface soil temperature. The $T_{1/2}$ is linked with the soil surface temperature $T_s$ through:

\[T_s=\left(1+\frac{z_{1/2}}{z_{3/2}-z_{1/2}}\cdot T^{1/2}_{t}-\frac{z_{1/2}}{z_{3/2}-z_{1/2}}\cdot T_{3/2}^t\right)\]

The heat flux is assumed to be zero at the bottom layer ($N=7$):

\[\frac{T_{N-1/2}^t-T_{N-1/2}^{t-\Delta{t}}}{\Delta{t}}=-\frac{\lambda}{C}\cdot\frac{1}{z_N-z_{N-1}}\cdot\left(\frac{T_{N-1/2}^t-T_{N-3/2}^t}{z_{N-1/2}-z_{N-3/2}} \right)\]

8.7.2. Soil properties: thermal conductivity#

Soil thermal conductivity and soil heat capacity are important parameters to determine the thermal evolution of the soil in depth. These parameters strongly depend on soil composition, including soil texture and soil moisture. The soil thermal conductivity ($\lambda_{soil}$) is calculated with the Johansen’s method () by interpolating between a dry and a saturated conductivity according to the Kersten number $\mathrm{Ke}$ described below.

\[\begin{split}\begin{align} &\lambda_{soil}(\theta)=\lambda_{dry}+\mathrm{Ke}(\theta)\cdot(\lambda_{sat}-\lambda_{dry})\\ % &S(\theta)=\frac{\theta-\theta^{w}}{\theta_f-\theta^{w}} \end{align}\end{split}\]

where $\lambda_{dry}$ and $\lambda_{sat}$ are the dry and saturated thermal conductivity, respectively (W/m/K).

The Kersten number $\mathrm{Ke}$ depends on the state of the water in the soil (liquid or solid) and on the degree of saturation $S_r$, corresponding to the fraction of porosity occupied by liquid water and ice ($S_r(\theta) = \frac{\theta_l+\theta_{ice}}{\theta_s}$), with $\theta_s$, $\theta_l$ and $\theta_{ice}$ being the porosity, and the volumetric liquid and solid water contents, respectively. $\theta_l$ and $\theta_{ice}$ are calculated in the hydrological module (section ??). For unfrozen soils ():

\[\begin{split}\begin{eqnarray} \mathrm{Ke} = \begin{cases} \log_{10}(S_r)+1, & \text{if}~S_r > 0.1~\text{and fine matter}\\ 0.7\log_{10}(S_r)+1, & \text{if}~S_r > 0.05~\text{and coarse matter}\\ 0, & \text{otherwise} \end{cases} \end{eqnarray}\end{split}\]

while for frozen soils, $\mathrm{Ke}=S_r$.

The dry conductivity $\lambda_{dry}$ is determined according to the soil’s porosity $\theta_s(st)$ that depends on the soil texture $st$ of the grid cell, and is estimated with the following equation:

(8.15)#\[\begin{align} \lambda_{dry}(st) = \frac{0.135\times (1-\theta_{s}(st))\times 2700 + 64.7}{2700(1-0.947\times (1-\theta_{s}(st))} \end{align}\]

Finally, the saturated conductivity $\lambda_{sat}(\theta,st)$ is a weighted average of solid conductivity $\lambda_{solid}(st)$, liquid water conductivity $\lambda_{liq}$ and ice conductivity $\lambda_{ice}$ when the soil is saturated ($S_r=1$).

(8.16)#\[\lambda_{sat}(\theta,st) = \lambda_{solid}(st)^{1-\theta_{s}} \lambda_{liq}^{\theta_{l}} \lambda_{ice}^{\theta_{s} - \theta_{l}}\]

For mineral soils, solid conductivity is determined by separating the quartz and the other minerals, as quartz has a very large conductivity compared to other minerals. The soil texture provides quartz contents $qz(st)$, enabling the calculation of the solid conductivity $\lambda_{solid}(st)$:

\[\begin{align} \lambda_{solid}(st)=\lambda_{quartz}^{qz(st)} \cdot \lambda_{other~minerals}^{1-qz(st)} \end{align}\]

8.7.3. Soil properties: heat capacity#

The soil heat capacity $C_P(\theta,st)$ is calculated as the sum of heat capacities of dry soil and liquid and solid water:

\[C_{P}(\theta,st)=C_{dry}(st)\cdot(1-\theta_s(st))+\theta_l\cdot C_{water}+ \theta_{ice} \cdot C_{ice}\]

where $C_{dry}$, $C_{water}$ and $C_{ice}$ are the volumetric soil heat capacities for dry soil, liquid water and ice, respectively (J/m$^3$/K). The parameters $C_{dry}(st)$ and $\theta_s(st)$ depend on the soil texture (values are given in Table 1). $\theta_l$ and $\theta_{ice}$ are the volumetric liquid and solid water contents, and are calculated in the hydrological module.

8.7.4. Impact of soil organic and mineral composition on thermal properties#

In ORCHIDEE, another parameterization () can be activated to include the impact of soil organic matter on soil thermal properties (heat capacity and thermal conductivity). This configuration accounts for the spatial diversity of soil textures and the different amounts of organic and mineral matter that can be present in the soils. It uses external maps of bulk density of fine matter, soil organic carbon concentration, and volume fraction of coarse matter as input, additionally to what is being used in the standard configuration.

In fact, the thermal properties of organic matter can differ widely from those of mineral matter, as organic matter has a low thermal conductivity associated with a high heat capacity compared to mineral matter and can be more than twice as porous as mineral matter (). To better estimate the respective impacts of each soil component, the volume they occupy in each grid cell and at each depth is estimated.

Two types of volume fraction are estimated for soil composition: a volume fraction that includes the porous space of the material, hereafter used with the notation “$V_{constituent}$” ; and a solid volume fraction, referring to the effective volume occupied by dense matter and excluding porous space, with the notation “$x_{constituent}$”. These fractions are estimated for organic matter (OM), fine mineral matter (MM) and coarse mineral matter (CM) from external maps: bulk density of fine matter ($BD_{FM}$), soil organic carbon mass fraction ($m_{SOC}$) and volume fraction of coarse matter ($V_{CM}$). Here, fine matter is composed of organic (OM) and fine mineral matter (MM), but does not include coarse matter (CM).

In a first step, it is possible to estimate the OM mass fraction $m_{OM}$ as a function of the organic carbon mass fraction of the soil, using the Van Bemmelen conversion factor () $\alpha_{VB}$, set at 0.58 according to the literature.

\[\begin{align} m_{OM} = \frac{m_{SOC}}{\alpha_{VB}} \end{align}\]

Then, the solid volume fractions can be calculated:

\[\begin{split}\begin{align} x_{OM} &= m_{OM} \cdot \frac{BD_{FM}}{\rho_{OM}} (1 - V_{CM})\\ x_{MM} &= (1-m_{OM}) \cdot \frac{BD_{FM}}{\rho_{MM}(st)}(1-V_{CM}) \\ x_{CM} &= V_{CM} \cdot (1-\theta_{s,MM}(st)) \end{align}\end{split}\]

where $\rho_{OM}$ and $\rho_{MM}(st)$ are the particle densities of organic and mineral matter, respectively, and $\theta_{s,MM}$ is the porosity of mineral matter.

The effective porosity of the soil is the remaining volume, and $\theta_s$ becomes:

\[\begin{align} \theta_s = 1 - x_{OM} - x_{MM} - x_{CM} \end{align}\]

This porosity is used to calculate the saturation ratio and the Kersten number defined in section ?? and is also used in equation (8.16).

The volume fractions $V$ including the pores can be calculated by assuming that the porosity of mineral matter $\theta_{s,MM}(st)$ remains the same within a soil texture, while the volume fraction of OM is estimated by removing the volumes occupied by fine ($V_{MM}$) and coarse mineral matter ($V_{CM}$):

\[\begin{split}\begin{align} V_{MM} &= \frac{x_{MM}}{1-\theta_{s,MM} (st)} \\ V_{OM} &= 1 - V_{MM} - V_{CM} \end{align}\end{split}\]

The heat capacity $C_{P}(\theta,st)$ is now a weighted arithmetic average of the capacity of each constituent, the solid volume fractions being used as weights.

(8.17)#\[\begin{align} C_{P}(\theta,st) = x_{OM,i} C_{OM} + (x_{MM,i} + x_{CM,i}) C_{MM}(st) + \theta_{l,i} C_{water} + \theta_{ice,i} C_{ice} + x_{air,i} C_{air} \end{align}\]

where $C_{OM}$, $C_{MM}(st)$, $C_{water}$, $C_{ice}$ and $C_{air}$ are the solid volumetric heat capacities of organic matter, mineral matter, liquid water, ice and air respectively ; and $\theta_l$ and $\theta_{ice}$ are the volumetric contents of liquid and solid water.

The methodology to calculate the soil thermal conductivity remains the same as the one described in section ??, but the calculation of $\lambda_{solid}$ and $\lambda_{dry}$ is modified.

The solid conductivity is a geometric average of the conductivities of OM $\lambda_{OM}$, quartz $\lambda_{quartz}$ and other solid minerals $\lambda_{solid,other~minerals}$, weighted by the solid volume fractions $x$ within the soil’s solids.

(8.18)#\[\begin{align} \lambda_{solid,i} = (\lambda_{OM}^{x_{OM,i}} \times \lambda_{quartz}^{x_{quartz,i}} \times \lambda_{solid,other~minerals}^{(x_{MM,i}+x_{CM,i}-x_{quartz,i})})^{1/(x_{OM,i}+x_{MM,i}+x_{CM,i})} \end{align}\]

with $x_{quartz}$ being determined via the quartz content $qz(st)$ and the solid volume fractions of fine and coarse mineral matter:

\[\begin{align} x_{quartz} = qz(st)\cdot(x_{MM}+x_{CM}) \end{align}\]

Finally, the dry soil conductivity corresponds to the thermal conductivity of the soil in the absence of liquid water and ice. The dry thermal conductivities are weighted by using the volume fractions $V$ of OM and coarse and mineral matter.

(8.19)#\[\begin{align} \lambda_{dry,i}=\lambda_{dry,OM}^{V_{OM,i}}\lambda_{dry, MM}^{V_{CM,i}+V_{MM,i}} \end{align}\]

where $\lambda_{dry,OM}$ is the dry conductivity of organic matter and $\lambda_{dry,MM}(st)$ is the dry conductivity of mineral matter, calculated with equation (8.15).

8.7.5. Snow thermal properties#

In the snow covered regions, the uppermost soil layers are replaced by snow according to the snow depth. The heat capacity and thermal conductivity of snow ($C_{snow}$ and $\lambda_{snow}$, Table 1) are used for the layers filled by snow. In the transition layer (mixed soil and snow), the heat capacity $C_{soil-snow}$ (or thermal conductivity $\lambda_{soil-snow}$) is parameterized by weighting the $C_{snow}$ (or $\lambda_{snow}$) and $C_{sat}$ (or $\lambda_{sat}$) according to the thickness of snow ($zl_{snow}$) and soil.

\[\begin{split}C_{soil-snow}(\theta)=\begin{cases} C_{snow}\cdot\frac{zl_{snow}}{zl_1}+C_{sat}\cdot\frac{zl_1-zl_{snow}}{zl_1}, & i=1\\ C_{snow}\cdot\frac{zl_{snow}-zl_{i-1}}{zl_i-zl_{i-1}}+C_{sat}\cdot\frac{zl_i-zl_{snow}}{zl_i-zl_{i-1}}, & i>1 \end{cases}\end{split}\]

The effective heat capacity is defined as

\[C_\mathsf{snow}(\theta)=snowxci\]

where xci is the specific heat of ice (2.106×103 J/Kg/K). The effective thermal conductivity of snow is from ISBA explicit snow scheme (), and it is computed by: snow(θ)=+bsnowsnow+max[0,(v+bv/(Tsnow+cv))P0/P] (10) where v represents vapor transfer in the snow; p is the atmospheric pressure in hPa, p0=1×103 hPa; ρ is the snow density (kg/m3). The coefficients were from and : αλ,v = -0.06023 W/m/K; bλ,v = -2.5425 W/m; cλ,v = -289.99 K. αλ = 0.02 W/m/K; bλ = 2.5 ×10-6 Wm5/kg2/K.

Verseghy, D. L., 1991: CLASS - A Canadian land surface schemefor GCMs. I. Soil Model. Int. J. Climatol., 11, 111–133.

When the soil is frozen, the $\lambda_{dry}$ and $C_{dry}$ values are the same with that of unfrozen case, while the $\lambda_{sat}$ and $C_{sat}$ are computed by taking into account the thermal properties of ice component ():

\[\begin{split}\begin{align} &\lambda_{sat,frz}=(\lambda_{soil}^{1-\theta_S})\lambda_{ice}^{(1-f_L)\cdot\theta_S}\lambda_{water}^{f_L\cdot\theta_S}\\ &C_{sat,frz}(\theta)=f_L\cdot C_{wet}+(1-f_L)\cdot C_{icy} \end{align}\end{split}\]

where $f_L$ is the fraction of the liquid water, and it is assumed to linearly varied from 1 to 0 between 0° C and -2° C in the “freezing window”; $\lambda_{soil}$, $\lambda_{ice}$, $\lambda_{water}$ are the thermal conductivity for solid soil, ice and water, respectively (W/m/K); $C_{wet}$ and $C_{icy}$ are the heat capacity for saturated unfrozen soil and saturated frozen soil, respectively (J/m3/K, Table X).

8.7.6. Surface litter and organic matter insulation#

Litter and groundcover vegetation (mosses, lichens) affect surface-atmosphere energy exchanges through their insulating properties. This effect has been shown to significantly impact the simulation of permafrost dynamics in the arctic when ORCHIDEE is coupled to the atmospheric model, LMDZ, []. It is not explicitly represented but instead, the thermal properties of the upper soil layers are modified to mimic the impact of such a surface organic layer (SOL). The surface organic layer is assumed to cover a fraction $f^{SOL}$ of each grid cell. This fraction is then weighted by the fraction of boreal PFTs, as mosses and lichens are almost ubiquitous in most Arctic ecosystems, but it can be extended to other PFTs.

To calculate the effect of this surface organic layer, a virtual column (not explicitly represented in the model) is defined, representing the aboveground surface organic layer (with a thickness defined by the parameter $d^{SOLT}$) and the soil layers whose thermal properties are modified (down to a depth defined by the parameter SOIL_INTEGRATION_DEPTH (SID)) (add schematic ?). The thermal capacity of the virtual column is a weighted average of the surface organic layer (C_SOL) and the soil (C_soil) thermal capacities:

\[\begin{split}\begin{align} \notag C_\text{virtual column} SID & = C_\text{SOL} SOLT + C_\text{soil} SID\\ \Leftrightarrow C_\text{virtual column} & = C_\text{SOL} \frac{SOLT}{SID} + C_\text{soil} \end{align}\end{split}\]

The total thermal capacity, also taking into account the fraction of the PFT not covered by the organic layer, is the weighted average of C_{virtual column} and C_soil :

\[\begin{split}\begin{align} \notag C_\text{tot}&=f_\text{SOL} C_\text{virtual column} + (1-f_\text{SOL}) C_\text{soil} \\ &= f_\text{SOL}C_\text{SOL}\frac{SOLT}{SID}+C_\text{soil} \end{align}\end{split}\]

where $C_\text{tot}$ is the total heat capacity (soil and surface organic layer), $C_\text{SOL}$ the thermal capacity of surface organic layer, $C_\text{soil}$ the soil thermal capacity (as calculated in section ??), $f_\text{SOL}$ the fraction of each grid point that contains the surface organic layer, SOLT the surface organic layer thickness and SID the soil integration depth, i.e. the depth above which soil organic layer properties are taken into account. We use $f_\text{SOL}=1$, SOLT=0.03m and SID=0.03m. SOLT is consistent with the moss thickness measured in . In the real world, the surface organic layer is above ground and does not directly influence soil thermal dynamics. Therefore, SID is chosen to be thin enough to allow the soil organic layer to influence surface-atmosphere energy exchanges, but to limit its effect on internal soil thermal transfers to the very top soil layers.

The approach for thermal conductivity is similar but takes into account that it is an intensive property. Therefore, the thermal conductivity of the surface organic layer column ($\lambda$_{virtual column}) is the equivalent thermal conductivity of the surface organic layer and soil layers in series:

\[\lambda_\text{virtual column} = \frac{\lambda_\text{SOL}\lambda_\text{soil}SID}{SOLT\lambda_\text{soil}+SID\lambda_\text{SOL}}\]

where $\lambda$_SOL is the thermal conductivity of the soil organic layer and $\lambda$_soil the thermal conductivity of the soil (as computed in section ??).

The total thermal conductivity is the equivalent thermal conductivity of the surface organic layer column and the soil column in parallel:

\[\begin{split}\begin{align} \notag \lambda_{tot}&=f_\text{SOL}\lambda_\text{virtual column} + (1-f_\text{SOL})\lambda_{soil}\\ &= f_\text{SOL}\lambda_\text{SOL}\lambda_\text{soil}\frac{SID}{SOLT\lambda_\text{soil}+SID\lambda_\text{SOL}}+(1-f_\text{SOL})\lambda_\text{soil} \end{align}\end{split}\]

In the model, from the surface down to SID, the mineral soil capacity and conductivity are replaced by $C_\text{tot}$ and $\lambda_\text{tot}$.

The thermal properties of the surface organic layer are parameterised using observations made on mosses. In particular, the surface organic layer thermal capacity and conductivity are linearly dependent on the water content of the soil layers down to SID []. The thermal capacity is set at 0.29$\times 10^6$ J.m$^{-3}$.K$^{-1}$ for a dry surface organic layer, 4.29$\times 10^6$ J.m$^{-3}$.K$^{-1}$ for a wet organic layer and 3.26$\times 10^6$ J.m$^{-3}$.K$^{-1}$ for a frozen surface organic layer []. The thermal conductivity is equal to 0.05 W.m$^{-1}$.K$^{-1}$ for dry surface organic layer, 0.56 W.m$^{-1}$.K$^{-1}$ for a wet surface organic layer and 1.40 W.m$^{-1}$.K$^{-1}$ for a a frozen surface organic layer [].

8.8. Implicit resolution of the energy budget when coupled to LMDZ atmospheric model#

To solve the energy budget at the surface, ORCHIDEE uses an approach proposed by . The numerical scheme is formally equivalent to the implicit resolution of a diffusion equation from the top of the atmosphere to the bottom of the soil but solved with two models totally independent. Using the continuity of the temperature and finite difference implicit numerical integration of the diffusion equation for the temperature the discretized surface energy balance reads

\[C_s^* \frac{ ( T^{1/2}_{t+1}-T^{1/2}_{t})} {\delta{t}}= LW^{net}_{t+1}+ SW^{net}_{t+1}+LE^{t+1}+H^{t+1}+F_s^{* (t+1)}\]

$\delta{t}$ is the time step for the numerical integration it is identical for the atmospheric model and for the land surface model. However, as the radiation scheme is very expensive computationally it is not called at every time step. Using the continuity of the temperature, the surface temperature extrapolated from the temperature a level 1/2 and 3/2 in the soil. The implicit solution for the temperature into the soil allows linearly express $T_{1/2}$ as function of $T_{3/2}$.

\[T_{3/2}^{t+1}=\alpha_1^{t} T^{1/2}_{t+1} +\beta_1^{t}\]

If $z_{n/2}$ denotes the depths for the level n below the surface and $\mu=\frac{ z_{1/2}} {z_{3/2}-z_{1/2}}$

The surface temperature reads

\[T_s=[ 1+ \mu(1-\alpha_1^t )] T^{1/2}_{t+1} -\mu \beta_1^{t+1}\]

Reporting the expression of Ts obtained in (6) in equation (4) on can write:

\[C_s^* \frac{ ( T^{s}_{t+1}-T^{s}_{t})} {\delta{t}} = LW^{net}_{t+1}+SW^{net}_{t+1}+LE^{t+1}+H^{t+1}+F_s^{ *(t+1)}\]

where t and t+1 stand for 2 successive time-steps. $C_s ^*$ is an effective heat capacity per unit area ($J m^{-2}K^{-1}$) and $F_s$ is an effective ground heat flux. These quantities together with $\alpha_1$ and $\beta_1$ are evaluated when the heat diffusion into the soil is solved using an implicit scheme as well

The important aspect here is that the latter quantities can be determined in advance at the time step t for the time step t+1. At this time, to solve the surface temperature at the new time step, one needs to find the values of the turbulent fluxes and the upward LW radiation at the new time step [].

$SW^{net}$ is computed by the GCM and is supposed not to depends from the surface temperature. For $LW^{net}$, the downward LW radiation is also supposed to be independent of the surface temperature. The upward LW radiation is estimated at the new time step with the help of a Taylor expansion.

\[LW_{up}^{t+1}=LW_{up}^t+4\epsilon\sigma T_s^{3 (t)}(T_s^{t+1}-T_s^t)\]

This approach allows to take into account surface temperature variations in the land surface model surface radiation budget even though the radiation scheme is not called at each time step. However it can lead to a small energy imbalance between the atmosphere and the surface which has been estimated to about 0.2 W/m$^2$ (from global multi-annual coupled simulations)

Once discretized in time and space the turbulent diffusion in the atmosphere leads to solving a linear spatio-temporal boundary values problem. The system is solved with an Euler implicit approach because the time steping requires large time steps (15mn) with respect to the typical time evolution of the turbulence. This involves the resolution of a tridiagonal system. As a consequence,

the value of the state variable X at the first level can be expressed as a function of the flux at the surface with the following expression: $X_1^t=A^x+B^xF_s^X\delta t$, where the coefficients $A^x$ and $B^x$ are function of $X_1^t$ and of the turbulent diffusion coefficient at the upper levels. They are independent of the value of X at the surface.

Using the expression of the turbulent fluxes at the surface resulting from the Monin Obukof similary theory(section xx), the potential enthalpy flux at (t+1) can be expressed as follows:

\[H_s^{t+1}=A_H^t+B_H^t(T_s^{t+1}-T_s^t)\]

And the latent heat flux can be expressed as

\[LE_s^{t+1}=A_q^t+B_q^t(T_s^{t+1}-T_s^t)\]

using a Taylor development of the saturation water vapor specific humidity as a function $T_s$. $T_s^{t+1}$ is then obtained by reporting the expression of the fluxes at the new time step into (7).

:::{figure} Figures/couplimplcit.png :name: implicit_coupling :width: 100% :align: center

Schematic representation of the implicit coupling between atmosphere and land surface for the energy budget :::

8.9. OK: Lakes energy budget#

In ORCHIDEE, two options are available to represent lakes. In the first option (default settings), lakes are considered bare soil surfaces with corresponding energy and water budgets. When coupled to an atmospheric model (LMDZ or WRF), the Great Lakes surface processes are either handled by the ocean model (NEMO) or constrained by climatic lake surface temperatures forced to sea surface temperature (SST) observations. In the second option, lakes are explicitly modeled, and a second energy budget is computed separately at the lake-atmosphere interface based on a one-dimensional Freshwater Lake model (FLake) [], allowing to compute lake thermodynamics and surface fluxes, including evaporation. However, the water balance is not maintained since the lake tile is considered an infinite water reservoir with static volume and surface area. This second option is, therefore, not available when ORCHIDEE is coupled with an atmospheric model.

8.9.1. OK: Lake model for the energy budget: principle#

FLake is a two-layer thermodynamic lake model representing a well-mixed bulk layer above a thermocline, both modeled through a self-similarity approach. The model solves the lake energy budget and calculates the energy vertical transfers and the water temperature profile. Snow, ice and sediment may be accounted for through specific parametrizations that can be activated or not. FLake has been implemented in ORCHIDEE by and evaluated against lake surface temperature observations over about 1000 lakes sampled in the GloboLakes database []. Lakes are represented as a separate tile within the grid cell with no connections to the vegetated part. FLake was implemented in the sechiba code since it is forced by the same meteorological variables as the vegetated part of the grid and runs at the same time step of a few minutes. The resolution of the bulk energy budgets of the mixed layer and the thermocline allows us to model the following prognostic variables: mixed layer temperature and depth, water bottom temperature, ice and snow layer top temperatures and thermocline shape factor. Compared to the original FLake model, a few refinements were brought to the shape factor and albedo parameterizations, well described in . More recently, proposed to account for partial ice cover to better simulate the lake surface albedo and ice phenology in cold weather. A parameterization of the ice fraction derived from was implemented and used to compute the surface albedo of the lake tile. This development contributes to better matching the ice phenology of 200 lakes, especially the ice melting phase with a reduction of the errors in the prediction of the ice-off time, up to 18 days.

8.9.2. OK: Lake model description#

Put main equations of the model + figure potentially ? + Check all variable notations

The Flake model implemented in ORCHIDEE can predict the seasonal evolution of the vertical temperature structure within lakes at the global scale. The model described in , was developed by A. Bernus during its PhD at LSCE. FLake is based on a self-similar parametric representation of the four mediums described within lakes: water column, sediment layer, snow, and ice layers if present, and for which, the heat budget equations are resolved. It is a bulk model where the energy budget and entrainment equations are resolved based on a two-layer representation of the mixed layer (ML) and thermocline temperatures, as well as for the sediment layer. Lake water is treated as a Boussinesq fluid (constant density except for the calculation of the buoyancy term representing natural convection). The model is fully described in many papers []. Here, we recall the main equations resolved in the ORCHIDEE version and the specificities linked to the model coupling and new parameterizations used. Figure 5 presents the lake temperature profiles and the 10 variables that are prognostically computed, i.e., the mixed layer temperature $T^{ML}$ and depth $H^{ML}$, the water bottom temperature at the water-sediment interface $T^{sed}$, $H$ the level in the sediment layer where the temperature gradient is null and $T^{H}$ the respective temperature, the snow temperature $T^{snow}$ at the snow-atmosphere interface, the ice temperature $T^{ice}$ at the snow-ice interface and the snow $H^{snow}$ and ice $H^{ice}$ depths. Besides, surface momentum and heat fluxes are diagnosed at each time step provided by the atmospheric forcing. In ORCHIDEE, the calculations are done at the time step of Sechiba (30 minutes).

:::{figure} Figures/Flake_temp2.png :name: fig:Flake_temp :width: 70% :align: center

Schematic representation of the temperature profile in the four mediums represented in Flake (snow, ice, water and sediment layers). :::

8.9.3. OK: Thermocline temperature profile#

The self-similarity parameterization adopted to resolve the temperature profiles assumes that the temperature follows a universal function $\Phi^{T}$ of the normalized depth $\zeta$ both given by:

\[\Phi^{T}=\frac{T^{surf}-T(z)}{T^{surf}-T^{sed}}\]

Where $T^{surf}$ is the lake surface temperature.

\[\zeta=\frac{z}{D}\]

Where $T(z)$ is the temperature at depth $z$ and $D$ is the mean depth of the lake.

\[\begin{split}\Phi^{T}(\zeta) = \begin{cases} 1-(1-\zeta)^3 & \text{if } dH^{ML}/dt > 0 \\ 1-4(1-\zeta)^3 + 3(1-\zeta)^4 & \text{if } dH^{ML}/dt \leq 0 \\ \end{cases}\end{split}\]

8.9.4. OK: Energy budget equations#

The calculation of the temperatures at the water-atmosphere and the water-sediment interfaces requires the resolution of two energy budgets: one for the mixed layer and the other for the whole lake layer:

\[H^{ML}\frac{dT^{surf}}{dt}=\frac{1}{\rho^{w}c^{w}}\left[LW_{\downarrow}+(1-\alpha^{w})SW_{\downarrow}-Q^{ML}-I(H^{ML})\right]\]

Where $\rho^{w}$ is the density of water, $c^{w}$ is the specific heat capacity of water, $LW_{\downarrow}$ is the incoming long wave radiation, $SW_{\downarrow}$ is the incoming short wave radiation, $Q^{ML}$ is the heat flux at the interface between the mixed layer and the thermocline, $\alpha^{w}$ is the albedo of water, and $I(H^{ML})$ is the solar radiation that reaches the mixed layer.

\[D\frac{dT}{dt}=\frac{1}{\rho^{w}c^{w}}\left[LW_{\downarrow}+(1-\alpha^{w})SW_{\downarrow}-Q^{sed}-I(D)\right]\]

Where $Q^{sed}$ is the heat flux at the lake–sediment interface, and $I(D)$ is the solar radiation that reaches the lake–sediment interface.

The resolution of the equations requires a final closure equation with the assumption that the vertical heat flux also follows a self-similarity law:

\[\Phi^{Q}(\zeta)=\frac{Q^{ML}-Q}{Q^{ML}-Q^{sed}}\]

The integration between the mixed layer depth and the lake depth allows to calculate the evolution of the water temperature at a given depth $z$ with the following equation:

\[\frac{(D-h)^2}{2}\frac{dT^{surf}}{dt}=\frac{d}{dt}\left[c^{TT}(D-h)^2(T^{surf}-T^{sed})\right] = \frac{1}{\rho^{w}c^{w}}\left[c^{Q}(D-h)(Q^{ML}-Q^{sed})+(D-h)I(h)-\int_{h}^{D}I(z)\,dz\right]\]

Where $c^{Q}$ and $c^{TT}$ are two shape factors with respect to the heat flux and to the temperature respectively.

8.9.5. OK: Turbulent transfers: mixed layer depth#

To calculate the mixed layer depth, two cases are considered: if the convection is driven by the water density gradient or if it is driven by the surface wind. In the first case, the evolution of the mixed layer is given by:

\[A+\frac{c_{c2}}{w^{*}}\frac{dH^{ML}}{dt}=c_{c1}\]

Where $c_{c1}$ and $c_{c2}$ are empirical dimensionless constants, $A$ is the entrainment ratio and $w^{*}$ the convective velocity scale. In the second case, the conservation of the turbulent kinetic energy (TKE) allows to define an equilibrium mixed layer depth. Then, the evolution of the mixed layer depth to this equilibrium is calculated according to a relaxation time scale depending on the surface friction velocity.

8.9.6. OK: Sediment layer#

The sediment layer is parameterized with a two-layer parametric representation similar to what is done for the thermocline following . Empirical polynomial expressions are used to calculate the temperature profiles and the heat flux at the water-sediment interface. The attenuation depth of the annual thermal wave $H(t)$ and its corresponding temperature are calculated whereas bottom temperature and total sediment depth are prescribed.

\[\begin{split}T(z,t) = \begin{cases} T^{sed}(t)-\left[T^{sed}(t)-T^{H}(t)\right]\Phi^{B1}(\zeta^{B1}) & \text{if } D \leq z \leq H(t) \\ T^{H}(t)-\left[T^{H}(t)-T^{L}\right]\Phi^{B2}(\zeta^{B2}) & \text{if } H(t) \leq z \leq L \\ \end{cases}\end{split}\]

8.9.7. OK: Ice and Snow cover#

The vertical temperature profile in the ice and snow layers are also parameterized with a self-similarity approach. The normalized temperatures are given by the following equations, assuming linear profiles called $\Phi^{ice}$ and $\Phi^{snow}$ for the ice and snow layers respectively.

\[\begin{split}T(z,t) = \begin{cases} T^{freeze}-\left[T^{freeze}-T^{ice}(t)\right]\Phi^{ice}(\zeta^{I}) & \text{if } -H^{ice}(t) \leq z \leq 0 \\ T^{ice}(t)-\left[T^{ice}(t)-T^{snow}(t)\right]\Phi^{snow}(\zeta^{S}) & \text{if } -\left[H^{ice}(t)+H^{snow}(t)\right] \leq z \leq -H^{ice}(t) \\ \end{cases}\end{split}\]

Where $T^{freeze}$ is the freezing temperature of water.

The resolution of the energy budgets for the two layers with the boundary conditions on the temperatures at all interfaces allows to calculate the depth of both layers and their temperatures:

\[\begin{split}\begin{split} \frac{d}{dt}\Big\{ &\rho^{ice} c^{ice} H^{ice} \left[T^{freeze}-C^{ice}(T^{freeze}-T^{ice})\right] \\ &+ \rho^{snow} c^{snow} H^{snow} \left[T^{ice}-C^{snow}(T^{ice}-T^{snow})\right] \Big\} \\ &- \rho^{snow} c^{snow} T^{snow} \frac{d}{dt}(H^{ice}+H^{snow}) \\ &= Q^{s} + I^{s} - I(0) + \lambda^{ice} \frac{T^{freeze}-T^{ice}}{H^{ice}} \Phi^{ice'}(0) \end{split}\end{split}\]

Where $\rho^{ice}$ and $\rho^{snow}$ are the ice and snow densities, $c^{ice}$ and $c^{snow}$ are the specific heat capacities of ice and snow, $\lambda^{ice}$ and $\lambda^{snow}$ are the thermal conductivities of ice and snow.

\[-\lambda^{ice}\frac{T^{freeze}-T^{ice}}{H^{ice}}\Phi^{ice'}(1) = -\lambda^{snow}\frac{T^{ice}-T^{snow}}{H^{snow}}\Phi^{snow'}(0)\]

\[L^{fusion}\frac{d\rho^{ice}H^{ice}}{dt} = Q^{w}+\lambda^{ice}\frac{T^{freeze}-T^{ice}}{H^{ice}}\Phi^{ice'}(0)\]

Where $L^{fusion}$ is the latent heat of fusion of water, and $Q^{w}$ is the heat flux at the water–ice interface.

\[L^{fusion}\frac{d\rho^{snow}H^{snow}}{dt} = - (Q^{s}+I^{s}) +I(-H^{ice})+L^{fusion}\left(\frac{dM^{snow}}{dt}\right)_{a}+c^{snow}T^{freeze}H^{snow}\frac{d\rho^{snow}}{dt}\]

Where $\left(\frac{dM^{snow}}{dt}\right)_{a}$ is the snow accumulation associated with precipitation.

In these equations, the snow and ice layer albedos are key parameters determining the amount of energy available to melt the snow and ice mediums. In ORCHIDEE-FLake, the lake surface albedo is calculated according to the surface temperature. For free water, the albedo is set to a value of 0.07 which is the standard value used in FLake. In the presence of ice, possibly covered by snow, the lake surface albedo ($\alpha$) depends on the temperature as suggested by and varies between two limits corresponding to wet and dry snow (in presence of snow), and to blue and white ice (without snow), based on the same equation:

(8.20)#\[\alpha(T^{surf})=\alpha_{max}+(\alpha_{min}-\alpha_{max})\exp{\frac{-c^{\alpha}\left(273.15-T^{surf}\right)}{273.15}}\]

where $\alpha_{min}$ and $\alpha_{max}$ are respectively the minima and maxima values for ice or snow, $T^{surf}$ is the snow or ice surface temperature in Kelvin, which is always lower than the water freezing point temperature, and $c^{\alpha}$ is a fitted coefficient equal to 95.6 []. In Flake, the minimum and maximum albedos are equal to 0.1 and 0.6, respectively, for snow and ice. They were revised by following and , and set to 0.15-0.5 and 0.50-0.87 for ice and snow, respectively.

8.9.8. OK: Ice cover fraction#

In the original Flake model, the whole lake surface freezes when the surface lake temperature falls below 0°C and the ice thaws above this temperature threshold, regardless of the lake’s size. Actually, large lakes may present partial ice coverage, a feature which is important to simulate, if one wants to represent correctly the lake surface temperatures and fluxes in cold weather. To better represent such conditions, introduced in ORCHIDEE-Flake, the parameterization proposed by to derive the ice cover fraction from the calculated ice thickness. This fraction varies linearly between 0 when the lake is free of ice, to 1 above a certain threshold set to a critical value dependent on the wind fetch, in order to represent the fact that ice is more likely to break under the action of wind stress until it grows to a critical thickness. This critical ice thickness value $H_{crit}$ may be written:

\[H_{crit}=\frac{\tau^{a}}{P^{*}L}\]

Where $\tau^{a}$ is the scale of the surface wind stress (set to 0.15 Pa), $P^{*}$ is the compressive strength of ice (set to 27.5 kPa) and $L$ is the lake fetch (in meters).

For each lake tile, the fetch is static and prescribed to the mean of the fetch of all the lakes falling in the tile. It is estimated at the lake level from the surface extent by assuming a circular shape and taking the diameter of this circle. Given that we did not consider lakes smaller than 0.1 $km^2$ due to the limitations of the HydroLAKES database, it means that the fetch values range between a few meters and a few hundred kilometers, leading to critical thicknesses ranging between 2 mm for the smallest lakes to 1.1 m for the larger ones.

The ice cover fraction of the lake tile, $f^{ice}$, is then derived from the modeled ice thickness $H^{ice}$ using:

\[f^{ice}=\frac{H^{ice}}{H_{crit}}\]

The ice fraction is used in ORCHIDEE-FLake to simulate the lake albedo through a supplementary equation, linearly linking the lake albedo to the ice fraction and accounting for the presence of snow, if any. The lake tile surface albedo $alb_{tile}$ is now given by:

\[\alpha_{tile}(T^{surf})=f^{ice}_{tile}\alpha(T^{surf})+(1-f^{ice}_{tile})\alpha^{w}\]

where $\alpha^{w}$ is the free water albedo, $\alpha(T^{surf})$ is the snow albedo in snowy conditions and the ice one otherwise, both derived from Eq. (8.20) and dependent on the surface temperature of the lake tile.

9. The water cycle#

9.1. Hydrological framework: Water fluxes in the soil - plant - atmosphere continuum#

Ancien text a revoir ou supprimer compte tenu de la section rajoute au tout debut (conceptual framework)

The transpiration sink $s(z,t)$ in equation 5.1 describes the interplay between the transpiration flux, $E_t$, the soil moisture profile, and the root density profile, which is assumed to decrease exponentially with depth, with a decay factor $c_j$ depending on the PFT (Table Y):

\[R(z)=\exp(-c_j z)\]

Bare soil evaporation $E_g$ is a parallel flux to transpiration, which originates from the entire bare soil column, and from the bare soil fraction of the other soil columns. It is described based on a supply/demand approach:

\[E_g=\min(E_p^{*},Q_{up})\]

where the demand is defined by $E_p^*$, the potential evaporation reduced following , and the supply by $Q_{up}$, the maximum amount of water that can be extracted from the soil given the moisture profile (thus at the soil column scale). To prevent from mass conservation violation, $Q_{up}$ is estimated by dummy integrations of equation 5.1 at the end of each time step, in which two boundary conditions are successively tested: firstly a flux condition favoring the demand ($E_g=E_p^*$); then, if the previous case leads to any node with $\theta_i<\theta_r$, a Dirichlet condition ($\theta_1=\theta_r$ at the top node), which strongly limits $E_g$. The resulting water stress factor $\beta m_{G}$ ($E_g=\beta m_{G} E_p$, equation 3.y) will control $E_g$ and the surface energy budget of the next time step. If the total moisture in the top 4 soil layers (assumed to be representative of the litter) is below the wilting point, this stress factor is arbitrarily reduced by a factor 2. Eventually, $E_g$ can proceed at the potential rate of , unless water becomes limiting, i.e. if upward diffusion to the top soil layer cannot provide enough water to sustain the demanded rate.

[AD] This needs to be made consistent with sections 3.1 and 3.4. The case of the bare soil “PFT” needs to be checked, especially for the root decay factor (=5 in the default values). Check if a weighting by the vegetation fractions fvj is required somewhere for transpiration (idem for 1- fvj and bare soil evaporation).

A separate soil water budget is performed in each soil column within a grid-cell. At each time step, the liquid water input to a soil column comprises through-fall and snowmelt, averaged over the contributing PFTs. Given the spatial average of bare soil evaporation and transpiration over the contributing PFTs, the soil hydrology scheme integrates the soil moisture variations over the time step, calculates the output water fluxes (surface runoff and drainage), and diagnoses the water stress factors to calculate the forthcoming evaporative fluxes and feed the STOMATE module (crossref). As detailed in and , soil water redistribution in the soil implies 1-D vertical water fluxes,

In all cases, $D(\theta)$ and $K(\theta)$ are linearized in the variation range of $\theta$ (between the residual and saturated values, $\theta_r$ and $\theta_s$), by means of 50 piecewise functions, respectively constant and linear in $\theta$ (details in section 1.1.4) This allows a first-order linearization of Equation CITE EQUATION, which can thus be solved by a tridiagonal matrix algorithm. The prognostic variables are the volumetric moisture contents $\theta_i$ of each node $i$ within each soil column. Diagnostic soil moisture layers are also defined, with limits equidistant between two consecutive nodes, so that each layer holds one node. The moisture content of the layers ($W_i$ in mm) is then diagnosed assuming a linear $\theta$ profile between two consecutive nodes.

9.2. OK: Canopy interception and througfall#

As described in Section ??, the canopy interception flux is calculated for each PFT. It is derived from potential evapotranspiration, moderated by a resistance factor specific to the interception process. This resistance factor incorporates both the aerodynamic and vegetation structural resistances, and is further weighted by the fraction of vegetation that holds intercepted water $f^{inter}$, defined as:

(9.1)#\[f^{inter} = f^{veg}\frac{M^{inter}}{M^{inter,max}}\]

Where $f^{veg}$ is the fraction of vegetation covered by the PFT, $M^{inter}$ is the amount of water intercepted by that PFT per unit of surface (kg m$^{-2}$), and $M^{inter,max}$ is the maximum amount of water that can be intercepted by that PFT per unit of surface (kg m$^{-2}$).

The maximum amount of water intercepted is controlled by the leaf area index $LAI$ (m$^{2}$ m$^{-2}$) and an arbitrary constant $c^{inter} = 0.02$, which transforms $LAI$ into the size of the interception reservoir (unitless), following:

\[M^{inter,max} = c^{inter} f^{veg} LAI\]

The vegetation canopy is assumed to form a continuous layer, with gaps represented by a supplementary bare soil fraction. For each PFT, the intercepted water storage is first updated by subtracting the interception loss flux over the timestep $F^{ET,inter}$ (Eq. CITE EQUATION Section ??):

(9.2)#\[\delta M^{inter} = - F^{ET,inter}\]

This term represents the loss of water from the interception reservoir due to wet-canopy evaporation. In principle, rainfall first wets the canopy, and only the excess drips to the soil from the intercepted reservoir. In that idealized case, the increase of canopy water storage from a rainfall input $P^{rain}$ would be proportional to the vegetation cover fraction of the PFT: $\delta M^{inter} = f^{veg} P^{rain}$

In reality, however, not all intercepted water remains on the canopy: some evaporates back to the atmosphere (Eq. (9.2)), and some drips to the soil. These processes occur on short time scales (minutes), while the model time step averages precipitation over longer intervals. As a result, the model cannot explicitly resolve the rapid balance between interception, dripping, and evaporation. To account for this limitation, ORCHIDEE prescribes a throughfall fraction $f^{throughfall} = 0.3$ that forces a fraction of rainfall to reach the soil directly. The increase in canopy water storage over the model timestep is then reduced accordingly:

\[\delta M^{inter} = f^{veg} (1 - f^{throughfall}) P^{rain}\]

As mentioned previously, canopy storage is limited by a maximum capacity $M^{inter,max}$. When the reservoir exceeds this maximum amount, the canopy water storage is capped and the excess drips into the soil $F^{drip}$:

\[F^{drip} = M^{inter} - \min\left(M^{inter},M^{inter,max}\right)\]

The input of water to the ground for each PFT $F^{ground}$, therefore includes three contributions: direct throughfall, drip from canopy reservoir overflow, and rainfall onto the unoccupied fraction of the PFT:

\[F^{ground} = f^{veg} f^{throughfall} P^{rain} + F^{drip} + (f^{veg,max} - f^{veg}) P^{rain}\]

Where $f^{veg,max}$ is the maximum fraction of the PFT (unitless).

The volumetric ratio $\frac{M^{inter}}{M^{inter,max}}$ is then a proxy for the actual fraction of the vegetation surface covered with water.

In addition to rainfall, the canopy can also intercept dew (provided it is not freezing). Dew is assumed to behave differently from rainwater: intercepted dew never drips to the soil but always evaporates back to the atmosphere. The fraction of dew that can be intercepted by leaves $g^{dew}$ (unitless), is expressed as a 5-degree polynomial function of $LAI$ CITATION :

(9.3)#\[g^{dew} = 0.887773 + 0.205673 * LAI - 0.110112 *LAI^2 + 0.014843 *LAI^3 - 0.000824 *LAI^4 + 0.000017 *LAI^5\]

This parametrization is only applied for vegetated PFTs with $LAI>1.5$, otherwise $g^{dew}$ is set to 1.

9.3. OK: Water status of plants and water transport inside the vegetation#

The multi-layer soil hydrology scheme presented section ?? is partly driven by the vegetation transpiration. This amount of water transpired by the different PFTs (and thus extracted from each soil tiles) is controlled by the rate of aperture of the stomata. This rate of aperture, hereinafter mentioned as stomatal conductance, is regulated by several environmental and physiological factors such as the sensitivity to vapour pressure deficit, leaf sugar concentration and plant hydraulics []. This latter stomatal conductance driver, plant hydraulics (and their link to soil moisture), is an essential process to take into account to regulate stomatal aperture and leaves-atmosphere exchanges. It is usually modeled either through an empirical function directly applied to the stomatal conductance formulation [] or a full representation of the water flow inside the soil-plant-atmosphere continuum which enables the representation of the plant water potentials and a direct control of the stomatal conductance by the leaf water potentials []. In the ORCHIDEE-trunk, both formulations are proposed: i) the historical empirical control of stomatal conductance by soil moisture, described in section ??; or ii) a plant hydraulics control of stomatal conductance following the implementation of Alléon et al. (submitted), described in section ??.

9.3.1. OK: Empirical dependence#

In its standard configuration, the stomatal conductance formulation follows (more details on the photosynthesis resolution in section ??). In this model, stomatal conductance is directly driven by two empirical functions which describe respectively the sensitivity of the stomata to leaf-to-air vapour pressure deficit (approximated here by the air vapour pressure deficit (VPD)) and the sensitivity to soil moisture deficit. For each LAI layer $i$, the stomatal conductance then follows:

\[g_{s,i}=(c^{g_0}+\frac{F^A_i+F^{R_d}_i}{p^{C_{int}}_i-(p^{\Gamma_*}-F^{R_d}_i/k^{g_m})} \cdot G^{VPD}) \cdot G^{hum}\]

where $p^{C_{int}}_i$ and $p^{\Gamma_*}-F^{R_d}_i/k^{g_m}$ represent the inter-cellular $CO_2$ partial pressure and the $CO_2$ compensation point in absence of day respiration ( mol.mol$^{-1}$), $F^A_i$ and $F^{R_d}_i$ represent the assimilation rate and the day respiration ( mol[CO$_2$].m$^{-2}$.s$^{-1}$), $c^{g_0}$ is the residual stomatal conductance (mol[CO$_2$].m$^{-2}$.s$^{-1}$). Finally, G$^{VPD}$ and G$^{hum}$ represent the sensitivities to VPD and soil moisture deficit. They follow respectively:

(9.5)#\[\begin{split}\begin{align} & G^{VPD} = \frac{1}{\left(\frac{1}{(c^{a_1}-c^{b_1}\cdot p^{VPD}_i)}-1\right)}\\ & G^{hum} = \sum_{i>1} g^{us}_i\\ & g^{us}_1 = 0 \\ & \text{for}\,i > \,1,\,g^{us}_i = f^{root}_i \cdot \max \left(0,\min \left(1,\frac{M^{sm}_i-M^{smw}}{ \left(M^{sm,nostress}-M^{smw} \right)} \right) \right)\\ & M^{sm,nostress} = c^{p_\%} \cdot (M^{smf}-M^{smw}) + M^{smw} \end{align}\end{split}\]

Where $c^{a_1}$ (unitless, set to 0.85) and $c^{b_1}$ (which equals 0.14 kPa$^{-1}$) drives the sensitivity to $p^{VPD}$, $f^{root}_i$ corresponds to the fraction of root located in soil layer $i$, with $\sum_i f^{root}_i = 1$ $M^{sm}_i$, $M^{smf}$ $M^{smw}$, represent respectively the soil water contents (kg/m$^2$): in layer $i$, at field capacity, and at the wilting point. $M^{sm,nostress}$ is the threshold water content under which stomatal conductance sensitivity $G^{hum}$ starts to decrease linearly and which, with $c^{p_\%}$ (unitless, set to 0.8), puts the start of tranpiration stress at $80\%$ on the segment from $M^{smw}$ to $M^{smf}$. $G^{hum}$ is the sum of the soil moisture stress $g_i^{us}$ across layers, weighted by the root profile.

A non-linear alternative soil water stress can be calculated with the following equation :

(9.5)#\[\begin{align} & \text{for}\,i > \,1,\,g^{us}_i = f^{root}_i \cdot \max \left(0,\min \left(1,\exp \left(-\alpha^{watstress} \cdot \frac{M^{smf}_i-M^{smw}}{M^{sm,nostress} - M^{smw}} \cdot \frac{M^{sm,nostress}-M^{sm}_i}{M^{sm}_i - M^{smw}} \right) \right) \right) \end{align}\]

ADDED material from Agnes paper

The transpiration sink $S_i$ in Eq. (9.6) describes the interplay between the transpiration flux, $E_t$, the soil moisture profile, and the root density profile.

