Prognostic Carbon Model

The prognostic carbon model (PrognosticCarbonModel) carries the live vegetation carbon in four prognostic pools and predicts biomass from the carbon the canopy fixes. It wraps an LAI model (PrescribedBiomassModel or ZhouOptimalLAIModel), which still sets the leaf, stem and root area indices: GPP and LAI are the same with or without the pools.

Pools and fluxes

The pools, in kg C m⁻² of ground, are the non-structural carbon (sugar) that GPP enters, and the leaf, wood (stem) and fine-root carbon it is allocated to:

\[\begin{align} \frac{dC_{sugar}}{dt} &= M_C\,\text{GPP} - R_m - G \\ \frac{dC_{leaf}}{dt} &= a f_{leaf} G - C_{leaf}/\tau_{leaf} \\ \frac{dC_{stem}}{dt} &= a f_{stem} G - C_{stem}/\tau_{stem} \\ \frac{dC_{root}}{dt} &= a f_{root} G - C_{root}/\tau_{root} \end{align}\]

where we have:

• the sugar (non-structural), leaf, wood and fine-root carbon, $C_{sugar}$, $C_{leaf}$, $C_{stem}$ and $C_{root}$ (kg C m⁻²)
• the gross primary production of the canopy, GPP (mol CO₂ m⁻² s⁻¹), from the photosynthesis model
• the molar mass of carbon, $M_C$ (kg C mol⁻¹), which converts GPP to kg C
• the maintenance respiration, $R_m$ (kg C m⁻² s⁻¹)
• the allocation to growth, $G$ (kg C m⁻² s⁻¹)
• the allocation fractions of leaves, wood and fine roots, $f_{leaf}$, $f_{stem}$ and $f_{root}$ (-), which sum to one
• the turnover times of leaves and fine roots, $\tau_{leaf}$ and $\tau_{root}$ (s)
• the wood turnover time, $\tau_{stem} = s\,\tau_{stem,0}$ (s), that of warm climates, $\tau_{stem,0}$, lengthened by the factor $s$ (-) in cold climates

The allocation becomes structure with efficiency $a$, and the rest, $(1-a)G$, is growth respiration. The turnover terms are the litter passed to the soil, $L = \sum_i L_i$ with $L_i = C_i/\tau_i$ (kg C m⁻² s⁻¹) for $i$ the leaves, wood and fine roots. The autotrophic respiration of the canopy is that of the pools, $R_a = R_m + (1-a)G$. The sections below give $R_m$, $G$, the fractions $f_i$ and $s$; the parameters, such as $a$ and the turnover times, are in the table at the end of the page.

Maintenance respiration

\[R_m = g\left(\frac{C_{sugar}}{C_{sugar,ref}}\right)\left[M_C R_d + Q_{10}^{(T - T_{ref})/10}\left(r_{stem} C_{sap} + r_{root} C_{root}\right)\right], \qquad C_{sap} = f_{sap} C_{stem}, \qquad f_{sap} = \frac{1}{1 + C_{stem}/C_{sap,half}}\]

where we have:

• the dark respiration of the leaves, $R_d$ (mol CO₂ m⁻² s⁻¹), from the photosynthesis model, which has its own temperature response
• the canopy temperature, $T$ (K)
• the sapwood carbon, $C_{sap}$ (kg C m⁻²), the living part of the wood, and its fraction of the wood, $f_{sap}$ (-)
• the ramp $g(x) = x^n/(1+x^n)$ (-)

The sapwood and the fine roots both respire at rates scaled by a $Q_{10}$ of the canopy temperature, so that the formulation is the same in a standalone canopy, which has no soil temperature. Only sapwood respires, and the wood of a large stem is mostly dead heartwood. The sapwood fraction $f_{sap}$ takes the simplest form with these limits: it is one in a small stem, half when $C_{stem} = C_{sap,half}$, and falls as $C_{sap,half}/C_{stem}$ in a large one. The sapwood carbon $C_{sap}$ thus grows linearly with a small stem and saturates at $C_{sap,half}$, a Michaelis-Menten form. The ramp shuts respiration down as the sugar pool empties, so the pool cannot become negative.

Allocation

Allocation draws the sugar pool toward a target proportional to the living biomass,

\[G = \frac{C_{sugar}}{\tau_{alloc}}\, g\left(\frac{C_{sugar}}{c_{nsc}\,(C_{leaf} + C_{sap} + C_{root})}\right),\]

so growth slows when reserves fall below the target, and the sugar pool oscillates seasonally around it.

There are no plant functional types: the allocation fractions are those of trees and of grasses, weighted by the tree share $f_{tree}$ of the vegetation, and only trees build wood,

\[f_{leaf} = f_{tree}\, f_{leaf,tree} + (1 - f_{tree})\, f_{leaf,grass}, \qquad f_{stem} = f_{tree}\, f_{stem,tree}, \qquad f_{root} = 1 - f_{leaf} - f_{stem}.\]

The tree share is that of the LAI model: with ZhouOptimalLAIModel, the one that sets its leaf cost, by default the climate tree share, a function of the seasonal maximum LAI a tree canopy would reach, of the number of dry months of the growing season and of its temperature; with prescribed LAI, it is given, e.g. the tree share of the natural vegetation of the CLM surface data (clm_tree_share). Where the dry season is long, the vegetation is grass and builds little wood, which separates savannas from forests under the same annual rainfall.