(9.6)#\[\begin{split}\begin{align} & S_i = g^{us}_i\, E_t / G^{hum} \\ & E_t = \sum_{i > 1} S_i \end{align}\end{split}\]

The resulting shapes of the transpiration stress factor are illustrated in Figure 6 for the various soil textures considered in ORCHIDEE.

:::{figure} Figures/agnes_adapted.png :name: fig:us :align: center

Variations of the transpiration stress factor $g^{us}_i$ as a function of mean volumeric water content in the soil layer $i$ (VWC), assuming a uniform VWC profile for simplicity. why is this assumption needed :::

Importantly, all above variables are actually defined at the PFT level, but the index of the PFT was omitted for simplicity. The aggregation at the grid-cell scale is additive, for both the transpiration flux and the total sink term $S_t$, which correspond to the same quantity by construction:

\[E_t = \sum_j E_t^j = \sum_j \sum_{i > 1} S_i^j = S_t\]

Eventually, in each PFT, the water stress factor $U_s$ is involved in two ways:

its value from the previous time step defines the root sink term thus the water budget of the current time step;
once updated at the end of the current time step, based on the corresponding soil moisture, it is used at the beginning of the following time step to calculate the stress function $\beta_3$ controlling the transpiration of the PFT thus the surface energy budget of the following time step.

9.3.2. Semi-empirical dependence#

Describe the Meridja function for humrel

9.3.3. OK: Plant hydraulics dependence#

ORCHIDEE-trunk now embarks a stomatal conductance dependence on leaf water potential. In this configuration, the photosynthesis model of is kept but the stomatal conductance formulation is revised and follows :

(9.7)#\[g_{s,i}=(c^{g_0}+\frac{F^A_i+F^{R_d}_i}{p^{C_{int}}_i-(p^{\Gamma_*}-F^{R_d}_i/k^{g_m})})G^{\psi^{leaf}}\]

Where $G^{\psi_{leaf}}$ (unitless) represents the sensitivity of stomata to a decrease in leaf water potential ($\psi^{leaf}$ in MPa):

(9.8)#\[G^{\psi^{leaf}} = \frac{1+exp(c^{s_f}.c^{\psi_{ref}})}{1+exp(c^{s_f}.(c^{\psi_{ref}} -\psi^{leaf}))}\]

This sigmoïdal function is driven by two PFT parameters: $c^{\psi_{ref}}$ (in MPa), the reference water potential which engenders approximately a 50 % stomatal closure and $c^{s_f}$ (in MPa$^{-1}$), the sensitivity to leaf water potential decrease.

The computation of $\psi^{leaf}$ relies on the representation of water transport from the soil-root interface towards the stomata. In this representation, two water storage pools (for the trunk and the leaves) are defined by capacitances and a set of fixed resistances is defined to model the water flow between each node (i.e root surfaces, trunk-root interface, trunk-trunk storage interface, trunk storage, trunk-leaf storage interface, leaf storage, stomatal cavities) via Ohm’s and Kirchoff’s current laws.

:::{figure} Figures/hydraulic_architecture_flow.jpg :name: fig:hydraulic_architecture:model :align: center

check whether the same symbols are used in the text and the figure. Adjust the figure to the text. Hydraulic Architecture resistance and capacitance network. For the resistances:(1) Mesophyll resistance ($R_{meso}$); (2) Trunk to leaf xylem resistance ($R_{trunk-leaf}$); (3) Root to trunk resistance ($R_{root-trunk}$); (4) Leaf storage resistance ($R_{leaf,store}$); (5) Trunk storage resistance ($R_{trunk,store}$); (6) Upper root system resistance ($R_{root,up}$); (7) Lower root system resistance ($R_{root,low}$). :::

9.3.4. Water storage pools#

Each water storage pool is defined as in through capacitances ($C^{X,store}$, in m$^3$.MPa$^{-1}$ with X representing either leaf or trunk storage pools) defined according to the storage pool water potential ($\psi^{X,store}$) by the following equation:

\[C^{X,store} = \frac{d M^{X,store}}{d \psi^{X,store}}\]

With,

\[\frac{M^{X,store}-M^{X,res}}{M^{X,max}-M^{X,res}} = \frac{1+exp(c^{\lambda} c^{\psi_0})}{1+exp(c^{\lambda} (-\psi^{X,store} + c^{\psi_0})}\]

Where, $M^{X,max}$ (m$^3$) is the maximum volume that can be stored in the pool (arbitrarily defined as 80 % of the biomass of carbon of the leaves and the sapwood as described in section ??), $M^{X,res}$ (m$^3$) is the residual volume of water of the pool (arbitrarily defined as 40 % of $M^{X,max}$ after personal communication with A. Tuzet), $c^{\lambda}$ (MPa$^{-1}$) is, according to a parameter depending on the morphological adaptation in different species and on environmental conditions experienced during growth and $c^{\psi_0}$ (MPa) is a reference water potential.

The dynamics of each storage pool rely on a differential equation (Eq. (9.9)) which links the storage compartment water potential ($\psi^{X,store}$) to the flux between the trunk-storage compartment interface and node above ($F^{up}$, in m$^3$.s$^{-1}$), to the resistances above and below the storage compartment ($R^{above}$ and $R^{below}$, in MPa.s.m$^{-3}$) and to the storage compartment resistance ($R^{X,store}$, in MPa.s.m$^{-3}$).

(9.9)#\[\frac{d \psi^{X,store}}{dt} = \frac{1}{C^{X,store}\cdot(R^{X,store} + R^{low})}\cdot(\psi^{low} - F^{up}\cdot R^{low}) - \frac{1}{C^{X,store}\cdot(R^{X,store} + R^{low})}\cdot\psi^{X,store}\]

Both storage pool associated differential equations are solved through a predictor-corrector scheme (a two-steps resolution where the state variable at time step $t+1$ is firstly predicted with a classical explicit resolution and then corrected by a second calculation involving the prediction). This resolution permits to avoid instabilities when calculating the new value of the storage water potential. In this resolution, the predictor term $\{\psi^{X,store}\}^{t,predict}$ is calculated and used as follows:

\[\{\psi^{X,store}\}_{t,predict} = \psi^{X,store}_t + \Delta t\cdot g(t,\psi^{X,store}_t)\]

\[\psi^{X,store}_{t+1} = \psi^{X,store}_t + \frac{\Delta t}{2}\cdot(g(t,\psi^{X,store}_t)+g(t,\{\psi^{X,store}\}_{t,predict}))\]

Where, $f(t,\psi^{X,store}_t)$ corresponds to the right-hand side of Eq. (9.9).

This predictor-corrector scheme enables to approximate the value of the storage pool water potential at the end of the time step which, along with the value at the beginning of the time step, leads to the calculation of the water fluxes coming from or inside the storage pool.

9.3.5. Water absorption by roots#

At the soil-root interface, the water absorption by roots is computed in two soil horizons (averaged from the multi-layer soil hydrology scheme presented in section ??). It can be represented by two different approaches.

In the simpler approach, a dynamic resistance is applied at the soil root interface. Its formulation (Eq. (9.10) for a soil layer $i$) follows the representation implemented by (Section A4.3, Equations A23 to A25).

(9.10)#\[R^{rootsurf-root}_i = \frac{ln(1 / (\pi L^{roots}_i)^{1/2} r^{roots}}{2\pi L^{roots}_i \Delta z_i k^s_i}\]

Where, $L^{roots}_i$ is the total root length per unit volume of soil in the soil horizon (m.m$^{-3}$ - calculated according to root biomass), $\Delta z_i$ is the soil horizon height (in m), $r^{roots}$ is the mean fine roots radius (which equals 0.5 mm) and $k^s_i$ is the hydraulic conductivity in layer $i$ (in mm.day$^{-1}$).

In this framework, the soil horizon water potential and hydraulic conductivity are calculated following either formulation or in order to avoid the numerical instabilities which can appear when the soil water content is close to the residual soil water content.

In the more complex framework, the water absorption by roots is represented following . In this mechanistic representation, the volume of soil in the horizon is considered cylindrically distributed around a fictive root of length $L_{roots_i}$. In this cylinder of soil, Richard’s equation (Eq. (9.11)) is solved radially which enables to compute the gradient of moisture content near the root.

(9.11)#\[\frac{\partial \theta}{\partial t} = \frac{1}{r}\cdot \frac{\partial}{\partial r}(r\cdot D(\theta)\cdot \frac{\partial \theta}{\partial r})\]

where $D(\theta)$ (mm$^2$.day$^{-1}$) is the soil diffusivity and $r$, the distance from the root axis. Richard’s equation is solved at a half hourly time step for $n^{muff}$ nodes along the radial axis. At the soil-root interface ($i=0$), the absorbed water flux ($F^{root,up}$ or $F^{root,low}$, in m$^3$.s$^{-1}$, respectively for the upper and lower soil horizons) is applied as a boundary condition. The radial resolution of Richards’ equation is performed using the same resolution framework as the vertical water diffusion presented before (see Section ??). This resolution enables the calculation of the water potentials at the root surfaces ($\psi^{rootsurface,up}$ and $\psi^{rootsurface,low}$), by considering a continuity between the soil water potential at the node $i=0$ and the root surface.

9.3.6. Numerical resolution#

During unstressed periods, the resolution of the hydraulic architecture is purely explicit. At the beginning of the time step, photosynthesis (section ??) is solved and $g^s_i$ is the first calculated variable, using Eq. (9.7) and the leaf water potential from the previous time step. Then, the resolution of the energy budget (sections ?? & ??) enables the calculation of the transpiration flux, $F^{transp}$ using $g^s_i$. The different water fluxes within the hydraulic system are computed from the top to the bottom imposing $F^{transp}$ flux as a boundary condition at the leaf level. The flux between each storage pool and the nearest node ($F^{trunk,store}$ or $F^{leaf,store}$) is assessed following the predictor-corrector scheme described in the previous section. Using Kirchhoff’s current law and the storage water fluxes, all water fluxes can be determined down to $F^{root-trunk}$, the flux between the trunk and the trunk-root interface, above the two root horizons. At this interface, the two fluxes of water uptake by roots, $F^{root,up}$ and $F^{root,low}$, are assessed from $F^{root-trunk}$ and the root water potentials of the previous time step with Eq. (9.12).

(9.12)#\[F^{root,low}_{t+1} = \frac{F^{root-trunk}_{t+1}+\frac{\psi^{rootsurf,low}_t - \psi^{rootsurf,up}_t}{R^{root,up}}}{1+ \frac{R^{root,low}}{R^{root,up}}}\]

Based on the two root water fluxes, water absorption by roots is solved. The water contents in the two soil layers are updated as well as the resulting root water potentials $\psi^{rootsurf,up}$ and $\psi^{rootsurf,low}$. The other water potentials of the hydraulic system are then updated, from the bottom to the top, using the previously calculated fluxes and the water potential of the compartment down the flux. Thus, the updated water potential at a given level ($\psi^{upper,level}$) is expressed as :

\[\psi^{upper,level} = \psi^{lower,level} - F^{level}\cdot R^{level}\]

Where $\psi^{lower,level}$ is the water potential at the level down the flux; and $F^{level}$ and $R^{level}$ are the flux and the resistance between the two levels, respectively.

During drought periods, the non linearity of the stomatal conductance formulation and the strong coupling between stomatal conductance, transpiration and leaf water potential can introduce strong feedbacks between the variables, further leading to numerical instabilities which would justify the use of an iterative resolution scheme (Alléon et al. (submitted), ). In ORCHIDEE-trunk, this drawback has been partly solved by implementing an adapted resolution scheme.

A first resolution of the hydraulic architecture model is performed using the explicit framework presented previously. This first calculation is purely diagnostic and enables the calculation of the variables at the current time step. A second resolution of the hydraulic architecture model is then launched. The objective of this second calculation is to approximate the value of the leaf water potential at the next time step in order to avoid a resulting stomatal conductance too far from reality at the next time step. To do this, an approximation of the transpiration flux at the next time step is performed. The transpiration equation can be summarized as:

(9.13)#\[F^{transp} = \frac{\rho\cdot\Delta t\cdot\Delta q}{R^{aero} + R^g}\]

where $\rho$ is the air density (kg.m$^{-3}$), $\Delta t$ is the time step length, $\Delta q$ is the difference of humidity between the surface and the atmosphere (kg.kg$^{-1}$), $R^{aero} = \frac{1}{\vert \overrightarrow{V}\vert C_d}$ is the aerodynamic resistance (m.s$^{-1}$) and $R^g = 1/\sum_i g^s_i\cdot f^{veg}_i$ is the stomatal resistance (m.s$^{-1}$). This equation can be simplified to:

(9.14)#\[F^{transp} = \frac{A}{B + \frac{1}{C\cdot g^{\psi}}}\]

The transpiration at time step $t$ and at time step $t+1$ can be expressed as:

(9.15)#\[F^{transp}_t = \frac{A_t}{B_t + \frac{1}{C_t\cdot g(\psi^{leaf}_{t})}} = \frac{A_t}{B_t + \frac{1}{C_t\cdot g^{\psi}_t}}\]

(9.16)#\[F^{transp}_{t+1} = \frac{A_{t+1}}{B_{t+1} + \frac{1}{C_{t+1}\cdot g(\psi^{leaf}_{t+1})}} = \frac{A_{t+1}}{B_{t+1} + \frac{1}{C_{t+1}\cdot f^{\psi}_{t+1}}}\]

The objective here is to estimate $F^{transp}_{t+1}$ from $F^{transp}_t$. Looking at the components of the transpiration:

$A_t$ and $A_{t+1}$ represent the evolution of $\Delta q$ over the day;
$B_t$ and $B_{t+1}$ represent the evolution of the aerodynamic resistance;
$C_t$ and $C_{t+1}$ represent the evolution of the components of $g^s_i$ that are not directly linked to $g^{\psi}$.

The following assumptions are made:

$A_{t+1}$ = $A_t \cdot \delta$;
$C_{t+1}$ = $C_t \cdot \delta$;
$B_{t+1}$ = $B_t$;
$\delta$ can be estimated by a sinusoidal function centered on solar noon.

The assumption is, hence, that in the absence of water stress, transpiration and stomatal conductance will follow a sinusoidal function centered on the solar noon, starting at sunrise and ending at sunset. Following this assumption, $\delta$ can be expressed as the ratio between the values of the sinusoidal function at time step $t$ and $t+1$. The amplitude of the sine function cancels out in the ratio.

Consequently, the ratio between $F^{transp}_{t+1}$ and $F^{transp}_t$ can be estimated as:

(9.17)#\[\frac{F^{transp}_{t+1}}{F^{transp}_t} = \frac{\frac{A_{t+1}}{B_{t+1} + \frac{1}{C_{t+1}\cdot g^{\psi}_{t+1}}}}{\frac{A_t}{B_t + \frac{1}{C_t\cdot g^{\psi}_t}}} = \delta \cdot \frac{1 + \frac{\vert \overrightarrow{V}\vert \cdot C_d}{\sum_i g^s_{i,t}}}{1+\frac{\vert \overrightarrow{V}\vert \cdot C_d\cdot g^{\psi}_t}{\delta \sum_i g^s_{i,t} g^{\psi}_t}}\]

This ratio, multiplied by $F^{transp}_t$, permits to have a prediction of $F^{transp}_{t+1}$ only with variables from the previous time step. This approximation enables the second solving of the hydraulic architecture and the estimation of the leaf water potential at the next time step. As the transpiration estimation is relying on the leaf water potential value at the same time step, several iterations are performed in order to make the resolution converge towards a unique leaf water potential value which will be sent to the photosynthesis module at the next time step. One can note that the scheme can be improved by improving the transpiration flux estimation.

9.4. OK: Snow and land ice#

The multi-layer snow scheme described and evaluated in is based on the ISBA-ES model () and has been implemented on the vegetated/biological fraction of the ORCHIDEE grid cell. Its extension to glaciers has been validated over Greenland ice-sheet () and replace the old single-layer scheme described in Chalita and Letreut, 1994. This multi-layer scheme is used to describe the hydro-thermal processes within the snowpack and solve the mass and energy balance equations.

To describe the snowpack evolution, a number of three layers was chosen, for which three prognostic variables (i.e., snow heat content, density and thickness) are calculated. Temperature and liquid water content are diagnosed at each time step and for each layer. The total snow mass and its mean density is then used to derive a snow cover fraction that enters the calculation of the surface albedo and roughness of each grid cell that contains snow, to account for the high albedo and smoothing impacts of snow .

The multi-layer snow scheme describes the energy and mass transfers between the snowpack and the atmosphere and the main internal processes at play such as snow settling, water percolation and refreezing. The calculations are done in the following way: at each time-step, the surface flux entering the top snow layer is computed from the surface energy balance equation. Snowfall updates the thickness, density, heat content, temperature and water content of the snow layers. Compaction is then modeled, modifying layer thicknesses, densities and heat content (equations (11)-(12) in ). Temperature is then updated, from top to bottom, taking into account the new surface temperature and the integration coefficients for the snow numerical scheme, similarly to the soil temperature scheme (ref Hourdin yyyy ?) to stick with the implicit scheme. Snowmelt is then calculated, followed by liquid water transfers and possible refreezing, energy transfers during these phase changes are accounted to update temperature, layer thickness, density and liquid water content. Sublimation is then calculated followed by snow aging, snow cover fraction and finally the surface albedo as presented in section ??

9.4.1. OK: Snow energy balance#

The surface flux entering the top snow layer is computed from the surface energy balance equation in the same way as for soil. At the surface of the snow, the energy budget equation is:

(9.18)#\[C_{surf} \frac{\partial T_s}{\partial t}=Frad+F1h+LF1+H1+G1\]

Where $G1$ is the soil heat flux due to heat conduction process; $H1$ is the energy released by rainfall (see Eq. (14) in ). $Frad$, $F1h$, and $LF1$ are the net radiation, sensible heat and latent heat flux respectively ($W\,m^{-2}$); $C_{surf}$ is the surface heat capacity of soil per unit area ($J\,m^{-2}\,K^{-1}$) and is computed as the sum of heat capacities for snow-covered and snow-free surfaces weighted by their respective grid cell fraction.

The heat conduction flux into the soil $G1$ is used to compute the surface temperature $T_{surf}$ of the grid cell at the next time step and provides the limit condition of the surface temperature at the snow-atmosphere interface for the calculation of the snow temperature profile.

Above snow-covered surfaces, when $T_{surf}$ is above the freezing temperature $T_0$ (273.15 K), the energy excess is first used to bring the snow temperature to $T_0$. A surface energy flux $F_{freezing}$ associated with the freezing temperature $T_0$ can be computed using a similar formulation to Eq. (9.18). The difference between $T_{surf}$ and $T_{freezing}$ is converted in an additional temperature expressed as:

\[T_{add\_snow} = T_{surf} - T_0 = \frac{G1 - G_{freezing}}{C_{soil}} dt\]

If $T_{add\_snow}$ is greater than $T_0$, the energy excess is used for melting snow, and $G1$ is further set to $G_{freezing}$ for energy conservation. If the new $G1$ value is greater than the total heat content of the snowpack, snow is entirely melted and the excess energy is transferred to the underlying soil. The energy released by snowfall is accounted for in the snowpack scheme to update the snow heat content of the snowpack after a snowfall event.

The sensible $F1h$ and latent heat $LF1$ fluxes computed for each grid cell are given respectively by:

\[F1h = \rho_{air}q_{cdrag}U(T_{surf} - T_{air})\]

\[LF1 = L_{s}\rho_{air}q_{cdrag}U(Q_{sat} - Q_{air})\]

where $\rho_{air}$ is the air density, $T_{surf}$ and $T_{atm}$ are the surface and the 2 m atmospheric temperatures, $Q_{air}$ and $Q_{sat}$ are the air specific humidity at 2 m and the saturated specific humidity at the surface, $L_s$ is the latent heat of sublimation (2.8345 106 J kg-1), U is the wind speed at 10 m and $q_{cdrag}$ is the drag coefficient computed as a function of the ice roughness length ($z0_{ice} = 0.001 m$), following the Monin-Obukhov turbulence similarity theory () and the parameterizations of the eddy fluxes proposed by .

9.4.2. OK: Snow layering#

In the initial version of ORCHIDEE explicit snow, the snow was distributed over 3 layers, with the first layer limited to a depth of 5cm and the second to 50cm. This vertical discretisation described in is still available but we now use a new discretisation with 12 layers. With this new version, the maximum thickness of the first 3 layers is respectively 1, 5 and 15cm, which limits the bias on the surface temperature. An application to the Greenland ice sheet has shown that this new 12 layers scheme provides a better simulation of snow evolution than the previous 3 layers model. The definition of snow layer thickness followed the layering scheme proposed by for ISBA-ES: In theses equations, $i$ identifies the snow layer and D the snow layer thickness

(9.19)#\[\begin{split}\begin{cases} D_i=min\left(\delta_i,\frac{D_{snow}}{12}\right) , & \forall i \; \leq 5 \; or \; \forall i \geq 9\\ D_6=0.3d_r - min(0,\; 0.3d_r - D_5)\\ D_7=0.4d_r + min(0,\; 0.3d_r - D_5) - min(0,\; 0.3d_r - D_9)\\ D_8=0.3d_r - min(0,\; 0.3d_r - D_9)\\ d_r=z_{snow} - \Sigma_{i=1}^{5} D_i - \Sigma_{i=9}^{12} D_i \end{cases}\end{split}\]

where the constants $\delta_i$ are the maximum snow depth for a layer $i$ with $\delta_1=0.01\,m$, $\delta_2=0.05\,m$, $\delta_3=0.15\,m$, $\delta_4=0.5\,m$, $\delta_5=1\,m$, $\delta_9=1\,m$, $\delta_{10}=0.5\,m$, $\delta_{11}=0.1\,m$, $\delta_{12}=0.02\,m$. For very thin snowpack (less than 0.1 m), each layer has the same thickness and when the snow depth is above 0.2 m, the first and last layers reach their constant values of 0.01 and 0.02 m. The layer thickness are updated if one of the first two layers or the bottom layer become too thin or too thick:

\[\begin{split}\begin{cases} D_i<\frac{1}{2}min\left(\delta_i,\frac{D_{snow}}{12}\right) , & or D_i>\frac{3}{2}min\left(\delta_i,\frac{D_{snow}}{12}\right)\\ \forall \; i = \{1,2,12\}. \end{cases}\end{split}\]

For example, if the thickness of the top layer becomes lower than 0.005 m or greater than 0.015 m, all the layer thicknesses are recalculated with Eq. (9.19) and the snow mass and heat content are redistributed accordingly.

9.4.3. Ok: Snow compaction#

When a snowfall occurs, the snow depth increases, but the depth decreases with compaction. This is why compaction is a particularly important process for the evolution of the density and depth of snow in each layer. Two processes are responsible for compaction: the weight of the overlying snow layers and the metamorphism of the snow. These two processes are taken into account in the following equation derived from :

(9.20)#\[\frac{1}{\rho_i} \frac{\partial \rho_i}{\partial t} = \frac{\sigma_i}{\eta_i(T_i,\rho_i)} + \xi_{i}(T_i,\rho_i)\]

The first term of the right-hand side of equation (9.20) represents the compaction due to snow load with $\sigma_i$ (Pa) being the pressure of the overlying snow and $\eta_i$ (Pa.s) the snow viscosity. The viscosity is an exponential function of snow temperature and density (equation (9.21), and ):

(9.21)#\[\eta_i = \eta_0 \; exp \left[ a_{\eta} (T_f - T_i) + b_{\eta} \rho_i \right]\]

with $\eta_0$=3.7 10$^{7}$ Pa s, $a_{\eta}$=8.1 10$^{-2}$ K$^{-1}$ and $b_{\eta}$=1.8 10$^{-2}$ m$^3$ kg$^{-1}$.

The second term $\xi$ of the right-hand side of equation (9.20) represents the effect of metamorphism:

\[\xi = a_{\xi} exp \left[ -b_{\xi} (T_f - Ti) - c_{\xi} max(0,\rho_i - \rho_{\xi})\right]\]

with a$_{\xi}$ = 2.8 10$^{-6}$ s$^{-1}$, b$_{\xi}$ = 4.2 10$^{-2}$ K$^{-1}$, c$_{\xi}$ = 460 kg m$^{-3}$ and $\rho_{\xi}$ = 150 kg m$^{-3}$. In the model, the density cannot exceed a threshold fixed at 750 kg m$^{-3}$. Compaction does not affect the total mass and the heat content of the snowpack but changes the layer thicknesses, this is the reason why the snow heat within the layers must therefore be updated.

9.4.3.1. OK: Heat content#

For processes linked to phase change, we use the equation of the heat content of snow (equation (9.22)). This allows to determine whether the snow is cold (dry) or warm (wet). The heat content of each snow layer is computed using the following equation:

(9.22)#\[H_i = D_i \left[ c_i (T_i - T_f) - L_f \rho_i \right] + L_f \rho_w W_{liq,i}\]

where L$_f$ is the latent heat of fusion and T$_f$ the triple-point temperature for water. H$_i$ is the total heat content, c$_i$ the snow heat capacity (J.m-2 K-1) and W$_{liq,i}$ liquid content for the i$^{th}$ layer. According to equation (9.22), heat content is used to diagnose the snow temperature assuming that there is no liquid water in the snow layer. If the calculated snow temperature exceeds the freezing point, snow temperature is set to T$_f$ and the liquid water content is diagnosed from equation (9.22).

9.4.3.2. OK: Melt and refreezing#

When melting occurs at the surface, the liquid water remains in the first layer if the maximum liquid water threshold has not been exceeded. When the amount of liquid water exceeds this threshold, the water percolates into the lower layer. The water can then either refreeze or be added to the existing liquid water, depending on the temperature of the snow layer. The maximum liquid water holding capacity is taken as a function of the snow layer density following .

\[W_{liq,i,max} = \left[ r_{min} + (r_{max} - r_{min} max\left( 0, \frac{\rho_t - \rho_i} {\rho_t} \right) \right] \frac{\rho_i}{\rho_w} D_i\]

with $r_{min}$=0.03, $r_{max}$=0.10 and $\rho_t$=200 kg m$^{-3}$.

When the liquid water reaches the bottom layer and exceeds the total liquid water content, it is then considered as runoff.

9.4.3.3. OK: Heat conduction#

The heat conduction from the surface to the bottom of the snowpack is described by a vertical diffusion equation relating the temporal evolution of the snow temperature in the snowpack at a depth z and the divergence of the snow heat flux F$_c$ and is solved using an implicit numerical scheme.

\[\frac{\partial T_i}{\partial t} = -\frac{1}{c_i} \frac{\partial F_c}{\partial z}\]

\[F_c = -\Lambda_i \frac{\partial T_i}{\partial z}\]

with $\Lambda$ the snow thermal conductivity (W m$^{-1}$ K$^{-1}$). At the snow-atmosphere interface, the boundary condition is given by the energy balance equation (9.18).

Along with the thermal gradient, a water vapor diffusive flux takes place from the warmer to the colder parts of the snowpack and sublimation or condensation may occur in the pore spaces depending on the water vapor saturation pressure. This process is particularly significant in the Arctic because of strong temperature gradients between soils and atmosphere and is in great part responsible for snow metamorphism. While it is explicitly accounted for in detailed snow models, in Explicit Snow, the effect of water vapor diffusion and phase changes is parameterized through the thermal conductivity (). The thermal conductivity $\Lambda$ is expressed as the sum of empirical formulations for snow thermal conductivity $\Lambda_{cond}$ (eq. (9.23)) and thermal conductivity from vapor transport $\Lambda_{vap}$ (eq. (9.24)) with:

(9.23)#\[\Lambda_{cond,i}=a_{\lambda c} + b_{\lambda c} \rho_{i}^2\]

(9.24)#\[\Lambda_{vap,i}=a_{\lambda v} + \frac{b_{\lambda v}}{c_{\lambda v}+T_i} \frac{P_0}{P}\]

where a$_{\lambda c}$ = 0.02 W m$^{-1}$ K$^{-1}$, b$_{\lambda c}$ = 2.5 10$^{-6}$ W m$^{-1}$ K$^{-1}$ (), a$_{\lambda v}$ = -0.06023 W m$^{-1}$ K$^{-1}$, b$_{\lambda v}$ = -2.5425 W m$^{-1}$ and c$_{\lambda v}$ = -289.99 K (). P is the atmospheric pressure in hPa and P$_0$ = 1000 hPa.

9.4.3.4. OK: Ice layers#

In previous versions of ORCHIDEE, ice-covered surfaces were simulated using the single layer snow scheme described in . Now, the multi layer snow model is used for all surface types. Furthermore, in order to be able to simulate the surface mass balance on glaciers and ice sheets, the treatment of ice-covered areas has been modified (). To take account of the presence of ice, we have added an ice module between the ground and the snow. This ice reservoir is made up of 8 layers of fixed thickness (0.01, 0.05, 0.15, 0.5, 1, 5, 10 and 50 m). A finer vertical spacing is imposed for the upper layers to better resolve heat conduction at the snow-ice or atmosphere-ice interface. The large thickness of the bottom layer allows it to have an almost constant temperature throughout the year as it has been observed at a few tens of meters depth (). The temperature in ice is obtained using the same numerical scheme as for the snow. The formulation of heat capacity $C_{ice}$ and thermal conductivity $\lambda_{ice}$ of the ice are coming from .

\[C_{ice} = \rho_{ice} \left(a_{c_ice}+b_{c_ice}(T_{ice} - T{0}) \right)\]

\[\lambda_{ice} = a_{\lambda_ice} exp(b_{\lambda_ice}T{0})\]

where $T_{ice}$ is the ice temperature, $a_{c_ice}$=2115.3 J K$^{-1}$ kg$^{-1}$, $b_{c_ice}$ = 7.79293 J K$^{-2}$ kg$^{-1}$, $a_{\lambda_ice}$= 6.727 W m$^{-1}$ K$^{-1}$ and $b_{\lambda_ice}$= -0.041 K$^{-1}$. Ice is considered as an impermeable medium, hence liquid water coming from melting ice is considered to runoff instantaneously with no possibility of refreezing.

9.4.3.5. OK: Snow cover fraction#

Snow cover fraction (SCF) is parameterized according to snow average thickness and density following Niu and Yang (2007) with revised values for minimal density of fresh snow $\rho_{min}$ (prescribed now to 50 kg/m2), ground roughness length $z_{0g}$ equal to 0.01 m , and a value of 1 for the adjustable parameter “m”.

\[SCF= tanh \left( \frac{D_{snow}}{2.5 \, z_{0g} \, \left(\frac{\rho_{snow}} {\rho_{new}}\right)^m} \right)\]

The thermal properties such as the heat capacity and thermal conductivity are also computed for each snow layer to describe the heat conduction within the snowpack and the temperature profile. (equation (5) in ). Finally, the integration coefficients for the snow thermal scheme are computed, just after the ones for the soil in the same bottom-up direction. This ensures that the multi-layer snow module completely respects the implicit scheme.

SCF is then used to weight the albedo and surface roughness of the grid cell containing glaciers. Snowmelt and runoff are calculated at each time step when surface temperature is above freezing. Soil thermics is calculated by replacing the uppermost soil layers by snow according to snow depth. Heat capacity and thermal conductivity of snow are used for the layers filled by snow.

9.5. Soil evaporation#

$Q_0$ (see XXX) is defined by the difference between infiltration into the soil (section ??) and soil evaporation, $E_g$.

The latter is a parallel flux to transpiration, which originates from the entire bare soil column, and from the bare soil fraction of the other soil columns. It is calculated using a supply/demand approach, assuming it can proceed at the potential rate (Eq. %s), unless water becomes limiting, i.e. if the upward diffusion to the top soil layer cannot provide enough water to sustain the required potential rate. If this is not the case, the diffusion needs to be recalculated assuming a lower soil evaporation.

*** Explain a bit the correction of Milly *** The potential rate is now calculated using the correction of . The corrected potential evaporation is further called $E^*_{\rm{pot}}$.

The limitation of bare soil evaporation by upward diffusion is estimated based on a dummy integration of Richards redistribution. *** better articlulate with the following

In each soil column $c$, we define the stress factor $\beta_g^c$ aiming at controlling the calculation of bare soil evaporation $E_g^c$, by

(9.25)#\[\begin{split}\begin{align} & \beta_g^c = E_g^c / E_{\rm{pot}}, \\ & E_g^c=\min(E^*_{\rm{pot}},Q^c_{\rm{up}}). \end{align}\end{split}\]

The demand is defined by $E^*_{\rm{pot}}$, the potential evaporation reduced following , and the supply by $Q^c_{\rm{up}}$, the maximum amount of water that can be extracted from the soil column given the moisture profile (thus at the soil column scale).

To prevent from mass conservation violation, $Q^c_{\rm{up}}$ is estimated by dummy integrations of the Richards redistribution equation at the end of each time step, in which two boundary conditions are successively tested:

firstly a flux condition favoring the demand ($E^c_g=E^*_{\rm{pot}}$),
then, if the previous case leads to any node with $\theta^c_i < \theta_r$, a Dirichlet condition ($\theta^c_1 = \theta_r$ at the top node), which strongly limits $E^c_g$.

In both cases, if the total moisture in the top 4 soil layers (assumed to be representative of the litter) is below the wilting point, $E_g^c$ thus $\beta_g^c$ are arbitrarily reduced by a factor 2. Eventually, $E^c_g$ can proceed at the potential rate of , unless water becomes limiting, i.e. if upward diffusion to the top soil layer cannot provide enough water to sustain the demanded rate. This is usually less frequent than the former case, but, as $E^*_{\rm{pot}} < E_{\rm{pot}}$ in most conditions, $\beta_g^c < 1$.

In ORCHIDEE-2.0, it is possible to reduce the demand, thus the soil evaporation, owing to a soil resistance following the formulation of :

(9.26)#\[\begin{align} & r_{\rm{soil}} = \exp(8.206 - 4.255 W_L/W_L^s), \end{align}\]

(9.27)#\[\begin{align} & E_g^c=\min(E^*_{\rm{pot}}/(1+r_{\rm{soil}}/r_a),Q^c_{\rm{up}}). \end{align}\]

$W_L$ is the soil moisture in the top 4 layers of the soil (litter zone, corresponding to 2.5 cm with the default vertical discretization, including soil ice), $W_L^s$ is the corresponding moisture at saturation, and $r_a$ is the aerodynamic resistance (Eq. %s).

The factor $\beta_g^c$ (thus the fluxes $E_g^c$ and $Q^c_{\rm{up}}$) is estimated after the update of the soil moisture profile as a function of the surface forcing (precipitation and evaporation fluxes required by the current time step’s energy budget). It will serve to control the bare soil evaporation and surface energy budget of the following time step, after spatial averaging towards the grid-cell scale, using the soil column areas as weights***

*** This leads to a single $\beta_4^b$, which is constrained to be smaller than $1-\beta_2^b-\beta_3^b)$, due to the fact that bare soil evaporation is possible together with transpiration and interception loss from the same PFT, and we should in no case have a higher than potential ETR rate. Priority is given to transpiration (and IL), which somehow balances the fact that $\beta_g^c$ is defined by a “dummy” integration of the soil water diffusion which neglects the root sink. Thus, we can consider that the dummy integration only deals with the evaporation from the bare soil fraction, independently from the transpiration, supposed to act on the vegetated fraction.***use notations for the fractions

*** Clarify and check the following: This expression uses the bulk stress functions $\beta^b$ (called vbeta in the code), which come from the local stress functions of section XXX after multiplication by the grid-cell fraction from which the flux originates ($A_v/A_L$ for the interception loss and transpiration, $A_g/A_L$ for the bare soil evaporation). Note that diffuco uses evapot ($E_\mathrm{pot}$) and evapot_penm ($E^*_\mathrm{pot}$) that were calculated during the previous time step (by enerbil which is called after diffuco in sechiba.f90), i.e. the same timestep at which $\beta^c_g$ and evap_bare_lim(ji) were calculated.

Eventually, we get:

\[\beta_4^b = \frac{A_g}{A_L} \beta_4 = \frac{A_g}{A_L} \min(\mathrm{supply}, \mathrm{demand})/E_\mathrm{pot}.\]

9.6. Wetlands#

Wetlands which impact surface hydrology are represented in the routing scheme. We distinguish two types of wetlands: inland wetlands which include floodplains, swamps, ponds and endorheic lakes and man-made wetlands, that is, irrigated lands. Global coverage of inland wetlands is derived from the Global Lakes and Wetlands Database map (GLWD, Lehner and Döll, 2004). Maximal fraction of each types of wetland is given by the aggregation of several fields of the database. The correspondence will be detailed further for each type of wetland.. The maximal fractions of irrigated areas are prescribed by the global map of estimated area of each 0.5° grid box equipped for irrigation around 1995 (Döll and Siebert, 1999, 2000, 2002) and up to 1999 for Europe and Latin America (Siebert and Döll, 2001).

Floodplains are land areas adjacent to streams that are subject to recurring inundation. The stream overflows its banks onto adjacent lands. This can deeply modify the discharge of rivers such as the Niger () or the Amazon (). Floodplains type in ORCHIDEE is the aggregation of three fields of the GLWD map: “Reservoir”, Freshwater marsh-Floodplain” and “Pan-Brackish/Saline wetland”. Over these areas, the streamflow from head waters of the reservoirs flows into a reservoir of floodplains instead of the stream reservoir of the next downstream.

Over some regions (Amazonia, Niger, Congo…), swamps can store water which saturates, infiltrates into the soil and does not return to the river. Swamps type in ORCHIDEE corresponds to the field “Swamp forests, flooded forests” of the GLWD map. Over these areas (), we compute a fraction of water (α = 0.2) that is uptaken from the stream reservoir and transferred into soil moisture without returning directly to the river.

Ponds type in ORCHIDEE corresponds to the field “Intermittent wetland/lake” of the GLWD map. To be developed next…

9.7. WIP (FK): Soil hydrology#

9.7.1. Water sources and sinks#

At the start of soil hydrology calculations, we calculate the sources and sinks of water for each soil tile. These are generally derived by splitting grid-cell average fluxes or by grouping fluxes that are stored per PFT.

9.7.1.1. Canopy throughfall#

The canopy througfall $F_i^\text{precisol}$ for soil tile $i$ (referenced to the soil tile area) is calculated by summing the contributions from PFTs belonnging to that soil tile. If the throughfall for PFT $j$ (referenced to the grid cell area) is $F_j^\text{precisol}$, the transformation is

\[F_i^\text{precisol} = \frac{1}{f^\text{tile\_cell}_i} \sum_{\text{PFT }j\text{ in tile }i} F_j^\text{precisol},\]

where $f^\text{tile\_cell}_i$ is the fraction of the grid cell covered by soil tile $i$.

This transformation ensures that area-integrated througfall flux for a grid cell, given by $F_i A^\text{cell} f^\text{tile\_cell}_i$ where $A^\text{cell}$ is the grid cell area, equals the sum of througfall contributions from each PFT in the soil tile.

9.7.1.2. Soil evaporation#

The soil evaporation $F^\text{ET,bare}_i$ for soil tile $i$ (referenced to the soil tile area) is calculated from the grid-cell soil evaporation $F^\text{ET,bare}$ (referenced to the grid cell area) as

\[F^\text{ET,bare}_i = F^\text{ET,bare} \frac{\beta_i}{\sum_{i'} \beta_{i'} f^\text{tile\_cell}_{i'}},\]

where $\beta_i$ are the per-tile soil evaporation resistances calculated as explained in Section . Weighing with $\beta_i$ accounts for the differences in the strength of evaporation between the soil tiles. This transformation ensures that the total soil evaporation flux from a grid cell, $\sum_i F^\text{ET,bare}_i A^\text{cell} f^\text{tile\_cell}_i$, equals $F^\text{ET,bare} A^\text{cell}$ as expected.

9.7.1.3. Transpiration#

The transpiration $F_i^\text{transpir}$ for soil tile $i$ (referenced to the soil tile area) is calculated from the per-PFT transpiration $F_j^\text{transpir}$ (referenced to the grid cell area) as

\[F_i^\text{transpir} = \frac{1}{f^\text{tile\_cell}_i} \sum_{\text{PFT }j\text{ in tile }i} F_j^\text{transpir}.\]

9.7.1.4. Root sink#

The root sink $F_{ik}^\text{rootsink}$ for soil tile $i$ and soil layer $k$ (referenced to the soil tile area) is calculated from the per-PFT transpiration as

\[F_{ik}^\text{rootsink} = \frac{1}{f^\text{tile\_cell}_i} \sum_{\text{PFT }j\text{ in tile }i} F_j^\text{transpir} \frac{f^\text{us}_{ijk}}{\sum_{i'k'} f^\text{us}_{i'jk'}},\]

where $f^\text{us}_{ijk}$ is the water stress index for transpiration in soil tile $i$, PFT $j$, and soil layer $k$ . Weighing with $f^\text{us}_{ijk}$ accounts for the differences in the strength of the root sink between the soil layers.

If plant hydraulic architecture is activated, the root sink is instead given by .

9.7.1.5. Routing return#

The return flow $F^\text{reinfiltration\_soil}$ (referenced to the soil tile area) is calculated from the routing fluxes $F^\text{returnflow}$ and $F^\text{returnflow}$ (both referenced to the grid cell area) by assuming they spread uniformly over the fraction $f^\text{vegtot}$ of the grid cell covered by PFTs (including bare soil),

\[F^\text{reinfiltration\_soil} = \frac1{f^\text{vegtot}} \left( F^\text{returnflow} + F^\text{reinfiltration} \right).\]

Since we assume that the amount of return flow received by each soil tile is proportional to its area, the flux does not depend on the soil tile area and has the same value for all tiles. (For throughfall, evaporation, etc. the total amount is fixed and thus the flux dependent on the size of the soil tile.)

9.7.1.6. Irrigation#

The irrigation flux is added to the water to infiltrate, at the top of the soil column. In the old irrigation scheme, the irrigation flux is spread uniformly over the PFT-covered grid cell area, just like the routing return flow. In the new irrigation scheme, the irrigation flux is also spread uniformly, but only soil tile 3 is irrigated, for which

\[F_3^\text{irrigation\_soil} = \frac{1}{f^\text{tile\_cell}_3} F^\text{irrigation},\]

where $F^\text{irrigation}$ is the irrigation flux referenced to the grid cell area.

9.7.1.7. Water balance of the soil surface#

After calculating the sources and sinks, we calculate the water balance of the soil surface for each tile. In the balance, we include any surface water carried over from the previous time step, plus throughfall, irrigation, return flow from the routing scheme, evaporation, and sublimation.

If water sources at the surface outweigh the sinks, water is present at the surface and will be infiltrated. If sinks at the surface outweight the sources, water present at the surface is zero and the amount of water needed to satisfy the sinks is saved as $F^\text{flux\_top}$ (as a positive value).

9.7.2. Infiltration#

The calculation of soil water propagation in ORCHIDEE consists of two main stages. First, net water input to the soil is distributed between the top soil layers using a spilling bucket approach and the amount of runoff is estimated. Then, the moisture present in the soil is redistributed by solving the Richards equation and drainage through the bottom of the soil column is calculated.

9.7.2.1. Surface slope and reinfiltration#

9.7.3. Redistribution#

9.7.4. Root sink#

9.7.5. Soil hydraulic properties#

9.7.6. Bottom of the soil column#

9.7.7. Soil moisture nudging#

9.7.8. Soil moisture metrics and averages#

9.7.9. Water conservation checks#

9.7.10. Infiltration and runoff#

9.7.10.1. Local scale#

The parametrization of infiltration into the soil is inspired by the model of , with a sharp wetting front propagating like a piston. A time-splitting procedure is used to describe the wetting front propagation during a time step as a function of its speed. To this end, the saturation of each soil layer is described iteratively from top to bottom. The time to saturate one layer depends on its water content, and on the infiltration rate from the above layer, which is saturated by construction (the top layer, with a 1-mm depth using the 11-mode discretization, is assumed to saturate instantaneously).

The input flux $I_{\rm{pot}}$ is composed of throughfall and snowmelt, plus the return flow from the routing scheme if the options allow for it. The procedure accumulates the time to saturate the soil layers from top to bottom, and if all the available water $I_{\rm{pot}}$ can infiltrate before the end of the time step, then no surface runoff is produced. Else, the part of $I_{\rm{pot}}$ that has not infiltrated during the time step becomes surface runoff, produced by an infiltration-excess, or Hortonian, mechanism.

For simplification, the effect of soil suction is neglected, which leads to gravitational infiltration fluxes. The infiltration rate is thus equal to the hydraulic conductivity at the wetting front interface, called $K_i^{\rm{int}}$, and defined as the arithmetic average of $K(\theta_i)$ in the unsaturated layer $i$ reached by the wetting front and at the deepest saturated node $(i-1)$:

\[\begin{eqnarray} K_i^{\rm{int}} = (K(\theta_i)+K^*_s(z_i))/2. \end{eqnarray}\]

9.7.10.2. Subgrid scale#

The parameterization also includes a sub-grid distribution of infiltration, which reduces the effective infiltration rate into each successive layer of the wetting front. In practice, the mean infiltrability over the grid-cell ($K_i^{\rm{int}}$ if we assume uniform properties) is spatially distributed using an exponential pdf, then compared locally to the amount of water to infiltrate ($I_{\rm{pot}}$). As a result, infiltration-excess runoff is produced over the fraction of the soil column where $I_{\rm{pot}}$ is larger than the local $k$ defined by the exponential distribution of the mean $K_i^{\rm{int}}$, with the following cdf:

\[F(k) = 1 - \exp(-k/K_i^{\rm{int}}).\]

A spatial integration is conducted for each soil layer $i$ that becomes saturated when the wetting front propagates, giving the mean infiltration excess runoff $R_i$ produced from the saturation of each soil layer i:

\[R_i = I_{\rm{pot}} - K_i^{\rm{int}} \,(1 - \exp(-I_{\rm{pot}}/K_i^{\rm{int}})).\]

Eventually, this leads to define an effective conductivity $K_i^{\rm{int}*}$ in each leayer:

\[\begin{eqnarray} K_i^{\rm{int}*} = K_i^{\rm{int}} \, (1 - \exp(-I_{\rm{pot}}/K_i^{\rm{int}}). \end{eqnarray}\]

Since this effective conductivity is smaller than the one of the uniform case ($K_i^{\rm{int}}$), the sub-grid distribution increases surface runoff, given by the sum of $R_i$ from all the layers saturated during the time step. This sub-grid distribution can be seen as the opposite to the parametrization of , detailed in , since the conductivity K rather than the precipitation rate is spatially distributed within the grid-cells.

Finally, a fraction $\gamma$ of surface runoff is allowed to pond in flat areas, to account for the effect of pond systems on infiltration []. This fraction is constant over time, but varies spatially, based on an input slope map, which gives the slope in % at the 0.25$^{\rm{o}}$ resolution and called cartepente2d_15min.nc. This file has been introduced by Tristan D’Orgeval during his PhD defended in 2006, and its headers refer to ETOPO, which suggests the 0.25$^{\rm{o}}$ slope has been calculated based on the DEM ETOPO5 (1988) or ETOPO2 (2001).

:::{figure} Figures/reinf_slope_half_deg.pdf :name: fig:reinf :align: center

Map of the reinfiltration fraction $\gamma$ produced by ORCHIDEE at the 0.5$^{\rm{o}}$ resolution. :::

In practice, the fraction $\gamma$ is defined in each grid-cell based on $S$, the weighted area mean of the high-resolution slope, and a threshold slope $S_{\rm{max}}$ (with a default value of 0.5%), such that local reinfiltration fraction decreases from 1 when $S=0$, to 0 when $S \ge S_{\rm{max}}$:

(9.28)#\[\gamma = 1 - \min(1,S/S_{\rm{max}}).\]

Surface runoff is then reduced by the fraction $\gamma$, which is temporarily stored in the variable water2infilt, kept to be infiltrated at the following time step with the following throughfall, snowmelt, and return flows from the routing scheme if any. The variable water2infilt thus conveys memory of water storage, and belongs to the prognostic variables. Note, however, that it is integrated to the total soil moisture variables tmc and humtot (sections %s and %s)

The parameter $S_{\rm{max}}$ is externalized and may be changed to modify the effectiveness of ponding. *** Cite Figure 8

9.7.10.3. Total runoff#

For numerical convenience, infiltration proceeds before soil water redistribution, and bare soil evaporation ($E_g$, see next section) is subtracted from the input water flux. If the former exceeds the latter, there is no infiltration, and the top boundary condition to water redistribution is a water demand amounting $E_g - I_{\rm{pot}}$.

Note that the runoff terms can be modified during the time step to correct the cases of “oversaturation” ($\theta>\theta_s$), which may arise from numerical errors when $\theta$ approaches saturation. If the drainage coefficient $F_D \leq 0.5$, or if soil freezing is allowed, all moisture above saturation is sent to surface runoff. Else, this “excess” moisture is sent to drainage.

*** provide an estimate of this error (mean and std, based on REF with PGF for IGEM, or Salma, or LS3MIP)

9.7.11. Water redistribution#

We assume unsaturated, 1D vertical, no lateral exchange *** Unless otherwise mentioned, the following text uses SI units. For instance, water flux variables $X$ are in kg.m$^{-2}$.s$^{-1}$.

A physically-based description of unsaturated soil water flow was introduced in ORCHIDEE by . It relies on a one-dimensional Richards equation, combining the mass and momentum conservation equations, but it takes the form of a Fokker-Planck equation, since the state variable is expressed as volumetric water content $\theta$ (m$^3$.m$^{-3}$) rather than pressure head.

Due to the large scale at which ORCHIDEE is usually applied, we neglect the lateral fluxes between adjacent grid cells. *** Applied in each soil column and not grid-cell *** We also assume all variables to be horizontally homogeneous, so that the mass conservation equation relating the vertical distribution of $\theta$ to its flux field $q$ (m.s$^{-1}$) is:

(9.29)#\[\frac{\partial \theta (z,t)}{\partial t} = -\frac{\partial q(z,t)}{\partial z} - s(z,t).\]

In this equation, $z$ is depth below the soil surface, and $t$ is time (in m and s respectively). The sink term $s$ (m$^3$.m$^{-3}$.s$^{-1}$) is due to transpiration and depends on the vertical profile of roots.

The flux field $q$ comes from the equation of motion known as equation in the saturated zone, and extended to unsaturated conditions by :

(9.30)#\[q(z,t) = -D(\theta(z,t)) \frac{\partial \theta(z,t)}{\partial z}+K(\theta(z,t)).\]

$K(\theta)$ and $D(\theta)$ are the hydraulic conductivity and diffusivity (in m.s$^{-1}$ and m$^2$.s$^{-1}$ respectively). The latter defines the link between the volumetric soil moisture $\theta$ and the matric potential $\psi$ (in m):

(9.31)#\[D(\theta(z,t)) = K(\theta(z,t)) \, \frac{\partial \psi}{\partial \theta}(\theta(z,t)).\]

The approach above was solved by making use of a finite difference integration. The Fokker-Planck equation defined by the combination of Eqs. (9.29)-(9.30) is solved using a finite difference method. To this end, the soil column is discretized using N nodes, defined by an index $i$ increasing from top to bottom, and where we calculate the values of $\theta$ (Figure 9). The middle of each consecutive couple of nodes represents the limit between two soil layers, except for the upper and the lower layers, for which the top/bottom interfaces are defined by the first/last nodes respectively. As a consequence, each soil layer holds only one node $i$, and we define layer $i$ as the layer holding node $i$. We can thus define the thickness $h_i$ of each layer by:

(9.32)#\[\begin{split}\begin{align} h_i &= [\,\Delta Z_i \, + \Delta Z_{i+1} \,]/2, \,\, i \in [2,N-1] \\ h_1 &= \Delta Z_2 /2 \\ h_N &= \Delta Z_N /2 \end{align}\end{split}\]

:::{figure} Figures/vertical_orc+temp.pdf :name: discret_zoom

Illustration of soil vertical discretization, in case of 4 soil nodes and soils layers: for soil hydrology in blue on the left, with links between node positions, their local volumetric water content $\theta_i$, the soil layers, their depth and integrated soil moisture $W_i$, in the simple case of four equidistant nodes. Link with code’s names in Table %s. The correspondence with the nodes for the soil thermics appears on the yellow column to the right.

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In these expressions, $\Delta Z_i$ is the distance between nodes $(i-1)$ and $i$, which have volumetric water contents $\theta_{i-1}$ and $\theta_i$ (Figure 9).

The total water content of each layer $i$, $W_i$ (m), is obtained by integration of $\theta(z)$, assumed to undergo linear variations between two consecutive nodes:

(9.33)#\[\begin{split}\begin{align} W_i &= [\,\Delta Z_i \, (3 \, \theta_i + \theta_{i-1}) + \Delta Z_{i+1} \, (3 \, \theta_i + \theta_{i+1})\,]/8, \,\, i \in [2,N-1] \\ W_1 &= [\,\Delta Z_2 \, (3 \, \theta_1 + \theta_2)\,]/8 \\ W_N &= [\,\Delta Z_N \, (3 \, \theta_N + \theta_{N-1})\,]/8 \end{align}\end{split}\]

Equation (9.29) can then be integrated between the nodes and over the time step $dt$, over which $q$ is assumed to be constantly equal to its value at $t+dt$ (implicit scheme). This allows defining the water budget of each layer $i$:

(9.34)#\[\frac{W_{i}(t+dt)-W_{i}(t)}{dt} = Q_{i-1}(t+dt)-Q_{i}(t+dt)-S_i,\]

where $S_i$ is the integrated sink term and $Q_i$ the water flux at the interface between layers $i$ and $(i+1)$ (both in m.s$^{-1}$).

The above flux is deduced from Equation (9.30), by approximating $\partial \theta/\partial z$ by the rate of increase between the equidistant neighbouring nodes $(i-1)$ and $i$, and this leads to:

(9.35)#\[Q_{i} = -\frac{D(\theta_{i-1})+D(\theta_{i})}{2} \, \frac{\theta_i-\theta_{i-1}}{\Delta Z_i}+ \frac{K(\theta_{i-1})+K(\theta_{i})}{2}.\]

To get this expression, we also approximate the values of $K$ and $D$ at the layers’ interfaces by the arithmetic average of their values at the neighbouring nodes.

To make Equation (9.35) linear with respect to $\theta$ such as to construct a tridiagonal matrix system to solve $\theta_i(t+dt)$, we discretize the interval $[\theta_r,\theta_s]$ in 50 regular subdomains where $K$ and $D$ are described by piecewise functions, respectively linear in $\theta$ and constant (Appendix %s).

Additional information is required to solve $\theta_1$ and $\theta_N$.

It consists of water fluxes $Q_0$ and $Q_N$ at the top and bottom of the soil column respectively. These fluxes need to be prescribed as boundary conditions at each time step, and thus drive the evolution of the soil moisture profile (section %s).

9.7.12. Drainage#

By default, the bottom boundary condition to water flow in the soil is defined by free gravitational drainage:

(9.36)#\[Q_N = K(\theta_N).\]

But two other options are possible. The first one, originally proposed by , consists in reducing the free drainage calculation by a coefficient $F_D$:

(9.37)#\[Q_N = F_D.K(\theta_N),\]

where $0 \leq F_D \leq 1$. This condition is equivalent to reducing the hydraulic conductivity $K$ under the bottom of the soil column, which could be achieved alternatively by enhancing the exponential decay of $K$ with depth. Setting $F_D=0$ makes the bottom totally impermeable as in the two-layer soil hydrology scheme of ORCHIDEE.

The second new boundary condition, proposed since , is to impose saturation under the calculation node $n_{sat}$ chosen by the user:

(9.38)#\[\theta_{i \geq n_{\rm{sat}}} = \theta_s.\]

This implies the presence of a water table inside the modeled soil column. To do so, we first solve the diffusion equation over the 2-m soil column assuming that $F=0$, then we adjust the resulting soil moisture to bring it back to saturation at nodes $n_{\rm{sat}}$ and below, if either upward diffusion or root absorption made it drop to unsaturated values. The required water flux is assumed to enter the soil column through the soil bottom interface, and thus represents negative drainage. *** violation of water conservation, mention also the new scheme GWF

9.7.13. Soil hydraulic properties#

9.7.13.1. Van Genuchten relationships for hydraulic conductivity and diffusivity#

The hydraulic parameters required by the diffusion equation solved in ORCHIDEE-2.0 are the hydraulic conductivity and diffusivity, $K$ and $D$, which depend on volumetric water content $\theta$. These relationships are given in ORCHIDEE-2.0 by the - model:

(9.39)#\[K(\theta) = K_{s} \sqrt{\theta_{f}} \left(1-\left(1-\theta_{f}^{1/m}\right)^m\right)^2,\]

(9.40)#\[D(\theta) = \frac{(1-m) K(\theta)}{\alpha m} \frac{1}{\theta-\theta_{r}} \theta_{f}^{-1/m}. \left(\theta_{f}^{-1/m} -1\right)^{-m},\]

which also writes:

(9.41)#\[D(\theta) = \frac{K(\theta)}{\alpha m n} \frac{1}{\theta-\theta_{r}} \theta_{f}^{-1/m}. \left(\theta_{f}^{-1/m} -1\right)^{-m}.\]

There, $K_s$ is the saturated hydraulic conductivity (m.s$^{-1}$), $\alpha$ (m$^{-1}$) corresponds to the inverse of the air entry suction, and $m$ is a dimensionless parameter, related to the classical Van Genuchten parameter $n$ by:

\[m=1 - 1/n.\]

The last Van Genuchten relationship defines the link between the matric potential $\psi$ (m), involved in the hydraulic diffusivity (Eq. (9.31)), and the volumetric water content $\theta$:

(9.42)#\[\psi(\theta) = - \frac{1}{\alpha} \left(\theta_{f}^{-1/m} - 1 \right)^{1/n}\]

These equations, illustrated in Figure 10, assume that $\theta$ varies between the residual water content $\theta_r$ and the saturated water content $\theta_s$, which leads to define relative humidity as $\theta_{f}=(\theta-\theta_r)/(\theta_s-\theta_r)$. The hydraulic conductivity $K$ decreases with soil moisture, and this effect dominates the variations induced by soil texture (section %s). In contrast, the matrix potential $\psi$, which corresponds to the retention forces by the unsaturated soil and is therefore negative, is stronger for weak soil moisture, at which it can efficiently counteract the gravitational forces controlled by $K$. Diffusivity results from both influences and decreases with soil moisture like $K$, but over a much smaller range owing to the counter influence of $\psi$.

[This was moved from somewhere else and still needs to be integrated – FK] Soil hydraulic properties for the USDA classes are determined using the pedotransfer function of . The hydraulic properties of oxisoils were obtained by averaging the values found in literature, resulting in saturated hydraulic conductivity much higher than that of regular clay and closer to that of sand [].

:::{figure} Figures/VG12.pdf :name: fig:vg

Van Genuchten relationship $K(\theta)$, $D(\theta)$, and $\psi(\theta)$, for $\theta$ in $[\theta_r, \theta_s]$, based on Eqs. (9.39), (9.41), and (9.42), for the 12 USDA texture classes *** a link must be made with texture and PTFs cf. section %s. The three thick lines show the three texture classes used with the simplified Zobler map (Table %s). The difference between the first two panels is that the second one uses a logarithmic axis for $K(\theta)$.

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9.7.13.2. Hydraulic conductivity decrease with depth due to compaction#

Following and , several modifications of $K_s$, thus $K(\theta)$, with depth are implemented in ORCHIDEE-2.0. Firstly, as suggested by , $K_s$ is assumed to decrease exponentially with depth (below the top 30 centimeters with the default parameters) to represent the effect of compaction:

(9.43)#\[K_s(z) = K_s^{\rm{ref}}\,F_K(z),\]

(9.44)#\[F_K(z) = \min(\max(\exp(-c_K(z-z_{\rm{lim}}), 1/F_K^{\rm{max}}),1).\]

Here, $z$ is the depth below the soil surface, $K_s^{\rm{ref}}$ is the reference value of $K_s$ based on soil texture (section ??). The other three parameters are constants, which are independent from the PFT, the soil column, and the soil texture: $z_{\rm{lim}}$ is the depth at which the decrease of $K_s$ starts; $c_K$ is the decay factor (in m$^{-1}$); and $F_K(z)$ cannot be smaller than $1/F_K^{\rm{max}}$. The default values are $c_K=2$ m, $z_{\rm{lim}}=0.3$ m, and $F_K^{\rm{max}}= 10$, so that $K_s$ is identical for all nodes below 1.45 m, and equal in this case to 0.1 $K_s^{\rm{ref}}$.

9.7.13.3. Hydraulic conductivity increase towards the surface due to bioturbation#

An increase of $K_s$ towards the soil surface is also taken into account, because the presence of vegetation tends to increase the soil porosity in the root zone and therefore to increase infiltration capacity []. This effect is tightly linked to the decrease of $K_s$ with depth, and could seem redundant, but it is described independently to introduce a dependency on the root density profile. Root density is assumed to decrease exponentially with depth, as defined by a root decay factor $c_j$ (in m$^{-1}$), which is a PFT characteristic (ranging from 0.4 m$^{-1}$ in tropical forests to 4 m$^{-1}$ in C3 cropland):

(9.45)#\[R_j(z)=\exp(-c_j z).\]

For each MC $j$ with vegetation (thus excluding the bare soil MC), a multiplicative factor $F_{Kj}$ is defined as a function of depth $z$ and $c_j$:

\[F_{Kj}(z) = \max\left(1,\left(\frac{K_s^{\rm{max}}}{K_s^{\rm{ref}}}\right)^{\frac{1-c_jz}{2}}\right).\]

$K_s^{\rm{max}}$ is a constant taken as the maximum $K_s$ given by [] for the sandy texture, so $K_s^{\rm{max}}$ = 7128.0 mm.d$^{-1}$. As shown by Figure 11, this formula leads to increase the saturated conductivity exponentially for $z< 1/c_j$ (which ranges between 0.25 m for grasses and crops, to 1.25 m for forests, when using the defaults values of $c_j$ for the multi-layer hydrology. It can be noted that the maximum $F_{Kj}$, at the surface, is independent from $c_j$ and the vegetation type, and is always $\sqrt{K_s^{\rm{max}}/K_s^{\rm{ref}}}$.

The resulting saturated conductivity for the soil column $c$ is eventually given by the following expressions:

\[K_s^*(z,c) = K_s(z) \, F_{K\rm{root}}(z,c),\]

(9.46)#\[F_{K\rm{root}}(z,c) = \prod_{j\in c} F_{Kj}(z)^{f^j/2} = \prod_{j\in c} \max\left(1,\left(\frac{K_s^{\rm{max}}}{K_s^{\rm{ref}}}\right)^{f^j(1-c_jz)/4}\right),\]

where $K_s(z)$ is given by Eq. (9.43), and $f^j$ is the fraction of MCs $j$ belonging to the soil column $c$.

The above equations assume that $K_s(z,c)$ is not modified by roots in the bare soil PFT ($j=1$), and were first been introduced by to obtain a reasonable evapotranspiration variability in Hapex-Sahel simulations. This work used $f_v^j$ instead of $f^j/2$, leading to variable $K_s^*(z,c)$ over time, since $f_v^j=f^j (1 - \exp(-k_{\rm{ext}} \, \rm{LAI}_j))$. It was changed to make $F_{K\rm{root}}$ constant if the PFT does not change.

9.7.13.4. Combined effects on hydraulic conductivity and other parameters#

Eventually, the vertical profile of the effective saturated hydraulic conductivity is given in each soil column $c$ by:

(9.47)#\[K_s^*(z,c) = K_s^{\rm{ref}} F_K(z) \, F_{K\rm{root}}(z,c).\]

The effect of each factor $F_K$ and $F_{K\rm{root}}$ is illustrated in Figure 11. In practice, these two factors are used to define : (i) the non-saturated hydraulic conductivity and diffusivity (as detailed in appendix %s) required for water redistribution based on the Richards equation; (ii) the saturated and non-saturated hydraulic conductivity to calculate infiltration (section ??). The non-saturated parameters also depend on the Van Genuchten parameters $n$ and $\alpha$, which are modified according to $F_K$ as explained in below.

:::{figure} Figures/ks_z_sandy_loam.pdf :name: fig:ksat :width: 1350% :align: center

Profiles of saturated hydraulic conductivity for a sandy loam soil. The final profiles are in orange for a grass PFT, and in yellow for a forest PFT. *** MUST BE CORRECRED TO GET A CONSTANT KS between 1.5m and 2m equal to $K_s^{\rm{ref}}$/10=106mm/day ***

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To introduce a consistency between $K_s$ and the parameters $\alpha$ and $n$, the latter can also be calculated, based on their log-log regression with $K_s$, using the values given by for the 12 USDA soil textures. The resulting regressions appear in Fig 4.14 of , and are defined by:

(9.48)#\[\begin{split}\begin{eqnarray} n(K_s) & = & n_0 + a_n (K_s)^{b_n} \\ \alpha(K_s) & = & \alpha_0 + a_\alpha (K_s)^{b_\alpha} \end{eqnarray}\end{split}\]

where $n_0$ = 0.95, $b_n$ = 0.34, $\alpha_0$ = 0.12 m$^{-1}$, and $b_\alpha$ = 0.53. The parameters $a_n$ and $a_\alpha$ equal 42 and 2500 respectively, but they vanish when combining equations.

Given the texture, we know $K_s^{\rm{ref}}$, and $K_s(z)=K_s^{\rm{ref}}.\,F_K$ (Eqs. (9.43) and (9.44)). From Eqs. (9.48) and %s, it comes:

\[\begin{split}\begin{eqnarray} n(K_s(z)) - n_0 & = & (n(K_s^{\rm{ref}})-n_0).(F_K)^{b_n} = (n^{\rm{ref}}-n_0).(F_K)^{b_n} \\ \alpha(K_s(z)) - \alpha_0& = & (\alpha(K_s^{\rm{ref}})-\alpha_0).(F_K)^{b_\alpha} = (\alpha^{\rm{ref}}-\alpha_0).(F_K)^{b_\alpha} \end{eqnarray}\end{split}\]

In ORCHIDEE-2.0, these relationships are used to define changes in $n$ and $\alpha$ resulting from the decrease of $K_s$ with depth below 30 cm (section ??), but they are not used to change $n$ and $\alpha$ as a result of $K_s$ increase with roots (section ??).

9.7.13.5. Link to soil texture maps#

Soil texture and related parameters are assumed to be horizontally uniform within each grid-cell. At the beginning of each simulation, a pre-processing step allows to extract the domimant soil texture (i.e the texture class occupying the largest fractional area of the grid-cell) given the resolution of the simulation and the chosen input soil map.

As detailed in sections %s and %s, most soil properties are inferred from this dominant soil texture. It is not the case, however, for the clay fraction, used in the STOMATE module for soil carbon calculations, and defined as the weighted-area average of the clay percent of each texture class inside a grid-cell. The mean percent of sand and silt particles is computed similarly for output. *** is percent sand used for thermal properties ?

In this framework, ORCHIDEE-2.0 accounts for the soil’s characteristics owing to the Van Genuchten parameters ($K_s$, $\theta_s$, $\theta_r$, $\alpha$ and $m$), as well as secondary parameters derived from the latter ones (section %s). *** Clay fraction for STOMATE and :

*** To this end, we use the look-up tables provided by to define one single central value for each class and parameters, which corresponds to the simplest pedotransfer function . *** Clay fraction is defined by the mean granumometric composition for each USDA class, as defined in .

The field capacity and wilting point are important threshold soil moistures, which control available water capacity, and are related to soil moisture stresses (section %s). The corresponding volumetric water contents, further noted $\theta_c$ and $\theta_w$, can be related to $n$ and $\alpha$ (Van Genuchten relationships), by means of characteristic soil matric potentials. These parameters are derived using the Van Genuchten relationships, based on a soil matric potential of -3.3 m (-1 m for sandy soils following ) for $\theta_c$, and -150 m (permanent wilting point) for $\theta_w$, so they vary with soil texture.

The resulting values of $\theta_c$ and $\theta_w$ are given in Table 3, with all the main properties of the 12 USDA soil textures, including the values of $d_2$ (if not modified to follow $K$) and $d_{50}$ (Appendix %s). Figure 12 further shows the spatial distribution of wilting point and field capacity based on the texture map of and an horizontal resolution of 144 $\times$ 143, used in many IPSL simulations for *** Add AWC*** Attention, le calcul est faux si on n’a pas alpha constant *** mettre les cartes pour Zobler et Reynolds

:::{figure} Figures/cartes_1_salma.pdf :name: fig:mapWPFC :align: center

The two soil texture maps available in ORCHIDEE-2.0 (top row), with important derived hydrologic properties : saturated hydraulic conductivity $K_s^{\rm{ref}}$ (values in mm.d$^{-1}$ tranformed with $\log_{10}$) (middle row), and available water content AWC (in mm) over the 2m-soil depth (bottom row). All the maps are produced at the 0.5° resolution, based on the dominant soil texture at this resolution. *** Zobler modified

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9.7.14. Soil moisture metrics and averages#

*** Since we solve one independent water budget per soil column, the number of times the diffusion scheme is used per grid-cell is defined by the number of soil columns.

Soil moisture can be quantified by different variables in ORCHIDEE (see Table %s for notations):

the volumetric water contents $\theta_i$, which give the local moisture at the calculation nodes, in m$^3$.m$^{-3}$ (Figure 9),
the total water contents $W_i$ of the soil layers defined in Figure 9, and calculated using Eqs (9.33)-%s, in kg.m$^{-2}$ or mm,
the total water content of the soil column: $W_t=\sum_1^N W_i$, in kg.m$^{-2}$ or mm. It is equivalent to the vertical integration of $\theta$.
the total water content of litter, defined in ORCHIDEE as the four top soil layers: $L=\sum_1^4 W_i$, in kg.m$^{-2}$ or mm.

All these variables are different in the different soil columns of one ORCHIDEE grid-cell, and we can define two types of spatial averages across the soil columns. In the following, we will indicate the simple weighted average across the different soil columns by an overbar: $\overline{W_t} = \Sigma_c \, g^c \, W_t^c$ , for the mean total soil moisture for instance. This value gives an average in kg per m$^2$ of $A_v+A_g=A_L-A_n$, and the conversion to kg per m$^2$ of $A_L$ defines $\widehat{W_t} = \overline{W_t} (A_v+A_g)$.

Finally, all these variables can be defined for total, liquid, and solid water, the latter two being identified by the exponents $^\mathrm{liq}$ and $^\mathrm{ice}$ in this document.

9.8. Routing to the oceans#

Eventually, the two runoff terms in ORCHIDEE are a surface runoff (Hortonian runoff minus reinfiltration), and drainage, gravitational by default, at the bottom of soil column. The input water flux feeding infiltration ($I_{pot}$) is the sum of through-fall and snowmelt during the time step, plus reinfiltration from the previous time step, and return flows from the routing scheme (from the flood plains and irrigation, see USE dynamic link: section 7.3), also from the previous time step. For numerical convenience, infiltration proceeds before soil water redistribution, and bare soil evaporation ($E_g$) is subtracted from the input water flux. If the former exceeds the latter, there is no infiltration, and top boundary condition to water redistribution is a water demand amounting $E_g-I_{pot}$.

The routing scheme calculates the lateral water flow through river networks across the continent into the ocean. The routing scheme of ORCHIDEE () was described in , and . It is based on a large-scale cell-to-cell methodology in which the watershed is represented as a single grid cell or a network of $n$ equal grid cells (Singh, 1989). Each of the interconnected grid cell is approximated as a cascade of linear reservoirs which do not interact with the atmosphere. When activated in ORCHIDEE, the routing scheme transforms every day ($\Delta{t}_{routing}$ = 86400 s) the runoff simulated by SECHIBA into river discharge and computes an hydrograph at any grid cell and not only at the outlet. The routing scheme is based on a parametrization of the water flow on a global scale (Miller et al., 1994; Hagemann and Dumenil, 1998). The global map of 6930 watersheds (; ; ) delineates the boundaries of the basins. By default, only the 50 largest basins are selected in the model ($N_B$ = 50). This map provides also single water flow direction among the 11 possibilities within each grid cell: 8 directions towards another grid cell (N, NE, E, SE, S, SW, W, NW), one direction towards the endorheic lakes, one direction towards the ocean and one direction accounting for the water from small rivers which flows in a disperse way into the ocean. The resolution of the basin map is 0.5°, coarser than usual resolution used when land surface models are applied. Therefore, we can have more than one basin in SECHIBA grid cell (sub-basins) and the water can flow either to the next sub-basin within the same grid cell or to the neighboring cell. No more than 7 sub-basins can be included in a grid cell ($n_{bas}^{\max}$ = 7). For a selected basin, cell-to-cell flow through the system of digital river networks is modelled using the continuity equation computed for each reservoir:

\[\frac{dV_i}{dt}=Q_i^{in}-Q_i^{out}\]

where $V_i$ (kg) is the water amount in the reservoir $i$ considered ($i$ = 1, 2 or 3) and $Q_i^{in}$ and $Q_I^{out}$ (both in kg/day) are respectively the total inflow and outflow of the reservoir $i$.

In each sub-basin, runoff simulated by SECHIBA is transformed into river discharge emanating from the so-called fast and slow reservoirs, designed to account for the delay and attenuation of overland flow and groundwater flow, respectively, at the grid-cell scale. The slow reservoir ($i$ = 3) collects the drainage $D$ (in kg/day), whereas the fast reservoir ($i$ = 2) collects the surface runoff $R_s$ (in kg/day). Outflow from these two reservoirs becomes streamflow at the outlet of the sub-basin, and feeds the stream reservoir ($i$ = 1) of the next downstream sub-basin, which also receives inflow from all upstream stream reservoirs:

\[\begin{split}\begin{align} &\frac{dV_1}{dt}=\sum{Q_1^{in}-Q_1^{out}}, \\ & \frac{dV_2}{dt}=R_s-Q_2^{out}, \\ & \frac{dV_3}{dt}=D-Q_3^{out} \end{align}\end{split}\]

The outflow of the reservoir $i$ is assumed to be related to the stored volume of water by a linear relationship:

\[Q_i^{out}=\frac{1}{T_i}\cdot V_i\]

Travel time within the reservoir $i$ depends on a characteristic timescale $T_i$ (day), which is the product of a topographical water retention index $k$ (in km) and a time constant $g_i$ (in day/km):

\[T_i=k\cdot g_i\]

$g_i$ does not vary horizontally but distinguishes the three reservoirs, while $k$ characterizes the impact of topography on travel time in each sub-basin, and is assumed to be the same in the three reservoirs of a given grid cell, even though it derives from stream-routing principles introduced by :

\[k=\frac{d}{\sqrt{\tan\beta}}\cdot 10^{-3}=\sqrt{\frac{d^3}{\Delta z}}\cdot 10^{-3}\]

This travel time is thus assumed to be proportional to stream length $d$ (in m) in the sub-basin, and inversely proportional to the square root of stream slope ($\sqrt{\tan\beta}$). The lengths and slopes are computed at the 0.5° × 0.5° resolution from the topographical map of Vörösmarty et al. (2000a, 2000b). This can be seen as a simplification of the Manning formula (), where the time constant $g_i$ compensates for the missing terms. The values of the time constants, $g_i$, were initially calibrated over the Senegal Basin then generalized for all the basins of the world (). The stream reservoir has the lowest constant ($g_1$ = 0.24 10$^{-3}$ day/km). $g$ value has been set to $g_2$ = 3.0 10$^{-3}$ day/km and $g_3$ = 25.0 10$^{-3}$ day/km in the fast and slow reservoirs respectively.

10. The carbon and nitrogen cycle#

10.1. DONE: Vegetation phenology#

10.1.1. DONE: Classification of plant status#

To accurately simulate seasonal ecosystem cycles it is essential to represent the biological life cycle of plants []. The timing of each of the plant phenological phases is dependent on a combination of physiological characteristics and climatic conditions. In ORCHIDEE, each PFT can progress through the following phases: establishment, bud presence, leaf onset, canopy presence, wood growth cessation (pre-senescence), senescence, dormancy, and death. The phenology models in ORCHIDEE control the timing of leaf onset and senescence. There are no leaf onset or senescence modules associated with evergreen ecosystems where leaf turnover is simply a function of climate and leaf age (equation (10.8)).

10.1.2. DONE: Leaf onset#

There are four different modules to determine leaf onset (Table 4), where leaf onset is determined by moisture conditions, or temperature conditions (including the number of growing days, the occurrence of chilling days, and growing degree days), or both temperature and moisture conditions.

:::{list-table} The phenological modules used to determine leaf onset. :header-rows: 0 :name: tab:pheno

- Module
- Leaf onset trigger
- PFTs affected
- NGD
- Number of growing days
- Boreal needleleaf deciduous forest
- GDD-NCD
- Growing degree days after cold period
- Temperate broadleaf deciduous forest, boreal broadleaf deciduous forest
- MOI
- Moisture conditions
- Tropical broadleaf raingreen forest
- MOIGDD
- Both temperature (GDD) and moisture conditions
- Boreal natural C3 grassland, temperate natural C3 grassland, tropical natural C3 grassland, natural C4 grassland, C3 cropland, C4 cropland :::

Within the number of growing days (NGD) module, which was originally proposed by for arctic and boreal biomes, the onset of leaves occurs when the number of growing days exceeds a PFT-specific critical value:

(10.1)#\[G^{n,grow} > c_0\]

where $G^{n,grow}$ refers to the number of growing days with a mean daily temperature above a threshold, 268.15 K in ORCHIDEE, calculated from the start of the midwinter (shortest day of the year) where, $c_0$ (days) represents a PFT-specific critical value (23.9 days for boreal needleleaf deciduous forests). To ensure it’s spring, temperatures must also be increasing:

(10.2)#\[T^{w} > T^{m}\]

where $T^{w}$ (K) and $T^{m}$ (K) represent the weekly and monthly mean temperatures, respectively.

The growing degree days - number of chilling days (GDD-NCD) module follows the general GDD model [], which is designed to simulate the accumulation of warm temperatures in the spring needed for budburst and is a common model applied to satisfy temperature conditions for leaf onset to occur []. As in the number of growing days model, temperatures must also be increasing (i.e., the weekly temperature must be greater than the monthly temperature - equation (10.2)). GDD is calculated as a sum of the mean daily temperatures for days that have a mean daily temperature above a given threshold, starting from a given time point. Leaf onset occurs once the GDD value has crossed a defined threshold. In the GDD-NCD module the GDD is calculated from midwinter (as for the NGD module) but the general GDD module described above is further adapted to account for PFTs that require a period of cold temperatures to trigger budburst [] by modifying the GDD threshold to account for the number of chilling days. Thus, in the GDD-NCD module the start of the growing season occurs when the growing degree days since midwinter exceeds a calculated GDD threshold:

(10.3)#\[G^{deg,grow} > G^{thres}\]

where $G^{deg,grow}$ (K) represents the growing degree days since midwinter and $G^{thres}$ (K) is a negative exponential that accounts for the number of chilling days with an additional three fixed parameters, following []:

(10.4)#\[G^{thres} = \frac{c_1}{e^{c_2 \cdot G^{n,chill}}} - c_3,\]

where $c_1$ (-), $c_2$ (-) and $c_3$ (-) are non PFT-specific model parameters that have been calibrated against satellite data, and are 964, 0.0058, and 12.8, respectively []. $G^{n,chill}$ is the number of chilling days since midwinter, where chilling days are defined as the number of days with a mean daily temperature below a PFT-specific threshold (278.15 K and 273.15 K for temperate and boreal broadleaved deciduous forests, respectively) following , , and .

Within the moisture module, the conditions determining the onset of leaf growth depend on either a soil moisture availability criterion (if hydraulic architecture is not used) or a vegetation stress proxy (if hydraulic architecture is used). This means that the growing season can begin if the last moisture minimum was a sufficiently long time ago (equation (10.5)) and either the soil moisture availability (or vegetation stress proxy) is increasing (equation %s) or if the monthly moisture availability (or monthly vegetation stress proxy) is high enough (equation %s).

(10.5)#\[\begin{split}\begin{align} &M^{nsm} > c_4,\\ &f^{Vst,w} > f^{Vst,m},\\ &f^{Vst,m} \ge c_5, \end{align}\end{split}\]

where $f^{Vst,w}$ (-) and $f^{Vst,m}$ (-) are proxies for weekly and monthly vegetation water stress, respectively, with values ranging from 0 (high moisture stress) to 1 (low moisture stress). $M^{nsm}$ (days) represents the number of days since the last moisture minimum, and $c_4$ (days) and $c_5$ (-) represent PFT-specific threshold parameters. For tropical broadleaf raingreen trees, $c_4$ and $c_5$ are 50 days and 1.0, respectively.

Vegetation water stress is calculated using the soil moisture availability criterion (essentially root profile weighted plant available relative soil moisture) that varies linearly from zero at wilting point to 1 at field capacity:

(10.6)#\[f^{Vst} = \sum_{i=1}^{nlsm} f^{root,fun}_{i} \cdot min\left( 1, max\left( 0, \frac{M^{sm}_{i} - M^{smw}_{i}}{M^{smf}_{i} - M^{smw}_{i}}\right)\right)\]

where $M^{sm}_{i}$ (kg $m^{-2}$) is the soil moisture of the $i$-th (m) soil layer (liquid phase), $M^{smf}_{i}$ (kg $m^{-2}$) soil moisture of each layer at field capacity, $M^{smw}_{i}$ (kg $m^{-2}$) is the soil moisture of each layer at wilting point, and $f^{root,fun}_{i}$ (-) is the normalized root mass/length fraction in each soil layer (0 - 1) []. The moisture availability criterion (equation (10.5) - equation %s) correspond to Model 4b in , which assumes that leaf onset in the dry tropics and semi-arid ecosystems occurs only after a certain amount of water has accumulated in the soil.

Within the moisture-growing degree day module (MOI-GDD), both the moisture conditions described above and a GDD threshold criterion must be met for leaf onset to occur. For the moisture conditions parameter $c_4$ is set to 35 days for grasses and 75 days for crops, and parameter $c_5$ is set to 0.98 for C3 grasses and 0.6 for C4 grasses and crops. The GDD test is similar to the general GDD module, described above, where the growing degree days since midwinter must exceed a calculated GDD threshold (i.e. equation (10.3)). In this version of the GDD model $G^{deg,grow}$ represents the growing degree days calculated using days with a mean daily temperature above 278.15 K since midwinter and the $G^{thres}$ (° C) is calculated using a second-degree polynomial using the long term mean annual air surface temperature []:

(10.7)#\[G^{thres} = c_6 + c_7 \cdot T^{3year} + c_8 \cdot {(T^{3year})}^2,\]

where $c_6$ (° C), $c_7$ (° C) and $c_8$ (° C) represent GDD PFT-specific parameters and $T^{3year}$ (° C) is the 3-year average 2 m air temperature. It is not possible to convert GDD between ° C and K and because these PFT parameters are used in ORCHIDEE in ° C, we have also written this relationship here in ° C. Where $c_6$ is 640 ° C, 400 ° C, 400 ° C, 450 ° C for c3 grasslands, c4 grasslands, c3 croplands, and c4 croplands. $c_7$ is 32.0 ° C, 0.0 ° C, 6.25 ° C, and 0.0 ° C for c3 grasslands, c4 grasslands, c3 croplands, and c4 croplands. $c_8$ is 0.1 ° C, 0.0 ° C, 0.0315 ° C, and 0.0 ° C for c3 grasslands, c4 grasslands, c3 croplands, and c4 croplands. As in the NGD and GDD-NCD models, temperatures must also be increasing (equation (10.2)). An alternative to the GDD threshold is that the monthly mean temperature must be above a certain PFT dependent threshold (defined as 283.15 K for all PFTs). Finally, for c4 grasses the monthly mean temperature must also exceed a pre-set temperature of 295.15 K.

10.1.3. DONE: Senescence#

Leaf senescence occurs due to age, the cessation of the growing season, or the occurrence of crop harvests (Table 5). Leaf senescence can also trigger fruit, root, or stem senescence. In the occurrence of leaf and root senescence, the nitrogen stored in leaves and roots is transferred into the labile nitrogen pool. Evergreen species do not experience seasonal senescence. Furthermore, if there is very low leaf mass during senescence, all leaves are lost.

:::{list-table} The phenological modules used to determine leaf senescence. :header-rows: 0 :name: tab:senescence

- Senescence module
- Senescence trigger
- PFTs affected
- Cold
- Cold temperatures
- Temperate broadleaf deciduous forest, boreal broadleaf deciduous forest, boreal needleleaf deciduous forest,
- Dry
- Water availability
- Tropical broadleaf raingreen forest
- Mixed
- Photosynthesis to plant respiration ratio
- Boreal natural c3 grassland, temperate natural c3 grassland, tropical natural c3 grassland, natural c4 grassland
- Crop
- Harvest
- c3 cropland, c4 cropland
- Age
- Leaf age
- All (except bare ground) :::

Even if the meteorological conditions are favorable for leaf maintenance, plants, and in particular, evergreen trees, must renew their leaves simply because the old leaves become inefficient. Therefore, the old leaves of all PFTs fall off regardless of climatic conditions, following.

(10.8)#\[\Delta M^{k}_{l} = M^{k}_{l} \cdot \min \left( 0.99,\frac{\Delta{t}} {S^{crit} \cdot \left( \frac{S^{crit}}{a^{leaf}_{i}} \right)^4} \right)\]

where $M^{k}_{l}$ is the carbon (g C m$^{-2}$ day$^{-1}$) and nitrogen (g N m$^{-2}$ day$^{-1}$) biomass of organs, $k$ (leaves, roots, fruits, and stems), of circumference class $l$. $\Delta M$ is the fraction of carbon and nitrogen biomass lost, $\Delta t$ is the timestep (1 day), $a^{leaf}_{i}$ (days) is the age of the leaf mass in age class $i$, and $S^{crit}$ (days) is the location-specific leaf longevity, dependent on the long-term temperature:

(10.9)#\[S^{crit} = \min \left( c_1 \cdot c_2, \max \left(c_1 \cdot c_3, c_1 - c_4 \cdot \left( T^{3year} - c_5 \right) \right) \right)\]

where $c_1$ (days) is the PFT-specific leaf longevity and is 180 days for raingreen trees, summergreen trees, and grasses, 730 days for broad-leaved evergreen trees, 910 days for temperate needleleaf trees, 2000 days for boreal needleleaf evergreen trees, and 200 days for crops. $c_2$ (-), $c_3$ (-), and $c_4$ (-) are PFT specific parameters linking leaf longevity to the reference temperature. $c_2$ is 1.5 for grasses and crops, 2.0 for boreal needleleaf evergreen trees, and 1.0 for all other PFTs. $c_3$ is 0.365 for boreal needleleaf evergreen trees, and 0.75 for grasses and crops, and 1.0 for all other PFTs. $c_4$ is 120 for boreal needleleaf evergreen trees, 10 for grasses and crops, and 0 for all other PFTs. $c_5$ (K) is a PFT-specific reference temperature for the calculation of leaf longevity and is 298.15K for tropical trees, 293.15K for temperate broad-leaved evergreen trees, C4 grasses and crops; 288.15K for temperate needleleaf evergreen trees, temperate broad-leaved summergreen trees, C3 grasses and crops; and 278.15K for boreal trees.

This means that older leaves fall more frequently than younger leaves and therefore the leaf age distribution must be recalculated after leaf senescence. The fraction of biomass in each leaf age class is updated using the following equation:

\[f^{leaf}_{i,t} = \frac {f^{leaf}_{i,t-1}\cdot M^{leaf}_{l,t-1} + \Delta M^{k}_{l}} {M^{leaf}_{l,t}}\]

where $f^{leaf}_{i,t}$ is the new leaf carbon and nitrogen fraction of leaf biomass in leaf age class $i$, $f^{leaf}_{i,t-1}$ is the old leaf fraction, $M^{leaf}_{l,t}$ is the new total leaf biomass of circumference class $l$, and $M^{leaf}_{t-1}$ is the old total leaf biomass. $\Delta M^{k}_{l}$ is the change in leaf carbon and nitrogen biomass in the considered class due to senescence.

Senescence at the end of the growing season, can be triggered by cooling temperatures, water availability limitations, or both. In the case of senescence at the end of the growing season, leaf senescence is only possible if leaves have reached a certain age (equation (10.10)).

(10.10)#\[a^{l,mean} > c0\]

where $a^{l,mean}$ (days) is the mean leaf age and $c0$ (days) is the PFT-specific parameter for the minimum leaf age for senescence. $c0$ is 90 days for tropical and temperate trees, 60 days for boreal trees, and 30 days for grasses and crops.

Leaf senescence due to cool temperatures at the end of the growing season is triggered by decreasing temperatures (equation (10.12)) and the fall of the monthly air temperature below a threshold temperature ($T^{m} < S^{T}$). The threshold temperature is defined using a second order polynomial of the long-term mean annual air surface temperature:

(10.11)#\[S^{T} = c_6 \cdot {T^{3year}}^{2} + c_7 \cdot T^{3year} + c_8\]

where $S^{T}$ (K) is the critical temperature for leaf senescence, $T^{3year}$ (K) is the long term annual mean temperature and $c_6$ (-), $c_7$ (-) and $c_8$ (-) are PFT-specific empirical constants in the quadratic temperature senescence equation. $c_6$ and $c_7$ are 0.00375 and 0.1, respectively, for C3 grasses, and 0 for all other PFTs. $c_8$ is 16.0 for temperate broad-leaved summergreen, 14.0 for boreal broad-leaved summergreen, 13.0 for boreal needleleaf summergreen trees, 5.0 for grasses, 12.0 for C3 crops and 13.0 for C4 crops. In addition, temperature must be decreasing, following:

(10.12)#\[T^{2w} < T^{m}\]

where $T^{2w}$ (K) and $T^{m}$ (K) refer to the mean daily temperatures calculated over the past fortnight and the past month, respectively. Alternatively, leaf senescence due to cooling temperatures is also triggered if the ratio of plant respiration to photosynthesis is above a PFT-specific threshold (equation (10.13)).

(10.13)#\[\frac{F^{rm,1week}}{F^{gpp,1week}} > c_9\]

where $F^{rm,1week}$ ($gC m^{-2} day^{-1}$) represents the mean of daily maintenance respiration and $F^{gpp,1week}$ ($gC m^{-2} day^{-1}$) represents the mean of daily photosynthesis calculated for the past week, respectively. $c_9$ represents the unitless PFT-specific minimum senescence ratio parameter and is set to 0.75 for C3 grasses, 0.89 for boreal needleleaf summergreen trees, 0.87 for boreal broad-leaved summergreen trees and 1.0 for all other PFTs.

Leaf senescence due to a lack of water availability is triggered by low soil moisture availability. As occurs in leaf onset (sect. ??), this can be calculated using either the soil moisture availability criterion, if hydraulic architecture is not used, or as a ratio between a proxy for stressed and unstressed ecosystem functioning if hydraulic architecture is used. Leaf senescence is triggered if either the weekly moisture availability (or vegetation stress) falls below a critical threshold ($V^{st,w} < V^{st,thres}$) or the ratio of plant respiration to photosynthesis is sufficient (equation (10.13)). The critical moisture availability is calculated as a function of last year’s soil moisture (or vegetation stress) minimum and maximum (or vegetation stress) using the following equation:

(10.14)#\[f^{Vst,thres} = min( max( f^{Vst,min1y} + c_10 \cdot (f^{Vst,max1y} - f^{Vst,min1y}), c_{11}), c_{12})\]

where $f^{Vst,thres}$ is the critical moisture availability threshold, $f^{Vst,min1y}$ and $f^{Vst,max1y}$ are the minimum and maximum moisture availabilities over the past year, $c_{10}$ is a critical relative moisture availability for phenology, $c_{11}$ is the PFT-specific critical moisture availability for senescence, and $c_{12}$ is the PFT-specific relative soil moisture availability above which there is no humidity-related senescence. $c_10$, $c_{11}$ and $c_{12}$ are 0.5, 0.3, and 0.8 for tropical broadleaf raingreen trees, respectively.

For grasses, senescence occurs if the ratio of plant respiration to photosynthesis is sufficient (equation (10.13)).

For crops, the moment of leaf senescence equates to the actual harvest and the entire plant is made senescent. To identify the harvest moment for crops, a second order polynomial of the long-term mean annual air surface temperature is used:

(10.15)#\[H^{crit} = \lvert T^{3year} - c_{12} \rvert \cdot (T^{3year} - c_{12}) + c_{13}\]

where $S^{crit}$ (days) represents the harvest time threshold, and $c_{12}$ (K) and $c_{13}$ (days) represent PFT-specific parameters. $c_{12}$ is 282K and $c_{13}$ is 160 days for all crop PFTs. For senescence to occur, the length of the growing season (number of days) must exceed $H^{crit}$. In addition, the growing degree days since the initiation of the growing season must be sufficient ($G^{init} > c_{14}$) where $G^{init}$ (° C) represents the growing degree days since the start of the growing season and $c_{14}$ (° C) represents the PFT-specific minimum number of growing degree days. $c_{14}$ is 2500 ° C for all crop PFTs.

For grasses, senescence at the end of the growing season is further extended to the whole plant (i.e. all carbon and nitrogen pools) except the carbohydrate reserve. For trees, senescence at the end of the growing season extends to the fine roots at the same rate as they lose their leaves. The rate of biomass loss of both fine roots and leaves (for trees) and leaves, roots, and fruits (for grasses) is prescribed through the equation:

(10.16)#\[\Delta M^{k}_{l} = M^{k}_{l} \cdot \frac{\Delta{t}} {c_{15}}\]

where $M$ is the carbon and nitrogen biomass, $\Delta M$ is the fraction of carbon and nitrogen biomass lost, $\Delta{t}$ is the times step, $c_{15}$ (days) is the PFT-specific time constant of meteorological determined turnover parameter. Where $c_{15}$ is 25 days for boreal trees, 15 days for temperate broad-leaved summergreen trees, and 5 days for temperate needleleaf evergreen trees and grasses.

All remaining leaves are shed when leaf biomass falls too low (below a PFT-specific parameter). For deciduous trees not only leaves but also fruits and fine roots are lost at this moment.

10.2. DONE: Photosynthesis#

The $CO_2$ assimilation rate ($F^A$, $\mu mol CO_2 m^{-2} s^{-1}$), stomatal conductance ($g_s$) and leaf intercellular $CO_2$ partial pressure ($p^{C_i}$) are calculated for each leaf layer, by jointly solving a set of three equations (see below). Both for c3 and c4 plants, these three equations can be manipulated into the form of a standard cubic equation for $F^A$, for which proposed an analytical solution.

\[(F^A)^3+p(F^A)^2+q(F^A)+r=0.\]

We implemented this analytical solution and the overall formalism for both c3 (Appendix B in ) and c4 photosynthesis (Appendix C in ).

10.2.1. DONE: C3 photosynthesis#

The Farquhar–von Caemmerer–Berry model [] predicts the net $CO_2$ assimilation rate as the minimum of the Rubisco-limited rate ($F^{A_c}$) and electron (e$^-$) transport-limited rate ($F^{A_j}$):

(10.17)#\[F^{A}=\min(F^{A_c},F^{A_j}).\]

These two parts of the model can be written in a single equation as:

(10.18)#\[F^A_i=\frac{(p^{C_c}-p^{\Gamma_*})\chi_1}{p^{C_c}+\chi_2}-F^{R_d}_i,\]

where for the Rubisco-limited part $\chi_1= k^{V_{cmax25}}$ and $\chi_2=k^{K_{mC}}(1+p^{C_o}/k^{K_{mO}})$, for the e$^-$ transport-limited part $\chi_1=F^J_i/4$ and $\chi_2=2p^{\Gamma_*}$. $p^{C_c}$ and $p^{C_o}$ are the carbon dioxide and oxygen partial pressures at the carboxylation sites, $p^{\Gamma_*} = 0.5 p^{C_o}/c$1, is the $CO_2$ compensation point in the absence of dark respiration ($F^{R_d}$). This latter case assumes a purely linear e$^-$ transport. $c_1$ is a constant representing the relative CO₂/O₂ specificity factor for Rubisco.

Following , the rate of e$^-$ transport ($F^J$) is described as a non-rectangular hyperbolic function of irradiance:

\[F^J_i = \frac{c^{\alpha_\mathsf{LL}}F^{I^{abs}_i} + k^{J_\mathsf{max}}-\sqrt{(c^{\alpha_\mathsf{LL}}F^{I^{abs}_i} + k^{J_\mathsf{max}})^2 - 4c^{\theta} k^{J_\mathsf{max}}c^{\alpha_\mathsf{LL}}F^{I^{abs}_i}}}{2c^{\theta}},\]

where $c^{\alpha_\mathsf{LL}}$ is the conversion efficiency of absorbed light into $J$ at strictly limiting light, $k^{J_\mathsf{max}}$ is the maximum capacity of e$^-$ transport, and $c^{\theta}$ is the convexity factor for response of $J$ to absorbed irradiance.

The leaf photosynthesis is coupled with the stomatal conductance as follows ():

(10.19)#\[g_{s,i}=c^{g_0}+\frac{F^A_i+F^{R_d}_i}{p^{C_{int}}_i-(p^{\Gamma_*}-F^{R_d}_i/k^{g_m})}G^{hum},\]

were $c^{g_0}$ is the residual stomatal conductance $(molCO_{2} m^{-2} s^{-1})$, $F^{R_{d}}$ $(\mu molCO_{2} m^{-2} s^{-1})$ is the dark respiration, $k^{g_m}$ is the mesophyll conductance and $G^{hum}$ is a function describing the sensitivity of the stomata to either vapour pressure deficit ($g^{VPD}$) or leaf water potential ($g^{\psi}$), depending on the selected drought scheme (see ??).

According to the first diffusion law of Fick, the $CO_2$ transfer from $p^{C_a}$ (ambient $CO_2$ partial pressure) to $p^{C_c}$ can be written as:

(10.20)#\[\begin{split}\begin{align} &p^{C_{int}}_i=p^{C_a}-F^A_i(1/c^{g_b}+1/g_{s,i}),\\ &p^{C_c}_i=p^{C_{int}}_i-F^A_i/k^{g_m}, \end{align}\end{split}\]

where $c^{g_b}$ is the leaf boundary-layer conductance.

The three equations (10.18), (10.19) and (10.20) with the three unknowns $F^A$, $g_s$ and $p^{C_{int}}$ form the system to be solved for c3 plants.

10.2.2. DONE: C4 photosynthesis#

For c4 plants, 2 is fixed by phosphoenolpyruvate (PEP) carboxylase in the mesophyll cell into a c4 compound, which then goes into a bundle-sheath cell, where it is decarboxylated to provide 2 to Rubisco. The 2 concentration in the bundle-sheath is high, thus preventing photorespiration.

\[F^A_i=F^{V_p}_i-F^L-F^{R_m}_i,\]

where $F^{V_p}$ is the rate of PEP carboxylation, $F^{R_m}$ is the mitochondrial respiration in the mesophyll and fixed to $0.5F^{R_d}$ and $F^L$ is the leakage of $CO_2$ from the bundle-sheath to the mesophyll through the bundle-sheath conductance $c^{g_{bs}}$:

(10.21)#\[F^L=c^{g_{bs}}(p^{C_c}-p^{C_{int}}).\]

$F^{V_p}$ can be limited by the PEP carboxylation rate or the e$^-$ transport rate. In the former case, we have:

\[F^{V_p}=\min(c^{k_p},p^{C_{int}},k^{V_{pmax}})\]

where $c^{k_p}$ is the initial carboxylation efficiency of the PEP carboxylase, and $k^{V_{pmax}}$ is the maximum rate of PEP carboxylation at the saturated $p^{C_{int}}$.

For the e$^-$ transport-limited case for c4 plants, we use:

\[\begin{split}\begin{align} &F^{V_p}_i(F^{J_2}_i)=\frac{c^{\chi} F^{J_2}_i(2+c^{f_Q}-k^{f_{cyc}})}{2c^h(1-k^{f_{cyc})}}=\frac{c^{\chi} F^{J_2}_ic^ z}{2}\\ &F^J_i=F^{J_2}_i\left(1-\frac{c^{f_{pseudo}}}{1-k^{f_{cyc}}}\right) \end{align}\end{split}\]

where $F^{J_2}_i$ is the rate of all e$^-$ transport through photosystem II (PSII), $k^{f_{cyc}}$ is a fraction of electrons that follows a cyclic e$^-$ transport (CET) around photosystem I (PSI), and $c^{f_{pseudo}}$ is a fraction of electrons at PSI that follow a pseudocyclic e$^-$ transport (PET). Both are not used for carbon reduction and photorespiration (which is negligible in c4 plants). Similarly, $c^{f_Q}$, a fraction of electrons at reduced plastoquinone that follow the Q-cycle, is not transferred to the plastocyanin. $c^{f_{pseudo}}$ and $c^{f_Q}$ are PFT-specific parameters. For $k^{f_{cyc}}$ we use:

\[k^{f_{cyc}}=1-\frac{4(1-c^{\chi})(1+c^{f_Q})+3c^hc^{f_{pseudo}}}{3c^h-4(1-c^{\chi})},\]

where $c^{\chi}$ is the fraction of e$^-$ transport rate partitioned to the mesophyll reactions, which is set at 0.4, and $c^h$ is the number of protons required to produce one ATP, which is set at 4.

As for c3 plants, the Rubisco-based assimilation of $CO_2$ in c4 plants can be limited by either Rubisco activity or by e$^-$ transport rate (equation (10.17)) following the equivalent to equation (10.18):

(10.22)#\[F^A_i=\frac{(p^{C_c}-\gamma_* p^{O_{bs}})\chi_1}{p^{C_c}+\chi_2 p^{O_{bs}}+\chi_3}-F^{R_d}_i,\]

where $p^{O_{bs}}$ is the oxygen partial pressure in the bundle-sheath, $\gamma_*=0.5/c^{S_{c/o}}$, $c^{S_{c/o}}$ is the relative CO₂/O₂ specificity factor for Rubisco. For the Rubisco-limited rate, we have: $\chi_1=k^{V_{cmax}}$, $\chi_2=k^{K_{mC}}/k^{K_{mO}}$, $\chi_3=k^{K_{mC}}$, and for the e$^-$ transport-limited rate: $\chi_1=(1-c^{\chi})F^{J_2}_i c^z/3$, $\chi_2=7p^{\Gamma_*}/3$, $\chi_3=0$.

The oxygen partial pressures between intercellular air-space and bundle-sheath are linked through the following equation:

\[p^{O_{bs}}=\frac{c^{\alpha} F^A}{0.047c^{g_{bs}}}+c^{O_{int}},\]

where $c^\alpha$ is the fraction of PSII activity in the bundle-sheath and 0.047 accounts for the diffusivities for O₂ and 2 in water and their respective Henry constants.

To enable an analytical solution [] propose an equation for c4 stomatal conductance that is slightly different than for c3:

(10.23)#\[g_{s,i}=c^{g_0}+\frac{F^A_i+F^{R_d}_i}{p^{C_s}-(p^{C_{x*}})}G^{hum},\]

with $C_{s}$, the $CO_2$ level at leaf surface and $p^{C_{s*}}$ the $C_{s}$-based $CO_2$ compensation point in the absence of $F^{R_d}$.

The three equations (10.22), (10.23) and (10.21) with the three unknowns $F^A$, $g_s$ and $p^{C_c}$ form the system to be solved for c4 plants.

10.2.3. DONE: Parameter dependency on temperature, nitrogen and water#

To account for the nitrogen dependency of $k^{V_{cmax25}}$ we use the nitrogen use efficiency (NUE, defined as $k^{V_{cmax25}}$ per leaf nitrogen content) as derived from observations by :

(10.24)#\[k^{V_{cmax25}}=k^{NUE}*M^{leaf,N}\]

where $k^{NUE}$ is NUE scaled with $f^{sugarload}_{t}$ (??, (10.50)) and leaf efficiency weighted for the fraction of leaves in each age class. Leaf efficiency is scaled by the leaf age of each leaf age class.

Most of the photosynthetic parameters are temperature-dependent and their temperature dependency in ORCHIDEE follows either an Arrhenius function ($F^{R_d}$, $p^{\Gamma_*}$, $k^{K_{mC}}$, $k^{K_{mO}}$, $c^{S_{c/o}}$) or a modified Arrhenius function, accounting for a decrease at high temperatures ($k^{J_{\max}}$, $k^{V_{c\max}}$, ). In addition, for c3 plants, we account for an acclimation term on the two entropy factors that are involved in the temperature function for $k^{V_{c\max}}$ and $k^{J_{\max}}$ and the ratio between $k^{J_{\max}}$ and $k^{V_{c\max}}$ (). These parameters are all function of the monthly temperature. Based on Kattge et al (2009), $k^{J_{\max}}$ is directly inferred from $k^{V_{c\max}}$ (scaled with the temperature dependency of $k^{J_{\max}}$) and the monthly temperature. There is no acclimation term on any of the parameters used for c4 plants.

The water limitation function $G^{hum}$ is impacting photosynthesis through $g_{s}$ ($G^{hum}$, see ??), but it is also limiting $k^{V_{c\max}}$, $k^{J_{\max}}$, $k^{g_m}$ and $F^{R_d}$.

$F^A$ is scaled from leaf to canopy level by multiplying with $d^{LAI}_{i}$ (see ??). Apart from the $I_{abs}$, the other environmental parameters that drive the photosynthesis (temperature, ambient 2, air humidity) are held constant along the canopy depth. We only account for a leaf nitrogen-related reduction of $k^{V_{c\max}}$ and $F^{R_d}$ along the canopy depth, depending on the fraction of absorbed light $f_{light}$, as calculated in the radiation transfer scheme (see ??).

The photon flux density absorbed by leaf photosynthetic pigments at canopy level $i$ ($I_{abs,i}$) is calculated as:

\[I_{abs,n}=\mathsf{SW}_{down}\cdot f_{light,i}\cdot c^{Wtomol}\cdot c^{RGtoPAR},\]

where $c^{Wtomol}$ is a conversion factor between units from $\mathsf{W m^{-2}}$ to $\mathrm{\mu mol\ photons\ m^{-2} s^{-1}}$, and $c^{RGtoPAR}$ is the fraction (=0.5) of $SW_{down}$ that is photosynthetically active.

10.3. DONE: Maintenance respiration#

Maintenance respiration refers to the carbon cost that occurs in living plant tissue to maintain a healthy and living state, e.g., resynthesis of enzymes and membrane lipids, maintenance of ion gradients across membranes, and acclimation to environmental changes []. ORCHIDEE does not distinguish the underlying processes and instead uses a bulk approach [] in which maintenance respiration of a specific PFT within a grid cell ($F^{rm}$; g C m$^{-2}$ s$^{-1}$) is the sum of maintenance respiration of the different plant organs. It is calculated as a function of biomass modulated by the temperature and nitrogen content of the vegetative and reproductive plant organs:

(10.25)#\[F^{rm} = \sum_{o=1}^{norgan}{c_1 \cdot M^{o} \cdot m^{o,rm,T} \cdot m^{o,rm,N}},\]

where $o$ denotes the plant organ, $norgan$ denotes the total number of plant organs (see ??), $c_1$ (s$^{-1}$) is a PFT-specific maintenance respiration coefficient prescribing the fraction of biomass that is lost during each time step and $M^{o}$ (g C m$^{-2}$) is the stand level nitrogen biomass of plant organ $o$. $c_1$ is set to zero for the above-ground and below-ground heartwood as well as the carbohydrate reserves as ORCHIDEE assumes that these organs have no maintenance respiration.

$m^{o,rm,T}$, is a temperature modulator (unitless) calculated for each plant organ $o$ as:

\[m^{o,rm,T} = 1 + (c_2 + c_3 \cdot T^{3year} + c_4 \cdot {T^{3year}}^2) \cdot T^{o},\]

where $T^{3year}$ (K) represents the mean air temperature above freezing over the 3 years preceding the time step under consideration. $c_2$, $c_3$, and $c_4$ are PFT-specific parameters. For the above-ground plant organs, the 2-meter air temperature ($T^{air}$; K) is used for $T^{o}$. For below-ground plant organs, $T^{o}$ represents the root-zone temperature $T^{soil,weighted}$ (K) calculated as the vertical profile of soil temperature (see ??) weighted by the vertical profile of the root biomass (see ??). When $T^{o}$ drops below 273.15 K, $m^{o,rm,T}$ is set to one, implying that maintenance respiration continues but levels off when freezing.

In equation (10.25), $m^{o,rm,N}$ (unitless) is a modulator for the N concentration of organs $o$ and is calculated as:

\[m^{o,rm,N} = \frac{f^{cn}_{k}}{{c_5}^{cn}_{k}} \cdot \max(\min( 1 + \left( 1 - \frac{f^{cn}_{k}} {{c_5}^{cn}_{k}} \right) \cdot c_6,\,1.2),\,0.8),\]

where $c_6$ is a global parameter, ${c_5}^{cn}_{k}$ is the reference carbon-to-nitrogen ratio for organ $o$, i.e., 45 for leaves, 52 for roots and fruits, and 517 for the sapwood and carbohydrates reserves. $f^{cn}_{k}$ (unitless) is the actual carbon-to-nitrogen ratio for organ $o$ which is truncated at 200 if the calculated carbon-to-nitrogen ratio exceeds this threshold.

The parameters $c_1$, $c_2$, $c_3$, $c_4$, but not $c_5$, have been tuned at the PFT-level and $c_6$ at the global level to simulate biomass production efficiencies within the ranges observed by and . Given that ORCHIDEE does not simulate the carbon fluxes from the vegetation to the mycorrhizae, the latter is included in $F^{rm}$. The simulated maintenance respiration flux ($F^{rm}$) should, therefore, not be compared against observed fluxes as their definitions are not identical.

10.4. DONE: Allocation and growth respiration#

10.4.1. DONE: Allocatable carbon and nitrogen#

Following bud burst, photosynthesis produces carbon compounds that are added to the labile carbon pool as long as the plant passed bud burst and is not in presenescence yet (see ??):

\[M^{lab,C}_{t} = M^{lab,C}_{t-1} + F^{gpp}_{t} \cdot \Delta{t},\]

where $M^{lab,C}_{t}$ (g C m$^{-2}$) is the labile carbon pool at the current time step $t$, $M^{lab,C}_{t-1}$ (g C m$^{-2}$) is the labile carbon pool at time step $t-1$, $F^{gpp}_{t}$ (g C m$^{-2}$ s$^{-1}$) is the photosynthetic carbon flux at the current time step, and $\Delta{t}$ is the length of the current time step which is one day given in seconds (s). If the plant is in presenescence or senescence (see ??), the photosynthetic flux is added to the reserve pool:

\[M^{res,C}_{t} = M^{res,C}_{t-1} + F^{gpp}_{t} \cdot \Delta{t},\]

where $M^{res,C}_{t}$ and $M^{res,C}_{t-1}$ are the carbon contained in the reserve pool at time step $t$ and $t-1$ (g C m$^{-2}$), respectively. The labile carbon pool is then split into an active and none-active pool. The size of the active pool is calculated as a function of plant phenology and temperature. If the plant passed bud break and is not in presenescence yet, the active part of the labile pool was formalized following , , and :

\[M^{totinc} = M^{lab,C} \cdot \exp\left(\frac{c_1}{c_2-c_3}-\frac{c_1}{T^{air}-c_3}\right),\]

where $c_1$, $c_2$ and $c_3$ are PFT-specific parameters representing respectively a growth temperature coefficient, the reference temperature above which all carbon in the labile pool is allocated to growth, and the minimum temperature for allocation. The parameters $c_1$, $c_2$ and $c_3$ were set such that no carbon is available for plant growth below -2 ° C, only 3 % of the labile pool is available for allocation at 0 ° C, and at 5 ° C, 100 % of the labile pool is available for allocation. $T^{air}$ is the 2-m air temperature at $t$, and $M^{totinc}$ (g C m$^{-2}$) is the total biomass increment at the current time step. If the plant passed presenescence, the active part of the labile pool is set to zero.

Prior to allocating, $M^{totinc}$ is used to sustain the maintenance respiration flux ($F^{rm}$; g C m$^{-2}$ s$^{-1}$; see ??) up to a maximum of 100 % of $M^{totinc}$. A first estimate of the growth respiration, i.e., the cost for producing new tissue excluding the carbon required to build the tissue itself [], is set aside. This first estimate assumes that all the available carbon will be allocated and is calculated as:

(10.26)#\[F^{rg,est} = \frac{ M^{totinc} \cdot c_4}{ (1 + c_4) \cdot \Delta{t}},\]

where $F^{rg,est}$ (g C m$^{-2}$ s$^{-1}$) is the estimated growth respiration and $c_4$ is the fraction of carbon respired to grow one unit of carbon, which is set to 28 % []. Hence, the amount of carbon that is available for biomass growth is:

\[M^{totinc} = M^{totinc} - (F^{rm}+F^{rg,est}) \cdot \Delta{t},\]

where $\Delta{t}$ is the length of the time step for the calculation of biomass allocation (see table 7). Once allocation is finalized, $F^{rg,est}$ will be recalculated (section ??) to obtain the exact growth respiration for the actual biomass increment.

Whether all of the available carbon can be allocated or not depends on the nitrogen availability. The labile N pool is calculated as:

\[M^{lab,N}_{t} = M^{lab,N}_{t-1} + F^{root,N}_{t} \cdot \Delta{t},\]

where $M^{lab,N}_{t}$ and $M^{lab,N}_{t-1}$ (g N m$^{-2}$) are the labile nitrogen pools at $t$ and $t-1$, respectively, and $F^{root,N}_{t}$ (g N m$^{-2}$ s$^{-1}$) is the nitrogen uptake by the plants at the current time step. Subsequently, $M^{totinc}$ is adjusted for the available nitrogen after estimating the nitrogen that would be required to allocate $M^{totinc}$ entirely. This estimate assumes that all tissue has the carbon-to-nitrogen ratio of leaves:

\[b_{1} = \min\left(\frac{1}{f^{cn,leaf}} - \frac{1}{c_5},0\right),\]

where $f^{cn,leaf}$ (unitless) is the actual carbon-to-nitrogen ratio of leaves for the PFT and grid cell under consideration and $c_5$ is the PFT-specific minimal carbon-to-nitrogen ratio for leaves. The maximal elasticity of the carbon-to-nitrogen ratio of leaves $b_{2}$ (unitless) is calculated as:

\[b_{2} = 1 -\exp\left(-\left(1.6 \cdot \frac{b_{1}}{\frac{1}{c_6} - \frac{1}{c_5}}\right)^{4.1}\right),\]

where $c_6$ is the PFT-specific maximal carbon-to-nitrogen ratio for leaves. Assuming all newly grown biomass has the actual carbon-to-nitrogen ratio of leaves, $b_{3}$ (g C m$^{-2}$) represents the maximum amount of carbon that could be allocated at time step $t$, given the available amount of nitrogen:

\[b_{3} = \frac{M^{lab,N} \cdot 0.9} {f^{cn,leaf}}\]

If there is enough nitrogen (thus $M^{totinc} < b_{3}$), $M^{totinc}$ can be allocated. If, however, there is not enough nitrogen to allocate the allocatable carbon entirely (thus $M^{totinc} > b_{3}$), the elasticity of the carbon-to-nitrogen ratio of leaves is used to adjust $M^{totinc}$:

\[\begin{split}\begin{align} &b_{4} = \frac{M^{lab,N} \cdot 0.9 \cdot f^{cn,leaf}} {M^{totinc}}, \\ &b_{5} = \min \left(\max \left(b_{4},1 - c_7 \cdot \left(1 - b_{2} \right) \right),1 \right), \\ &b_{6} = \max \left(\min \left(\frac{b_{5}}{f^{cn,leaf}}, \frac{1}{c_5} \right), \frac{1}{c_6} \right), \\ &b_{7} = \min \left(M^{lab,N} \cdot 0.9, M^{totinc} \cdot b_{6} \right) \\ &M^{totinc} = \min \left(M^{totinc}, \frac {b_{7}} {b_{6}} \right), \end{align}\end{split}\]

where $c_7$ is the maximal elasticity of foliage N concentrations and set to 0.25 and $M^{totinc}$ is the carbon that will be allocated either for restoring plant allometry or for plant growth as described in the following sections.

10.4.2. DONE: Plant allometry#

Biomass is allocated to leaves, roots, sapwood, heartwood, and fruits. Allocation to leaves, roots and wood respects the pipe-model theory [] and thus assumes that producing one unit of leaf mass requires a proportional amount of sapwood to transport water from the roots to the leaves as well as a proportional fraction of roots to take up the water from the soil.

The scaling parameter between leaf and sapwood mass is derived from:

(10.27)#\[d^{leaf} = k^{ls} \cdot m^{water} \cdot d^{sap},\]

where $d^{leaf}$ (m$^{2}$) is the one-sided leaf area of an individual plant, $d^{sap}$ (m$^{2}$) is the sapwood area of an individual plant, $k^{ls}$ a calculated parameter (unitless) linking leaf area to sapwood area and, $m^{water}$ (unitless) is the water stress as calculated in section ??. Alternatively, leaf area can be written as a function of leaf mass ($M^{leaf}$) and the specific leaf area ($k^{sla}$; m$^{2}$ g C$^{-1}$):

(10.28)#\[d^{leaf} = M^{leaf} \cdot k^{sla}\]

Sapwood mass $M^{sap}$ can be calculated from the sapwood area $d^{sap}$ as follows:

(10.29)#\[M^{sap} = d^{sap} \cdot d^{h} \cdot c_8,\]

where $d^{h}$ is the tree height (m) and $c_8$ is the sapwood density (g C m$^{-3}$). Following substitution of equations (10.28) and (10.29) into equation (10.27), leaf mass can be written as a function of sapwood mass:

(10.30)#\[M^{leaf} = \frac{M^{sap} \cdot f^{KF}} {d^{h}},\]

where

(10.31)#\[f^{KF} = \frac{k^{ls} \cdot m^{water}} {k^{sla} \cdot c_8},\]

where $k^{ls}$ is calculated as a function of the gap fraction as supported by site-level observations []:

(10.32)#\[k^{ls} = c_9 + f^{Pgap} \cdot (c_{10} - c_9).\]

$c_9$ is the minimum observed leaf area to sapwood area ratio, $c_{10}$ is the maximum observed leaf area to sapwood area ratio and $f^{Pgap}$ is the actual gap fraction of the canopy (see ??). By using the gap fraction as a driver of $k^{ls}$ more carbon will be allocated to the leaves until canopy closure is reached.

Following , sapwood mass and root mass ($M^{root}$) relate as follows:

(10.33)#\[M^{sap} = c_{11} \cdot d^{h} \cdot M^{root},\]

where the parameter $c_{11}$ (m$^{-1}$) is calculated according to (their equation (17)):

(10.34)#\[c_{11} = \sqrt{\frac{c_{12} \cdot c_{14} \cdot c_8} {c_{13} \cdot c_{15}}},\]

where $c_{12}$ is the hydraulic conductivity of roots (m$^{3}$ kg$^{-1}$ s$^{-1}$ MPa$^{-1}$), $c_{13}$ is the hydraulic conductivity of sapwood (m$^{2}$ s$^{-1}$ MPa$^{-1}$), $c_{14}$ is the longevity of sapwood (days) and $c_{15}$ is the root longevity (days). Following substitution of equation (10.33) into (10.30) and rearrangement, leaf mass can be written as a function of root mass:

(10.35)#\[M^{leaf} = f^{LF} \cdot M^{root},\]

where

(10.36)#\[f^{LF} = c_{11} \cdot f^{KF}.\]

Parameter values used in equations (10.27) to (10.35) are PFT-specific. The allometric relationships between the plant components and the hydraulic architecture of the plant are both based on the pipe-model theory, hence, the same parameter values for the hydraulic conductivity of the plant components, i.e., $c_{12}$, $c_{13}$, are used in their calculations (section ??).

10.4.3. DONE: Restoring plant allometry#

The different biomass pools have different turnover times (section ??), are affected differently by the phenological processes, and are managed differently. As a consequence, the actual biomass components may no longer respect the allometric relationships (see ??). Plant allometry is restored by making use of allometric relationships.

(10.37)#\[\begin{split}\begin{align} & M^{leaf,target}_{l} = \max \left( \frac{f^{KF} \cdot M^{sap}_{l}}{d^{h}_{l}}, M^{root}_{l} \cdot f^{LF} , M^{leaf}_{l} \right), \\ & M^{sap,target}_{l} = \max \left( \frac{M^{leaf}_{l}\cdot d^{h}_{l}}{f^{KF}} , \frac{M^{root}_{l} \cdot f^{LF} \cdot d^{h}_{l}}{f^{KF}}, M^{sap}_{l} \right),\\ & M^{root,target}_{l} = \max \left( \frac{M^{leaf,target}_{l}}{f^{LF}}, \frac{f^{KF} \cdot M^{sap}_{l}}{d^{h}_{l} \cdot f^{KF}} ,M^{root}_{l} \right), \end{align}\end{split}\]

where $M^{leaf,target}_{l}$, $M^{sap,target}_{l}$, and $M^{root,target}_{l}$ are, respectively, the leaf, sapwood, and root mass (g C plant$^{-1}$) for circumference class $l$ following the allometric relationships. The carbon required to restore the allometric relationships is calculated as:

(10.38)#\[\begin{split}\begin{align} & M^{linc}_{l} = M^{leaf,target}_{l} - M^{leaf}_{l}, \\ & M^{sinc}_{l} = M^{sap,target}_{l} - M^{leaf}_{l}, \\ & M^{rinc}_{l} = M^{root,target}_{l} - M^{leaf}_{l}, \\ & M^{inc}_{l} = M^{linc}_{l} + M^{rinc}_{l} + M^{sinc}_{l}, \end{align}\end{split}\]

where $M^{linc}$, $M^{sinc}$, and $M^{rinc}$ are the increment of the leaf, sapwood, and root biomass (g C plant$^{-1}$) for circumference class $l$ and $M^{inc}_{l}$ (g C plant$^{-1}$) is the total biomass that is allocated to circumference class $l$.

In ORCHIDEE forests are modeled to have ${ncirc}$ circumference classes with $d^{ind}$ (plants m$^{-2}$) identical trees in each circumference class. Hence, the allocatable biomass needs to be distributed across all individuals in ${ncirc}$ circumference classes. Following the restoration of the allometric relationships, the biomass remaining to be allocated is calculated as:

(10.39)#\[M^{totinc} = \max \left( \sum_{l=1}^{ncirc} d^{ind}_{l} \cdot M^{inc}_{l}, 0 \right)\]

Within a single time step, biomass is thus allocated until the allometic relationships are restored or $M^{totinc}$ is consumed. It may take several time steps, especially at the start of the growing season, to restore the allometric relationships.

10.4.4. DONE: Plant growth#

Once the allometric relationships are restored, the pipe-model can be used to allocate biomass to leaves, sapwood and roots during plant growth. Fruit allocation is calculated first as it does not follow allometric relationships:

(10.40)#\[M^{finc} = c_{16} \cdot M^{totinc},\]

where $c_{16}$ is a PFT-specific parameter representing the share of carbon that is allocated to the fruits. $M^{finc}$ (g C m$^{-2}$) is calculated at the PFT-level and then distributed over the circumference classes $l$.

Equations (10.30) and (10.35) can be rewritten as:

(10.41)#\[\begin{split}\begin{align} &\frac{M^{leaf}_{l} + M^{linc}_{l}}{ M^{sap}_{l} + M^{sinc}_{l}} = \frac{f^{KF}}{d^{h}_{l} + d^{hinc}_{l}}, \\ &M^{leaf}_{l} + M^{linc}_{l} = (M^{root}_{l} + M^{rinc}_{l}) \cdot f^{LF}, \end{align}\end{split}\]

and allometric relationship is used to describe the relationship between tree height and basal area:

(10.42)#\[d^{h}_{l} = k^{height} \cdot \left( \frac{4}{\pi} \cdot d^{ba}_{l} \right)^{\frac{c_{17}}{2}},\]

where $k^{height}$ (m) is a grid cell specific parameter giving the tree height for a tree with a diameter of 1 meter (see ??), $c_{17}$ is a PFT-specific parameter for the relationship between basal area and tree height, and $d^{ba}_{l}$ (m$^{2}$) is the basal area of an individual tree in circumference class $l$. The change in height is then calculated as a function of $d^{ba}_{l}$ and $d^{bainc}_{l}$ where $d^{bainc}_{l}$ (m$^{2}$) is the basal area increment:

(10.43)#\[d^{hinc}_{l} = \left(k^{height} \cdot \left( \frac{4}{\pi} \cdot \left( d^{ba}_{l} + d^{bainc}_{l} \right) \right) ^{\frac{c_{17}}{2}}\right) - d^{h}_{l},\]

The distribution of carbon across the $ncirc$ circumference classes depends on the basal area of the model tree within each circumference class $l$. Trees with a large basal area are expected to have a higher leaf area and photosynthesis and are, therefore, assigned more carbon for wood allocation than trees with a smaller basal area, according to the rule of :

(10.44)#\[d^{bainc}_{l} = \frac{k^{\gamma}}{2} \cdot \left( d^{circ}_{l} - c_{18} \cdot k^{\sigma} + \left( \left(c_{18} \cdot k^{\sigma} + d^{circ}_{l} \right)^{2} - \left(4 \cdot k^{\sigma} \cdot d^{circ}_{l} \right) \right)^{\frac{1}{c_{19}}} \right),\]

where $d^{bainc}_{l}$ and $k^{\gamma}$ (m) are unknown. $d^{circ}_{l}$ (m) is the circumference of the model tree in circumference class $l$, $c_{18}$ is a fixed parameter (unitless), $c_{19}$ ranges between 2 and 3.5 in function of $d^{dia,mean}$, and $k^{\sigma}$ (m) is a calculated parameter. $k^{\sigma}$ is a function of the diameter distribution of the stand at a given time step:

(10.45)#\[k^{\sigma} = c_{20} + c_{21} \cdot d^{circ,med},\]

where $c_{20}$ and $c_{21}$ are parameters (unitless) and $d^{circ,med}$ is the median circumference of the $ncirc$ circumference classes for the PFT and grid cell under consideration.

Equations (10.36) and (10.40) to (10.45) need to be simultaneously solved. An iterative scheme was avoided by linearizing equation (10.43), which was found to be an acceptable numerical approximation as allocation is calculated at a daily time step, and hence, the changes in height are small and the relationship is locally linear:

(10.46)#\[d^{hinc}_{l} = \frac{d^{bainc}_{l}}{f^{s}},\]

where $f^{s}$ is the slope of the locally linearized equation (10.43) for all $ncirc$ circumference classes and is calculated as:

(10.47)#\[f^{s} = \frac{c_{22}}{\left( k^{height} \cdot \left(\frac{4}{\pi} \cdot \left( d^{ba}+c_{22} \right) \right)^{\frac{c_{17}}{2}} - k^{height} \cdot \left(\frac{4}{\pi} \cdot d^{ba} \right)^{\frac{c_{17}}{2}} \right)},\]

where $c_{22}$ represents a typical increase in basal area for a single time step. Following linearization of equation (10.43), equations (10.36) and (10.40) to (10.47) can be substituted and rearranged to obtain a quadratic equation of which the positive root is used as a solution for $k^{\gamma}$. $k^{\gamma}$ distributes photosynthates across the different circumference classes and as such controls the intra-species competition within a stand. $k^{\gamma}$ thus depends on the total allocatable carbon and needs to be optimized at every time step. Once $k^{\gamma}$ has been calculated, $d^{bainc}_{l}$, $M^{linc}_{l}$, $M^{rinc}_{l}$ and $M^{sinc}_{l}$ are calculated. Unallocated carbon, typically in the order of 10$^{-9}$ to 10$^{-15}$, is added back to $M^{lab}$ to preserve carbon.

For grasses and crops the height is calculated as a function of their biomass instead of function of basal area as is the case for forest. Likewise, a single circumference class is considered. Hence, intra-specific competition is not accounted for.

$M^{linc}_{l}$, $M^{rinc}_{l}$ and $M^{sinc}_{l}$ propose how the carbon should be allocated to jointly respect plant allometry and intra-specific competition in the case of forest PFTs and plant allometry in the case of grassland and cropland PFTs, supposing that there is enough nitrogen to sustain the proposed growth. Prior to allocating these estimates to their respective pools, it is checked whether there is sufficient labile nitrogen to sustain the proposed allocation. When nitrogen is limiting, the allocation will be linearly adjusted (see below). Given the non-linearity of the pipe-model (section ??), such adjustments should be avoided and when they occur should be kept small. ORCHIDEE reduces the need to adjust the proposed allocation by estimating, at the start of the allocation process (section ??), the mass of carbon that can be allocated given the available nitrogen. Now that all the required information is available, the subsequent calculations finalize and refine the initial estimate.

Nitrogen allocation starts with the calculation of the allocation fractions:

\[\begin{split}\begin{align} &f^{leaf,alloc} = \frac{\sum_{l=1}^{ncirc} M^{linc}_{l} \cdot d^{ind}_{l}}{M^{totinc}}, \\ &f^{sap,above,alloc} = \frac{ c_{23} \cdot \sum_{l=1}^{ncirc} M^{sinc}_{l} \cdot d^{ind}_{l}}{M^{totinc}}, \\ &f^{sap,below,alloc} = \frac{ (1-c_{23}) \cdot \sum_{l=1}^{ncirc} M^{sinc}_{l} \cdot d^{ind}_{l}}{M^{totinc}}, \\ &f^{root,alloc} = \frac{ \sum_{l=1}^{ncirc} M^{rinc}_{l} \cdot d^{ind}_{l}}{M^{totinc}}, \\ &f^{fruit,alloc} = \frac{ \sum_{l=1}^{ncirc} M^{finc}_{l} \cdot d^{ind}_{l}}{M^{totinc}}, \end{align}\end{split}\]

where $f^{leaf,alloc}$, $f^{sap,alloc}$, $f^{root,alloc}$, and $f^{fruit,alloc}$ (unitless) are PFT-level fractions of the available carbon that are proposed to be allocated to the leaf, sapwood, root and fruit pools, respectively. The allocation within the sapwood pool is divided into above- and belowground sapwood, where the fraction allocated aboveground is given by the PFT-specific parameter $c_{23}$. Following the pipe-model approach that states that increasing the leaf mass has to be complemented by an increase in sapwood and roots to sustain the hydraulic continuity of the plant, the nitrogen cost is calculated as the total nitrogen mass required to allocate 1 g of N to the leaves []:

\[f^{N,cost} = f^{leaf,alloc} + c_{24} \cdot f^{sap,alloc} + c_{25} \cdot (f^{root,alloc} + f^{fruit,alloc}),\]

where $c_{24}$ is the carbon-to-nitrogen ratio of wood relative to the carbon-to-nitrogen ratio of leaves and set to 0.087, and $c_{25}$ is the carbon-to-nitrogen ratio of roots relative to the carbon-to-nitrogen ratio of leaves and set to 0.86. Given the actual carbon-to-nitrogen ratio of the leaves $f^{cn,leaf}$, the carbon that could be allocated $b_{6}$ (g C m$^{-2}$) is calculated as:

\[b_{6} = \frac{M^{lab,N} \cdot f^{cn,leaf}}{f^{N,cost}}\]

In case there is not enough nitrogen to sustain the proposed biomass increments, i.e., $M^{totinc}$ is greater than $b_{6}$), $M^{totinc}$ will be reduced to meet the nitrogen availability by taking into account the elasticity of the carbon-to-nitrogen ration. The carbon that can be allocated $M^{totinc}$, given the maximal carbon-to-nitrogen elasticity $b_{8}$ (unitless) and the available nitrogen $M^{lab,N}$ (g N m$^{-2}$), is calculated as:

\[\begin{split}\begin{align} &b_{6} = \frac{M^{lab,N} \cdot f^{N,cost}}{M^{totinc} \cdot f^{cn,leaf}}, \\ &b_{7} = \min \left( \max \left( b_{6},1-b_{2} \right),1 \right), \\ &b_{8} = \min \left( M^{lab,N}, M^{totinc} \cdot f^{N,cost} \cdot \max \left( \min \left( \frac{b_{7}}{f^{cn,leaf}},\frac{1}{c_5} \right), \frac{1}{c_6} \right) \right), \\ &M^{totinc} = \min \left( M^{totinc}, \frac{b_{8}} {f^{N,cost} \cdot \max \left( \min \left( \frac{b_7}{f^{cn,leaf}} ,\frac{1}{c_5} \right), \frac{1}{c_6} \right) } \right) \end{align}\end{split}\]

In case there is sufficient nitrogen to sustain the proposed carbon allocation, $M^{totinc}$ does not need to be recalculated, but the carbon-to-nitrogen ratio will be decreased to use as much nitrogen as possible until the maximum elasticity of the carbon-to-nitrogen ratio is reached:

\[\begin{split}\begin{align} &b_{9} = \min(\max(b_{6},1),1+b_{2})), \\ &b_{10} = \min(M^{lab,N}, M^{totinc} \cdot f^{N,cost} \cdot \max \left( \min \left (\frac{b_{9}}{f^{cn,leaf}}, \frac{1}{c_5} \right), \frac{1}{c_6} \right) \end{align}\end{split}\]

The nitrogen constraint is calculated at the PFT-level, and if nitrogen limitation requires a reduction in $M^{totinc}$, $M^{linc}_{l,t}$, $M^{sinc}_{l,t}$, and $M^{rinc}_{l,t}$ are all linearly reduced.

The biomass increments for both carbon and nitrogen are taken from the labile pools and allocated to their respective biomass pools:

\[\begin{split}\begin{align} &M^{leaf}_{l,t} = M^{leaf}_{l,t-1} + M^{linc}_{l,t}, \\ &M^{sap}_{l,t} = M^{sap}_{l,t-1} + M^{sinc}_{l,t}, \\ &M^{root}_{l,t} = M^{root}_{l,t-1} + M^{rinc}_{l,t}, \\ &M^{lab}_{t} = M^{lab}_{t-1} - M^{finc}_{t} - \sum_{l=1}^{ncirc}{(M^{linc}_{l,t} - M^{sinc}_{l,t} - M^{rinc}_{l,t}) \cdot d^{ind}_{l,t}} \end{align}\end{split}\]

Sapwood contains mostly physiologically active cells that transport water and minerals from the roots to the leaves through the stem []. As the sapwood ages, the cells nearest to the center of the trunk accumulate chemical compounds which makes theses cells to give up their transport function to become what is known as heartwood []. In ORCHIDEE the carbon and nitrogen contained in the sapwood is turned over into heartwood with a PFT-specific time constant that represents the longevity of the sapwood. Contrary to the other turnover processes in ORCHIDEE sapwood turnover is not added to the litter pool as the biomass is not lost, but converted to heartwood:

(10.48)#\[\begin{split}\begin{align} &M^{sap}_{l,t} = M^{sap}_{l,t-1} \cdot \left( \Delta t - \frac{1}{c_7} \right) ,\\ &M^{heart}_{l,t} = M^{heart}_{l,t} + \frac{M^{sap}_{l,t-1}}{c_7}. \end{align}\end{split}\]

The initial estimate of the growth respiration $F^{rg,est}$ (equation (10.26)) can be recalculated now that the final allocation has been decided on:

\[\begin{split}F^{rg} = \frac{c_4 \cdot M^{totinc}} {\Delta{t}} \\\end{split}\]

The carbon that was set aside for growth respiration but not used, i.e., the difference between the initial estimate ($F^{rg,est}$; equation (10.26)) and the final calculation ($F^{rg}$; equation (10.48)) is moved back into the labile carbon pool:

\[M^{lab,C} = M^{lab,C} + (F^{rg,est} - F^{rg}) \cdot \Delta{t}\]

10.4.5. DONE: Labile and reserve pools#

The target carbon mass for the labile pool $M^{lab,target}$ is calculated as a function of the nitrogen biomass of the living plant organs and the actual 2-meter air temperature:

\[\begin{split}\begin{align} &M^{lab,target} = c_{26} \cdot \exp(\frac{c_1}{c_2-c_3}-\frac{c_1}{T^{air}-c_3}) \cdot (M^{leaf,N} + M^{root,N}+ M^{fruit,N} + M^{sap,N}), \\ &M^{lab,target} = \max(M^{lab,target}, 10 \cdot F^{gpp,week} \cdot \Delta{t}), \end{align}\end{split}\]

where $c_{26}$ is a PFT-specific parameter, $M^{o,N}$ (g N m$^{-2}$) is the nitrogen mass of plant organ $o$, and $F^{gpp,week}$ is the mean photosynthetic flux over the past week.

The target carbon mass for the reserve pool is calculated as a function of the phenology type of the PFT. For evergreen forests the target reserve pool is calculated as:

\[M^{res,target} = \min(c_{27} \cdot M^{sap,C}, M^{leaf,target,C} \cdot (1+\frac{c_{28}}{f^{LF}})),\]

where $c_{27}$ and $c_{28}$ are PFT-specific parameters representing the share of the sapwood and roots, respectively, to the reserve pool, $M^{res,target}$ is the target mass for the reserve carbon pool, and $M^{leaf,target,C}$ (g C m$^{-2}$) is the leaf carbon mass for a canopy that is in allometric balance with the sapwood carbon mass. For deciduous forest the target reserve pool is calculated as:

\[M^{res,target} = c_{29} \cdot M^{sap,C},\]

where $c_{29}$ is a PFT-specific parameter representing the share of the sapwood to the reserve pool. The value of $c_{29}$ also depends on the plant phenology. The value is higher when the plant is in (pre)senescence. For grassland PFTs the target reserve pool is calculated as:

\[M^{res,target} = \min(c_{29} \cdot (M^{r,C} + M^{sap,C}), M^{leaf,target,C} \cdot (1+\frac{c_{28}}{f^{LF}}).\]

ORCHIDEE fills the reserve and labile pools at the same time but with different (arbitrary) speeds. In ORCHIDEE there are two important differences between the reserve and the labile pool: (1) only the labile pool is used in the allocation but storing carbon and nitrogen in the labile pool comes with a carbon cost, i.e., maintenance respiration, and (2) the reserve pool comes without maintenance respiration but the carbon contained in it cannot be allocated. These differences imply that the plant should store as much carbon and nitrogen as needed for growth in its labile pool but no more than that to limit the respiration cost. This trade-off is formalized as:

(10.49)#\[M^{res,target} \cdot x^{2} + M^{lab,target} \cdot x = M^{res} + M^{lab},\]

For simplicity it was assumed that carbon can move freely between both pools. The optimal allocation to both pools can be calculated by solving equation (10.49) for $x$.

\[\begin{split}\begin{align} &M^{res} = \max(0.1 \cdot (M^{res,target}+M^{lab,target}), \min((M^{res,target} \cdot x^{2}), 0.9 \cdot (M^{res,target}+M^{lab,target}))), \\ &M^{lab} = \max(0,M^{res,target}+M^{lab,target} - M^{res}). \end{align}\end{split}\]

As long as $M^{res} + M^{lab}$ is below the target, $x$ is less than one, implying that the available carbon will be preferentially stored in the labile pool. Once the target has been reached, $x$ exceeds one, implying that the available carbon will be preferentially stored in the reserve pool.

Following carbon allocation to the reserve and labile pool, nitrogen is allocated:

\[\begin{split}\begin{align} &M^{lab,target,N} = \frac{M^{lab,target,C}}{f^{cn,leaf}} \cdot f^{N,cost},\\ &b_{11} = M^{lab,N} - M^{lab,target,N},\\ &M^{res,N} = M^{res,N} + b_{11}. \end{align}\end{split}\]

ORCHIDEE does not control the carbon-to-nitrogen ratio of the the reserve pool; even if nitrogen is limiting plant growth, carbon may accumulate in the reserve pool. Rather than respiring this excessive carbon [], ORCHIDEE decreases its photosynthesis to reduce the further accumulation of carbon. Photosynthesis is reduced when a PFT has more labile and reserve carbon than the target mass of the labile and reserve pools. Reduced photosynthesis when reserve and labile carbon start to accumulate is justified by the observation that too much sugars in the leaves increases sap viscosity which in turn hampers the sap flow in the phloem. Viscosity can be reduced by closing the stomata and transpiring less of the sap flow from the xylem. By closing the stomata, photosynthesis will be reduced []. Because ORCHIDEE does not calculate the phloem sap flow, turgor, or viscosity yet, a ratio $f^{sugarload}$ is used to reduce the nitrogen use efficiency used in the calculation of photosynthesis (see ??). If the plant has less carbon in its labile and reserve pools than targeted, the nitrogen use efficiency is increased. Up and down regulation of nitrogen use efficiency is only accounted for between bud break and the start of (pre)senescence.

(10.50)#\[f^{sugarload}_{t} = \frac{M^{res,target,C} + M^{lab,target,C}}{M^{res,C} + M^{lab,C}},\]

where $f^{sugarload}_{t}$ (unitless) is a proxy for sugar loading at time step $t$. The sugar loading used to regulate the nitrogen use efficiency is the average sugar loading over the last 7 days to avoid sudden changes in the nitrogen use efficiency.

10.5. DONE: Vegetation characteristics#

10.5.1. DONE: Leaf age#

ORCHIDEE distinguishes four age classes ($nlage$) where the length (days) of an individual age $i$ class is PFT-specific and calculated as:

\[l_{i} = \frac{b_{2}}{nlage},\]

where $b_{2}$ is the PFT-specific, location specific critical leaf longevity (days) calculated as:

\[\begin{split}\begin{align} &b_{1} = \max(c_1 \cdot c_3, c_1 - c_4 \cdot (T^{3year} - c_5) ),\\ &b_{2} = \min(c_1 \cdot c_2, b_{1}) \end{align}\end{split}\]

where $c_1$ is the parameter prescribing the longevity of a typical leaf or needle at the reference temperature ($c_5$, K) for the PFT under consideration. $c_2$, and $c_3$ are parameters to calculate the upper and lower boundaries of the critical leaf age. The parameter $c_4$ describes the temperature dependency of the critical leaf age in between its lower and upper boundary. The temperature dependency makes $b_{1}$ location specific in addition to $c_1$, $c_2$, $c_3$, $c_4$, and $c_5$ being PFT-specific. The temperature dependency enables simulating a range in longevity within in a single PFT.

Newly allocated leaf biomass ($M^{linc}$) goes into the first age class:

\[M^{leaf}_{1} = f^{leaf}_{1} \cdot M^{leaf} + M^{linc},\]

where $f^{leaf}_{i}$ is the fraction of leaf mass in leaf age class $i$ and $f^{leaf}_{1}$ is thus the fraction of leaf mass in the first age class. The fractions for the youngest leaf age class is then calculated:

\[f^{leaf}_{1} = \frac{M^{leaf}_{1}} {(M^{leaf} + M^{linc})}.\]

The fractions for the remaining leaf age classes are then calculated as:

\[f^{leaf}_{i} = \frac{f^{leaf}_{i} \cdot M^{leaf}} {(M^{leaf} + M^{linc})}.\]

Each time step, the age of the leaves ($a^{leaf}_{i}$) in age class $i$ increases with the number of days of the time step ($\Delta{t}$):

\[a^{leaf}_{i,t} = a^{leaf}_{i,t-1} + \Delta{t}\]

The turnover between the leaf age classes $i$ is calculated as:

\[\Delta{f^{leaf}_{i}} = \frac{f^{leaf}_{i-1} \cdot \Delta{t}} {l_{i}}\]

The fraction of the leaves in each age class is then updated:

\[\begin{split}\begin{align} &f^{leaf}_{i-1} = \min(0,f^{leaf}_{i-1} - \Delta{f^{leaf}_{i}}) \\ &f^{leaf}_{i} = f^{leaf}_{i} + \Delta{f^{leaf}_{i}} \end{align}\end{split}\]

The leaf age in each age class $i$ is updated from its previous value:

\[a^{leaf}_{i} = \frac{(f^{leaf}_{i} - \Delta{f^{leaf}_{i+1}}) \cdot a^{leaf}_{i} + \Delta{f^{leaf}_{i}} \cdot a^{leaf}_{i-1}} {f^{leaf}_{i} + \Delta{f^{leaf}_{i}} - \Delta{f^{leaf}_{i+1}}}\]

As there is no turnover in the last age class ($nlage$), leaf age is updated as follows:

\[a^{leaf}_{nlage} = \frac{f^{leaf}_{nlage} \cdot a^{leaf}_{nlage} + \Delta{f^{leaf}_{nlage}} \cdot a^{leaf}_{nlage-1}} {f^{leaf}_{nlage} + \Delta{f^{leaf}_{nlage}}}\]

The leaf age fractions are then normalized to avoid numerical issues. Mean leaf age ($a^{leaf}$) is used in the calculation of photosynthesis (section ??) and senescence (section ??) and calculated as the mean of all $nlage$ age classes weighted by their mass fractions:

\[a^{leaf} = \sum_{i=1}^{nlage}{f^{leaf}_{i} \cdot a^{leaf}_{i}}\]

10.5.2. DONE: Specific leaf area#

The specific leaf area $k^{sla}$ (m$^{2}$ g$^{-1}$) is needed to convert leaf mass, which is a prognostic variable in ORCHIDEE, to leaf area which is used in the calculation of, among others, photosynthesis, interception, albedo, and canopy cover. The specific leaf area of a PFT at a given location is calculated as a function of the leaf biomass [] to better account for the observed variation in specific leaf area []:

\[k^{sla} = \frac{\ln(1 + c_1 \cdot M^{leaf} \cdot c_2)} {c_1 \cdot M^{leaf}},\]

where $c_1$ is the PFT-specific extinction coefficient of the vertical nitrogen distribution in the canopy [] and $c_2$ is the initial value for the PFT-specific specific leaf area. The user choose to use a fixed PFT-specific specific leaf area as an alternative to the dynamic approach described in this section.

10.5.3. DONE: Tree height#

In ORCHIDEE the diameter and height of an individual tree are calculated from its tree mass by assuming: (1) a PFT-specific form factor to account for the fact that trunks are conical rather than cylindrical, (2) a PFT-specific wood density, and (3) a PFT-specific relationship between the diameter and height of the tree (i.e., equation (10.42)). The relationship between tree diameter and height makes use of the parameter $k^{height}$ which represents the height of a tree with a diameter of 1 meter.

The calculation of $k^{height}$ as a function of precipitation is based on the principles laid out in . In that paper, transpiration ($F^{Tr}$) of an individual tree scales to tree height (equation 3 in ); likewise, the root radius $d^{r,root}$ of a tree scales to tree height (equation 4 in ), and the water available to an individual tree ($F^{w,avail}$) is then calculated as a function of its root radius (equation 5 in ), shown as below:

(10.51)#\[\begin{split}\begin{align} &F^{Tr} = c_1 \cdot {d^{h}}^{c_2}, \\ &d^{r,root} = c_3^{\frac{1}{4}} \cdot d^{h}, \\ &F^{w,avail} = c_3 \cdot \pi \cdot ({d^{r,root}}^{2}) \cdot P \end{align}\end{split}\]

The maximum tree height will be reached when all the water available to the tree is used for transpiration. Following substitution of equations (10.51) to %s, maximum tree height can be written as a function of precipitation ($P$) and $k^{height}$ is then calculated as:

\[k^{height} = \max(c_6, c_4 \cdot {P^{3year}}^{c_5}),\]

where $c_4$ and $c_5$ are global parameters that were parameterized by fitting the annual precipitation sum from the CRUJRA climate forcing [] to the GEDI-derived 75 % tree heights [] for the year 2019. After assuming that the diameter of the trees that reached the 75 % percentile height was 0.5 m, the fitted $c_5$, and observed 75 % height were used to recalculate $c_4$ for a tree with a diameter of 1.0 m in line with the definition of $k^{height}$. $c_6$ represents the minimal value for $k^{height}$ and is used at the start of the simulation when the accumulated precipitation is still unrepresentative for the long-term precipitation of a given location, $P^{3year}$ is the long-term precipitation represented by the annual precipitation over the last 3 years. Hence, due to its dependency on precipitation, $k^{height}$ is location-dependent rather than PFT-specific.

10.5.4. DONE: Vegetation density#

For croplands and grasslands, ORCHIDEE defines an individual as all plants that occupy a square metre of land. The maximum density of a cropland or grassland in ORCHIDEE is therefore 10,000 individuals per hectare. The vegetation density of croplands, defined as the number of individuals per unit of area, is calculated as:

\[d^{ind} = c_1\]

where d$^{ind}$ is the density of the cropland (ind m$^{-2}$), and c$_1$ is a specific parameter of the PFT that prescribes the density. The fixed density reflects the practice in agriculture to plant at the final density to avoid individual plant mortality before harvest. Although a temporally fixed density for croplands is justifiable, cropland PFTs in ORCHIDEE are global and only distinguish C3 and C4 plants. Maize grown in, for example, humid temperate and semi-arid climates have the same density in ORCHIDEE although the biomass of an individual may differ.

Unlike croplands, the actual density of grasslands is not fixed to a prescribed parameter, but instead is calculated. Grassland density is decreased based on both the reserve and labile carbon, and the adjustment is calculated daily when the reserve and labile carbon in the plant drop below their respective target values and this condition persists over a longer time period. The downwards adjusted density of a grassland is estimated in accordance with the available resources by redistributing the resources among a maximum number of individuals capable of surviving under the given environmental conditions. By decreasing the number of individuals and redistributing the labile and reserve carbon among the surviving plants, their chances of future survival increase. The new, decreased number of individuals is calculated as:

\[d^{ind}_{t+1} = \frac{M^{tot,C}_{t} \cdot d^{ind}_{t} - M^{res,target}_{t} - M^{lab,target}_{t}}{M^{tot,C}_{t} - M^{res}_{t} - M^{labile}_{t}}\]

where M$^{res,C}_{t}$ (gC ind$^{-1}$), M$^{labile,C}_{t}$ (gC ind$^{-1}$), and M$^{tot,C}_{t}$ (gC ind$^{-1}$) are the carbon biomasses of the reserve, the labile pool, and the sum of all plant compartments of an individual. M$^{res,target,C}_{t}$ (gC ind$^{-1}$) and M$^{lab,target,C}_{t}$ (gC ind$^{-1}$) refer to the target pools for reserve carbon and labile carbon, respectively, which are calculated as explained in Section ??. The subscript t and t+1 refer to the value before and after the density adjustment, hence, d$^{ind}_{t}$ (ind m$^{-2}$) is the initial density, and d$^{ind}_{t+1}$ (ind m$^{-2}$) is the downwards adjusted density.

Note that the carbon of other compartments, i.e., leaf, aboveground sapwood, root, and fruit in each individual remains constant when the number of individuals is decreased. The nitrogen content of each compartment in an individual is updated by multiplying the previous nitrogen content by the ratio of the previous density to the current density. A minimum density of 0.05 is set to avoid numerical errors. In ORCHIDEE, grassland densities is only decreased during the following phenological stages: pre-senescence, senescence, and dormancy (for a more detailed description of the phenological stages see Section ??).

When the carbon content in both reserve and labile pools exceeds the target, and there is carbon presents in the fruit pool, grass density will be increased. The carbon from the fruit, reserve, and labile pools will be used to create new individuals. After updating the number of individuals, the labile and reserve pools equal their target values:

\[d^{ind}_{t+1} = \frac{M^{tot,C}_{t} \cdot d^{ind}_{t} - M^{res,target,C}_{t} - M^{lab,target,C}_{t}}{M^{tot,C}_{t} - M^{res,C}_{t} - M^{labile,C}_{t} - M^{fruit,C}_{t}}\]

where M$^{fruit,C}_{t}$ (gC ind$^{-1}$) is the biomass stored in the fruit pool. The density is only increased during the phenological stage labelled as growth (for a more detailed description of the phenological stages see Section ??). This approach for increasing grassland density reflects grass recruitment through asexual means, which is suited to treat perennial plants []. Note that the carbon of other compartments, i.e., leaf, aboveground sapwood and roots, in each individual remain constant during the density in increased. Nitrogen is treated using the same method as that applied to decreasing density.

In ORCHIDEE the simulated grasslands density accounts for spatial and temporal dynamics by accounting for the carrying capacity of the location. For grasslands, the reserve and labile carbon pools are used as a proxy for the carrying capacity of a grid cell. When the density of a grassland increases, recruitment through vegetative reproduction is simulated, but the biomass of the new individuals is identical to the biomass of the already established individuals. Vegetation density of grasslands is an emerging property of the model, as there are no parameters describing the density.

For forests, the resource availability at a given location and PFT determines the maximum number of individuals ($d^{ind,max}$; m$^{-2}$) that the site can support. Compared to a site with a low carrying capacity, sites with higher carrying capacities can maintain higher stand densities for the same quadratic mean diameter ($d^{qmdia}$). When the maximum carrying capacity is reached, individual trees die and the resources that become available as a result of this mortality event are used by the remaining trees. This process is known as self-thinning and formalized in the commonly observed self-thinning relationship [].

(10.52)#\[\begin{split}\begin{align} &d^{ind,max} = \left(\frac{d^{qmdia}}{k^{\alpha}}\right)^{\frac{1}{c_1}},\\ &d^{qmdia} = \sqrt{\sum_{l=1}^{ncirc}{\frac{{d^{dia}_{l}}^{2} \cdot d^{ind}_{l}} { \sum_{l=1}^{ncirc}{d^{ind}_{l}}}}}, \end{align}\end{split}\]

where is $c_1$ is the PFT-specific exponent of the self-thinning relationship and expected to be between -0.66 [] and -0.75 []. $k^{\alpha}$ represents the carrying capacity for a given grid cell and PFT. ORCHIDEE contains two approaches to simulate the carrying capacity. It is prescribed and thus fixed at the PFT-level, or the carrying capacity depends on the climatic conditions, available plant water, light, and nutrients []. In ORCHIDEE the calculation of photosynthesis accounts for all these factors and could therefore be considered as a modifier for the carrying capacity.

During the spinup of the model, $k^{\alpha}$ is calculated as:

(10.53)#\[k^{\alpha} = c_2 \cdot \max(\min\left(\frac{F^{gpp,ref}} {c_3}, 1.25 \right), 0.75)\]

where $c_2$ is the PFT-specific reference carrying capacity, $c_3$ is the PFT-specific pre-industrial reference photosynthesis, and $F^{gpp,ref}$ (g C m$^{-2}$ s$^{-2}$) is the pre-industrial reference photosynthesis for a given grid cell and PFT. $F^{gpp,ref}$ is calculated as the decadal mean photosynthetic flux. As the purpose of a model spinup is to bring the carbon, nitrogen and water pools into equilibrium, the temporal dynamics are deliberately kept to the minimum. Therefore, the spinup establishes the spatial variation in $k^{\alpha}$. During transient, historical, and future simulations, the carrying capacity, and thus the stand density, evolves over time as the growing environment is changing due to global changes such as climate change and nitrogen deposition. The temporal dynamics in $k^{\alpha}$ are then calculated as:

(10.54)#\[k^{\alpha} = k^{\alpha} \cdot \frac{F^{gpp,decade}}{F^{gpp,ref}}\]

where $F^{gpp,ref}$ is the mean photosynthesis over the previous decade. As an alternative to the calculated $k^{\alpha}$, a prescribed PFT-specific parameter can be used to calculate $d^{ind,max}$ in equation (10.52). If a PFT-specific parameter is used, the carrying capacity is no longer a function of global changes.

The relative density index ($f_{RDI}$, unitless) expresses the fraction of the actual density against the maximum stand density given the carrying capacity of the location and PFT. Although the concept of relative density is rooted in forest management [], it can also be applied to unmanaged forest as it has been observed that the actual stand density of unmanaged forests is also below the maximum density based on their carrying capacity []. Nevertheless, the processes responsible for the reduction in stand density differ between managed and unmanaged forests. In managed forest, forestry and natural disturbances are the main drivers of the density reduction whereas in unmanaged forests competition between individual trees and natural disturbances are the main drivers.

ORCHIDEE uses three different relative densities in its calculations: the actual ($f^{RDI,act}$), the lower target ($f^{RDI,low}$), and the upper target relative density ($f^{RDI,upp}$). If the actual relative density exceeds the upper target for relative density, the stand density ($d^{ind}$) is reduced such that it matches the lower target for relative density (sections ?? and ??. The processes driving the reduction in stand density depend on the forest management strategy applied to the PFT and grid cell (sections ?? and ??). The relative densities are calculated as follows:

(10.55)#\[\begin{split}\begin{align} &f^{RDI,act} = \frac{d^{ind}}{d^{ind,max}},\\ &f^{RDI,low} = \sum_{i=1}^{ndeg}({c^{low}_{i} \cdot {d^{qmdia}}^{i}}),\\ &f^{RDI,upp} = \sum_{i=1}^{ndeg}({c^{upp}_{i} \cdot {d^{qmdia}}^{i}}), \end{align}\end{split}\]

$c^{low}_{i}$ and $c^{upp}_{i}$ are obtained from fitting a polynomial through five management-specific parameters. The five key parameters that define the targeted relative density are: (1) relative density of a newly planted stand, (2) the maximum relative density, (3) the relative density when the diameter reaches its maximum, (4) the largest tree diameter, and (5) diameter above which the initial relative density starts to increase. These parameters are defined for each management strategy in ORCHIDEE as well as for unmanaged forests, and may vary by PFT. $ndeg$ is the degree of the polynomial function which in turn depends on the forest management strategy (section ??. For unmanaged forest a two degree polynomial is used, for the other management strategies the polynomial is fitted with three degrees.

In ORCHIDEE the forest density calculation accounts for spatial and temporal dynamics by accounting for the carrying capacity of the grid cell. For forests the carrying capacity is estimated from photosynthesis. Recruitment is calculated as a separate demographic process (Section ??) and recruits have a lower biomass than the already established vegetation in the stand. Although the parameter representing the carrying capacity is adjusted to the environmental conditions of each grid cell, the shape-parameter of the self-thinning relationship is held constant within the PFT for the length of the simulation. Stand density in forests, therefore, remains strongly guided by the self-thinning relationship.

10.5.5. DONE: Circumference distribution#

Whenever the model is set-up to run with more than one circumference class ($ncirc$ > 1), the total stand density based on the self-thinning relationship (section ??), has to be distributed over the different circumference classes. Equation (10.44) of the carbon and nitrogen allocation scheme requires that, at all times, more than one circumference class contains individuals. In ORCHIDEE the number of circumference classes is fixed ($ncirc$) to limit the memory use, but the boundaries and number of individuals in each circumference class are updated as the simulation proceeds. Given its flexibility and resemblance with observed circumference distributions [], a Weibull distribution is used to calculate the distribution in both managed and unmanaged forests.

The Weibull distribution is a continuous distribution implying that it also estimates occurrence probabilities of unrealistically large trees. In ORCHIDEE, these unrealistically large trees are avoided by introducing the parameter $c_1$ at which the Weibull is truncated. $c_1$ is prescribed and depends on the forest management strategy. The shape parameter ($k^{shape}$, unitless) of the Weibull distribution is calculated or prescribed depending on the forest management strategy:

(10.56)#\[\begin{split}k^{shape}= \begin{cases} \text{if rotational even-aged, }\ &\frac{d^{qmdia} \cdot c_2}{c_3} + 0.5\\ \text{else,}\ &c_2 \end{cases}\end{split}\]

where $d^{qmdia}$ (m) is the quadratic mean stand diameter, $c_2$ is a prescribed parameter that depends on PFT, and $c_3$ (m) is the prescribed PFT-specific tree diameter above which a stand is harvested. $c_3$ is the same parameter as $c_1$ in section ??). The Weibull cumulative distribution function ($f^{ind}_{l}$; unitless) for circumference class $l$ is:

(10.57)#\[\begin{split}\begin{align} &b_{1,l} = ( 1 - \exp(-c_4^{k^{shape}})) - ( 1 - \exp(-(c_4 - \frac{c_1}{ncirc})^{k^{shape}}))\\ &f^{ind}_{l} = \frac{b_{1,l}}{\sum_{l=1}^{ncirc}{b_{1,l}}}\\ &d^{ind}_{l} = f^{ind}_{l} \cdot d^{ind} \end{align}\end{split}\]

where ($c_4$; m) is the center diameter of circumference class $l$. This approach allows the circumference distribution to change dynamically, over the course of several decades, as the stand structure evolves. Under rotational even-aged management the distribution is expected to evolve, from a stand structure dominated by many small individuals to a structure centered around the diameter targeted for exploitation ($c_3$). Under continuous cover forestry as well as for unmanaged stands, a structure with many small individuals and fewer large individuals is maintained through time. The circumference class distribution is updated daily after accounting for mortality (section ??), forestry (section ??), land cover change (section ??), and disturbances (section ??).

10.5.6. DONE: Forest structure#

The simulation of carbon and nitrogen cycling within ORCHIDEE is based on mass flow with a rigorous mass balance check, hence, the mass of the different components are among the prognostic variables. Structural characteristics are diagnostic variables that are, whenever needed, calculated from the mass of the relevant components. Structural characteristics of an individual tree can be calculated from its mass by assuming: (1) a PFT-specific form factor ($f^{ff}$) to account for the fact that trunks are conical rather than cylindrical, (2) a PFT-specific wood density, and (3) a relationship between the diameter and height of the tree that depends on both precipitation and PFT (i.e., equation (10.42)):

(10.58)#\[\begin{split}\begin{align} &M^{stem}_{l} = M^{sap}_{l} + M^{heart}_{l},\\ &M^{stem}_{l} = d^{vol}_{l} \cdot c_1, \\ &d^{vol}_{l} = d^{ba}_{l} \cdot d^{h}_{l} \cdot f^{ff}, \\ &d^{ba}_{l} = \frac{\pi}{4} \cdot {d^{dia}_{l}}^{c_2}, \\ &d^{h}_{l} = k^{height} \cdot {d^{dia}_{l}}^{c_2}, \end{align}\end{split}\]

where $M^{stem}_{l}$ (g C plant$^{-1}$) is the stem mass of a tree in circumference class $l$ calculated as the sum of the sapwood ($M^{sap}_{l}$; g C plant$^{-1}$) and heartwood mass ($M^{heart}_{l}$; g C plant$^{-1}$). The parameter $c_1$ in equation %s is the wood density (g C m$^{-3}$) and thus identical to $c_1$ in equation (10.29). The parameter $c_2$ in equations %s and %s are identical to $c_{10}$ in equation (10.42). Note that equation (10.42) can be derived from equation %s by substituting and rearranging equations %s and %s.

Substitution of equations (10.58) to %s and rearranging results in expressions of individual tree height and diameter as a function of stem mass:

(10.59)#\[\begin{split}\begin{align} &d^{h}_{l} = \left(\left(\frac{4}{\pi} \cdot \frac{M^{stem}_{l}}{f^{ff} \cdot c_1}\right)^{\frac{c_2}{2}} \cdot k^{height}\right)^{\frac{3}{c_2}}, \\ &d^{dia}_{l} = {\left(\frac{4}{\pi} \cdot \frac{M^{stem}_{l}}{f^{ff} \cdot c_1 \cdot k^{height}}\right)}^{\frac{1}{2+c_2}},\\ &d^{ba}_{l} = \left(\frac{\pi}{4} \cdot \left(\frac{M^{stem}_{l}}{f^{ff} \cdot c_1 \cdot k^{height}}\right)^{\frac{2}{c_2}}\right)^{\frac{c_2}{c_2+2}} \end{align}\end{split}\]

The width of a tree ring is calculated daily for each circumference class $l$ as:

(10.60)#\[d^{trw}_{l,t} = \sqrt{\frac{d^{ba}_{l,t}}{\pi}} - \sqrt{\frac{d^{ba}_{l,t-1}}{\pi}},\]

where $d^{trw}_{l,t}$ (m) is the ring width for a tree in circumference class $l$ at $t$, and $d^{ba}_{l,t}$ and $d^{ba}_{l,t-1}$ (m$^{2}$) are the basal areas of a tree in circumference class $l$ at $t$ and $t-1$ respectively.

10.5.7. DONE: Canopy structure#

Tree crowns are formalized as prolate spheroids which are ellipsoids rotated along their major which, in ORCHIDEE is the vertical axis. The crown of an individual tree belonging to circumference class $l$ is thus characterized by a vertical ($d^{cdia,ver}_{l}$; m) and horizontal crown diameter ($d^{cdia,ver}_{l}$; m) which can be calculated from the tree height and thus depends on the stem mass of an individual tree:

(10.61)#\[\begin{split}\begin{align} &d^{cdia,ver}_{l} = c_3 \cdot d^{h}_{l}, \\ &d^{cdia,hor}_{l} = c_4 \cdot d^{cdia,ver}_{l}, \end{align}\end{split}\]

where $c_3$ is a PFT-specific parameter that links the crown depth to the tree height and $c_4$ is a PFT-specific parameter that prescribed the relationship between the horizontal and vertical crown diameter. The crown diameters can then be used to calculate the crown volume ($d^{cv}_{l}$; m$^{3}$ plant$^{-1}$) and projected crown area ($d^{cn}_{l}$; m$^{2}$ plant$^{-1}$):

(10.62)#\[\begin{split}\begin{align} & d^{cn}_{l} = \frac{\pi}{6} \cdot d^{cdia,ver}_{l} \cdot {d^{cdia,hor}_{l}}^{2}, \\ & d^{cn}_{l} = \frac{\pi}{4} \cdot {d^{cdia,hor}_{l}}^{2} \end{align}\end{split}\]

The crown volume at the stand level ($d^{cn}$; m$^{3}$ m$^{-2}$) is calculated as:

(10.63)#\[d^{cn} = \sum_{l=1}^{ncirc}{(d^{cn}_{l} \cdot d^{ind}_{l})}\]

Because the parameters $c_3$ and $c_4$ are not density dependent, crown packing ($c^{cp}$) may exceed unity. A crown packing of more than 1, indicates that the total crown volume at the stand level exceeds the total available space which is the volume confined by the maximum tree height in the stand and the surface area of the stand. If crown packing exceeds 1, the horizontal crown diameters in ORCHIDEE are until a crown packing of 0.74 is obtained. The target of 0.74 represents the maximum possible packing efficiency of equal spheres []. The adjusted crown diameters can then be calculated as:

(10.64)#\[\begin{split}\begin{align} &b_{1} = \left({{\frac{6 \cdot c^{cp} \cdot d^{h}_{ncirc}}{\pi \cdot {c_4}^{2} \cdot \sum_{l=1}^{ncirc}d^{ind}_{l} \cdot {d^{cdia,ver}}^{3}}}}\right)^{\frac{1}{3}},\\ & d^{cdia,ver}_{l} = b_{1} \cdot c_3 \cdot d^{h}_{l}, \\ & d^{cdia,hor}_{l} = c_4 \cdot d^{cdia,ver}_{l} \end{align}\end{split}\]

Stand structure controls the amount of light that penetrates to a given depth in the canopy. Light penetration is used in the calculation of albedo (section ??), photosynthesis (section ??), allocation (section ??), and recruitment (section ??). Prior to calculating light penetration (section ??) the canopy is discretized in $nlev$ canopy layers and the leaf area within each layer $i$ is calculated.

For forest PFTs the height of the top of the canopy ($z_{top}$; m) is given by equation (10.64) and the bottom ($z_{bot}$; m) is calculated as the difference between $b_{1}$ in equation (10.64) and the maximum vertical crown diameter (max($d^{cdia,ver}_{l}$)). The canopy height of grasses and crops, which is the height of the top of the canopy ($z_{top}$), is assumed linear to leaf area by making use of a PFT-specific parameter. The height of the bottom of their canopy ($z_{bot}$) is placed at an arbitrary 0.0001 m above the soil. The height of canopy layer $i$ ($z_{i}$; m) is then calculated as:

\[\begin{split}\begin{align} &b_{3} = \frac{z_{top} - z_{bot}}{nlev}, \\ &z_{i} = \sum_{i=1}^{nlev}(z_{bot} + b_{3} \cdot i) \end{align}\end{split}\]

For forest PFTs, the partial crown volume within each canopy layer is calculated for each of the $nlev$ layers that were defined at the stand level and each of the $ncirc$ circumference classes. It is first checked whether level $i$ crosses the spheroid:

\[\begin{split}\begin{align} &b_{4} = z_{top}-\frac{d^{cdia,ver}_{l}}{2},\\ &b_{5} = z_{i} - b_{4},\\ &b_{6} = z_{i+1} - b_{4}. \end{align}\end{split}\]

Level $i$ crosses the spheroid of circumference class $l$ if $b_{5}$ is greater or equal to $d^{cdia,ver}_{l}/2$ or $b_{6}$ is less or equal to $-d^{cdia,ver}_{l}/2$. If level $i$ crosses the spheroid, the calculations continues as:

\[\begin{split}\begin{align} &b_{7}=\min\left(z_{i+1}-b_{4},\frac{d^{cdia,ver}_{l}}{2}\right), \\ &b_{8}=\max\left(z_{i}-b_{4},\frac{-d^{cdia,ver}_{l}}{2}\right), \\ &d^{cv}_{l,i} = \pi \cdot \left ({\frac{d^{cdia,hor}_{l}}{2}}\right)^{2} \cdot (b_{7}-b_{8}-\left(\frac{{b_{7}}^{3}-{b_{8}}^{3}}{3 \cdot \left({\frac{d^{cdia,ver}_{l}}{2}}\right)^{2}}\right), \end{align}\end{split}\]

where $d^{cv}_{l,i}$ (m$^{3}$ plant$^{-1}$) represents the partial crown volume for circumference class $l$ at canopy level $i$. Note that $b_{7}$ and $b_{8}$ have their centers at the origin and that the horizontal and vertical crown radius is used rather than the crown diameters. The partial crown volume at layer $i$ ($d^{cv}_{i}$, $m^{3} m^{-2}$) is calculated by summing over the circumference classes:

\[d^{cv}_{i} = \sum_{l=1}^{ncirc}{d^{cv}_{l,i} \cdot d^{ind}_{l}}\]

For grassland and cropland PFTs, the partial crown volume within each canopy layer is calculated for each of the $nlev$ layers that were defined at the stand level. Note that the canopy of grasslands and croplands is formalized as a cuboid contrary to the spheroid assumed for the forest PFTs. The crown volume within a canopy layer is:

\[d^{cv}_{i} = (z_{i+1} - z_{i}) \cdot d^{A}\]

where $d^{A}$ is the surface area of the PFT (m$^{2}$) which is set to 1 m$^{2}$ in ORCHIDEE For all vegetative PFTs, the leaf area within each canopy layer is then calculated as:

(10.65)#\[d^{LAI}_{i} = \frac{d^{cv}_{i} \cdot d^{LAI}}{d^{cv}}\]

where $d^{LAI}/d^{cv}$ is the foliage area volume density ($m^{2} m^{-3}$) which is assumed to be constant across the circumference classes $l$ and the canopy layers $i$. Likewise to forest structure (section ??), canopy structure and characteristics are diagnostic variables that, whenever needed, are calculated from the mass of the relevant components.

10.5.8. DONE: Gap fraction#

The gap fraction, which is the basic information in calculating light penetration at different depths in the canopy, is calculated following the statistical approach of . Following minor adaptations, the implementation of was incorporated in ORCHIDEE . The gap model represents the canopy by a statistical height distribution with varying crown sizes and stem diameters for each circumference class $l$. The crown canopies are treated as spheroids containing homogeneously distributed single scatterers. Although this approach to model canopy gaps can explicitly include trunks, ORCHIDEE excludes them, as the spectral parameters for our radiation model (section ??) are extracted from remote sensing data without distinguishing between leafy and woody masses.

The gap probability for trees $f^{Pgap,trees}$ (unitless) is calculated as a function of height ($z$; m) and solar zenith angle ($\theta$; radians):

(10.66)#\[f^{Pgap,trees}_{\theta,z} = \exp{(- {d^{c}_{\theta,z} \cdot (1 - f^{Pwc}_{\theta,z}) \cdot d^{ind}})},\]

where $d^{c}_{\theta,z}$ (unitless) is the projected crown area for an opaque canopy at height $z$ for solar angle $\theta$ and is calculated from the geometric relationships detailed in Appendix A in . $f^{Pwc}_{\theta,z}$ (unitless) is the mean crown porosity at height $z$ for solar angle $\theta$. The mean refers to the mean over the tree distribution over the $l$ circumference classes. $f^{Pwc}_{\theta,z}$ is calculated as:

\[b_{9} = z - d^{h}_{l} + \frac{d^{cdia,ver}_{l}}{2}\]

If $b_{9}$ falls between $-d^{cdia,ver}_{l}/2$ and $d^{cdia,ver}_{l}/2$, then the partial volume of a spheroid above height $z$ ($b_{10}$, m$^{3}$ plant$^{-1}$), is calculated as the difference of the volume of the whole spheroid and the volume of a partial spheroid below height $z$:

\[\begin{split}\begin{align} b_{10} & = \frac{4}{3} \cdot \pi \cdot \left({\frac{d^{cdia,hor}_{l}}{2}}\right) ^ {2} \cdot {\frac{d^{cdia,ver}_{l}}{2}} - \nonumber \\ &\frac{d^{cdia,ver}_{l}}{3} - b_{9} + \frac{{b_{9}}^{3}}{3 \cdot \left({\frac{d^{cdia,ver}_{l}}{2}}\right)^{2}} \cdot \pi \cdot \left({\frac{d^{cdia,hor}_{l}}{2}}\right)^{2}, \end{align}\end{split}\]

If $b_{9}$ exceeds $d^{cdia,ver}_{l}/2$ the whole volume of the spheroid is above height $z$ and therefore $b_{10}$ is calculated as:

\[b_{10} = \frac{4}{3} \cdot \pi \cdot \left({\frac{d^{cdia,hor}_{l}}{2}}\right) ^ {2} \cdot {\frac{d^{cdia,ver}_{l}}{2}}\]

Finally $d^{s}_{\theta,z,l}$ (m) is the mean path length through all canopy levels above height $z$ for solar angle $\theta$ for circumference class $l$ and is calculated as:

\[d^{s}_{\theta,z,l} = \frac{temp{10}}{\cos(\theta) \cdot d^{c}_{\theta,z}}\]

Since a tree crown is not opaque, the crown porosity is calculated by considering the projection of the leaf area in the direction of the incoming light. A spherical leaf area distribution is assumed which makes the projected leaf area in the direction of the incoming light constant and equal to 0.5 []. The crown porosity is then calculated as:

\[f^{Pwc}_{\theta,z,l} = \exp\left(\frac{-0.5 \cdot d^{LAI} \cdot d^{s}_{\theta,z,l} \cdot c_5}{ d^{cv}}\right),\]

where $c_5$ is a PFT-specific parameter prescribing the leaf clumping at the level of the plant shoots. By considering the stand density in each circumference class $l$, stand level porosity which is used in equation (10.66), is calculated as:

\[f^{Pwc}_{\theta,z} = \sum_{l=1}^{ncirc}{f^{Pwc}_{\theta,z,l} \cdot d^{ind}_{l}}\]

As ORCHIDEE also simulates crops, grasses, and bare soil, the calculation of $f^{Pgap}$ was adjusted for these PFTs as well. For grasses and crops, the same formulation is used:

\[f^{Pgap,gc}_{\theta,z} = \exp{\left(\frac{-0.5 \cdot d^{LAIabove} \cdot c_5}{ cos(\theta)}\right)}\]

where $d^{LAIabove}$ (unitless) is the total amount of LAI above height $z$, and $c_5$ accounts for the fact that grasses and crops are treated as homogeneous blocks of vegetation with no internal structure. For bare soil, there is no vegetation to intercept radiation, and therefore $f^{Pgap,bs}_{\theta_{z},z}$ is always unity. Depending on the calculations, $f^{Pgap}$ is either used for a specific height $z$ and solar angle $\theta$ (e.g., albedo in section ??) or is integrated over all layers (e.g. light reaching the forest floor in section ??).

10.5.9. DONE: Effective leaf area#

The effective LAI for canopy layer $i$ with its bottom at height $z_{i}$ for solar angle $\theta$ is computed to be used with the two-stream albedo model (section ??) by using equation 25 in :

\[d^{LAIeff}_{i} = -2 \cdot \cos(\theta) \cdot \log(P^{gap}_{\theta,z_{i}}),\]

Calculation of $d^{LAIeff}_{i}$ (unitless) is a computationally expensive part of ORCHIDEE due to the loops over grid cells, PFTs, circumference classes, canopy levels, and solar angles in addition to using trigonometric functions. Moreover, a value for $d^{LAIeff}_{i}$ is needed every time step so, up to 48 times a day. Computational costs were reduced by calculating $d^{LAIeff}_{i}$ for eight solar angles (i.e., 5 °, 15 °, 25 °, 35 °, 45 °, 55 °, 65 ° and 75 °) exactly, and then fitting a function (i.e., $c_6 + c_7 \cdot \theta + c_8 \cdot \theta^{2} + c_9 \cdot \theta^{3} + c_{10} \cdot \theta^{4}$) to preset solar angles to predict the value of $d^{LAIeff}_{i}$ at any solar angle. A least square method is used to fit the parameters $c_6$, $c_7$, $c_8$, $c_9$, and $c_{10}$. This method works well for low solar zenith angles, which are the most important since they will have the most incoming solar radiation.

10.5.10. DONE: Root distribution#

ORCHIDEE calculates two vertical root profiles: a structural and a functional. Each root profile gives the share of the root biomass in each soil layer. The structural root profile is constant over time and is used in processes where the depth of the roots is one of the drivers, i.e., soil water infiltration along roots (section ??), input of soil carbon and nitrogen at depth due to the turnover of roots (section ??), maintenance respiration (section ??), and resistance to wind throw (section ??). The functional root profile changes over time and is used in processes where the vertical profile roots activity, rather than the vertical profile of root mass, is considered an important driver, i.e., transpiration ((8.1)).

The maximum rooting depth ($z_{root}$) is calculated by considering two constraints: (1) roots do not extend into frozen soils, and (2) crops are assumed to root to 0.8 m, grasslands are assumed to root no deeper than 1 m, and forests are assumed to root down to 2 m. Note that in ORCHIDEE the maximum rooting depth does not depend on the root biomass, hence, plants root down to their maximum rooting depth from the start of the simulation.

The structural root profile is calculated as an exponentially decreasing root mass with depth where the shape of the profile is determined by the PFT-specific parameter $c_1$ []. The total surface area under an exponential curve between $z_top$ (m) and $z_{root}$ (m) is calculated:

\[b_{1} = \frac{1}{\exp(-z_{top} \cdot c_1) - \exp(-z_{root} \cdot c_1 )}\]

The structural root profile (unitless) is then calculated as:

\[f^{root,str}_{i} = b_{1} \cdot ( \exp(-z_{i-1} \cdot c_1) - \exp(-z_{i} \cdot c_1 ))\]

where $z_{i}$ (m) is the depth of the layer $i$ and $z_{i-1}$ (m) is the depth of the previous layer. The calculation is repeated for all layers $i$ between $z_{top}$ and $z_{root}$.

The functional root profile gives the share of the root biomass in each soil layer based on the soil water in each layer. This results in a dynamic root profile that is changing over time. The total active root mass is calculated as:

\[b_{2} = \sum_{i=z_{top}}^{z_{root}}\max(0,M^{sm}_{i}-M^{smw}_{i})\]

The functional root profile (unitless) is then calculated as:

\[f^{root,fun}_{i} = \frac{\max(0,M^{sm}_{i}-M^{smw}_{i})}{b_{2}}\]

10.6. DONE: Natural mortality#

10.6.1. DONE: Defining plant mortality#

ORCHIDEE distinguishes between plant and PFT mortality. Plant mortality is defined as an event in which part of the individuals within a PFT die without changing the ground area of this PFT. The causes of natural plant mortality accounted for in ORCHIDEE are described in section ?? and in section ?? for anthropogenic plant mortality. ORCHIDEE defines PFT mortality as an event in which all individuals within a PFT die without changing the ground area of that PFT. The causes of natural PFT mortality accounted for in ORCHIDEE are described in section ??. Anthropogenic causes of the mortality of forest PFTs is described in section and for agricultural PFTs in section ??). Events in which all or part of the individuals of a PFT die and the ground area of the PFT changes also distinguish natural and anthropogenic causes. The natural causes of stand replacing disturbances are described in section ??. The anthropogenic process ORCHIDEE accounts for and that results in an area change is land cover change which is described in section ??).

10.6.2. DONE: Natural plant mortality#

Natural plant mortality, i.e., an event in which part of the individuals within a PFT die without changing the ground area, is considered when the PFT has not yet been marked for killing due to PFT mortality (section ??. Two natural sources of plant mortality are considered: (1) self-thinning due to resource competition in forests, and (2) background mortality due to environmental stressors. The daily calculation of the self-thinning mortality for unmanaged forests starts with calculating three relative densities, i.e., the actual ($f^{RDI,act}$), the lower target ($f^{RDI,low}$), and the upper target relative densities ($f^{RDI,upp}$), according to equations (10.55) to %s. If the actual relative density ($f^{RDI,act}$) is less than its upper target ($f^{RDI,upp}$), the actual stand density is within the carrying capacity of the site and is, therefore, maintained. The carrying capacity is exceeded if the actual relative density exceeds its upper target ($f^{RDI,upp}$). A reduction in stand density ($d^{ind}$) is then calculated such that the actual relative density matches its lower target for ($f^{RDI,low}$):

\[b_{1} = d^{ind} - f^{RDI,low} \cdot d^{ind,max},\]

where $b_{1}$ is the stand density (m$^{-2}$) that has to be removed to reach the lower target for relative density and $d^{ind,max}$ is the maximum stand density the site can support given the actual quadratic mean diameter of the stand. Since in ORCHIDEE the number of trees are given per unit area, the total number of trees and stand density are identical and thus interchangeable. The use of circumference classes adds realism to the ORCHIDEE simulations, but it raises the questions which trees should be targeted by self-thinning mortality? For self-thinning, the mortality distribution ($f^{dist,l}$) over the circumference classes $l$ is considered to follow the diameter distribution (section ??). The targeted mortality in each diameter class is then calculated as:

\[d^{ind,kill}_{l} = f^{dist,l} \cdot b_{1},\]

where $d^{ind,kill}_{l}$ (m$^{-2}$) is the stand density that is killed in each circumference class. If self-thinning mortality occurs, environmental mortality is not accounted for. If there is no self-thinning, environmental mortality is accounted for. The model user can choose from two approaches to calculate mortality from environmental stress. In the first default approach the environmental mortality is considered constant:

\[M^{kill} = \frac{\sum_{l=1}^{ncirc}{M^{plant}_{l} \cdot d^{ind}_{l}}} {c_1},\]

where $M^{kill}$ is the total biomass that needs to be killed, $M^{plant}_{l}$ the biomass of trees in circumference class $l$, $d^{ind}_{l}$ the stand density in circumference class $l$, and $c_1$ is the PFT-specific maximum age (days) for the PFT under consideration.

In the second approach the daily mortality from environmental stress is a function of the productivity of the PFT:

\[\begin{split}\begin{align} &b_{2} = \frac{F^{npp,3year}} {d^{LAI}},\\ &b_{3} = \max\left(c_4, \frac{c_2}{1+c_3\cdot b_{2}}\right),\\ &M^{kill} = \sum_{l=1}^{ncirc}{b_{3} \cdot M^{plant}_{l} \cdot d^{ind}_{l}}, \end{align}\end{split}\]

where $c_2$ is the maximum mortality, $c_3$ is the reference mortality and $c_4$ is minimum mortality. $c_2$, $c_3$, and $c_4$ are in year$^{-1}$ and are all PFT-specific. $F^{npp,3year}$ (g C m$^{-2}$ s$^{-1}$) is the 3-year mean net primary production. $M^{kill}$ is the total biomass that needs to be killed but ORCHIDEE requires the stand density that has to be killed in each circumference class ($d^{ind,kill}_{l}$; m$^{-2}$). Where for self-thinning mortality, mortality was assumed proportional to the stand density in each circumference class, environmental stress was considered to preferentially affect the larger trees, being more prone to cavitation, wind damage, lightening, and heart rot. While killing the larger trees also trees in the other diameter classes were killed following []:

\[\begin{split}\begin{align} &f^{kill}_{l}=f^{kill}_{l-1}\cdot c_5^{1-(ncirc-1)},\\ &f^{kill}_{l} = \frac{f^{kill}_{l}} {\sum_{l=1}^{ncirc}{f^{kill}_{l}}},\\ &d^{ind,kill}_{l} = \frac{f^{kill}_{l} \cdot M^{kill}} {M^{plant}_{l}}, \end{align}\end{split}\]

where $c_5$ is the death distribution factor, which is the factor by which the smallest and largest circumference classes differ e.g., a $c_5$ of 10 means that the largest circumference class will lose ten times as much biomass as the smallest circumference class as a result of the mortality.

ORCHIDEE accounts already for several sources of mortality, i.e., forestry for managed forests, self-thinning for unmanaged forests, storms and insects for forests and fire for all PFTs and will be further developed to include even more sources. The end goal of these developments is to replace the pure empirical environmental mortality by a more mechanistic approach towards mortality. This is an ongoing process [] and until this goal has been reached, environmental mortality is kept as a fail-safe approach but the value of $c_1$ is increased every time a new source of mortality is added to the model. Likewise, with the addition of new sources of mortality, the current self-thinning mortality is evolving from a poorly defined mixture of sources of mortality towards pure competition driven mortality.

10.6.3. DONE: Natural PFT mortality#

PFT mortality. i.e., an event in which all individuals within a PFT die without changing the ground area of that PFT, occurs in two cases: carbon starvation and density-driven mortality. In both cases, all individuals in the PFT are marked for killing and PFT ground area does not change.

If at the end of the day, there is no carbon available in the carbohydrate reserve, leaf and labile pools, all individuals in the PFT will be marked for killing. This condition is the result of carbon starvation and typically takes 3 to 5 years to develop in ORCHIDEE . In this situation, the plants are not able to grow new leaves which are essential to assimilate new carbon from the atmosphere to initiate plant recovery.

In case of high intensity management or natural disturbances, the stand density could become so low that unrealistically large tree diameter would be simulated. In ORCHIDEE this situation is prevented through density-driven PFT mortality . Density-driven PFT mortality is not an ecological process but a fail-safe approach to keep stand density and tree diameter within observed boundaries.

10.6.4. DONE: Moving biomass to litter pools#

For each circumference class either none, part, or all of the individuals are marked for killing ($d^{ind,kill}_{l}$; m$^{-2}$) (sections ?? and ??). Subsequently, the biomass of the individuals that were marked for killing, is moved into the appropriate biomass pools. This section describes mortality in unmanaged PFTs only but the mortality could be caused by: carbon starvation (section ??), density-driven (section ??), self-thinning (section ??), environmental stress ??), bark beetles ??), or wind storms ??). Trees marked for killing are moved into the litter pool and the stand density is updated:

(10.67)#\[\begin{split}\begin{align} &M^{lit}_{t} = M^{lit}_{t-1} + M^{plant}_{l,t} \cdot d^{ind,kill}_{l,t}, \\ &d^{ind}_{l,t} = d^{ind}_{l,t-1} - d^{ind,kill}_{l,t}, \end{align}\end{split}\]

where $M^{lit}_{t}$ and $M^{lit}_{t-1}$ are the litter mass at respectively time step $t$ and $t-1$ for a single PFT within a grid cell, and $M^{plant}_{l}$ (g plant$^{-1}$) is the plant mass in circumference class $l$. This calculation is repeated for the carbon and nitrogen biomass.

For grasslands and croplands fewer sources of mortality are accounted for in ORCHIDEE compared to forest. Moreover, the density of a grassland or cropland is constant in ORCHIDEE. When partial mortality occurs, the individual plant mass is adjusted rather than the number of individuals as is the case in equation %s.

10.7. DONE: Litter decomposition#

10.7.1. DONE: Adding biomass to the litter pools#

The litter is formalized as four pools: a metabolic, structural, woody, and a snag, i.e., standing dead trees, pool. These four pools further distinguish an above-ground and a below-ground component with the exception of the snags that only have an aboveground pool. The metabolic litter is the labile or rapidly decomposing fraction of the litter, where the structural litter is the more resistant and, therefore, slowly decomposing fraction. This discretisation thus accounts for differences in the decomposition rates between cytoplasmic compounds of plant cells, and cell walls []. The woody pool represents the coarse woody debris, even slower decomposing than the structural pool [], and the snag pool containing the dead standing trees, and assumed to be the slowest pool to decompose []. In ORCHIDEE the snag pool is optional. When the snag pool is not accounted for, all dead woody biomass is transferred to the woody litter pool.

The litter pools receive dead biomass as an input: each of the nine dead biomass pool is moved to one or two specific litter pools: the share of leaves, fruit and root biomass that is moved to the metabolic pool is calculated as:

(10.68)#\[ \begin{align}\begin{aligned}\begin{align}\\ &\ f^{o,metab} = c^{metabfrac} - c_1 \cdot c^{o,LC} \cdot f^{cn,leaf}, \end{align}\end{aligned}\end{align} \]

where $f^{o,metab}$ is the fraction of the input biomass of organ $o$ that goes into the metabolic pool, with $o$ being fruit, leaf or root biomass, $c^{metabfrac}$ is a reference fraction along with $c_1$ an empirical coefficient. $c^{o,LC}$ (g lignin g C$^{-1}$) is the lignin to carbon ratio of the organ $o$ of the input biomass and $f^{cn,leaf}$ is the carbon-to-nitrogen ratio of the input biomass. The remainder, i.e., 1-$f^{o,metab}$, is added to the structural pool. Following mortality, leaves and fruit biomass is entirely added to the above-ground litter pool whereas the mass of dead roots is added to the below-ground litter. The labile and reserve biomass are moved to the above-ground metabolic litter pool. Sixty-eight percent of the heartwood and sapwood biomass that dies is moved into the woody litter pool, the remaining 32 % into the snag litter pool but this proportion needs to be refined as it is based on observed snag proportions [], which also results from the decay rates of the pools. Above-ground and below-ground heartwood and sapwood are already distinguished at the plant level, so following mortality, these mass components are moved in the matching above-ground and below-ground wood litter pools.

The lignin content of the different litter pools is calculated according to:

(10.69)#\[ \begin{align}\begin{aligned}\begin{align}\\ &M^{in,lignin,i} = \sum_{o=1}^{norgan} c^{o,LC} \cdot F^{in,o,i}, \end{align}\end{aligned}\end{align} \]

where $M^{in,lignin,i}$ (g C m$^{-2}$ s$^{-1}$) the input of lignin to the litter pool $i$, $c^{o,LC}$ is the PFT-specific lignin to carbon ratio of the biomass pool $o$, $norgan$ the number of plant organs and thus biomass pools, and $F^{in,o,i}$ (g C m$^{-2}$ s$^{-1}$) the carbon input from plant organ $o$ into litter pool $i$.

When a PFT is fertilized with manure, the nitrogen contained in manure is read from a fertilisation map (section ??) and added to the soil mineral nitrogen pools. The carbon contained in the manure is calculated from the prescribed carbon to nitrogen ratio of manure and added in the metabolic above-ground litter pool. By doing so, we mimic the application of manure on top soil by farmers that brings C to soil and since manure is considered to be a labile material with high turnover rates, we added it into the metabolic above ground pool. (ADD RATIONALE TO DO DO SO)

10.7.2. DONE: Dynamics of the litter pools#

The dynamics of the litter pools is based on the CENTURY first order decay model [] extended by coarse woody debris and snags, which follow similar dynamics []. Snags start as standing dead wood but due to decay at their stem base near the ground, snags will fall well before they are decomposed []. During snag fall, the mass of the snag is transferred to the above-ground woody litter pool, according to a fall rate with no control of moisture or lignin content (contrarily to the litter pools decay). Each of the pools are divided between an above and a below pool, which decompose in the same way with exception to the decomposition moderator functions (accounting for temperature, decomposer presence, and humidity effects). The litter dynamics are described as:

(10.70)#\[\begin{split}\begin{align} &\frac{\partial M^{lit}_{i}}{\partial t} = \sum_{k=1}^{norg} F^{in,o,i}-c_{i}\cdot M^{lit}_{i} \cdot m^{temp}_{i} \cdot m^{moist}_{i}, \\ &\frac{\partial M^{lit}_{i}}{\partial t} = \sum_{k=1}^{norg} F^{in,o,i}-c_{i}\cdot M^{lit}_{i} \cdot m^{temp}_{i} \cdot m^{moist}_{i} \cdot m^{lignin}_{i},\\ &\frac{\partial M^{lit}_{i}}{\partial t} = \sum_{k=1}^{norg} F^{in,o,i}-c_{i}\cdot M^{lit}_{i} \cdot m^{temp}_{i} \cdot m^{moist}_{i} \cdot m^{lignin}_{i}+c_{f} \cdot M^{lit}_{i},\\ &\frac{\partial M^{lit}_{i}}{\partial t} = \sum_{k=1}^{norg} F^{in,o,i}-c_{i}\cdot M^{lit}_{i}\cdot m^{temp}_{i} \cdot m^{moist}_{i} \cdot m^{lignin}_{i}-c_{f} \cdot M^{lit}_{i} \end{align}\end{split}\]

where $M^{lit}_{i}$ (g C m$^{-2}$ or g N m$^{-2}$) the carbon and nitrogen mass of the litter pool $i$. In equations (10.70), eq:struc_litter, eq:wood_litter, eq:snag_litter $i$ stands respectively for metabolic, structural, woody and snag. $F^{in}_{i}$ (g C m$^{-2}$ s$^{-1}$ or g N m$^{-2}$ s$^{-1}$) are the inputs, $c_{i}$ the decomposition rates, $c_{f}$ the snags fall rate, $m^{moist}_{i}$ and $m^{temp}_{i}$ the rate modifiers respectively accounting for the effect of soil moisture and soil temperature on decomposition. $m^{lignin}_{i}$ is a modifier that represents the moderating effect on decomposition of the lignin content in the litter pool ($m^{lignin}_{i}$; g lignin g C$^{-1}$). The rate modifiers follow the equations below :

(10.71)#\[\begin{split}\begin{align} & m^{lignin}_{i} = \exp(-3 \cdot M^{lignin}_{i}),\\ & m^{moist,j}_{i} = max(\theta_{min}, min(1, -c_{1} \cdot \theta_{j}^{2} + c_{2} \cdot \theta_{j} -c_{3})),\\ & m^{temp,j}_{i} = min(1, \exp(c_{Q10soil} \cdot \frac{[T^{j}-T_{soil}^{ref}]}{c_{Q10}})),\\ \end{align}\end{split}\]

j is above or below : when j is above $\theta_{j}$ is $\theta$ the humidity relative to field capacity, and $T^{j}$ is $T^{surf}$, the surface temperature, and for j being below $\theta_{j}$ is $\theta^{decomp}$ and $T^{j}$ is $T^{decomp}$, the relative humidity and temperature averaged over soil layers and accounting for the decomposers’ presence, both calculated according to (10.76). $c_{Q10soil}$, $c_{Q10}$, $T_{soil}^{ref}$, $c_{1}$, $c_{2}$, $c_{3}$ are empirical parameters which values can be seen in equations (10.75), eq:som_decomp_moisture_modifier which are the same as the equations for the below litter pools.

10.7.3. DONE: Heterotrophic respiration from litter decomposition#

When litter is decomposed, carbon dioxide is released in form of heterotrophic respiration. Heterotrophic respiration is calculated as a fraction of the carbon fluxes going from the litter to the soil carbon pools (section ??).

(10.72)#\[\begin{split}\begin{align} &\ b_{1} = \sum_{i \not = metab}\sum_{j=1}^{N_{soil}} \frac{(1 - f^{soil}_{i,j}) \cdot M^{declitter}_{i} \cdot (1 - m^{lignin}_{i})}{\Delta{t}},\\ &\ b_{2} = \sum_{j=1}^{N_{soil}} \frac{(1 - f^{soil,metab}_{j}) \cdot M^{declitter,metab}}{\Delta{t}},\\ &\ F^{resp,het,litter} = b_{1} + b_{2}, \end{align}\end{split}\]

where $b_{1}$ (g C m$^{-2}$ s$^{-1}$) is the heterotrophic respiration in the pools which involve a lignin fraction thus all but the metabolic litter pool, $b_{2}$ is the heterotrophic respiration in the metabolic litter, $N_{soil}$ the number of soil pools, $j$ the index of the soil carbon pools, $M^{declitter}_{i}$ the carbon from the decayed litter pool $i$, $m^{lignin}_{i}$ the lignin content in g lignin g C$^{-1}$ of the litter pool $i$, $f^{soil,metab}_{j}$ the fraction of the decayed litter going to the soil (and not the atmosphere via heterotrophic respiration) and $F^{resp,het,litter}$ (g C m$^{-2}$ s$^{-1}$) is the total heterotrophic respiration from the litter decay. All nitrogen released during the decay of the litter pools goes initially into the mineral nitrogen pool of the soil, where it is available for plant uptake, or other nitrogen processes (section ??).

10.8. Soil carbon decomposition, nitrogen transformations, and heterotrophic respiration#

The representation of soil carbon in ORCHIDEE is based on the CENTURY model []. In each grid cell and PFT, the soil carbon is split between surface, active, slow, and passive pools. The surface pool has been added in ORCHIDEE to . Optionally (see Sec. ??) the pools can be further split into depth layers, allowing for the representation of the vertical diffusion of organic matter due to cryo- and bioturbation.

The representation of soil nitrogen in ORCHIDEE is based on…

10.8.1. DONE: Adding litter to soil carbon and nitrogen pools#

:::{figure} Figures/litter-som.png :name: fig:litter_som :align: center

Partitioning of decomposed litter between the surface, active, and slow soil organic carbon pools. The passive soil organic carbon pool (not shown here) does not receive litter, but only decomposed organic matter from the other pools. :::

DSG:
For respiration you separate litter and SOC although its the same shape of equations. (FK: TODO, will see what we can do.)
Using f for a parameter will confuse readers. (FK: f is both a fraction and a parameter, so both ’f’ and ’c’ would be appropriate. I agree with whoever made the choice of ’f’, the fraction-nature seems more important to me here than the fact that it’s a constant.)

In each time step, the mass $M^\text{dec\_litt}_\text{C}$ of carbon in the decomposed litter is split between the surface, active, and slow soil organic carbon pools as shown in Fig. 13. The fraction of decomposed carbon from litter pool $i,j$ ($i \in \{\text{above},\text{below}\}, j \in \{\text{metab},\text{struct},\text{wood},\text{snag}\}$) that enters the soil organic matter pool $k$ is controlled by the parameter $f^\text{litt\_SOM\_2D}_{i,j,k}$. In addition to this parameter, the distribution of decomposed structural, woody, and snag litter takes into account the PFT-dependent lignin content $f^\text{lignin}_{i,j}$ and allocates the non-lignin fraction to the active and surface pools and the lignin fraction to the slow pool. The resulting input fluxes $F^\text{litt\_SOM\_2D}_{\text{C},k}$ of soil organic carbon pools are

(10.73)#\[\begin{split}\begin{align} F^\text{litt\_SOM\_2D}_{\text{C},\text{surface}} &= \left[f^\text{litt\_SOM\_2D}_{\text{above},\text{metab},\text{surface}} + \sum_{j \in \{\text{structural},\text{woody},\text{snag}\}}f^\text{litt\_SOM\_2D}_{\text{above},j,\text{surface}} \left(1 - f^\text{lignin}_{\text{above},j}\right)\right] M^\text{dec\_litt}_\text{C} / \Delta{t} \\ F^\text{litt\_SOM\_2D}_{\text{C},\text{active}} &= \left[f^\text{litt\_SOM\_2D}_{\text{below},\text{metab},\text{active}} + \sum_{j \in \{\text{structural},\text{woody},\text{snag}\}}f^\text{litt\_SOM\_2D}_{\text{below},j,\text{active}} \left(1 - f^\text{lignin}_{\text{below},j}\right)\right] M^\text{dec\_litt}_\text{C} / \Delta{t} \\ F^\text{litt\_SOM\_2D}_{\text{C},\text{slow}} &= \sum_{i \in \{\text{above},\text{below}\}, j \in \{\text{structural},\text{woody},\text{snag}\}} f^\text{litt\_SOM\_2D}_{i,j,\text{slow}} f^\text{lignin}_{i,j} M^\text{dec\_litt}_\text{C} / \Delta{t} \end{align}\end{split}\]

10.8.2. DONE: Dynamics of soil carbon and nitrogen pools#

:::{figure} Figures/som.png :name: fig:som_dynamics :align: center

Flow of soil organic carbon between the pools. Each connection shows the fraction of the decomposed organic matter in the source pool that enters the destination pool. In all cases, the remaining decomposed organic matter is respired. Clay and silt refer to soil textural fractions (0–1). :::

ORCHIDEE keeps track of the concentrations $M^\text{SOM\_2D}_{l,k}$ (in $\unit{g/m^2}$) of element $l$ (carbon or nitrogen) in pool $k$ in each grid cell and PFT (indices suppressed). In each time step, these concentrations are updated according to the following equation combining the litter input flux $F^\text{litt\_SOM\_2D}_{l,k}$, the decomposition output flux $F^\text{SOM\_decomp}_{l,k}$, and fluxes between the pools,

\[\frac{\Delta M^\text{SOM\_2D}_{l,k}}{\Delta t} = F^\text{litt\_SOM\_2D}_{l,k} - F^\text{SOM\_decomp}_{l,k} + \sum_{i \ne k} f^\text{SOM}_{l,i,k} F^\text{SOM\_decomp}_{l,i},\]

where the last term is the flux from pool $i$ to pool $k$, calculated as a fraction $f^\text{SOM}_{l,i,k}$ of the decompositon flux of pool $i$. The default values of the fractions $f^\text{SOM}_{l,i,k}$ are shown in Fig. 14.

The decomposition flux for pool $k$ is calculated assuming linear kinetics with turnover time $c^\text{som\_turn}_{k}$, which is modified to account for soil temperature, moisture, and clay content, as well as for tillage in crop PFTs,

(10.74)#\[F^\text{SOM\_decomp}_{l,k} = c^\text{som\_turn}_{k} m^\text{temp} m^\text{moist} m^\text{clay}_k m^\text{tillage} M^\text{SOM\_2D}_{l,k}.\]

The default turnover times are $c^\text{som\_turn}_\text{active} = 7.3 \,\unit{yr}$, $c^\text{som\_turn}_\text{surface} = 6 \,\unit{yr}$, $c^\text{som\_turn}_\text{slow} = 0.1 \,\unit{yr}$, and $c^\text{som\_turn}_\text{passive} = 0.0025 \,\unit{yr}$.

The temperature and moisture modifiers [],

(10.75)#\[\begin{split}\begin{align} m^\text{temp} &= \min\left(1,\exp\left(0.069 (T^\text{decomp}-303.15)\right)\right), \\ m^\text{moist} &= \max\left(0.25, \min\left(1, 1.1 \left(\theta^\text{decomp}\right)^2 + 2.4 \theta^\text{decomp} + 0.29\right)\right), \end{align}\end{split}\]

depend on $T^\text{decomp}$ and $\theta^\text{decomp}$ which are the temperature and volumetric water content of the current soil layer (if vertical dicretization of soil organic matter is activated) or averages over the soil profile. The averaging is done by integrating temperature or moisture over depth with a weighing factor reflecting the assumed concentration of decomposers close to the surface. This weighing factor is exponential with a characteristic depth of $z^\text{decomp}$, equal to 0.2  by default. By assuming the moisture is uniform in each soil layer, the integration for soil moisture $\theta^\text{decomp}$ gives

(10.76)#\[\begin{split}\begin{align} % TWO LINES % \theta^\text{decomp} =& % \left[ % \exp{\left(z^\text{lb\_hydro}_0 / z^\text{decomp}\right)} - % \exp{\left(z^\text{lb\_hydro}_{N^\text{soil}} / z^\text{decomp}\right)} % \right]^{-1} \\ % & \cdot \sum_{i = 1}^{N^\text{soil}} \theta_i \left[ % \exp{\left(z^\text{lb\_hydro}_{i-1} / z^\text{decomp}\right)} - % \exp{\left(z^\text{lb\_hydro}_{i} / z^\text{decomp}\right)} % \right], % % ONE LINE FRAC \theta^\text{decomp} = \frac{\sum_{i = 1}^{N^\text{soil}} \theta_i \left[\exp{\left(z^\text{lb\_hydro}_{i-1} / z^\text{decomp}\right)} - \exp{\left(z^\text{lb\_hydro}_{i} / z^\text{decomp}\right)}\right] }{ \exp{\left(z^\text{lb\_hydro}_0 / z^\text{decomp}\right)} - \exp{\left(z^\text{lb\_hydro}_{N^\text{soil}} / z^\text{decomp}\right)} }, % % ONE LINE INLINE % \theta^\text{decomp} = \left[ % \exp{\left(z^\text{lb\_hydro}_0 / z^\text{decomp}\right)} - % \exp{\left(z^\text{lb\_hydro}_{N^\text{soil}} / z^\text{decomp}\right)} % \right]^{-1} % \sum_{i = 1}^{N^\text{soil}} \theta_i % \left[ % \exp{\left(z^\text{lb\_hydro}_{i-1} / z^\text{decomp}\right)} - % \exp{\left(z^\text{lb\_hydro}_{i} / z^\text{decomp}\right)} % \right], \end{align}\end{split}\]

where $\theta_i$ is the soil moisture in layer $i$, $z^\text{lb\_hydro}_i$ is the depth of the lower boundary of layer $i$ as used in the soil hydrology calculations (with $z^\text{lb\_hydro}_0$ being the depth of the upper bondary of layer 1). The average temperature $T^\text{decomp}$ is calculated analogously.

As in the original CENTURY model [], the decomposition rate in Eq. (10.74) is decreased in proportion to the clay content of soil to represent the slowing down of decomposition due to association of organic matter with minerals. The clay modifier is equal to 1 for all but the active pool, for which it is determined by the clay content $c^\text{clay}$ (0–1),

(10.77)#\[m^\text{clay}_\text{active} = 1 - 0.75 \, c^\text{clay}.\]

Finally, the decomposition rate is increased to account for the effect of tillage in crops. This modifier is based on and given by

(10.78)#\[\begin{split}m^\text{tillage} = \begin{cases} 1.2 & \text{for c3 crop PFT}, \\ 1.4 & \text{for c4 crop PFT}, \\ 1 & \text{otherwise}. \end{cases}\end{split}\]

10.8.3. DONE: Heterotrophic respiration from soil carbon decomposition#

Heterotrophic respiration from soil (F$^\text{resp,het,soil}$) is obtained by summing the fractions of decomposition fluxes $F^\text{SOM\_decomp}_{\text{C},i}$ which do not enter any other soil organic carbon pool,

(10.79)#\[F^{resp,het,soil} = \sum_i \left(1 - \sum_k f^\text{SOM}_{\text{C},i,k} \right) F^\text{SOM\_decomp}_{\text{C},i}.\]

All nitrogen released during the decay of the som pools goes into the mineral nitrogen pool of the soil, where it is available for plant uptake, or other nitrogen processes (section ??). Please note that when soil organic matter is discretized the mineral N pool is not therefore all the minerazlied N is going into a single mineral N pool.

10.8.4. DONE: Vertically discretized carbon and nitrogen pools#

When soil organic matter discretization is activated, the active, slow, and passive pools are divided into depth layers identical to the ones used in the soil thermodynamics calculations. In each layer of pool $k$, ORCHIDEE keeps track of the volumetric concentration $M^\text{SOM\_3D}_{l,k}$ (in $\unit{g/m^3}$) of element $l$ (carbon or nitrogen). When soil organic matter discretization is activated, the surface soil organic matter pool is not used.

The areal (per $\unit{m^2}$) litter input flux $F^\text{litt\_SOM\_2D}_{l,k}$ of element $l$ to pool $k$ is split into volumetric (per ) fluxes $F^\text{litt\_SOM\_3D}_{l,k,i}$ entering layer $i$ following a dimensionless profile $f^\text{litt\_SOM\_3D}_i$,

\[F^\text{litt\_SOM\_3D}_{l,k,i} = f^\text{litt\_SOM\_3D}_i \frac{F^\text{litt\_SOM\_2D}_{l,k}}{z^\text{lb\_thermo}_i - z^\text{lb\_thermo}_{i-1}},\]

where $z^\text{lb\_thermo}_i$ is the depth of the lower boundary of layer $i$ of soil thermodynamics and $z^\text{lb\_thermo}_0 = 0$. The profile $f^\text{litt\_SOM\_3D}_i$ is normalized to sum to 1 and extends down to a grid-cell- and PFT-dependent depth $z^\text{intdep}$. Above that depth, it is proportional to the integral of a piecewise constant exponential with a grid-cell- and PFT-dependent characteristic depth $z^\text{litt}$, giving

\[\begin{split}f^\text{litt\_SOM\_3D}_i \propto \begin{cases} \exp{\left(-z^\text{lb\_thermo}_{i-1} / z^\text{litt}\right)} - \exp{\left(-z^\text{lb\_thermo}_i / z^\text{litt}\right)} & z^\text{lb\_thermo}_i < z^\text{intdep}, \\ 0 & z^\text{lb\_thermo}_i \geq z^\text{intdep}. \end{cases}\end{split}\]

By default (runtime flag new_carbinput_intdepzlit set to false), $z^\text{litt}$ and $z^\text{intdep}$ are both be set to last year’s maximum active-layer thickness or to $0.5~\unit{m}$, whichever is larger. In the new approach (new_carbinput_intdepzlit set to true), $z^\text{intdep}$ is set to the maximum depths root occur at ($z^\text{root\_depth}$), as calculated by the soil hydrology scheme. At the same time, $z^\text{litt}$ is set to the fine rooting depth of a given PFT or to a value proportional to the last year’s maximum active-layer thickness, whichever is smaller.

Following the distribution of litter, the decomposition of soil organic matter proceeds independently in each layer using the same formulation as the bulk soil organic matter dynamics described previously. Following decomposition, vertical transport of soil organic matter by cryoturbation or bioturbation is simulated. Both processes are assumed to follow Fick’s law of diffusion and differ only in the value of the depth-dependent diffusion constant $D(z)$. That is, for each element $l$ and pool $k$ the vertical soil organic matter flux is

(10.80)#\[F^\text{vertical}_{l,k}(z) = - D(z) \frac{\partial M^\text{SOM\_3D}_{l,k}}{\partial z}.\]

Cryoturbation takes place in grid cells and PFT tiles in which last year’s maximum active layer thickness is larger than or equal to a parameter set to $1\,\unit{cm}$ by default. In tiles where last year’s maximum active layer thickness is larger than or equal to a parameter set to $3\,\unit{m}$ by default, bioturbation takes place instead.

There are 6 alternative parametrizations of the depth-dependent cryoturbation diffusion constant. In the default (or old) parametrization and in the new parametrizations numbered 1, 2, 4, and 5, the cryoturbation diffusion constant is equal to a parameter $c^\text{cryotb\_diff}= 0.001\,\unit{m^2/yr}$ down to the layer corresponding to last year’s maximum active layer thickness $z^\text{altmax\_lastyear}$. In the layers below, the diffusion constant decreases in a way that depends on the parametrization. In the old parametrization, the diffusion constant in the layers immediately below $z^\text{altmax\_lastyear}$ is equal to, respectively, $c^\text{cryotb\_diff}/10$, $c^\text{cryotb\_diff}/100$, and 0 all the way down to the bottom of the soil column. In the new parametrizations numbered 1, 4, and 5, the diffusion constant decays linearly reaching zero either at $2 z^\text{altmax\_lastyear}$ (parametrization 1), at $3 z^\text{altmax\_lastyear}$ (parametrization 4), or at $3\,\unit{m}$ (parameterization 5). In parametrizations 2 and 3, the diffusion constant decays exponentially with a characteristic depth of $z^\text{altmax\_lastyear}$. The decay starts below $z^\text{altmax\_lastyear}$ in parametrization 2 and at the soil surface in parametrzation 3.

The bioturbation diffusion constant has a unique parametrization. It is equal to a parameter $c^\text{biotb\_diff} = 0.0001\,\unit{m^2/yr}$ down to a constant depth $c^\text{biotb\_depth}$, equal to $2\,\unit{m}$ by default, and 0 below.

The diffusion equation (10.80) is solved using the same (finite-difference, implicit) numerical scheme as used for the soil heat transport. The scheme is described in detail by and by (Annex A).

10.8.5. Nitrogen fluxes from soil dynamics#

Where should we describe the impose_cn option? Here? or in allocation where it is most impactful or in phenology where it is also applied? I described impose_cn = n as “a dynamic nitrogen cycle allowing nitrogen limitation” in the description of the model configuration (section ??). I have not yet describe impose cn = y but in line with the above it would become “a dynamic nitrogen cycle preventing (or avoiding?) nitrogen limitation”. Feel free to change the wording. If changed, please also change in section ??.

Mineral nitrogen is represented by five pools: ammonium (NH$_3$ / NH$^+_4$) , nitrate (NO$^-_3$), nitrogen oxides (NO$_x$), nitrous oxide (N$_2$O) and dinitrogen (N$_2$) soil pools. The fate of mineral N in the five pools is controlled by the internal N flows associated with nitrification (the oxidation of NH$_3$ / NH$^+_4$ in NO$^-_3$) and denitrification (the reduction of NO$^-_3$ up to the production of N$_2$) processes and by incoming and outgoing mineral fluxes. Incoming fluxes which are accounted for in ORCHIDEE are mineralization, biological nitrogen fixation, fertilization and atmospheric nitrogen deposition. Outgoing fluxes are emissions of N$_2$O, NO$_x$, N$_2$ and NH$_3$, plant nitrogen uptake of NH$^+_4$and NO$^-_3$ and leaching. Figure XXX summarizes the mineral nitrogen pools and fluxes considered in ORCHIDEE.

10.8.6. Incoming mineral N fluxes#

Biological nitrogen fixation (BNF). By default, BNF is not computed into ORCHIDEE and is assumed to be an input data. The BNF input map used currently in ORCHIDEE set the BNF rate as a function of the annual latent heat flux based on the work of Cleveland et al. (1999). It provides BNF rate varying in space but constant in time. In ORCHIDEE, the reactive N produces by BNF feeds the NH$^+_4$ pool.

Synthetic fertilization. Synthetic fertilisation is provided through input maps at a yearly time step. It assumes that reactive nitrogen is applied uniformly in time over the year over cropland and grassland area. Several datasets are available with different specificities: information for cropland and grassland, or for C3 and C4 crops only; as total N (to be splitted internally in NO$^-_3$ / NH$^+_4$ using the parameter RATIO_NH4_FERT) or specifically as NO$^-_3$ and NH$^+_4$
Atmospheric N deposition Information on Atmospheric N deposition is updated at a yearly time-step with monthly- or annual-scale data. It provides information on total NO$^-_y$ and NH$_x$ that feed NO$^-_3$ and NH$^+_4$ pools respectively; or information on subcategories (wet and dry NO$^-_y$ and NH$_x$)

10.8.7. Outgoing mineral N fluxes#

Emissions of N2O, NOx, N2 and NH3
Plant nitrogen uptake of NH4 and NO3
Leaching of NH4 and NO3

10.8.8. Internal mineral N fluxes#

Nitrification
Denitrification

10.9. DONE: BVOC emissions#

ORCHIDEE calculates the emissions of volatile organic compounds (VOCs) from vegetation, also known as biogenic VOCs (BVOCs). BVOCs are generally characterized by a strong chemical reactivity in the atmosphere, together with high natural emissions, and therefore play a key role in the atmospheric chemical composition. Implementing BVOC emission schemes into the land surface model ORCHIDEE enables studying the interactions between the terrestrial biosphere, atmospheric chemistry, and climate. The original emission scheme [] has been extensively updated by , based on the model developed by . It includes an extended list of biogenic emitted compounds, updated emission factors, and a dependency on light for almost all compounds.

A wide range of BVOCs are simulated, including isoprene, monoterpenes (as a family of compounds and also with eight speciated monoterpenes), sesquiterpenes, methanol, acetone, acetaldehyde, formaldehyde, acetic acid and formic acid. The compounds accounted for by ORCHIDEE solely consist of carbon, hydrogen, and oxygen, hence, there is no interaction between the production of BVOCs in ORCHIDEE and the nitrogen cycle. Moreover, the BVOC emission calculation is a supplementary diagnostic and is not accounted for in the carbon mass balance.

The PFT-specific emission flux ($F_{j,i}$; $\mu$g C m$^{-2}$ h$^{-1}$) of the specific biogenic compound $j$, at canopy layer $i$ is calculated as:

\[F_{j,i}=\frac{d^{LAI}_{i} \cdot c_{1,j} \cdot m^{clim}_{j,i} \cdot m^{age}_{j}}{k^{sla}}\]

where $d^{LAI}_{i}$ is the leaf area of canopy layer $i$, $k^{sla}$ is the PFT-specific leaf area (m$^{2}$ g$^{-1})$, $c_{1,j}$ ($\mu$g C g$^{-1}$ h$^{-1}$) is the PFT-specific basal emission at the leaf level for an individual emitted compound $j$ at a reference temperature of 303.15 K and a photosynthetically active radiation of 1000 $\mu$mol m$^{-2}$ s$^{-1}$. $m^{clim}_{j,i}$ (unitless) is a compound-specific modulator that reflects the deviation from the standard conditions of temperature and photosynthetically active radiation, and $m^{age}$ (unitless) is a modulator that accounts for the change in emission capacity depending on the age of the leaves.

The temperature and radiation modulator is calculated separately for each compound $j$ and canopy layer $i$:

\[m^{clim}_{j,i} = (1-c_{2,j}) \cdot b_{1,j} + c_{2,j} \cdot b_{2} \cdot b_{3,i}\]

The purpose of $m^{clim}_{j,i}$ is to represent the light-dependent and light-independent processes that lead to emissions. c$_{2,j}$ is the prescribed fraction of emission that is, for one particular BVOC compound, light-dependent, $b_{2}$ and b$_{1,j}$ are the dependencies of emissions to temperature, for light-dependent and light-independent fractions respectively, and $b_{3,i}$ is the radiation-dependency relationship, considered for the light-dependent fraction of emissions. With

\[b_{1,j} = exp \left(c_{3,j} \cdot \left(T^{air}-303.15 \right) \right)\]

where c$_{3,j}$ is a prescribed temperature dependency for each compound $j$, and T$^{air}$ is the temperature of the air.

\[b_{2} = \frac{b_{4}} {\left(un + b_{5}\right)}\]

where

\[b_{4} = exp \left(\frac{c_4 \cdot \left(T^{air}-303.15 \right)}{c^R \cdot 303.15 \cdot T^{air}} \right)\]

and

\[b_{5}(ji) = exp \left(\frac{c_5 \cdot \left(T^{air}-314.15 \right)}{c^R \cdot 303.15 \cdot T^{air}} \right)\]

with c$^R$ the ideal gas constant, i.e., 8.314 J mol$^{-1}$ K$^{-1}$.

\[b_{3,i} = \frac{c_6 \cdot F^{PAR\downarrow}_{i}}{ \sqrt{1+c_7^{2} \cdot \left(F^{PAR\downarrow}_{i} \right)^{2}}}\]

where F$^{PAR\downarrow}_{i}$ is the incoming photosynthetic active radiation in canopy layer $i$ calculated following the scheme for direct and diffuse light provided by and . Emissions are thus calculated for each canopy layer $i$ considering the sunlit and shaded leaf fractions and the light extinction and light diffusion through canopy. Note that vertical discretisation of the canopy and the light transfer calculated to simulate the BVOC emissions are not necessarily consistent with the canopy discretisation (section ??) and the light transfer ?? used in the rest of the model.

The leaf age modulator is calculated for isoprene and methanol only, for which a change in the emission capacity with leaf age has been demonstrated [] and is equal to unity for other BVOCs. Regarding isoprene and methanol, it is calculated in all leaf age classes, based on the fraction of leaves within each class f$^{leaf}$ calculated in ORCHIDEE (Section ??, and considering a different emission activity for each class, with a 50% reduction in emissions capacity c$_{8,j}$ for leaf age classes 1 and 4 in the case of isoprene and for leaf age classes 3 and 4 in the case of methanol.

\[m^{age}_{j} = f^{leaf} \cdot c_{8,j}\]

Finally, the total emission per grid cell is obtained by summing $F_{j,i}$ over the layer $i$ and adding the emission contribution of each individual PFT, weighted by PFT fractional land coverage. Every values of emission factor and light-dependent fraction for each BVOC considered in the model can be found in .

10.10. DONE: Vegetation demography#

Since vegetation canopy acts as the interface between the land and the atmosphere, forest stand structure has implications beyond the carbon and nitrogen budgets of forest management and disturbances, and has been shown to influence albedo, transpiration, photosynthesis, soil temperature, roughness length, and recruitment []. Three demography processes are defined in ORCHIDEE. Succession refers to the aging of vegetation and describes how stands biomass is dynamically allocated to age and circumference classes following growth, management or mortality events. Recruitment refers to cases where small-scale mortality events are compensated by the growth of saplings and happens when management is unmanaged or continuous-cover forestry or in case of a disturbance affecting less than 30% of the stand basal area. Establishment refers to the growth of saplings in an area where all biomass is dead or non-existent, for example after harvest or after a disturbance affecting more than 30% basal area.

10.10.1. DONE: Succession within established vegetation#

To better represent forest succession and its impacts on element, energy, and water fluxes, ORCHIDEE distinguishes between different age classes ($nage$). Each age class also includes circumference classes to enable simulating forest management, forest disturbances, and more refined simulation of canopy structure. For each PFT, processes such as establishment, phenology, photosynthesis, maintenance respiration, allocation and growth respiration, mortality, land use, disturbances, litter decomposition, and soil decomposition are calculated as described in sections ?? to ??. Throughout these processes, the multiple circumference classes are considered, with trees in different classes experiencing preferential growth and mortality rates driven by different processes. In long enough simulations or under high mortality conditions, stand density in some circumference classes may eventually drop to zero, reducing the numerical resolution of the carbon and nitrogen allocation scheme (specifically equation (10.44)). When only one populated circumference class remains, the scheme loses its meaning, as all the newly produced biomass is allocated to the sole remaining class. To ensure maintaining the same level of detail throughout the simulation, individuals and their biomass are redistributed across all circumference classes at the end of each day.

The daily redistribution of biomass across circumference classes largely recalculates the structural impact of thinning, disturbances, and recruitment. During the redistribution, individuals are allocated to the $ncirc$ circumference classes according to the Weibull distribution, with the shape depending on the forest management strategy and the quadratic mean diameter of the stand for rotational even-aged management (equations (10.56) to %s). Once the number of individuals in each circumference class is known, their corresponding biomass is moved to the respective class. The new total biomass of each circumference class is then divided by its new number of individuals to obtain the mean biomass of the model tree in each class. With the wood biomass now determined, the circumference of each model tree is calculated, and the boundaries of the circumference classes are updated.

10.10.2. DONE: Recruitment within established vegetation#

Recruitment of saplings within an existing forest PFT compensates for the loss of individuals due to density-driven mortality (section ??) and other sources of tree mortality. By default, recruitment is calculated once per year and only applies to unmanaged or continuous-cover forestry stands. The number of recruits added each year ($d^{ind,rec}$; m$^{-2}$) in the first age class is given by:

\[d^{ind,rec} = c_1 ^{(\log{(c_2 \cdot d^{ind}) \cdot \sqrt{\log{(\max(100 \cdot f^{Pgap,trees,season},un)}}})}\]

where $c_1$ and $c_2$ are PFT-specific parameters, $d^{ind}$ is the stand density (m$^{-2}$) before recruitment, and $f^{Pgap,trees,season}$ is the fraction of light reaching the forest floor (equation (10.66)) averaged over the growing season. Recruitment density increases with lower initial stand densities and sparser canopies.

The height of an individual sapling is determined by a PFT-specific parameter $c_3$ (m) and is used to calculate the initial diameter (m) using equation (10.42). Height and diameter are then used in equation (10.29) to calculate the sapwood mass ($M^{sap}$; g plant$^{-1}$) for an individual recruit. Subsequently, leaf mass ($M^{leaf}$; g plant$^{-1}$) and root mass ($M^{root}$; g plant$^{-1}$) are calculated using equations (10.30) and (10.35), respectively, with the allocation parameters $f^{KF}$ (equation (10.31)) and $f^{LF}$ (equation (10.36)) recalculated to account for the effect of light stress on allocation.

If recruitment occurs outside the growing season in a deciduous PFT, the leaf and root masses are transferred to the reserve pool ($M^{res}$; g plant$^{-1}$). Finally, the newly recruited biomass and individuals are added to the first circumference class of the existing PFT as follows:

\[\begin{split}\begin{align} &M^{plant}_{1} = \frac{M^{plant}_{1} \cdot d^{ind} + M^{rec} \cdot d^{ind,rec}}{d^{ind}_{1} + d^{ind,rec}}\\ &d^{ind}_{1} = d^{ind}_{1} + d^{ind,rec} \end{align}\end{split}\]

Recruitment is mono-specific and follows the PFT in which it occurs. In other words, ORCHIDEE does not explicitly simulate multiple species within a single PFT. However, the PFT parameters can be set to represent a multi-species stand, as in the case of species-rich tropical forests represented by the MTC labelled as Tropical broad-leaf evergreen forest (Table 6).

10.10.3. DONE: Establishment of new vegetation#

In ORCHIDEE new vegetation is established following a stand-replacing event, which can be natural mortality (section ??), natural disturbance (section ??), land cover change (section ??), forest management (section ??), or crop harvesting (section ??). In cases of land cover change (section ??), the newly established vegetation has a different PFT than that of the previous vegetation. Similarly, for a change in forest PFT (section

), the newly established vegetation may have a different PFT. In all other cases, the newly established vegetation retains the same PFT as the previous vegetation.

The fraction of the newly established PFT within the grid cell ($f^{veg,max}$; unitless) is read from an external annual vegetation distribution map (section ??) or derived from internal areal bookkeeping in the case of natural disturbances (section ??). For forests, the size distribution of saplings across circumferences classes in the first age class is calculated, including the diameter ($d^{dia}_{l}$; m$^{-2}$), height ($d^{h}$; m) and biomass of model individuals ($M^{plant}_{l}$; g plant$^{-1}$) as well as the number of saplings in each circumference class. The approach formalized in ORCHIDEE introduces a circular dependency because the density distribution across circumference classes $l$ depends on the quadratic mean diameter (section ??), while the quadratic mean diameter itself depends on the density distribution (section %s). This circularity is resolved through a single iteration that begins by calculating the representative diameter for each circumference class using the PFT-specific minimum diameter ($c_2$; m) and maximum diameter ($c_3$; m):

(10.81)#\[d^{dia,init}_{l} = \frac{c_2 + (l-1) \cdot (c_3-c_1)}{ncirc-1}\]

An initial estimate of the frequency distribution of individuals in each circumference class $l$ is derived from the Weibull distribution (equations (10.57) to %s) with distribution parameters depending on PFT and management strategy $d^{qmdia,init}_{l}$ as the central circumference class diameter, the diameter at which the continuous Weibull distribution is truncated ($c_1$ in equation (10.57)) depending on the forest management strategy and the shape ($k^{shape}$; unitless) parameter of the Weibull distribution is either calculated or prescribed depending on the forest management strategy (equation (10.56)).

The total number of individuals is determined using equation (10.52) with the quadratic mean diameter ($d^{qmdia, init}$; m). The frequency distribution in each circ class and total number of individuals ($d^{ind}$; m$^{-2}$) are then used to calculate the number of individuals in each circumference class $l$.

The representative diameters in each circumference class derived from the Weibull distribution are also used to calculate the biomass for individual trees in each circumference class $l$ following the same allocation principles (section ??). The diameter of the representative sapling in each circumference class is determined using equation (10.81) and it is then used to calculate the corresponding tree height via the allometric equation (10.42). Height and diameter are subsequently used in equation (10.29) to calculate the sapwood mass ($M^{sap}$; g plant$^{-1}$) of an individual sapling. Then, leaf mass ($M^{leaf}$; g plant$^{-1}$) and root mass ($M^{root}$; g plant$^{-1}$) of a sapling are calculated using Equations (10.30) and (10.35), respectively, where the allocation parameters $f^{KF}$ (equation (10.31)) and $f^{LF}$ (equation (10.36)) are calculated assuming that there is no light stress at the time of vegetation establishment. For deciduous PFTs, if recruitment occurs outside the growing season, leaf and root masses are moved into the reserve pool ($M^{res}$; g plant$^{-1}$). Since the allocation equations and the self-thinning relationship are not suited for seedlings, initial diameters (i.e., $c_2$ and $c_3$) are prescribed to exceed $\sim$ 0.02 m. The initial forests thus better represent the sapling than the seedling stage.

For grassland and cropland PFTs, the stand density is prescribed and fixed. These PFTs are initialized by prescribing the initial height, which is converted into a leaf area using a single prescribed parameter. After calculating the allocation parameters $f^{KF}$ (equation (10.31)), $f^{LF}$ (equation (10.36)), and the specific leaf area (equation ??), leaf mass ($M^{leaf}$; g plant$^{-1}$) is computed as follows:

\[M^{leaf} = \frac{d^{LAI}}{k^{sla}}\]

With the leaf mass and allocation parameters determined, root ($M^{root}$; g plant$^{-1}$) and stem mass ($M^{sap}$; g plant$^{-1}$) are calculated using Equations (10.30) and (10.35). All grassland and cropland PFTs are assumed to be deciduous, with crops established on the day of bud break. As with forest vegetation, if the grassland PFT is established outside the growing season, the leaf and root mass are transferred into the reserve pool ($M^{res}$; g plant$^{-1}$) until bud break.

During vegetation establishment, the carbon necessary to build the initial biomass is taken from the atmosphere. Nitrogen needed for establishment is drawn from the soil mineral nitrogen pool, which is replenished through mineralisation (section ??), atmospheric deposition and nitrogen fertilisation (section ??). If the mineral nitrogen pool cannot satisfy the initial demand, nitrogen is supplied from the organic soil nitrogen pools (section ??). If the latter pools cannot meet the demand either, nitrogen is finally taken from the atmosphere.

11. Land use#

11.1. DONE: Land cover change#

11.1.1. DONE: Land cover change in ORCHIDEE#

ORCHIDEE accounts for land cover change by reading time series of PFT maps that document the changes in the fraction of each PFT within each grid cell ($f^{veg,max}$), with a default annual time step. The annual PFT maps are generated by making use of a cross-walking table that links ORCHIDEE PFTs with discrete land cover types, whose time-dependent distributions are derived based either on remote sensing (, ) or reconstructions ().

The carbon and hydrological impacts of land cover changes in ORCHIDEE are accounted for based on the net land cover changes between two consecutive years. This means that the local-scale, bi-directional flows between two land cover types, known as gross land cover change, are ignored. At the usual coarse resolution of global scale simulations, i.e., between 0.25 ° x 0.25 ° and 2 ° x 2 °, simulating net instead of gross land cover changes is likely to underestimate the 2 emissions from land cover changes []. When ORCHIDEE is run at a high spatial resolution, issues from this simplification could be somehow mitigated.

Within ORCHIDEE, land cover change is defined as the change over time in the PFT fractions covering a grid cell, with some PFTs losing land cover, other PFTs gaining land cover. The change in land cover for each PFT is obtained by calculating the difference between two consecutive PFT maps, the sign of which determines whether each PFT is losing or gaining land cover.

\[\begin{align} &f^{veg,delta} = f^{veg,max}_{y+1}-f^{veg,max}_{y} \end{align}\]

The basic approach to account for the effects on the carbon and nitrogen cycle of land cover change is to transfer the fresh litter, together with existing litter and soil organic matter, from the shrinking PFTs to expanding PFTs through an temporary, intermediate litter and soil bank.

11.1.2. DONE: Land cover loss of a PFT#

When a PFT is losing land cover, its carbon, nitrogen and water-related vegetation stress pools are temporarily stored in pool banks. When using forest circumference classes, all circumference classes will lose their areas in proportion to their current areas. Depending on the PFT losing land cover, the biomass pools being transferred will differ. If it is forest, above-ground woody biomass is first harvested and allocated to different components of wood product pool according to the diameter of harvested stem, with different life spans of 1, 17 and 50 years (section ??). The unharvested above-ground forest biomass components, including branches, leaves and fruits, and the unharvested below-ground roots, are then transferred to the soil and litter bank pools. For the loss of non-forest PFTs, all biomass components are transferred to the soil and litter bank pools. Each pool bank consists of the average of shrinking pools weighted by their contribution to the total loss in land cover.

(11.1)#\[M^\text{bank,o} = \frac{\sum_k M^\text{o}_k \cdot f^{veg,delta}}{\sum\limits_{f^{veg,delta}<0} f^{veg,delta}}\]

11.1.3. DONE: Land cover gain of a PFT#

Patches of PFTs gaining land cover, with their $f^{veg,max}$ being equal to their areal expansion ($f^{veg,delta}$), will be first established, following ??, and then merged with existing identical PFTs if ther are any. The final pool density of a given PFT ($M^{update}$), with a gaining area fraction of $f^{veg,delta}$, after integrating the transferred pools from shrinking PFTs, is thus determined as:

\[M^{update,o}_{k}= \frac{ M^{o}_{k} \cdot f^{veg,max} + M^\text{bank,o} \cdot f^{veg,delta}} {f^{veg,max}+f^{veg,delta}}\]

In the case where circumference classes are used, the vegetation characteristics for the land cover gain are initialized following ??. The gained land cover will be added to the youngest age class. The patch of the youngest circumference class will be first established, as described above, and then merged with the patch of the existing youngest age class when there is one.

11.1.4. DONE: Land cover change of non-biological land#

When the land cover change involves a gain or loss of non-biological land, the handling of soil carbon and nitrogen has to be adjusted. For gain of non-biological land (i.e. urbanization), soil organic matter is considered as being buried and thus set aside from decomposition, assuming an inert carbon pool. For loss of non-biological land (i.e. glacier retreat or lake dry-off), the initial soil organic matter for the newly vegetated land cover will be taken from the soil and litter bank filled up by corresponding land cover losses.

11.2. Irrigation#

In ORCHIDEE the crop irrigation is optional. When irrigation is simulated, the model calculates XXX

11.3. DONE: Crop and grass harvest#

In ORCHIDEE crop harvest is simulated either as a daily turnover throughout the growing season (See ??) or as single harvest event at the end of the growing season. When harvest a simulated at the end of the growing season, a fraction ($c_1$) of the above-ground carbon and nitrogen mass of cropland PFTs is moved into the harvest pool. The remaining biomass pools are moved into the litter pool:

\[\begin{split}\begin{align} &M^{har,crop}_{t} = M^{har,crop}_{t-1} + c_1 \cdot (M^{leaf}_{l,t} + M^{res}_{l,t} + M^{lab}_{l,t} + M^{fruit}_{l,t} + M^{sap}_{l,t} )) \cdot d^{ind,kill}_{l,t}, \\ &M^{lit}_{l,t} = M^{lit}_{l,t-1} + (M^{root}_{l,t} + (1-c_1 ) \cdot (M^{leaf}_{l,t} + M^{res}_{l,t} + M^{lab}_{l,t} + M^{fruit}_{l,t} + M^{sap}_{l,t} )) \cdot d^{ind,kill}_{l,t}. \end{align}\end{split}\]

Finally, plant biomass and number of individuals are reset to zero until the PFT is replanted in the next growing season (section ??):

\[\begin{split}\begin{align} &M^{plant}_{t} = 0,\\ &d^{ind}_{t} = 0, \end{align}\end{split}\]

In ORCHIDEE grasslands are simulated as unmanaged ecosystems with a constant plant density. When mortality occurs, the dead biomass is moved to the litter pools. Given that grasslands are simulated as deciduous PFTs, a dormant grassland is replanted the next day. Bud-break and subsequent growth requires suitable PFT-dependent environmental conditions (section ??).

11.4. DONE: Forest management#

11.4.1. DONE: Forest management strategies#

ORCHIDEE uses spatially and temporally resolved management reconstructions to prescribe the forest management strategy to each PFT and grid cell. Where the European management reconstruction [] supports this level of detail, the current global reconstruction (section ??) prescribes a single management strategy to the entire grid cell. ORCHIDEE relates each management strategies to a set of rules that make forest management dependent on biomass production, diameter, and stand density but not age. As a consequence, forest management in ORCHIDEE evolves as the environmental conditions change and the wood harvest is an emerging model outcome rather than a prescribed model input.

The default management has no human intervention and stand structure is determined by disturbances (section ??), natural mortality (section ??), and recruitment (section ??). In addition, four management systems with human intervention have been implemented: (a) rotational even-aged management, in which human intervention is restricted to thinning operations and occasional clear cuts []; (b) continuous cover forest management, where recruitment restores stand density following a thinning []; (c) coppice management, in which stand density is maintained through shoots sprouting from the root system following a thinning []; and (d) short rotation forestry, in which a limited number of thinning and re-sprouting cycles occur before the stand is clear cut [].

11.4.2. DONE: Forest management rules#

The state of the forest is described through: (1) two production related indicators, i.e., mean annual wood increment over the life time of the forest ($F^{npp,wood,lt}$) and mean periodic wood increment ($F^{npp,wood,10year}$) both in g C m$^{-2}$ s$^{-1}$; (2) two density related indicators, i.e., relative density index ($f^{RDI,act}$) and the target relative density index ($f^{RDI,upp}$) both unitless; and (3) four structural indicators, i.e., the quadratic mean diameter ($d^{qmdia}$), the quadratic mean diameter of the 50 % largest trees of the stand ($d^{qmdia,50\%}$), the height of the tallest 100 individuals ($d^{h,100trees}$), and the stand density that exceed the PFT-specific clear cut diameter ($d^{ind,ccdia}$). Diameters and height are expressed in m, the stand density in m$^{-2}$. $d^{qmdia}$, $f^{RDI,act}$ and $f^{RDI,upp}$ are calculated following equations (10.55), %s and %s, respectively. Note that $f^{RDI,upp}$ is a function of the management strategy. The remaining indicators are calculated as:

\[\begin{split}\begin{align} &F^{npp,wood,lt}_{t} = F^{npp,wood,lt}_{t-1} \cdot (lt-1) + F^{npp,wood,lt}_{t},\\ &F^{npp,wood,10year}_{t} = F^{npp,wood,10year}_{t-1} \cdot 9 + F^{npp,wood,10year}_{t},\\ &d^{qmdia,50\%} = \sqrt{\frac{4 \cdot \sum_{n=0.5 \cdot d^{ind}}^{d^{ind}}{{d^{dia}_{n}}^{2}}}{\pi \cdot 0.5 \cdot d^{ind}}},\\ &d^{h,100trees}= \frac{\sum_{n=d^{ind}-100}^{d^{ind}}{d^{h}_{n}}}{100}, \end{align}\end{split}\]

where $t-1$, $t$, and $lt$ denote respectively the previous year, the current year, and the life time or age of the forest (years). $d^{ind}$ is the stand density (m$^{-2}$), $d^{dia}_{n}$ the diameter of individual $n$, and $d^{h}_{n}$ the height of individual $n$. In addition to these indicators, the PFT-specific tree diameter above which a stand is cut ($c_1$; m), the PFT-specific tree diameter above which a stand is coppiced ($c_2$; m), the PFT-specific stand density below which a stand is cut ($c_3$; m$^{-2}$), the PFT-specific minimum age at which a stand is cut ($c_4$; years) are used to decide if management measures are to be taken at a given PFT and grid cell.

For an unmanaged forest the following rules decide whether the forest has to be self-thinned, converted, or left growing:

\[\begin{split}\text{unmanaged} \begin{cases} % thinning of unmanaged stands is described in stomate_mark_to_kill.f90 \text{self-thin, if}\ &f^{RDI,act} > f^{RDI,upp}\ \text{and}\ d^{qmdia,50\%} > 0.66 \cdot c_1\\ % density driven mortality is described in stomate_kill.f90 \text{replant or convert, if}\ &d^{ind} < c_3\\ \end{cases}\end{split}\]

Contrary to other management strategy, an unmanaged forest is killed and then replanted. Where in the case of unmanaged forests, “killing” and “replanting” are ORCHIDEE terminology for, an unspecified stand-replacing disturbance and natural regeneration, respectively. Because the stand is unmanaged, the same PFT as before the stand-replacing disturbances is replanted.

For a forest under rotational even-aged management the following rules decide whether the forest has to be thinned, cut, converted, or left growing:

\[\begin{split}\text{manage} \begin{cases} \text{thin from above, if}\ &f^{RDI,act} > f^{RDI,upp}\ \text{and}\ d^{qmdia,50\%} > 0.66 \cdot c_1\\ \text{thin from below, if}\ &f^{RDI,act} > f^{RDI,upp}\ \text{and}\ d^{qmdia,50\%} < 0.66 \cdot c_1\\ \text{cut or convert, if}\ &F^{npp,wood,lt}_{t} > F^{npp,wood,10year}\ \text{and}\ lt > c_4\\ \text{cut or convert, if}\ &d^{ind,ccdia} > c_3\ \text{and}\ d^{qmdia,50\%} > c_1\\ \text{cut or convert, if}\ &d^{ind} < c_3 \end{cases}\end{split}\]

If changes in forest management are accounted for in the simulation (section

) abandoning rotational even-aged management in favor of unmanaged forest occurs when one of the criteria for a cut are met and, at that time, replaces the final cut.

For a forest under continuous cover management the following rules decide whether the forest has to be thinned, converted, or left growing:

\[\begin{split}\text{manage} \begin{cases} \text{thin from above, if}\ &f^{RDI,act} > f^{RDI,upp}\ \text{and}\ d^{qmdia,50\%} > 0.66 \cdot c_1\\ \text{thin from below, if}\ &f^{RDI,act} > f^{RDI,upp}\ \text{and}\ d^{qmdia,50\%} < 0.66 \cdot c_1\\ \text{convert, if}\ &F^{npp,wood,lt}_{t} > F^{npp,wood,10year}\ \text{and}\ lt > c_4\\ \text{convert, if}\ &d^{ind} < c_3 \end{cases}\end{split}\]

For a forest under coppice management, the stand density at which the first cut happens ($d^{ind,cop}$; m$^{-2}$) is recorded and the following rules decide whether the forest has to be thinned, cut, converted, or left growing:

\[\begin{split}\text{manage} \begin{cases} \text{thin, if}\ &f^{RDI,act} > f^{RDI,upp}\\ \text{first cut or convert, if}\ &d^{qmdia,50\%} > c_2\\ \text{subsequent cut or convert, if}\ &d^{qmdia,50\%} > c_2\ \text{and}\ d^{ind} > d^{ind,cop}\\ \end{cases}\end{split}\]

For a short rotation coppice forest, the number of rotations ($n_{rot}$; unitless) is calculated from a PFT-specific parameter for rotation length ($c_5$; years), the maximum number of rotations is prescribed ($c_6$; unitless) and the following rules decide whether the forest has to be thinned, cut, converted, or left growing:

\[\begin{split}\text{manage} \begin{cases} \text{thin, if}\ &f^{RDI,act} > f^{RDI,upp}\\ \text{first cut or convert, if}\ &n_{rot} = 1\\ \text{subsequent cut or convert, if}\ &n_{rot} > 1\ \text{and}\ n_{rot} < c_6\\ \text{final cut or convert, if}\ &n_{rot} = c_6 \end{cases}\end{split}\]

When a forest meets one of the criteria to be cut, converted, or replanted, all trees are marked for harvesting ($d^{ind,kill}_{l}$; m$^{-2}$):

\[d^{ind,kill}_{l} = d^{ind}_{l},\]

The management strategy determines which biomass pools of the marked trees are harvested and which not (section ??). One exception to this general rule is when management is abandoned and the forest management is changed to unmanaged (section

). In that case the biomass for the PFT and grid cell under consideration is preserved reflecting the logic that setting aside older forests with a high biomass provide suitable initial condition to restore the exceptional provision of ecosystems services by intact forests [].

When a forest meets one of the criteria to be thinned, only part of the trees are marked for harvesting. Marking depends on the thinning approach ($c_7$; unitless) which in ORCHIDEE is PFT-specific and varies continuously from a thinning from below (indicated by a positive value for $c_7$) to a thinning from above (indicated by a negative value for $c_7$). The probability that circumference class $l$ is thinned ($p^{thin}_{l}$; unitless) is calculated as:

\[\begin{split}p^{thin}_{l} \begin{cases} \text{if}\ c_7>0,\ &\frac{c_8+(c_9-c_8) \cdot (\max(d^{circ})-d^{circ}_{l})}{(\max(d^{circ})-\min(d^{circ}))^{c_7}}\\ \text{else,}\ &\frac{c_8+(c_9-c_8) \cdot (d^{circ}_{l}-\min(d^{circ})}{(\max(d^{circ})-\min(d^{circ}))^{|c_7|}}\\ \end{cases}\end{split}\]

where $c_8$ (unitless) is the PFT-specific minimum probability that a circumference class is thinned, $c_9$ (unitless) is the PFT-specific maximum probability that a circumference class is thinned, $d^{circ}$ (m) is the circumference of each model tree, and $d^{circ}_{l}$ is the circumference of class $l$. Thinning is calculated following an iterative approach:

\[\begin{split}\begin{align} &b_{1} = p^{thin}_{l} \cdot (d^{ind}_{l} - d^{ind,thin}_{l}),\\ &d^{ind,kill}_{l} = d^{ind,thin}_{l} + b_{1},\\ &f^{RDI,pot} = \frac{\sum_{l=1}^{ncirc}{d^{ind}_{l} - d^{ind,kill}_{l}}}{d^{ind,max}} \end{align}\end{split}\]

The probability $p^{thin}_{l}$ enables estimating the stand density that is thinned ($b_{1}$), this estimate is added to previous estimates to record the total stand density marked for killing ($d^{ind,kill}_{l}$), which in turn can be used to calculate the relative density index following the thinning. Iterations continue until $f^{RDI,pot}$ equals $f^{RDI,low}$ for the management strategy under consideration. In case $f^{RDI,pot}$ is less than $f^{RDI,low}$, the stand density marked for thinning is reduced. The management strategy determines which biomass pools are harvested and which not (section ??).

11.4.3. DONE: Moving biomass to litter and harvest pools#

For each circumference class either none, part, or all of the individuals are marked for killing ($d^{ind,kill}_{l}$; m$^{-2}$) because of forest management (sections ??. Subsequently, the biomass of the individuals that were marked for killing, is moved into the appropriate biomass pools. Following the thinning or the cut, the stand density is updated. For forest under rotational even-aged management the above-ground wood is harvested and the remainder of the biomass is added to the litter for both a thinning and a cut:

(11.2)#\[\begin{split}\begin{align} &M^{har,forest}_{t} = M^{har,forest}_{t-1} + (1-c_{10}) \cdot M^{stem}_{l,t} \cdot d^{ind,kill}_{l,t}, \\ &M^{lit}_{t} = M^{lit}_{t-1} + (M^{plant}_{l,t} - (1 - c_{10}) \cdot M^{stem}_{l,t}) \cdot d^{ind,kill}_{l,t}, \\ &d^{ind}_{l,t} = d^{ind}_{l,t-1} - d^{ind,kill}_{l,t}, \end{align}\end{split}\]

where $c_{10}$ is the PFT-specific branch fraction, $M^{har,forest}_{t}$ and $M^{har,forest}_{t-1}$ are the harvest pools at respectively time step $t$ and $t-1$ for a single PFT within a grid cell, $M^{lit}_{t}$ and $M^{lit}_{t-1}$ are the litter mass at respectively time step $t$ and $t-1$ for a single PFT within a grid cell, $M^{stem}_{l,t}$ is the above-ground stem biomass (g plant$^{-1}$), and $M^{plant}_{l}$ (g plant$^{-1}$) is the plant mass in circumference class $l$. This calculation is repeated for the carbon and nitrogen biomass.

Continuous cover forests are only subjected to thinning for which the above-ground wood is harvested and the remainder of the biomass is added to the litter following equations (11.2) to %s.

Under coppice management the root system is preserved for both thinning and cutting:

(11.3)#\[\begin{split}\begin{align} &M^{har,forest}_{t} = M^{har,forest}_{t-1} + (1-c_{10}) \cdot M^{stem}_{l,t} \cdot d^{ind,kill}_{l,t}, \\ &M^{lit}_{t} = M^{lit}_{t-1} + (M^{plant}_{l,t} - (1 -c_{10}) \cdot M^{stem}_{l,t} - M^{stem,below}_{l,t}) \cdot d^{ind,kill}_{l,t}, \end{align}\end{split}\]

When the stand is thinned, the number of individual stems which share the same root system is updated:

(11.4)#\[\begin{split}d^{ind}_{l,t} \begin{cases} \text{if thin,}\ &d^{ind}_{l,t} = d^{ind}_{l,t-1} - d^{ind,kill}_{l,t},\\ \text{if cut,}\ &d^{ind}_{l,t} = d^{ind,cop} \end{cases}\end{split}\]

Under short rotation coppice, harvest, litter and biomass pools are treated in the same way (i.e., equations (11.3) to (11.4)) as if the forest was coppiced. At the last cycle of the rotation the entire forest is cut and the harvest, litter and biomass pools are treated as a clear cut following equations (11.2) to %s.

11.5. DONE: Forest PFT and management change#

Most land surface models, including previous versions of ORCHIDEE are developed to work with a historical land cover and land management reconstruction, hence, the PFT and its management are prescribed and independent of the stand characteristics. When, for example, the reconstruction prescribed a land cover in the year 1950, this change will be implemented irrespective of whether the stand was thinned the year before or whether the stand was only 10 years old. A difference between ORCHIDEE and previous versions of the model is the optional functionality to account for anthropogenic changes in forest PFTs and forest management as a function of the stand dynamics. If changes in forest PFTs and forest management are simulated, ORCHIDEE calculates stand growth and forest management but when the stand has reached maturity and needs to be harvested the stand can be replaced by: (1) a stand of a different PFT with under the same forest management, (2) the same PFT under different forest management, (3) a different forest PFT under different forest management, or (4) replanted with the same PFT under the same management as before.

In reality, the species distribution of managed forest is the outcome of human preferences within a climate envelope rather than a natural climate-driven process. Therefore, the ORCHIDEE user can define the forest PFT that is planted at each grid cell if an opportunity emerges. In ORCHIDEE opportunities for a change in forest PFT are: (1) a die-back, (2) a clear cut, or (3) a stand replacing disturbance of the current PFT. At present, PFT changes (this section) and land cover changes (section ??) cannot be combined in a single simulation. The functionality to change to prescribed PFTs has been used to quantify the climate impact of future changes in species preferences [].

Similarly, the ORCHIDEE user can opt to simulate changes in forest management strategy when an opportunity arises. In ORCHIDEE opportunities for forest management changes are: (1) a die-back, (2) a clear cut, and (3) a stand replacing disturbance. To avoid that all unmanaged forests are taken into management in the first year of the simulation, the model waits until the unmanaged forest has reached a stand diameter that would make it qualify for a clear cut if it would have been managed. Forest management strategy changes has been used to quantify the climate impact of changes in forest management policies [].

ORCHIDEE treats age classes of the forest MTC as different PFTs. Following a change in forest management, different age classes of the same species can thus be under different management strategies, i.e., the youngest age class will follow the new strategy whereas the older age classes will still follow the previous forest management. When biomass from different age classes has to merged into a single age class, the management of the youngest age class of this merge is applied to the entire merged age class to maintain the intention to change forest management. Forests are replanted as different PFTs regardless if they died from human intervention (i.e., a clearcut for high stand management or any harvest with coppice) or from natural causes. If a forest died from a natural cause it is not replanted until January 1st of the following year.

11.6. DONE: Forest litter raking#

Towards the end of the middle ages, farmers began to keep their cattle inside during winter, which led to a demand of forest litter to absorb animals’ wastes []. In spring, the waste-soaked litter would then be spread on the fields as a form of fertilizer. From 1750 throughout the 1800s litter demand increased []. The expanding railroad network, however, made straw more easily available for areas without grain production, and forest litter collection was abandoned towards the end of the 1800s and beginning of the 1900s.

In ORCHIDEE litter raking is optional. If litter raking is accounted for in the simulations, the maps of annual litter raking give an estimate of the amount of litter to be removed from each grid cell at the end of every year. If a grid cell does not have enough litter to cover the demand, all is removed but no litter is taken from surrounding grid cells. Subsequently the carbon and nitrogen contained in the raked litter is added to the litter pools of the agricultural PFTs on the same grid cell. The main result of litter raking is thus that forest carbon and nitrogen are diverted to croplands.

11.7. DONE: Products use and decay#

The product pool distinguished three types of biomass products: short-lived ($M^{prod,s}$) , medium-lived ($M^{prod,m}$), and long-lived products ($M^{prod,l}$) with a default longevity of 1 ($lshort)$, 17 ($lmedium$), and 50 ($llong$) years [], respectively. Once per year, at the end of the calendar year, the harvest pool is allocated to the three product pools according to one out of two approaches, i.e., a prescribed or dynamic allocation. Once the biomass is allocated to the different product pools, their decomposition follows the same approach irrespective of the allocation approach.

The prescribed approach allocates all harvested biomass from grassland and cropland PFTs to the short-lived products. The biomass harvested from forest is distributed over the short, medium and long-lived pools according to the parameters $c_1$ and $c_2$ which denote the fraction of forest harvest allocated to the short and medium-lived pool respectively:

(11.5)#\[\begin{split}\begin{align} &M^{prod,s}_{1} = M^{har,crop} + M^{har,grass} + c_1 \cdot M^{har,forest} \\ &M^{prod,m}_{1} = c_2 \cdot M^{har,forest} \\ &M^{prod,l}_{1} = (1 - c_1 - c_2) \cdot M^{har,forest} \end{align}\end{split}\]

The dynamic allocation moves all harvested biomass from grassland and cropland PFTs to the short-lived products as well as the biomass from the forest with a diameter of less than 0.2 m ($M^{har,forest,<0.2m}$).

(11.6)#\[\begin{split}\begin{align} &M^{prod,s}_{1} = M^{har,crop} + M^{har,grass} + M^{har,forest,<0.2m} \\ &M^{prod,m}_{1} = \frac{c_2}{c_2 + c_3} \cdot M^{har,forest,\ge0.2m} \\ &M^{prod,l}_{1} = \frac{c_3}{c_2 + c_3} \cdot M^{har,forest,\ge0.2m} \end{align}\end{split}\]

In both approaches, $M^{prod,s}_{t}$, $M^{prod,m}_{t}$, and $M^{prod,l}_{t}$ are expressed in gram per grid cell. This overcomes the need to store the harvest areas at the PFT and grid cell for the longevity of the long-lived products. $t$ refers to the number of years prior to the current year. For example, $M^{prod,l}_{40}$ would represent the remainder of the biomass added 40 years ago to the long-lived pool.

Both allocation approaches calculate the decomposition of the product pools and the remaining product pools in the same way. First, the annual decomposition rates of the harvested biomass added to this year’s age class ($t$=1) ($F^{prod,s}_{1}$, $F^{prod,m}_{1}$, and $F^{prod,l}_{1}$; g year$^{-1}$) are calculated:

(11.7)#\[\begin{split}\begin{align} &F^{prod,s}_{1} = \frac{M^{prod,s}_{1}}{lshort}\\ &F^{prod,m}_{1} = \frac{M^{prod,m}_{1}}{lmedium}\\ &F^{prod,l}_{1} = \frac{M^{prod,l}_{1}}{llong} \end{align}\end{split}\]

Given the crude assumptions made in the allocation of the woody biomass to the products pools and the global parametrization of the longevity of the product pools, their decomposition is kept simple by considering a linear decomposition rate although an exponential decomposition rate might be more realistic especially for the short-lived product pool []. This year’s decomposition rate, together with the decomposition rates calculated in the previous years are summed to obtain the total decomposition of each product pool ($F^{prod,s}$, $F^{prod,m}$, and $F^{prod,l}$; g year$^{-1}$). The number of previous years considered in this calculation depends on the longevity of the product pool:

(11.8)#\[\begin{split}\begin{align} &F^{prod,s} = \sum_{t=1}^{lshort}{F^{prod,s}_{t}}\\ &F^{prod,m} = \sum_{t=1}^{lmedium}{F^{prod,m}_{t}}\\ &F^{prod,l} = \sum_{t=1}^{llong}{F^{prod,l}_{t}} \end{align}\end{split}\]

$F^{prod,s}$, $F^{prod,m}$, and $F^{prod,l}$ represent the carbon and nitrogen that was once stored in a product pool but now returns back to the atmosphere due because part of the product pool reached the end of its live. Due to these emissions, the carbon and nitrogen that remains in the different age classes of each product pools is calculated as follows:

(11.9)#\[\begin{split}\begin{align} &M^{prod,s}_{t} = M^{prod,s}_{t} - F^{prod,s}_{t} \\ &M^{prod,m}_{t} = M^{prod,m}_{t} - F^{prod,m}_{t}\\ &M^{prod,l}_{t} = M^{prod,l}_{t} - F^{prod,l}_{t} \end{align}\end{split}\]

Owing to the assumption that the decomposition of the product pools is linear, each year, the following applies:

(11.10)#\[\begin{split}\begin{align} M^{prod,s}_{lshort} = F^{prod,s}_{1},\\ M^{prod,m}_{lmedium} = F^{prod,m}_{1},\\ M^{prod,l}_{llong} = F^{prod,l}_{1}, \end{align}\end{split}\]

Because of equalities (11.10), %s, and %s, $M^{prod,s}_{lshort}$, $M^{prod,m}_{lmedium}$, and $M^{prod,l}_{llong}$ are zero after applying equations (11.9) to %s. This implies that $F^{prod,s}_{lshort}$, $F^{prod,m}_{lmedium}$, and $M^{prod,l}_{llong}$ are no longer needed. Because the last age classes are now empty or no longer needed, the mass or fluxes contained in each age class is moved to the one year older age class which frees the first age class to receive next year’s harvest in $M^{prod,s}_{1}$, $M^{prod,m}_{1}$, and $M^{prod,l}_{1}$ and its annual decomposition in $F^{prod,s}_{1}$, $F^{prod,m}_{1}$, and $F^{prod,l}_{1}$.

12. OK: Disturbances#

12.1. DONE: Drought#

In ORCHIDEE one out of two approaches to calculate plant water stress needs to be chosen by the user: a soil-atmosphere or a soil-plant-atmosphere approach. If the soil-atmosphere approach (section ??) is selected, a soil water stress logistic function, using a soil water potential at wilting point $\theta^{w}$, is used as a proxy for plant water stress. The direct impact of soil water stress on the photosynthesis is accounted for by reducing the maximum carboxylation rate. A relationship between soil water and maximum carboxylation rate is not supported by observational evidence. If the soil-plant-atmosphere approach (section ??) is selected, the plant water stress is calculated, by explicitly calculating the potentials and resistances of the plant organs, and controls jointly photosynthesis and transpiration through its control on stomatal conductance, which is function of the leaf water potential. The relationship between soil water and stomatal conductance is supported by observational evidence []. Irrespective of the approach, ORCHIDEE only accounts for the direct effects of soil water stress on photosynthesis. Plant strategies and their associated carbon and nitrogen costs to avoid plant water stress, e.g., leaf shedding, or to cope with the impacts of plant water stress, e.g., cavitation, or legacy effects of droughts have not yet been implemented. At present only long sustained droughts result in vegetation mortality in ORCHIDEE through carbon starvation.

12.2. OK: Wind throw#

In ORCHIDEE the calculation of wind throw and subsequent tree mortality is optional. If wind throw is simulated, the model calculates the critical wind speed based on the principles applied in ForestGALES [], and storm damage based on the approach developed and tested by . The wind throw module is composed of several components. The first step is the detection of a storm (section ??). Once a storm is detected, the calculation of wind speed and gustiness provide proxies for potential damage (sections ?? and ??). Next, the susceptibility of the grid cell to wind damage is assessed by considering the soil and root characteristics, forest gaps, and forest edges (section ?? and ??). Finally, storm damage is calculated and the damaged biomass is move to either the litter or harvest pools depending on the forest management (sections ?? and ??).

12.2.1. OK: Storm detection#

The storm detection algorithm performs three tasks: (1) mimicking vegetation adaptation in areas frequently exposed to strong winds, (2) distinguishing between long-lasting storms and shorter gusts, and (3) linking events occurring at consecutive time steps of different days.

Particularly in coastal areas, high wind speeds persist. In response to frequent wind exposure, vegetation growing in these areas tend to grow adaptive capacities by adjusting the height and shape, reducing their vulnerable to strong wind conditions []. However, ORCHIDEE does not yet account for such adaptations (section ?? “Evolutionary assumption”). To address this issue, the ratio of the actual wind speed to the mean wind speed at each location is used instead of the actual wind speeds. By calculating the relative wind speed, generally high wind-speed grid cells require a much higher wind speed to experience damage compared to grid cells in areas with lower mean wind speeds, thus implicitly mimicking the effect of vegetation adaptation.

The duration of high relative wind speed is a defining characteristic of storminess. It is assumed that trees exposed to strong winds for extended periods are more likely to be uprooted or broken. To account for this effect, the relative half-hourly wind speed is summed over the course of the day (ORCHIDEE is driven by half-hourly wind fields) for every time step it exceeds its threshold. This approach yields a daily count between zero and 48 half hours exceeding the threshold wind speed ratio. This count is used to classify the daily wind conditions as a storm or not.

The model calculates wind damage at the last time step of the day, with the original module using the daily maximum wind speed to compute the damage []. However, if a storm event spans multiple days in the forcing, the model would repeatedly calculate wind damage, leading to an overestimation of the damage. This issue required a more dynamic approach to detecting the start and end of a storm event. ORCHIDEE now calculates the daily sum of the relative wind speed over three days. If the 3-day sum exceeds the threshold, and the daily maximum wind speed surpasses the threshold, the module recognizes the beginning of the storm.

Once the module detects the start of a storm, it waits for the daily maximum wind speed to drop below the threshold. Once the wind speed decreases, the module waits until it remains below the threshold for 5 days. If the wind speed exceeds the threshold within 5 days, the process of detecting the end of the storm restarts. If the wind speed does not exceed the storm threshold for 5 days after a storm is detected, the storm is considered over, and the damage caused by the storm is calculated. This approach results in a 5 day delay between the end of the storm and the moment the damage is accounted for.

12.2.2. OK: Critical wind speeds#

The presence or absence of storm damage in a forest stand is modeled using the concept of critical wind speed []. If the wind speed exceeds the critical wind speed of a forest, the forces applied may be sufficient to overturn the whole tree or break its stem. The exact value of the critical wind speed depends on the canopy structure [], the tree species, the soil properties and the root profiles []. The physics formalized in ForestGALES [], a hybrid mechanistic forest wind damage risk model, is included in ORCHIDEE This approach simulates the critical wind speeds of all forest stands for two types of damage: tree uprooting and stem breakage. The critical wind speed for uprooting is calculated as:

(12.1)#\[u^{cws,uproot} = \dfrac{1}{\kappa \cdot d^{D,trees}} \cdot \sqrt{ \frac{c_1 \cdot M^{stem,fresh}}{c_2 \cdot f^{gust} \cdot d^{h}_{0}} } \cdot \sqrt{ \frac{1} {f^{CW} \cdot f^{edge}} } \cdot ln\left( \dfrac{d^{h}-d^{h}_{0}}{z^{m}_{0}} \right)\]

Where $u^{cws,uproot}$ (m s$^{-1}$) is the critical wind speed for uprooting. $\kappa$ (unitless) is the von Karman constant [] and $d^{D,trees}$ (m) is the inter-tree spacing. $c_1$ (N m kg$^{-1}$) is a regression coefficient that was derived from tree pulling experiments [], where N stands for Newton. $M^{stem,fresh}$ (kg) is the green mass of the bole of the tree. $M^{stem,fresh}$ is calculated by multiplying the simulated above-ground biomass with green density for different tree species []. $c_2$ is air density (kg m$^{-3}$) and $f^{gust}$ is a unitless gust factor described in section ??. $f^{CW}$ (unitless) is the enhanced momentum caused by the overhanging displaced mass of the canopy. In ORCHIDEE $f^{CW}$ is set to 1.136, taken from the results of extensive tree-pulling data []. $f^{edge}$ is a unitless factor to account for the edge effect on gustiness (section ??). $d^{h}$ (m) is the tree height, $d^{h}_{0}$ (m) is the the displacement height, $z^{m}_{0}$ (m) is the roughness length (section ??). The critical wind speed for stem breakage is calculated as:

(12.2)#\[u^{cws,break} = \dfrac{1}{\kappa \cdot d^{D,trees}} \cdot \sqrt{ \frac{\pi \cdot c_4 \cdot {d^{dia}}^3 }{32 \cdot c_2 (d^{h}_{0}-1.3)} } \cdot \sqrt{ \frac{c_3} {f^{CW} \cdot f^{edge} } } \cdot ln\left( \dfrac{d^{h}-d^{h}_{0}}{z^{m}_{0}} \right)\]

Where $u^{cws,break}$ (m s$^{-1}$) is the critical wind speed for stem breakage, $c_4$ (Pa) is the PFT-specific modulus of rupture of green wood []. $d^{dia}$ (m) is the tree diameter at breast height as simulated by ORCHIDEE and $c_3$ is a unitless factor to reduce wood strength due to the presence of knots.

The roughness of the canopy for momentum and displacement height is calculated follow the analytical relationships proposed by []:

(12.3)#\[\begin{split}\begin{align} &z^{m}_{0} = \left( d^{h}-d^{h}_{0} \right) \cdot \exp(- \kappa \cdot b_{1} + c_5),\\ &b_{1} =\frac{1}{\sqrt{0.003+0.15 \cdot \frac{d^{cdia,hor} \cdot d^{cdia,ver}} {{d^{D,trees}}^{2}}} }, max \left(\frac{d^{cdia,hor} \cdot d^{cdia,ver}} {{d^{D,trees}}^{2}}\right),\\ &c_5 = ln(2)-1 + \frac{1}{2}, \end{align}\end{split}\]

where, $b_{1}$ (unitless) is dependent on canopy characteristics (equation %s) and $c_5$ (unitless) is the atmospheric stability correction function [] and $d^{h}_{0}$ is calculated as:

(12.4)#\[\begin{split}\begin{align} &d^{h}_{0} = \cdot \left( 1 -\frac{1-\exp{- \sqrt{ c_6 \cdot b_{2} \cdot \frac{d^{cdia,hor} \cdot d^{cdia,hor}} {{d^{D,trees}}^{2}} } } } { \sqrt{ c_6 \cdot b_{2} \cdot \frac{d^{cdia,hor} \cdot d^{cdia,ver}} {{d^{D,trees}}^{2}} } } \right),\\ &b_{2} = c_7 \cdot {u^{h}}^{-c_8} \end{align}\end{split}\]

Where $u^{h}$ (m s$^{-1}$) represents wind speed at the top of the canopy, $c_6$ (-) is a constant , $d^{cdia,hor}$ (m) is crown width and $d^{cdia,ver}$ (m) is crown depth. Tree crowns, branches and stems are considered as porous and flexible materials that will streamline and change their shape with changing wind speeds ($u^{h}$). Streamlining was parameterised through the parameters $c_7$ and $c_8$ (-), which are reported for wind tunnel experiments with different tree species []. $b_{2}$ represents a reduction in the drag coefficient from a reduction in canopy area due to streamlining its minimum values is set at 10 $m\,s^{-1}$ and the maximum at 25 $m\,s^{-1}$ where these limits are based on the wind speed range reported in . The species specific streamlining effect for a wind speed outside this range was calculated by holding $u^{h}$ constant, using the lower or upper threshold. $d^{h}_{0}$ and $z^{m}_{0}$ depend on the wind speed at canopy height hence iterations are required to solve equation (12.1) and (12.2).

Critical wind speeds are calculated as the solutions of a non-linear set of equations for uprooting, i.e., equations (12.1), (12.4) and (12.3), and another set of equations for stem breakage, i.e., equations (12.2), (12.4) and (12.3). An initial wind speed ($u^{h}$) of 25 $m\,s^{-1}$, is applied to equation (12.4) and equation (12.3) to obtain an approximation for $u^{cws,uproot}$ by applying equation (12.1). Similarly, an initial wind speed ($u^{h}$) of 25 $m\,s^{-1}$, is applied to equation (12.4) and equation (12.3) to obtain an approximation for $u^{cws,break}$ by equation (12.2). Subsequently, $u^{h}$ is set to the value of $u^{cws,uproot}$ (or $u^{cws,break}$) to estimate the aerodynamic parameters ($d^{h}_{0}$ and $z^{m}_{0}$) for the next iteration. The iteration process is stopped if the difference in $u^{cws}$ between two iteration falls below 0.01 $m\,s^{-1}$ or the number of iterations exceeds 20.

Whereas ORCHIDEE is designed to simulate both even-aged and uneven-aged stands (see section ??), the original model [] is limited to simulating the critical wind speeds for even-aged forests. Although this design difference is expected to have minimal impact, it becomes important in the calculation of the ratio between tree height and tree spacing (known as inter-tree spacing, $d_{D,trees}$). In even-aged stands, both tree height and tree spacing are uniform and well-defined at the stand level, whereas in uneven-aged stands, tree height is no longer uniform. In ORCHIDEE , the largest diameter class represents the dominant trees that form the canopy and hold the majority of the stand’s biomass. Therefore, the critical wind speed is calculated only for this dominant diameter class. To calculate the inter-tree spacing for this class, the total woody biomass at the stand level is calculated. This total biomass is then divided by the biomass of the modeled trees in the largest diameter class. The resulting value is used as the virtual inter-tree spacing, $d_{D,tree}$, in equation(12.1) and equation(12.2) to calculate the critical wind speeds.

ORCHIDEE calculates the critical wind speed for both breakage and uprooting, based on the vegetation structure parameters. Two critical wind speeds are computed for each forest in each grid cell, and the lower of the two is used to determine the type of damage to the PFTs. Biomass loss is then calculated by multiplying the damage rate ($f^{wind,damage}$; see section ??) by the total biomass of the PFTs, with the biomass loss accounted for from the largest to the smallest diameter class. The number of damaged trees is calculated based on the biomass loss rate.

12.2.3. DONE:Gustiness and edge effect#

ORCHIDEE is driven by half-hourly wind fields. Such a time step averages out the extreme wind gusts that occur within a half-hour. Because storm damage is primarily influenced by extreme wind gusts rather than the average wind speed, this scaling issue is addressed by considering gustiness. Unlike the original module from , ORCHIDEE uses a simple modifier to account for gustiness ($m_{G}$):

(12.5)#\[u^{max} = m_{G} \cdot max(u^{h})\]

$max(u^{h})$ (m s$^{-1})$) represents the maximum wind speed over the last three days (see section ??), and $u^{max}$ (m s$^{-1})$ is the extreme wind gust that is compared with the critical wind speed to calculate damage.

The edge effect is considered at the landscape level. Within each grid cell, forests are separated into two regions, i.e., the inner area and the outer area. For the area bordering the gap (see section ??), the effect of vegetation structure on wind speed is accounted for using the edge factor $f^{edge}$ in equation(12.1) and equation(12.2). The calculation of $f^{edge}$ follows the approach proposed by :

(12.6)#\[f^{edge} = \dfrac{ (2.7193(\frac{d^{D,trees}}{d^{h}})-0.061) + (-1.273 (\frac{d^{D,trees}}{d^{h}})+0.9701) \cdot (1.1127(\frac{d^{D,trees}}{d^{h}})+0.0311^{\frac{d^{edge}}{d^{h}}}) }{ (0.68(\frac{d^{D,trees}}{d^{h}})-0.0385) + (-0.68 (\frac{d^{D,trees}}{d^{h}})+0.4785) \cdot (1.7239(\frac{d^{D,trees}}{d^{h}})+0.0316^{\frac{d^{edge}}{h}})}\]

where $d^{edge}$ represents the distance of the forest to the nearest forest edge. For the forest area away from the gap, the edge effect is negligible such that $f^{edge}$ is set to 1.0.

12.2.4. DONE: Soil characteristics#

Although ORCHIDEE distinguishes 13 soil classes (see section ??), the current approach to simulating soil water hydrology, assumes all soils are free-draining at the bottom of the soil layers. This differs from the original model [], which distinguishes four soil classes with varying drainage properties: freely draining mineral soil, gleyed waterlogged and oxygen-deficient mineral soil, peaty mineral soil, and deep peat []. Currently, ORCHIDEE only uses parameters for freely draining mineral soils. As a result, ORCHIDEE is expected to overestimate the critical wind speed and thus underestimate damage in areas with shallow or wet soils.

Additionally, the original model formulation in ForestGALES [] differentiates between shallow, medium, and deep rooting species. This classification is applied in ORCHIDEE via the parameter that describes the vertical root profile. The storm damage calculation make use of the structural root profile (see section ??) that follows a truncated exponential decay from the top to the bottom of the soil layers, independent of site conditions or stand age. PFTs are considered shallow-rooted if 90 % of their total root mass is above a depth of 2 m. Under the current settings, the wind throw module assumes that all PFTs are shallow-rooted. The effect of rooting depth on critical wind speeds is accounted for by using different regression coefficients, $c_1$, for shallow and deep-rooting species. When the soil is frozen, the model only simulates windstorm damage from stem breakage, meaning no tree uprooting is possible. The soil is deemed frozen based on the temperature at 0.8 m below the surface, which serves as the threshold.

12.2.5. DONE: Forest gaps#

Vegetation structure is simulated at both the landscape level and the stand level. At the landscape level the simulations distinguish between forests with a newly formed forest edge and forest with established edges. Edges result from natural or anthropogenic stand replacing disturbances. First, the surface area of stand replacing disturbances is accumulated over the last 5 years ($S^{cut,5year}$ ($m^2$)), a time horizon corresponding to the time required for forests nearby newly formed edge to adapt to the increased gustiness []. By prescribing the average gap size ($c_9$; $m^2$) to 2 ha (20000 m$^{2}$), and assuming gaps are square shaped and the gustiness is affected over a distance of 9 times the canopy height ($d^{h}$) [], the forest area that experiences an increased gustiness due to proximity of recent gaps ($S^{border}$ ($m^2$)) is calculated as:

(12.7)#\[S^{border} = \frac{1}{4} \cdot \left(\left(\sqrt{c_9} + 2 \cdot 9 \cdot d^{h} \right)^2 - c_9 \right) \cdot \left( \frac{S^{cut,5year}}{c_9} \right)\]

Where the factor of $\frac{1}{4}$ accounts for the fact that only the downwind edge perpendicular to the wind will experience an increased gustiness. The second term is the area bordering a single gap and the third term is the total number of gaps in the grid cell. The forest area that has no edges in its proximity ($S_{away}$; $m^2$) is calculated as the residual:

(12.8)#\[\begin{split}S_{away} = \begin{cases} S^{gridcell} - \left( S^{cut,5year} + S^{border} \right) \textrm{, when } S^{cut,5year} + S^{border} < S^{gridcell} \\ 0 \textrm{ and } S^{border} = S^{gridcell} \textrm{, when } S^{cut,5year} + S^{border} \geq S^{gridcell} \\ \end{cases}\end{split}\]

Where $S^{gridcell}$ (m$^{-2}$) is the area of the simulated grid cell.

12.2.6. DONE: Storm damage #

When wind speeds approach the critical wind speed, damage such as defoliation and branch damage become more likely. Once the wind speed exceeds the critical wind speed, uprooting and stem breakage are possible but their likelihood increases with further increasing wind speeds. A sigmoid damage function is applied to simulate the rate of storm damage to a forest, which was proposed and tested by for estimating storm damage as a function of the daily maximum wind speed. This relationship is formalized as:

(12.9)#\[f^{wind,damage} = f^{wind,damage,max} \left(\frac{1}{1+\exp{(-\frac{u^{max} - u^{cws}}{c_{10}})}} - \frac{1}{1+\exp{(\frac{u^{cws}}{c_{10}})}}\right)\]

Where $f^{wind,damage}$ (unitless) is the damage rate and thus the share of trees that will be killed, $f^{wind,damage,max}$ (unitless) is an observed maximum damage rate which was set to 0.8 but scales to the size of an individual grid cell in ORCHIDEE. Where a damage rate of 0.8 is likely at the hectare scale, it is unlikely for a 2 ° pixel. $c_{10}$ is a relaxation parameter to adjust the damage rate given by a certain wind speed below the model calculated critical wind speed, and a value 6.0 was applied for all PFTs. $u^{max}$ is the maximum daily wind speed from the forcing or the atmospheric model. Subsequently, the lowest out of the six calculated critical wind speeds (see section ??), is used to determine the damage type for each diameter class. The number of damaged trees in each diameter class ($d^{ind,kill}_{l}$) is then calculated by multiplying the damage rate ($f^{wind,damage}$) with the tree numbers within each diameter class. The total stand density damaged by a storm is, therefore, the number of damaged trees per unit of ground area summed across each diameter class.

12.2.7. DONE: Moving biomass to litter and harvest pools#

Damaged trees due to storms are left on site in unmanaged forests, however, salvage logging is often carried out for a managed forest in order to recover some of the economic losses and avoid large scale insects outbreaks triggered by wind disturbance []. When dealing with the effects of wind damage on the biomass pools of forests, the subsequent anthropogenic response is accounted for. ORCHIDEE distinguishes managed and unmanaged forest (see section ??). In unmanaged forests all carbon contained in trees killed by wind storms end up in the litter pools following equations (10.67) and %s.

For managed forests, salvage logging is implemented according to equations (11.2) to %s where $c_{10}$, i.e., the branch ratio is replaced by $c_{11}$ which prescribes the ratio of the stem that is salvaged logged. During salvage logging not only the branches are left on site but also part of the stems is left because they no longer have an economic value due to stem breakage [], hence $c_{11}$ is lower than $c_{10}$.

In species that are prone to bark beetle attacks following wind throw (see section ??), the volume left on site is very small. In Sweden, a maximum volume of 5 $m^{3}\,ha^{-1}$ newly damaged logs is allowed []. However, following large-scale storm damage this threshold has been temporarily lowered to 3 $m^{3}\,ha^{-1}$ in order to reduce the risk of spruce bark beetle outbreaks []. Given that the current implementation of storm damage was designed to deal with large wind storms, with a fair risk for subsequent bark beetle outbreaks [], applying a very high efficiency for salvage logging, i.e., 99 %, appears justified .

12.3. DONE: Bark beetles#

12.3.1. DONE: Life cycle of a bark beetle outbreak#

In ORCHIDEE the calculation of bark beetle outbreaks and subsequent tree mortality is optional. If bark beetle disturbances are calculated, the life cycle of an outbreak includes the following stages (Fig. 15):

a) the endemic stage at which the forest stand experiences low bark beetle pressure enabling the forest to maintain a pseudo-climax or climax depending on whether the stand is managed or not (shown as stage 1).
b) The build-up stage is characterised by a rapid increase in the bark beetle population due to an event that weakened part of the trees but without visible impact on healthy trees (stage 1 & 2). Although in reality wind storms and drought are typical events to initiate the start of the build-up stage, ORCHIDEE only considers wind storms as a possible trigger for an increase in the beetle population.
c) During the epidemic stage bark beetles are so numerous that they can successfully attack healthy trees causing a change in leaf colour (stage 2 & 3).
d) In the post-epidemic stage a significant reduction in the bark beetle population occurs due to a lack of substrate for feeding and breeding (stage 3 & 4). Stage 4 : In the gray stage infected trees that retain their leaves and remain standing, gradually die turning into so-called snags. Stage 5 : in the ecological transition stage degradation from wind throws and bark beetles result in openings in the canopy reducing-between tree competitions. In Stage 6 bark beetles return to their initial population level resulting in a new endemic stage during which recruitment may help the forest to reach a (pseudo-)climax stage.

:::{figure} Figures/Life_cycle_bark_beetles.png :name: fig:barkbeetle:lifecycle :align: center

Life cycle of a bark beetle outbreak and subsequent dynamics of a forest stand :::

12.3.2. DONE: Implicit representation of bark beetle populations#

The bark beetle breeding indicator of the current year ($i^{beetle,generation}$; unitless) is calculated from a logistic function, which depends on the number of generations a bark beetle population can sustain within a single year:

(12.10)#\[i^{beetle,generation} = \frac{1}{1 + \exp\left(-c_1 \left( \frac{T^{eff,gdd}}{T^{ref,gdd}} - c_2 \right)\right)}\]

Where ($c_1$) and ($c_2$) are tuning parameters for the logistic function, $T^{eff,gdd}$ represents the sum of effective temperature for bark beetle reproduction in ° C day$^{-1}$, while $T^{ref,gdd}$ denotes the thermal sum of degree days for one bark beetle generation in ° C day$^{-1}$. Saturation of $i^{beetle,generation}$ represents the lack of available breeding substrate when many generations develop over a short period.

Bark beetles, which tend to breed earlier when winter and spring were warmer, can produce multiple generations in the same calendar year []. This pattern is accounted for in the calculation of $T^{eff,gdd}$, which is calculated from January 1$^{\text{st}}$ until the first generation’s diapause, triggered when day length exceeds 14.5 h (e.g., April 27th for France). Each day prior to this diapause with a daily average temperature around the bark above 8.3 degree C ($T^{min}$) and below 38.4°C ($T^{max}$) is accounted for in the summation of $T^{eff,gdd}$:

(12.11)#\[T^{eff,gdd} = \sum_{i=1}^{ndia} \left( c_3 - c_4 \right) \cdot \left( \exp\left(c_5 \cdot T^{bark}_{t}\right) - \exp\left(c_5 \cdot c_6 - \frac{c_6 - T^{bark}_{t}}{c_7} \right)\right) - c_8\]

Where $t$ is the day, $ndia$ is the number of days between January 1st and the day of the diapause. The optimal bark temperature for beetle development $c_3$ is set at 30.3° C and $c_4$ is the bark temperature below which beetle development stops. $T^{bark}_{t}$ is the average daily bark temperature, calculated as the daily average air temperature minus 2° C. All parameter values are taken from [].

The bark beetle pressure indicator $i^{beetle,pressure}$ (unitless) is formulated based on two components: (1) the $i^{beetle,generation}$ and (2) an indicator of the loss of tree biomass in the previous year due to bark beetle infestation ($i^{beetle,activity}$; unitless) which is thus a proxy of the previous year’s bark beetle activity. The expression accounts for the legacy effect of bark beetle activities by averaging activities over the current and previous years. In this approach, the susceptibility indicator ($i^{beetle,survival}$; unitless) serves as an indicator for increased bark beetle survival which could result from favorable conditions for beetle demography (see next section).

(12.12)#\[i^{beetle,pressure} = i^{beetle,survival} \cdot \left( i^{beetle,generation} + i^{beetle,activity} \right)/2\]

The bark beetle activity of the previous year ($i^{beetle,activity}$; unitless) is calculated as:

(12.13)#\[i^{beetle,activity} = \frac{1}{1 + exp\left(-c_9 \cdot \left( \frac{M^{kill}_{t-1}}{M^{total}} - c_{10} \right)\right)}\]

Where $M^{kill}_{t-1}$ (g m$^{-2}$) is the biomass killed by bark beetles during previous year, $M^{total}$ (g m$^{-2}$) is the total biomass of the stand, and $c_9$ and $c_{10}$ are parameters that drive the intensity of this negative feedback.

During the build-up of the epidemic i.e stage in which bark beetles population is high enough to mass attack healthy tree, the population of bark beetles can either return to its endemic i.e. stage in which beetles population is at the lowest, if tree defense mechanisms are preventing bark beetles from successfully attacking healthy trees, or evolve into an epidemic stage, if the tree defense mechanisms fail. During the post-epidemic stage, the forest is still subject to higher mortality than usual, but signs of recovery appear []. Recovery may help the forest ecosystem to return to its original state or switch to a new state with different species or a change in forest structure depending on the intensity and frequency of the disturbance []. ORCHIDEE was shown of being capable of simulating a change in forest structure [] but cannot simulate a species change.

12.3.3. DONE: Bark beetle survival#

The capability of bark beetles to survive the winter in between two breeding seasons is critical in simulating epidemic outbreaks. During regular winters, winter mortality for bark beetles is around 40 % for the adults and 100 % for the juveniles []. In our representation, this mortality rate is implicitly accounted for in the calculation of the bark beetle survival indicator ($i^{beetle,survival}$; unitless). A lack of data linking bark beetle survival to anomalous winter temperatures justifies an implicit approach and prevented including this information as a modulator of $i^{beetle,survival}$. The latter explains why winter temperatures do not appear in equation (12.11). Instead, the model simulates the survival as a function of the abundance of suitable tree hosts, which decreases the competition for shelter and food:

(12.14)#\[i^{beetle,survival} = \max\left(i^{host,dead}, i^{host,alive} \right)\]

The availability of woody necromass from trees that died recently, particularly following windstorms, plays a critical role in bark beetle survival and proliferation. In the year following a windstorm, uprooted and broken trees may offer an ideal breeding substrate for bark beetles, facilitating their population growth. In , an empirical correlation between windthrow events and bark beetle susceptibility was used. ORCHIDEE enhances realism by considering the actual suitable hosts, i.e., living or recently died trees, as the primary driver of bark beetle survival. To avoid overestimating bark beetle population growth, the parameter $c_{11}$ (unitless) which corresponds to the maximum fraction of the living biomass that could be killed by bark beetles, is introduced. Any addition of dead trees beyond $c_{11}$ is considered ineffective in affecting the bark beetle population. This ensures that an excess of breeding substrate does not artificially inflate beetle numbers. This relationship is quantitatively represented in ORCHIDEE through the dead host indicator, $i^{host,dead}$, which is driven by the availability of recent dead trees. The formulation of $i^{host,dead}$ is as follows:

(12.15)#\[i^{host,dead} = \min \left( \frac{M^{dead,wood} / M^{wood}}{c_{11}}, 1 \right)\]

where, $M^{dead,wood}$ (g m$^{-2}$) represents the quantity of woody necromass from the current year, $M^{wood}$ (g m$^{-2}$) is the total living woody biomass in the stand, and $c_{11}$ is the threshold at which the ratio $M^{dead,wood}/M^{wood}$ is at the maximum level.

This indicator captures the immediate increase in dead trees suitable for bark beetle breeding following a windthrow event. However, Dead wood is most suitable for bark beetle breeding within the first year after tree death. During this period, the wood retains higher moisture content and nutritional quality, which are critical for beetle reproduction. This is accounted for by excluding woody necromass that is older than 1 year from the calculation of $i^{host,dead}$.

The parameter $c_{12}$ is a scale dependent parameter. The mortality rate of trees that will trigger an outbreak is very different across spatial scales. Where a relatively high share of dead wood is needed to trigger an outbreak at the patch-scale, a much lower share of dead wood suffices at the landscape-scale to trigger a widespread bark beetle outbreak. So these parameters must be set according to the spatial resolution of the simulation experiment. Currently, the $c_{11}$ parameter is fixed to 0.2 based on the evaluation of large European simulation scale.

The alive host indicator $i^{host,alive}$ is driven by two factors: (1) the abundance of weak trees, which can be more easily infected by bark beetles, and (2) the ability of bark beetles to attack healthy trees. $i^{host,alive}$ denotes the survival of bark beetles, which is facilitated by the abundance of suitable trees, reducing the competition among bark beetles for breeding substrates and therefore increasing their survival.

(12.16)#\[i^{host,alive} = i^{beetles,massattack} \cdot i^{host,susceptibility}\]

The indicator $i^{beetles,massattack}$ represents the ability of bark beetles to attack healthy trees when the number of bark beetles is large enough. This indicator only depends on the size of the bark beetle population $i^{beetle,pressure}$ (equation (12.12)):

(12.17)#\[i^{beetles,massattack} = \frac{1}{1 + \exp\left(c_{12} \cdot i^{beetle,pressure} - c_{13} \right)}\]

$c_{12}$ controls the steepness of the relationship, while $c_{13}$ is the bark beetle pressure indicator at which the population is moving from endemic to epidemic stage where mass attacks are possible. The epidemic stage corresponds to the capability of bark beetles to mass attack healthy trees and overrule tree defenses []. At this point in the outbreak, all trees are potential targets irrespective of their health. Three causes have been suggested to explain the end of the epidemic phase: (1) the most likely cause is high interspecific competition among beetles for tree hosts when the density is decreasing [], resulting in a decreasing $i^{host,alive}$ in ORCHIDEE; (2) a series of (very) cold years will decrease the ability of the beetles to reproduce which is simulated in ORCHIDEE through a decreasing $i^{beetle,generation}$; and (3) a rarely demonstrated increasing population of beetle predators []. In ORCHIDEE the first two causes are represented but the last, i.e., the predators, is not.

ORCHIDEE does not explicitly represent weak trees, but tree health is thought to decrease with increasing stand density given the stand diameter. The indicator for host suitability is thus calculated by making use of the relative density index $f^{RDI}$:

(12.18)#\[i^{host,susceptibility} = \frac{1}{1 + \exp\left(c_{14} \cdot \left( f^{RDI} - c_{15} \right)\right)}\]

$c_{14}$ represents the steepness of the relationship, and $c_{15}$ is a parameter for the relationship between stand rdi and beetle susceptibility. $f^{RDI}$ is used to estimate the average competition between trees at the stand level. At an $f^{RDI}$ (unitless) of 1, the forest is expected to be at its maximum density given the carrying capacity of the site, implying the highest level of competition between trees. The calculation of $f^{RDI}$ follows equation (10.55) but is limited to the spruce PFTs:

(12.19)#\[\begin{split}\begin{align} &f^{RDI} = \sum_{a=1}^{nage} \frac{d^{ind,spruce}_{a}}{d^{ind,max,spruce}} \cdot \frac{f^{spruce}_{a}}{f^{spruce}}\\ &f^{RDI}_{a} = \frac{d^{ind}_{a}}{d^{ind,max}} \end{align}\end{split}\]

Where $d^{ind,spruce}_{a}$ (unitless) is the current tree density of an age class $a$, $d^{ind,max,spruce}$ (unitless) represents the maximum stand density of a stand given its diameter (see equations (10.52) and %s), $f^{spruce}_{a}$ (unitless) is the fraction of spruce in the grid cell that resides in this age class and $f^{spruce}$ is the fraction of spruce within a grid cell. The number of age classes ($nage$) within a vegetation meta class (MTC) is four when demographic succesion is activated (section ??).

12.3.4. DONE: Host attractivity#

The stand attractivity indicator ($i^{host,attractivity}$) represents how interesting a stand is for a new bark beetle colony and is calculated as follow: . When $i^{host,attractivity}$ tends to 0, the stand is constituted mainly by healthy trees which are less attractive for beetles, whereas an $i^{host,attractivity}$ approaching 1 represents a highly stressed spruce stand suitable for colonization by bark beetles.

(12.20)#\[i^{host,attractivity} = \max\left(i^{host,competition}, i^{host,defense}\right) \cdot i^{host,share}\]

Where $i^{host,competition}$ (unitless) indicates the stress the trees experience due to within-stand resource competition and $i^{host,defense}$ (unitless) represents the ability of the host to set up a defense against a bark beetle attack. Factors that contribute to the stress of a forest and its ability to guard off beetle attacks include nitrogen availability, limited carbohydrate reserves, age structure, and tree species diversity. Trees experiencing extended periods of environmental stress are expected to have fewer carbon and nitrogen reserves available for defense compounds, making them vulnerable to bark beetle attacks even at relatively low beetle population densities []. Nonetheless, the reserve pools in ORCHIDEE have not yet been evaluated, so the $i^{host,defense}$ and $i^{host,competition}$ is calculated by using proxies such as 1-year drought modulator ($m^{water,1year}$; unitless) and a relative density index ($f^{RDI}$):

(12.21)#\[\begin{split}\begin{align} &i^{host,defense} = \frac{1}{1 + \exp\left(c_{16} \cdot (1 - m^{water,1year}\cdot \frac{f^{spruce}_{a}}{f^{spruce}}- c_{17}\right)},\\ \end{align}\end{split}\]

Where $m^{water,1year}_{t}$ is the average daily plant water stress indicator over the growing season for the spruce stand. When $m^{water,1year}_{t}$ is equal to 0 or 1, plants are highly stressed or not stressed, respectively. $c_{16}$ represents the steepness of the relationship, and $c_{17}$ is the plant water stress below which the health of the stand will be affected.

In addition to drought, overstocking may also decrease the overall healthiness of a spruce stand ($i^{host,competition}$).

(12.22)#\[i^{host,competition} = \frac{1}{1 + \exp\left(c_{18} \cdot (f^{RDI} - c_{19}) \right)}\]

Where $c_{18}$ represents the steepness of the relationship, and $c_{19}$ represents the limit at which the bark beetle outbreak starts to decline because of a lack of suitable host trees. The severity of bark beetle-caused tree mortality decreases when going from the stand to the landscape scale. At the landscape scale, covering areas of $\sim$ 10$^{3}$ to $\sim$ 10$^{4}$ km$^{2}$, the duration of mortality may be longer and the severity lower because beetles disperse across the landscape and cause mortality at different times. This distinction is important for interpreting model results, particularly when considering parameters like $c_{19}$ in the ORCHIDEE model. $c_{19}$ describes the proportion of trees surviving after an outbreak and should therefore be adjusted for the spatial scale of a grid cell in ORCHIDEE. In a model set-up where a grid cell represents a single stand ($\sim$ 1 ha), $c_4$ should be close to 0, indicating that nearly all trees may be killed. However, in a simulation with grid cells representing 2500 km$^{2}$, not all trees will be killed, which is reflected in setting $c_{19}$, in this example, to 0.4. Equation (12.22) is close to %s, but the parameter $c_{18}$ is reduced by a factor of two in order to reflect that $i^{host,susceptibility}$ is more sensitive to $f^{RDI}$ than $i^{host,competition}$.

The indicator $i^{host,share}$ (used in equation (12.20)) takes into account that in a mixed tree species landscape, even a few non-host trees may chemically hinder bark beetles in finding their host trees [], explaining why insect pests, including Ips typographus outbreaks, often cause more damage in pure compared to mixed stands []. ORCHIDEE does not simulate multi-species stands but does account for landscape-level heterogeneity of forests with different PFTs. The bark beetle model in ORCHIDEE assumes that within a grid cell, the fraction of spruce over other tree species is a proxy for the degree of mixture:

(12.23)#\[\begin{split}\begin{align} &i^{host,share} = \frac{1}{1 + \exp\left(c_{20} \cdot \left( f_{host} - c_{21} \right)\right)},\\ &f_{host} = \frac{f^{deciduous}}{f^{deciduous}+f^{coniferous}}, \end{align}\end{split}\]

where $c_{20}$ represents the steepness of the relationship, and $c_{21}$ are prescribed parameters TO EXPLAIN

12.3.5. DONE: Bark beetle damage#

The biomass of trees killed by bark beetles in one year and one grid cell ($M^{beetle,kill}$; g m$^{-2}$) is calculated as the product of the probability of a successful attack ($f^{success}_{l}$; see next section) averaged over the number of spruce age classes $l$, the biomass of trees attacked by bark beetles ($M^{beetle,attack}$; g m$^{-2}$) and weighted by their fraction $\left(f^{spruce}_{a}/f^{spruce}\right)$. The approach assumes that a successful beetle colonization results in the death of the attacked trees, which is a simplification from reality .

(12.24)#\[M^{beetle,kill} = \sum_{l=1}^{nage} f^{success}_{l} \cdot M^{beetle,attack} \cdot \frac{f^{spruce}_{a}}{f^{spruce}}\]

During the endemic stage, the biomass of attacked trees ($M^{beetle,attack}$) and killed trees ($M^{beetle,kill}$) are at their lowest level and the damage from bark beetles has little impact on the structure and functioning of the forest. Biomass losses from bark beetles can be considered to contribute to the background mortality.

$M^{beetle,attack}$ is the outcome of bark beetles that successfully overcame the tree defenses and succeeded in boring holes in the bark in order to reach the sapwood. $M^{beetle,attack}$ is calculated at the grid cell by multiplying the actual stand biomass of spruce ($M^{total}$) and the probability that bark beetles attack spruce trees in the grid cell ($f^{attack}$; unitless).

(12.25)#\[M^{beetle,attack} = M^{total} \cdot f^{attack}\]

$f^{attack}$ represents the ability of the bark beetles to spread and locate new suitable spruce trees as hosts for breeding. $f^{attack}$ is calculated by the product of two indicators: (1) the stand attractivity indicator ($i^{host,attractivity}$; see equation (12.20)) which reflects the ability of the forest to resist an external stressor such as bark beetle attacks and (2) the beetle pressure indicator ($i^{beetle,pressure}$; see equation (12.12)) which is a proxy for the bark beetle population.

(12.26)#\[f^{attack} = i^{host,attractivity} \cdot i^{beetle,pressure}\]

12.3.6. DONE: Tree mortality from bark beetle infestation#

When bark beetles attack a tree, the success of their attack will depend on the capability of the tree to defend itself from the attack. Trees defend themselves against beetle attacks by producing secondary metabolites []. The high carbon and nitrogen costs of these compounds limit their production to periods with environmental conditions favorable for growth []. The probability of a successful bark beetle attack is driven by the size of the bark beetle population ($i^{beetle,pressure}$) and the health of each tree. ORCHIDEE, however, is not simulating individual trees but rather circumference classes within an age class. An indicator of tree health for each age class ($i^{host,health}_{a}$; unitless) was calculated as:

(12.27)#\[f^{success}_{l} = (i^{host,health}_{a} + i^{beetle,pressure})/2\]

Except during mass attacks, trees rarely die solely from bark beetle damage as female beetles often carry blue-stain fungi, which colonize the phloem and sapwood, blocking the water-conducting vessels of the tree []. This results in tree death from carbon starvation or desiccation. As ORCHIDEE does not simulate the effects of changes in sapwood conductivity on photosynthesis and the resultant probability of tree mortality, the indicator of weakened trees ($i^{host,health}_{a}$) makes use of two proxies adjusted to be calculated only for one age class at a time:

(12.28)#\[\begin{split}\begin{align} &i^{host,health}_{a} = \left( i^{host,competition}_{a} + i^{host,defense}_{a} \right)/2,\\ &i^{host,competition}_{a} = \frac{1}{1 + \exp \left(c_{18} \cdot \left( f^{RDI}_{a} - c_{19} \right) \right)},\\ &i^{host,defense}_{a} = \frac{1}{1 + \exp\left(c_{16} \cdot \left( 1 - m^{water}_{t,a} - c_{17} \right)\right)} \end{align}\end{split}\]

To assess the bark beetle damage rate ($f_{damage,beetles}$; unitless), $M^{beetle,kill}$ is divided by $M^{total}$. The number of dead trees from bark beetles attack $d^{ind,kill}_{l}$ is updated from $M^{beetle,kill}$ where only trees with a diameter above 0.07 m can be kill.

12.3.7. DONE: Moving biomass to litter and harvest pools#

Trees attacked by bark beetles are left on site in unmanaged forests, however, salvage logging is common in managed forest in order to recover some of the economic losses REF. When dealing with the effects of bark beetle outbreaks on the forest biomass, the anthropogenic response to is accounted for. ORCHIDEE distinguishes managed (section ??) and unmanaged forest (section ??). In unmanaged forests all carbon contained in trees killed by bark beetles ends up in the litter pools following equations (10.67) and %s. For managed forests, salvage logging is implemented according to equations (11.2) to %s.

:::{figure} Figures/Bark_beetles_flow_chart.png :name: fig:barkbeetle:flow :align: center

Is this useful? It is already outdated and it will be a pain to keep this up-to-date? A reference to Marie et al might be enough? If useful, this figure needs to be cited in the text Order of the calculations. The dotted line boxes represent the five main processes of the outbreak model as described in sections ?? to ??. The numbers correspond to the equation numbers in []. The variable names are listed in Table 1. :::

12.4. DONE Fire#

In ORCHIDEE the simulation of fires and subsequent tdiree and grass mortality is optional. If fires are simulated, the approach in ORCHIDEE is based on the SPITFIRE (SPread and InTensity of FIRE) model, initially developed by and integrated into an earlier version of ORCHIDEE []. Only key simulation processes are described below. For detailed information, please refer to and .

Unlike models that simulate the native physical and chemical processes of vegetation combustion, such as chemical reactions during the combustion process, the associated energy processes during combustion, including radiation, and conduction and convection heating of unburned fuel, the SPITFIRE simulates surface fires through an empirical prognostic approach: (1) the model assumes a uniformly mixed fuel bed by integrating fuel characteristics of forest and grassland, rather than simulating burning process for each PFT (or forest age class) individually. Hence, no change in fire behavior is accounted for when fire front passes the boarder of different vegetation types. (2) Multi-day burning events are not explicitly modeled. Instead, SPITFIRE operates on a daily time step, with fires being started and extinguished over the same day. Fire weather and fire behavior are mostly determined by daily mean values of meteorological variables including wind speed, and hence any diurnal dynamics in fire behavior are ignored. (3) The daily burned area within a model grid cell is calculated as the product of fire number and fire size, assuming that all fires are independent burning events sharing the same size as determined by the steps of (1) and (2). The fire size is determined by fire spread rate and fire duration, assuming that fires start and extinguish within the same day. (4) The simulation of fire spread is based on an empirical surface fire spread model. Crown fire spread processes are not considered. (5) The surface fire spread model primarily applies to natural burning processes. However, because it is often implemented on a coarse grid, the effects of fire barriers—such as firebreaks, rivers, or bare soil separating cropland—are largely neglected in the simulation of fire spread. (6) For this reason, cropland burning is currently not included. (7) As the simulation of fire spread and resulting fire size is based exclusively on active burning, the associated estimates of carbon and trace gas emissions account only for emissions from active combustion; smoldering-phase emissions are mostly ignored.

The biogeochemical and biogeophysical impacts of fires are accounted for by simulating fire-induced forest and grassland mortality as well as the resulting carbon emissions. Part of dead trees and grasslands, and surface litter, are combusted directly by fire causing carbon and trace gas emissions from land to the atmosphere. Un-combusted biomass is transferred to litter. Following fire-induced mortality, it is mostly assumed that the pre-fire PFT will establish as explained in the section ??

12.4.1. DONE: Fire weather and surface fuel moisture#

Surface fuel, namely above-ground litter, is categorized into different classes based on the scale of time needed to reach a moisture level in equilibrium with the ambient environment: 1-hour (1h), 10-hour (10h), 100-hour (100h), and 1000-hour (1000h). The first three classes are ‘surface fine fuel’ whose moisture status determines fire likelihood and the resulting fire behaviour. Surface fuel moisture is determined through a prognostic relationship using meteorological variables. The Nesterov Index (NI) (unit: ℃²) is a meteorological dryness indicator calculated as (Thonicke et al. 2010):

\[NI = \sum T^{\max} \cdot \left(T^{\max} - T^{\mathrm{dew}}\right)\]

where $T^{\mathrm{dew}}$ is the dew point temperature, and $T^{\max}$ is the daily maximum temperature. The summation is accumulated for the period when daily precipitation remains lower than 3mm. The moisture content of surface fine fuel ($\omega^{\mathrm{o}}$, unitless, value range : 0–1) then depends on fuel composition and meteorological dryness (Thonicke et al. 2010):

\[\omega^{\mathrm{o}} = \exp\left(-\left(\sum_{i=1}^{3} \alpha_{i} \cdot \frac{{w^{\mathrm{o}}_{i}}}{w^{\mathrm{o}}}\right) \cdot NI \right)\]

where ${w^{\mathrm{o}}_{i}}$ (gC m^-2) represents the fuel load of each fuel type i of the surface fine fuel, $w^{\mathrm{o}}$ is the total fuel load of surface fine fuel, and $\alpha_{i}$ (℃^-2) is proportional to the surface-area-to-volume ratio of different fuel types. The surface fine fuel moisture is then compared with the fuel moisture of extinction ($\omega^{\mathrm{e}}$, unitless), parameterized for each PFT, above which the surface fuel is too wet to allow any fire burning, in order to derive the fire danger index (FDI):

\[FDI = \max\left(0, \left(1 - \frac{\omega^{\mathrm{o}}}{\omega^{\mathrm{e}}} \right)\right)\]

:::{figure} Figures/fire_flow_chart.png :name: fig:fire:flow :align: center

Is this useful? If so, this figure needs to be cited in the text Flow and dependencies of the fire calculations. :::

12.4.2. DONE: The number of fires#

Potential fire ignitions include human-induced incidents and lightning-induced natural ignitions (Yue et al.2014). The total number of potential ignitions per km² and per day ($n^{\mathrm{ig}}$) is calculated as:

\[n^{\mathrm{ig}} = n_{l}^{\mathrm{ig}} + n_{h}^{\mathrm{ig}}\]

where $n_{h}^{\mathrm{ig}}$ and $n_{l}^{\mathrm{ig}}$ represent potential ignitions caused by human activity and lightning strikes, respectively. The monthly climatology of total lightning strikes, including both cloud-to-cloud and cloud-to-ground lightnings, derived from the Lightning Imaging Sensor–Optical Transient Detector (LIS/OTD), was utilized as the lightning input data (Yue et al. 2014). The number of potential lightning ignitions ($n_{l}^{\mathrm{ig}}$) is calculated as :

\[n_{l}^{\mathrm{ig} }= f^{\mathrm{cg}} \cdot l^{\mathrm{efficiency}} \cdot (1 - h^{\mathrm{suppression}})\]

where $f^{\mathrm{cg}}$ represents the fraction of cloud-to-ground lightnings, $l^{\mathrm{efficiency}}$ represents the fraction of lightnings with sufficient energy to ignite a fire, and $h^{\mathrm{suppression}}$ represents the fraction of lightning-ignited fires that are suppressed by human.

The human suppression on lightning fires ($h^{\mathrm{suppression}}$) is calculated as a function of population density ($P^{\mathrm{D}}$) following Li et al. (2012):

\[{h^{\mathrm{suppression}}} = 0.99 - 0.98 \cdot \exp\left(-0.025 \cdot P^{\mathrm{D}} \right)\]

Human-induced potential ignitions ($n_{h}^{\mathrm{ig}}$) is also approximated as a function of population density ($P^{\mathrm{D}}$):

\[n_{h}^{\mathrm{ig}} = P^{\mathrm{D}} \cdot \exp\left(-0.5 \cdot \sqrt{P^{\mathrm{D}}} \right) \cdot a(N^{\mathrm{D}})\]

where $P{^\mathrm{D}}$ represents the population density (individuals km^-2), and $a(N^{\mathrm{D}})$ denotes the potential number of fire ignitions per person per day. The parameter $a(N^{\mathrm{D}})$ has regional specific values as parameterized in Thonicke et al. (2010). The role of humans to suppress fire ignitions is implicitly accounted for in the equation above, which prescribes a maximum number of ignitions when population density reaches 16 individuals km^-2.

Apart from human suppression, fuel availability is also assumed to limit the success chance of a potential ignition through the ignition efficiency ($i^{\mathrm{efficiency}}$):

\[\begin{split}i^{\mathrm{efficiency}} = \begin{cases} 0, & w^{\mathrm{o}} \leq w^{\mathrm{o}}_{lower} \\ \frac{w^{\mathrm{o}} - w^{\mathrm{o}}_{lower}}{w^{\mathrm{o}}_{upper} - w^{\mathrm{o}}_{lower}}, & w^{\mathrm{o}}_{lower} < w^{\mathrm{o}} < w^{\mathrm{o}}_{upper} \\ 1, & w^{\mathrm{o}} \geq w^{\mathrm{o}}_{upper} \end{cases}\end{split}\]

where $w^{\mathrm{o}}$ represents fuel load, $w^{\mathrm{o}}_{lower}$ denotes the lower threshold of fuel load, below which fuel availability is too small to allow any burning, and $w^{\mathrm{o}}_{upper}$ denotes the upper threshold of fuel load, above which fuel availability no longer limits ignition success. Between these two thresholds, the ignition efficiency linearly increases with fuel availability.

The number of fires ($n^{\mathrm{fire}}$) within a model grid cell for a given day, including both lightning- and human-caused ignitions and accounting for the influence of fire weather and fuel availability, is then calculated as:

\[n^{\mathrm{fire}} = FDI \cdot i^{\mathrm{efficiency}}\cdot { n^{\mathrm{ig}}} \cdot area^{\mathrm{veg}}\]

where $area^{\mathrm{veg}}$ represents the area covered by forests and grasslands within a grid cell because cropland fires are currently excluded.

12.4.3. DONE: Fire size#

The simulation of fire size assumes that, on a daily time step, the burned patch developed from a single ignition source can be approximated by an ellipse with one of its two focal points being the ignition source. The major axis length of the burned ellipse can be calculated as the product of the fire spread rate, which is the sum of both forward and backward spread rates, and fire duration time. The length-to-breadth ratio of the ellipse is a function of surface wind speed.

The simulation of the forward surface fire spread rate ($ROS_{f}^{\mathrm{surface}}$, $m\ minute^{-1}$) consistent with the prevailing daily wind direction is based on a quasi-empirical surface fire spread model (Rothermel 1972), which is based on the conservation of energy described by a heat source divided by a heat sink:

\[ROS_{f}^{\mathrm{surface}} = \frac{I^{\mathrm{R}} \cdot \xi \cdot (1 + \Phi^{\mathrm{w}})}{\rho^{\mathrm{b}} \cdot \varepsilon \cdot Q^{\mathrm{ig}}}\]

where $I^{\mathrm{R}}$ represents the reaction intensity, namely, the rate of energy release per unit area of the fire front $(kJ \, m^{-2} \, min^{-1})$; $\xi \$ represents the propagating flux ratio (unitless), which is the proportion of the reaction intensity that heats neighboring fuel particles to the point of ignition under no wind conditions; $\Phi^{\mathrm{w}}$ is a unitless multiplier that accounts for the effect of wind in increasing the propagating flux ratio; $\rho^{\mathrm{b}} \$ is the bulk density $(kg \, m^{-3})$ of surface fine fuel, $\varepsilon \$(unitless) is the effective heating number indicating the proportion of a fuel particle that is heated to ignition temperature at the time flaming combustion starts; $Q^{\mathrm{ig}}$ is the heat of pre-ignition $(kJ \, kg^{-1})$, which is the quantity of heat necessary to ignite a specific fuel mass (Rothermel 1972, Thonicke et al., 2010). The calculation of each of these variables is further detailed in Thonicke et al. (2010).

The rate of backward surface fire spread ($ROS_{b}^{\mathrm{surface}}$, $m\ minute^{-1}$), in a contrary direction to the daily prevailing wind, is calculated as:

\[ROS_{b}^{\mathrm{surface}} = ROS_{f}^{\mathrm{surface}} \cdot \exp\left(-0.012 \cdot U_{\text{forward}} \right)\]

where $U_{\text{forward}}$ ($m\ minute^{-1}$) represents the daily wind speed.

Fire duration ($t^{\mathrm{fire}}$, in $minute$) is simulated to increase with FDI but with a maximum fire duration time ($t^{\mathrm{max}}$) , currently set as 241 minutes which represents the time limit of active afternoon burning, which cannot exceed a single day given that fire is simulated on a daily time step:

\[t^{\mathrm{fire}} = \frac{t^\mathrm{max}}{1 + 240 \cdot \exp\left(-11.06 \cdot FDI \right)}\]

The distances (in $m$) of forward ($d_{f}$) and backward ($d_{b}$) surface fire spread are calculated as:

\[\begin{split}\begin{align} &d_{f} = ROS_{f}^{\mathrm{surface}} \cdot t^{\mathrm{fire}},\\ &d_{b} = ROS_{b}^{\mathrm{surface}} \cdot t^{\mathrm{fire}} \end{align}\end{split}\]

The length of the major axis of the burned ellipse is given by the sum of $d_{f}$ and $d_{b}$.

The length-to-breadth ratio ($L^{\mathrm{B}}$) of the elliptical fire scar is calculated as:

\[\begin{split}L^{\mathrm{B}} = \left\{ \begin{array}{ll} 1, & \text U_{\text{forward}} < 16.67 \\ \min(8, f_{tree} \cdot L^{\mathrm{B}}_{tree} + f_{grass} \cdot L^{\mathrm{B}}_{grass}), & \text U_{\text{forward}} \geq 16.67 \end{array} \right.\end{split}\]

where $f_{tree}$ and $f_{grass}$ represent the relative fractions of forests and grasslands, and $L^{\mathrm{B}}_{tree}$ and $L^{\mathrm{B}}_{grass}$ represent the values $L^{\mathrm{B}}$ of forests and grasslands, respectively:

\[\begin{split}\begin{align} &L^{\mathrm{B}}_{tree} = 1.0 + 8.729 \cdot (1 - \exp\left(-0.03 \cdot0.06\cdot U_{\text{forward}} \right))^{2.155},\\ &L^{\mathrm{B}}_{grass} = 1.1 +0.06 \cdot U_{\text{forward}}^{0.0464} \end{align}\end{split}\]

The size of the fire patch (in $hectare$) is then calculated as follows:

\[\bar{a} = \frac{ \dfrac{\pi}{4 \cdot L^{\mathrm{B}}} \cdot \left(d_{f} + d_{b}\right)^{2} }{10000}\]

Thus, the daily burned area($BA^{\mathrm{daily}}$) (in $hectare$) within a model grid cell is the product of the fire number and the (average) fire size:

\[BA^{\mathrm{daily}} = n^{\mathrm{fire}} \cdot \bar{a}\]

12.4.4. DONE: Carbon and trace gas emissions #

During active burning, the combustion of biomass fuel results in carbon and other trace gas emissions. Emissions from smoldering are in principle excluded because this phase of burning is not explicitly simulated. The carbon emissions from surface dead fuel, on a per burned area basis, are calculated as:

\[E_{surface} = \sum_{i=1}^{4} CF_{i} \cdot w^{\mathrm{o}}_{i}\]

where $w^{\mathrm{o}}_{i}$ and $CF_{i}$ represents fuel load (gC m^-2) and fuel combustion fraction (unitless) for different fuel classes (1h, 10h, 100h, and 100h).

The $CF$ for 1h surface dead fuel is given by:

\[\begin{split}CF_{1h} = \left\{ \begin{array}{lcl} 1.0, & \frac{\omega_{1h,l}}{\omega^{\mathrm{e}}} \leq 0.18 \\ 1.11 - 0.62 \cdot \frac{\omega_{1h,l}}{\omega^{\mathrm{e}}}, & 0.18 < \frac{\omega_{1h,l}}{\omega^{\mathrm{e}}} \leq 0.73 \\ 2.45 - 2.45 \cdot \frac{\omega_{1h,l}}{\omega^{\mathrm{e}}}, & \frac{\omega_{1h,l}}{\omega^{\mathrm{e}}} > 0.73 \\ \end{array} \right\}\end{split}\]

where ${\omega_{1h,l}}$ represents the fuel moisture content (unitless) by combining 1h dead fuel with live grass fuel.

The $CF$ values for surface dead fuel of 10h, 100h and 1000h are calculated as:

\[\begin{split}CF_{10h} = \left\{ \begin{array}{ll} 1.0, & \frac{\omega_{10h}}{\omega^{\mathrm{e}}} \leq 0.12 \\ 1.09 - 0.72 \cdot \frac{\omega_{10h}}{\omega^{\mathrm{e}}}, & 0.12 < \frac{\omega_{10h}}{\omega^{\mathrm{e}}} \leq 0.51 \\ 1.47 - 1.47 \cdot \frac{\omega_{10h}}{\omega^{\mathrm{e}}}, & \frac{\omega_{10h}}{\omega^{\mathrm{e}}} > 0.51 \end{array} \right\}\end{split}\]

\[\begin{split}CF_{100h} = \left\{ \begin{array}{ll} 0.98 - 0.85 \cdot \frac{\omega_{100h}}{\omega^{\mathrm{e}}}, & \frac{\omega_{100h}}{\omega^{\mathrm{e}}} \leq 0.38 \\ 1.06 - 1.06 \cdot \frac{\omega_{100h}}{\omega^{\mathrm{e}}}, & \frac{\omega_{100h}}{\omega^{\mathrm{e}}} > 0.38 \end{array} \right\}\end{split}\]

\[CF_{1000h} = -0.8 \cdot \frac{\omega_{1000h}}{\omega^{\mathrm{e}}} + 0.8\]

Additionally, the combustion fraction for 100h and 1000h surface dead fuels are further limited to a maximum value of 0.9 following Yue et al. (2014).

The combusted surface fuel is removed from litter pool, with proportional losses in nitrogen according to the C:N ratio of surface litter. Nitrogen emissions of various species from active surface fire burning is thus not accounted for, except for that NOx emissions are provided as a diagnostic variable (see below).

Carbon emissions in SPITFIRE includes forest crown scorching that is accounted for using a diagnostic approach (??). Other trace gas emissions($E^{\mathrm{tg}}$), including CO, CH4, VOCs, total particular matter (TPM), and NOx, are diagnostic variables derived using emissions factors:

\[E^{\mathrm{tg}} = E^{\mathrm{carbon}}_{s,c} \cdot ef^{\mathrm{tg}}\]

where $E^{\mathrm{carbon}}_{s,c}$ is carbon emissions from fire including those from both surface fuel and crown scorching; $ef^{\mathrm{tg}}$ is emission factor that quantifies the ratio between trace gas emissions and carbon emissions.

12.4.5. DONE: Forest mortality#

Fire-induced tree mortality includes two aspects: cambial damage, and crown damage caused by flame crown scorching. Although only surface fire spread is simulated, the surface fire flame can reach a height comparable to tree crown and cause crown damage. The flame scorching height ($SH$, in $m$) scales with surface fire intensity ($I_{{surface}}$, $kW m^{-1}$):

\[SH = c \cdot I_{{surface}}^{{0.667}}\]

where $c$ is a PFT-specific constant parameter. Surface frontline fire intensity measures energy release per length of fire front and is derived empirically as:

\[I_{{surface}} = h \cdot \frac{\sum_{i=1}^{3} FC_{i}}{ f^{\mathrm{burned}}} \cdot \frac{ROS_{f}^{\mathrm{surface}}}{60}\]

where $h$ represents the heat content ($kJ kg^{-1}$) of surface fine fuel (1h, 10h and 100h), $FC_{i}$ denotes the combusted surface fuel in terms of dry matter calculated using $CF_{i}$ and $w^\mathrm{o}_{i}$, $f^\mathrm{burned}$ means the fraction of vegetation area that’s been burned, $ROS_{f}^\mathrm{surface}$ is forward surface fire spread rate ($m\ minute^{-1}$) which is converted to $m\ s^{-1}$ by being divided by 60.

As ORCHIDEE represents forest stand structure using tree density and circumference classes and as tree diameter ($d^{\mathrm{dia}}_{l}$), height ($d^{\mathrm{h}}_{l}$) and crown length (or crown vertical diameter $d^{\mathrm{cida,ver}}_{l}$) differ among circumference classes (see ?? and ??), the derivation of fire-induced tree mortality is thus made for each circumference class. Within each circumference class, the number of individual dead trees is determined by multiplying the derived mortality rate with its individual density.

The fraction of flame crown scorching ($f^{\mathrm{cs}}_l$) for the circumference class $l$ is derived by comparing the flame scorching height with tree height and canopy base height ($d^{\mathrm{h}}_{l}-d^{\mathrm{cida,ver}}_{l}$):

\[\begin{split}f^{\mathrm{cs}}_{l} = \left\{ \begin{array}{ll} 1.0, & d^{\mathrm{h}}_{l} \leq SH \\ \frac { S H -{ (d^{\mathrm{h}}_{l}-d^{\mathrm{cida,ver}}_{l})} } { d^{\mathrm{cida,ver}}_{l} }, & (d^{\mathrm{h}}_{l}-d^{\mathrm{cida,ver}}_{l})<SH< d^{\mathrm{h}}_{l} \\ 0 &(d^{\mathrm{h}}_{l}-d^{\mathrm{cida,ver}}_{l}) \geq SH \end{array} \right\}\end{split}\]

Post-fire mortality from crown scorching ($P^{m,cs}_{l}$, unitless) for a given circumference class $l$ is calculated as :

\[P^{\mathrm{m,cs}}_{l} = c_1 \cdot (f^{\mathrm{cs}}_ {l})^{c_2}\]

where $c_1$ and $c_2$ are PFT-specific parameters.

Post-fire mortality from cambial damage is derived by comparing fire flame residence time ($\tau^{\mathrm{flame}}_l$, minute) with tree-dependent critical residence time ($\tau^{\mathrm{critical}}_l$, minute). The fire flame residence time is computed as:

\[\tau^{\mathrm{flame}}_l = 2.0 \cdot \left( \frac{CF^{\mathrm{fine\_fuel}}_{1h,10h,100h}}{\Gamma} \right)\]

where $CF^{\mathrm{fine\_fuel}}_{1h,10h,100h}$ represents the combustion fraction for surface fine litter fuel, including 1 h, 10 h and 100 h fuel classes, $\Gamma$ is the reaction velocity ($minute ^{-1}$) derived in simulating fire spread rate.

The critical residence time for fire fame is derived as:

\[\tau^{\mathrm{critical}}_{l} = 2.9 \cdot B T_{l}^{2}\]

where $B T_{l}$ is the thickness of bark for trees of a given circumference class $l$, which is calculated using the diameter at breast height ($d^{\mathrm{dia}}_{l}$):

\[B T_{l} = {par_1} \cdot d^{\mathrm{dia}}_{l} \cdot100 + par_2\]

where $par_1$ and $par_2$ are PFT-specific parameters.

Forest mortality due to cambial damage is then calculated as:

\[\begin{split}P^{\mathrm{m,cd}}_{l} = \left\{\begin{array}{ll} 0, & \frac{\tau^{\mathrm{flame}}_l}{\tau^{\mathrm{critical}}_l} \leq 0.22 \\ 0.563 \cdot \frac{\tau^{\mathrm{flame}}_l}{\tau^{\mathrm{critical}}_l} - 0.12386, & 0.22 < \frac{\tau^{\mathrm{flame}}_l}{\tau^{\mathrm{critical}}_l} < 2.0 \\ 1, & \frac{\tau^{\mathrm{flame}}_l}{\tau^{\mathrm{critical}}_l} \geq 2.0 \end{array}\right.\end{split}\]

Finally, post-fire forest mortality for a given circumference class $l$, combing from both crown scorching and cambial damage, is derived by assuming that the two mortality causes operate independently:

\[P_{l}^{\mathrm{m}} = P^{\mathrm{m,cd}}_{l}+ P^{\mathrm{m,cs}}_{l} - P^{\mathrm{m,cd}}_{l}\cdot P^{\mathrm{m,cs}}_{l}\]

The stand density killed by fire ($d^{\mathrm{ind,kill}}_{l}$) is derived as:

\[d^{\mathrm{ind,kill}}_{l} = d^{\mathrm{ind}}_{l}\cdot P^{\mathrm{m}}_{l}\]

where $d^{\mathrm{ind}}_{l}$ represents the stand density (individual $m^{-2}$) for a given circumference class $l$.

Due to crown scorching, part of live biomass, in particular small branches and leaves, are considered as being combusted, leading to carbon emissions to the atmosphere. The fractions of different biomass components that are directly combusted are prescribed in the model. The uncombusted dead biomass are transferred to corresponding litter pool.

12.5. DONE: Herbivory#

In ORCHIDEE herbivory is optional. If herbivory is simulated, herbivore activity reduces the biomass of leaves and fruits of forest PFTs and leaves, fruits, and stalks of grass PFTs. In ORCHIDEE, cropland PFTs are not affected by herbivores and herbivory does not modify leaf age structure. To estimate edible biomass and herbivore consumption, the mean long-term leaf production is used :

\[\begin{split}\begin{align} &b_{1} = c_2 \cdot {(c_1 \cdot T^{npp,3year})}^{c_3}, \\ &\tau^{her} = \frac{ c_4 \cdot \Delta{t} \cdot {(c_1 \cdot T^{npp,3year})}} {b_{1}}, \end{align}\end{split}\]

where $b_{1}$ (g C m$^{-2}$ s$^{-1}$) is the consumption of biomass by herbivores, $c_1$ (unitless) is a fixed fraction of the leaves that is available to the herbivores, $c_2$ (g C m$^{-2}$ s$^{-1}$ K$^{-3}$) and $c_3$ (unitless) are PFT-specific constants to calculate the herbivore activity as a function of 3-year mean net primary production ($T^{npp,3year}$; g C m$^{-2}$ s$^{-1}$). Herbivory is calculated daily, so $\Delta{t}$ (s) is the number of seconds in a day and $c_4$ (-) is a constant that differs between deciduous and evergreen PFTs to account for the length of the growing season. $\tau^{her}$ (s) is the time constant of the probability of a leaf to be eaten by a herbivore.

The consumption of biomass by herbivores and the resultant herbivore activity are calculated following:

\[\frac{\Delta{M^{o}}}{\Delta{t}} = 1 - \frac{\Delta{t}} {\tau^{her}},\]

where $o$ are the plant organs affected by herbivory - the leaves and fruits of forest PFTs and the leaves, fruits and stalks of grassland PFTs. Finally, the consumed biomass ($M^{o}$) is removed from the plant biomass and added to the litter pool.

12.6. DONE: Individual tree vs. stand replacing disturbances#

Damages from different disturbances are implemented as continuous functions that range from zero to the entire stand. If few individuals are damaged, the stand density is reduced. If, on the other hand, the damage exceeds 30 % of the total biomass, a new cohort is established in the youngest age class of the plant functional type when multiple age classes are present. Over several decades, disturbances contributes to the coexistence of different age classes within the same plant functional type. The 30 % threshold is somewhat arbitrary but based on observations that forests recover quickly after thinning 30 % of the basal area []. Currently, this scheme is applied only to abrupt disturbances, such as wind damage, but will be extended to more gradual disturbances, like drought stress and bark beetle damage, in future versions of the model.

13. OK: Quality control and quality assurance of the model code#

13.1. DONE: Mass balance checks for carbon and nitrogen#

An overarching daily mass balance check consists of comparing the pool-based net biome production [] calculated as the changes in carbon and nitrogen in the vegetation, soil, litter and product pools with the flux-based net biome production calculated as the sum of all the carbon and nitrogen fluxes from and to the atmosphere, and between the vegetation, litter, soil, and products. The residual of this check is written to the output files.

For each subroutine related to the calculation of the carbon or nitrogen cycle in ORCHIDEE , changes in the pools involved in the calculations of that subroutine are compared to the sum of the fluxes involved in that subroutine. If the residual exceeds a threshold, the model is stopped.

13.2. OK: Water balance checks#

An overarching water balance check is run at the same temporal resolution as the calculation of the water pools and fluxes, i.e., sub-daily (section ??). Mass changes in the three water reservoirs, i.e., the soil water, the canopy interception, and the snow pack, are compared against the difference between the water sources and sinks. The sources considered are, rain and snow precipitations, water from irrigation returning to soil moisture, fluxes out of floodplains, routed water which comes back into the soil from the bottom, and routed water which comes back into the soil at the top. The sinks considered are surface runoff, drainage, interception loss, transpiration, bare soil evaporation, snow and evaporation, floodplain evaporation. The residuals of this check are written to the output files.

Note that the current approach to calculate the residuals of the water budget does not include the stream, fast and slow routing reservoirs. XXX.

Depending on the model configuration, routing may be calculated on a different grid than the other water pools and fluxes (section ??). If this is case, global water conservation is enforced every time the calculations change grids.

13.3. DONE: Surface area checks#

In ORCHIDEE the surface area of each PFT within a grid cell is represented by the share of that PFT within the grid cell. At the start of a simulation, the share of each PFT within each grid cell is read from a vegetation distribution map (section ??). When the information from the map is transferred to ORCHIDEE, the residual between the sum of the shares of PFTs and unity is declared as no biological fraction (currently treated as glacier in ORCHIDEE). Therefore, the sum of the vegetation, bare soil and no biological fractions is unity for each grid cell. If the simulation accounts for land cover changes, an annual vegetation map is read for each year of the simulation and the aforementioned check is repeated annually.

Area conservation is also checked at the grid cell for each subroutine in which the surface area of the PFTs may change, e.g, land cover change, disturbances, and age class dynamics. Within these subroutines the sum of shares and the no biological fraction should remain unity.

Due to long term effects of land cover change, disturbances, age class dynamics, or a combination of these processes, the share of a PFT may become too small to justify its computational costs. If that happens, the biomass at the PFT is moved into the harvest pool, and the litter pools, soil pools, and the area share are moved to the PFT with the largest share or the bare soil. Although the bare soil PFT is not initialized with any carbon or nitrogen pools, this process could result in carbon and nitrogen pools and subsequent fluxes from the bare soil in ORCHIDEE .

13.4. DONE: Technical quality control of the code#

After committing code changes to the versioning server, a series of technical tests are launched automatically during the night. This includes 14 technical tests with a stand-alone configuration of ORCHIDEE as well as 7 technical tests with the coupled land-atmosphere configuration of ORCHIDEE called LMDZOR. These 21 tests in total enable checking whether: (1) the model compiles and runs both in debug and production mode; (2) the exact same results (bit-by-bit) are obtained irrespective of the number of processors used; (3) the exact same results (bit-by-bit) are obtained irrespective of the restart frequency of the model; (4) the exact same results (bit-by-bit) are obtained with the land-atmosphere configuration for runs with different parallelization schemes; (5) the model can run with different climate forcing files; (6) the model can run with both the previous and current driver to read and interpolate the climate forcing files; (7) the model runs with settings that are still under development or testing but that should become the default setting in the future. The full series of tests are run at the most powerful server whereas a subset is run as well at a smaller server. Running the tests at both servers is by itself the 22^nd test.

14. OK: Default model configuration#

The flexibility of the ORCHIDEE model in terms of the source of the climate, the boundary files, the processes that are accounted for, and for some processes the way they are calculated, results in hundreds of possible model configurations. This section describes the default configurations for stand-alone and land-atmosphere simulations.

The stand-alone configuration requires the use of a climate forcing. By default, the 6-hourly CRUJRA climate forcing regridded to 2 degrees is used (Section ??). In addition, the following boundary files are used: (1) A soil properties map following a USDA classification [] that was adjusted to represent 13 texture classes (See ??);

(2) A hydrological digital elevation model based on to calculate the river network (See ??); (3) A spatially explicit time series of historical to present distribution of the 15 PFTs of ORCHIDEE (See ??); (4) A spatially explicit time series of historical and present forest management (see ??);

(5) A spatially explicit time series of nitrogen deposition, and application of nitrogen fertiliser as well as a map for biological nitrogen fixation (See ??); (6) A background albedo map for the visible and near infrared wavelengths (See ??); and

(7) A global time series for the historical and present atmospheric 2 concentration (See ??).

The model is configured to simulate 15 PFTs: (1) Bare Soil, (2) Tropical Broadeaf Evergreen, (3) Tropical Broadleaf Raingreen, (4) Temperate Needleleaf Evergreen, (5) Temperate Broadleaf Evergreen, (6) Temperate Broadleaf Summergreen, (7) Boreal Needleleaf Evergreen, (8) Boreal Broadleaf Summergreen, (9) Boreal Larix Sp., (10) Tropical C3 Grass (11) Temperate C3 Grass, (12) Boreal C3 Grass Boreal, (13) Global C4 Grass, (14) Global C3 Agriculture, and (15) Global C4 Agriculture. All PFTs are represented by a single age class (See ??) and the forest PFTs make use of three diameter classes (See ??) to represent the stand structure within the PFT. Emissions from biologic volatile organic compounds are not simulated.

For several of its process calculations, ORCHIDEE includes two or more approaches. These approaches often represent the initial well-tested approach and a more recent, more refined or more realistic approach that is, however, less well tested. Over time, the refined or more realistic approach should become the only remaining approach. For the time being, one the approaches needs to be selected:

For the energy budget, the following approaches are selected: (1) the background albedo is read from a map (See ??)

(2) roughness length for momentum is calculated as a function of leaf area and tree height, the roughness length for heat is calculated as constant fraction of the roughness length for momentum (See ??);

(3) mean height is used in the calculation of the roughness length (See ??);

(4) heat conductivity in the soil is calculated as a function of soil carbon pools (See ??)

for the water budget, the following approaches are selected: (1) infiltration along plant roots is a function of the root profile (See %s);

(2) river routing (See ??) is calculated by the hybrid approach (See ??);

(3) plant water stress is calculated as a linear response between field capacity and wilting point (See ??);

(4) additional surface resistance from, for example, litter, mosses, and crusts, are not accounted for in the calculation of bare soil evapotranspiration (See ??);

(5) the multi-layer snow scheme with 12 layers (See ??);

For the carbon and nitrogen budgets, the following approaches are selected: (1) a dynamic nitrogen cycle that allows for nitrogen limitations (See ???);

(2) the specific leaf area is calculated as a function of leaf mass (See ??);

(3) the carrying capacity of the forest PFTs varies over time and space as a function of the environmental conditions (See ??);

(4) the different components of the soil carbon pool are simulated as bulk pools (See ??).

For the demographic calculations, the following approaches are selected: (1) the PFT distribution is read from annual maps;

(2) a constant rate for background mortality is used in addition to self-thinning (See ??)

For land use, the following approaches are selected: (1) for the product pools the dynamic allocation is used implying that the diameter of the wood harvest determines whether the harvest will be used for a short-lived product or a medium and long-lived products (See ??);

(2) at the end of the growing season, a harvest event, removes the aboveground biomass from croplands (See ??); and

(3) A parameter set tuned towards global use is selected for forest management, wind storms and bark beetle outbreak because these processes are scale dependent. Clear cuts, for example, are common at the site-level but are unlikely at the scale of a 50 x 50 km$^{2}$ pixel as that would imply a 2500 km$^{2}$ clear cut.

Features that are still under development or that have been tested and evaluated only in regional applications are by default deactivated in the stand-alone configuration:

For the energy budget, the following processes are deactivated: (1) multi-layer energy budgets (See ???);

(2) heat storage in lakes (See ??);

(3) the calculation of soil thermal conductivity as a function of XXX (See ?? and the calculation of soil heat capacipty as a function of XXX ??)

For the water budget, the following processes are deactivated: (1) hydraulic architecture and related functionality such as vessel mortality and water storage in the vegetation (See ??)

(2) ice sheets are not simulated (See ??);

(3) re-infiltration from ponds and floodplains (See ??);

For the carbon and nitrogen cycle, the following processes are deactivated: (1) grassland density as a function of plant growth (See ??)

For land use, the following processes are deactivated: (1) crop irrigation (See ??),

(2) PFT changes after a stand replacing disturbance (See ??),

(3) forest management changes when an opportunity arises (See

),

(4) litter raking (See ??)

For disturbances, the following processes are deactivated: (1) mortality from droughts (See ??),

(2) mortality from wind storms (See ??),

(3) mortality from insect outbreaks (See ??),

(4) mortality from fire (See ??),

(5) loss of biomass from herbivory (See ??), and

(6) snags as a pool for coarse woody debris (??).

The stand-alone experiment uses the default values of global and PFT-specific parameters that are included in the model code. This stand-alone configuration is used in the TRENDY attribution experiment [] that consists of four treatments (i.e., S0 to S3). Each treatment includes a different combination of climate forcing and boundary files but applies the same model configuration and model parameters. Likewise, the stand-alone configuration is used in the technical quality control of the code (See ??) and the reference simulations used to evaluate the evolution of the ORCHIDEE model.

The land-atmosphere configuration makes use of an atmospheric model for its climate conditions rather than one of the climate forcings (See ??). In addition, drag is calculated by the atmospheric model instead of ORCHIDEE (See

) and because land-atmosphere simulations have a higher temporal resolution than stand-alone simulations, throughfall is parametrized differently (See %s).

15. Model results for the main outputs#

Provide standard composite figures for all major model outputs with a map of the annual mean value et possibly a comparison with existing benchmark PLUS a time series for the mean seasonal cycle and the mean diurnal cycle including different curves for different regions.

15.1. Results for the energy budget#

15.2. Results for the water budget#

15.3. Results for the carbon-nitrogen budgets#

15.4. Results for land use#

16. Table: MTC and PFT#

:::{list-table} List of the meta classes (MTCs) and plant functional types (PFTs) defined in ORCHIDEE . :header-rows: 0 :name: tab:pft

- Metaclass (MTC)
- MTCnumber
- Plantfunction type (PFT)
- PFTnumber
- Bare ground
- 1
- Bare ground
- 1
- Tropical broadleaf evergreen forest
- 2
- Tropical broadleaf evergreen forest
- 2
- Tropical broadleaf raingreen forest
- 3
- Tropical broadleaf raingreen forest
- 3
- Temperate needleleaf evergreen forest
- 4
- Temperate needleleaf evergreen forest
- 4
- Temperate broadleaf evergreen forest
- 5
- Temperate broadleaf evergreen forest
- 5
- Temperate broadleaf deciduous forest
- 6
- Temperate broadleaf deciduous forest
- 6
- Boreal needleleaf evergreen forest
- 7
- Boreal needleleaf evergreen forest
- 7
- Boreal broadleaf deciduous forest
- 8
- Boreal broadleaf deciduous forest
- 8
- Boreal needleleaf deciduous forest
- 9
- Boreal needleleaf deciduous forest
- 9
- c3 grassland
- 10
- Boreal natural c3 grassland
- 15
- c3 grassland
- 10
- Temperate natural c3 grassland
- 10
- c3 grassland
- 10
- Tropical natural c3 grassland
- 14
- c4 grassland
- 11
- Natural c4 grassland
- 11
- c3 cropland
- 12
- c3 cropland
- 12
- c4 cropland
- 13
- c4 cropland
- 13 :::

17. Table: spatial and temporal discretization#

:::{list-table} Spatial and temporal resolution of the processes calculated in ORCHIDEE . Make sure to use the same labels as in the previous table :header-rows: 0 :name: tab:spacetime

*   **Minutes**

*   **Daily**

*   **Annual**

- PFT
- Albedo
  
  Heterotrophic respiration
  
  Biogenic volatile organic compound emissions
  
  Litter carbon
  
  Maintenance respiration
  
  Photosynthesis
  
  Roughness
  
  Soil carbon
- Carbon allocation
  
  Growth respiration
  
  Phenology
  
  Senescence
  
  Turnover
  
  natural mortality
- Biogeography
  
  Grass and crop harvest
  
  Land cover change
  
  Product use
  
  Vegetation distribution
  
  Wood harvest
- Soil column
- Soil hydrology
  
  Soil temperature
- grid cell
- Energy budget
  
  Snow dynamics
  
  Snow temperature
- Watershed
- Routing

:::

18. Table: Variable description#

18.1. Table: fluxes#

:::{list-table} Variable description. Variables were grouped as follows: $F$=flux, $f$=fraction, $M$=pool, $m$=modulator, $d$=stand dimension, $T$=temperature, $p$=pressure, $R$=resistance, $q$=humidity, $g$=function. :header-rows: 0

- Symbol in text
- Unit
- **Symbol in ORCHIDEE **
- Description
- $F^A$
- $\mu$
  
  mol 2 m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- assimi
- $CO_2$
  
  assimilation rate
- $F_{gpp}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- gpp
- Photosynthesis
- $F^{gpp,ref}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- pre_indust_ref_gpp
- Pre-industrial reference photosynthesis
- $F^{gpp,st}$
- gC m
  
  $^{-2}$
  
  day
  
  $^{-1}$
- stressed_daily
- Stressed daily photosynthesis
- $F^{gpp,us}$
- gC m
  
  $^{-2}$
  
  day
  
  $^{-1}$
- unstressed_daily
- Unstressed daily photosynthesis
- $F^{gpp,1week}$
- gC m
  
  $^{-2}$
  
  day
  
  $^{-1}$
- gpp_week
- mean of daily photosynthesis calculated for the past week
- $F_{npp,3year}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- npp_long-term
- Net primary production averaged over the previous 3 years
- $F^{npp,wood,10year}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- pai
- Net primary wood production averaged over the previous 10 years
- $F^{npp,wood,1year}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- mai
- Net primary wood production averaged over the life time of the forest
- $F^{rm}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- resp_maint
- Maintenance respiration
- $F^{rg}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- resp_growth
- Growth respiration
- $F^{rm,1week}$
- gC m
  
  $^{-2}$
  
  day
  
  $^{-1}$
- resp_maint_week
- mean of daily maintenance respiration calculated for the past week
- $F^{Tr}$
- m s
  
  $^{-1}$
- transpir
- Amount of water that a tree transpires
- $F^{w,avail}$
- m s
  
  $^{-1}$
- -
- Amount of water available to the tree
- $F^{resp,het,litter}$
- g C.m
  
  $^{-2}$
  
  .s
  
  $^{-1}$
- resp_hetero_litter
- Total heterotrophic respiration from the litter decay
- $F_{in,lignin}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- lignin_
  
  $pool$
  
  _inc
- Input of lignin to the litter pools,
  
  $pool$
  
  being struc, metab, wood, or snag
- $F_{in,o,i}$
- g C m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- bm_to_litter, turnover
- Litter carbon input from plant organ
  
  $o$
  
  into litter pool
  
  $i$
- $F_{bvoc,i}$
- gC m
  
  $^{-2}$
  
  s
  
  $^{-1}$
- ???
- BVOC emissions from canopy layer
  
  $i$
- $F^{prod,s}_{t}$
- gC grid cell
  
  $_{-1}$
  
  year
  
  $_{-1}$
- flux_prod_l
- Decomposition rate of age class
  
  $t$
  
  in the short-lived products
- $F^{prod,m}_{t}$
- gC grid cell
  
  $_{-1}$
  
  year
  
  $_{-1}$
- flux_prod_m
- Decomposition rate of age class
  
  $t$
  
  in the medium-lived products
- $F^{prod,l}_{t}$
- gC grid cell
  
  $_{-1}$
  
  year
  
  $_{-1}$
- flux_prod_l
- Decomposition rate of age class
  
  $t$
  
  in the long-lived products
- $F^{prod,s}$
- gC grid cell
  
  $_{-1}$
  
  year
  
  $_{-1}$
- flux_l
- Annual decomposition of the short-lived products
- $F^{prod,m}$
- gC grid cell
  
  $_{-1}$
  
  year
  
  $_{-1}$
- flux_m
- Annual decomposition of the medium-lived products
- $F^{prod,l}$
- gC grid cell
  
  $_{-1}$
  
  year
  
  $_{-1}$
- flux_l
- Annual decomposition of the long-lived products :::