Wood turnover

Trees live longer in cold climates. Below a reference mean annual temperature $T_{\tau}$, the wood turnover time $\tau_{stem} = s\,\tau_{stem,0}$ is lengthened by

\[s = \min\left(q^{\max(T_{\tau} - T_{annual},\, 0)/10},\ 10\right),\]

and a factor $q \le 1$ disables the scaling ($s = 1$). The mean annual temperature $T_{annual}$ (K) is a running mean of the air temperature $T_{air}$ (K) with a memory of two years, a time-integrated variable:

\[\frac{dT_{annual}}{dt} = \frac{T_{air} - T_{annual}}{\tau_{climate}}.\]

Soil carbon

In LandModel and SoilCanopyModel, the litter makes the soil organic carbon prognostic: $dSOC/dt = S_{litter}(z) - S_m$, with the microbial respiration $S_m$ of the Soil Biogeochemistry model. Leaf and stem litter enter on a shallow exponential profile and root litter follows the root distribution, each normalized by its integral over the soil column so the soil receives exactly the litter the pools shed:

\[S_{litter}(z) = \left(L_{leaf} + L_{stem}\right)\frac{p(z; d_{litter})}{\int p(z; d_{litter})\,dz} + L_{root}\,\frac{p(z; d_{root})}{\int p(z; d_{root})\,dz}, \qquad p(z; d) = \frac{e^{z/d}}{d},\]

where we have:

• the soil organic carbon, SOC (kg C m⁻³), and the microbial respiration, $S_m$ (kg C m⁻³ s⁻¹)
• the litter input to the soil organic carbon, $S_{litter}$ (kg C m⁻³ s⁻¹), whose integral over the soil column, $\int S_{litter}\,dz = \sum_i L_i$, is the litter of the pools: what leaves the pools enters the soil
• the leaf, wood and fine-root litter of the pools, $L_{leaf}$, $L_{stem}$ and $L_{root}$ (kg C m⁻² s⁻¹)
• the depth coordinate, $z$ (m), negative below the surface, the integrals being over the soil column
• the litter e-folding depth, $d_{litter}$ (m), and the rooting depth of the LAI model, $d_{root}$ (m)

Vegetation and soil carbon together then change by $M_C\,\text{GPP} - R_a - \int S_m\,dz$. These models include the coupling (PrognosticSoilOrganicCarbon) when the canopy carries the pools; without them, SOC is held at its initial condition.

Equilibrium and spin-up

The pools follow GPP and do not feed back on it. Under constant forcing, or periodic forcing averaged over its period, each structural pool is its allocation times its turnover time,

\[C_{leaf} = a f_{leaf} \tau_{leaf} G, \qquad C_{stem} = k G, \qquad C_{root} = a f_{root} \tau_{root} G, \qquad k = a f_{stem} \tau_{stem},\]

and the allocation is the GPP left after maintenance respiration, $G = M_C(\text{GPP} - R_d) - f_T\,(r_{stem} C_{sap} + r_{root} C_{root})$, with $f_T$ the mean of $Q_{10}^{(T - T_{ref})/10}$. The limitation of respiration by an empty sugar pool ($g \approx 1$) is neglected. As the sapwood saturates with the wood pool, $G$ solves

\[(1 + \rho)\, G + \frac{f_T\, r_{stem}\, k G}{1 + k G / C_{sap,half}} = M_C\,(\text{GPP} - R_d), \qquad \rho = f_T\, r_{root}\, a f_{root} \tau_{root},\]

a quadratic in $G$ (Canopy.equilibrium_carbon_pools). The sugar pool is the one whose allocation is $G$, found by Newton's method (Canopy.equilibrium_sugar).

The pools start empty, and wood takes decades to fill. spin_up_carbon_pools! replaces that transient: it advances a simulation over a spin-up period, averaging the drivers of the pools (GPP, $R_d$, $f_T$ at the canopy temperature, the air temperature and the tree share), then sets the leaf, wood and fine-root pools to their equilibrium under them, and $T_{annual}$ to the mean air temperature, which is also the $T_{annual}$ of the equilibrium. The sugar pool, which follows the fluxes of the last days, keeps its state and settles over weeks after such a jump, so a second spin-up over the following year leaves a state without transients.

Usage

lai_model = Canopy.ZhouOptimalLAIModel{FT}(surface_domain, toml_dict)
biomass = Canopy.PrognosticCarbonModel{FT}(lai_model, toml_dict)
land = LandModel{FT}(forcing, toml_dict, domain, Δt; biomass, prognostic_land_components)
simulation = LandSimulation(start_date, stop_date, Δt, land)
for years in 1:2
    ClimaLand.Simulations.spin_up_carbon_pools!(simulation, start_date + Year(years))
end
solve!(simulation)

The pools then take the tree share of the LAI model; around a prescribed LAI, pass one, e.g. tree_share = Canopy.clm_tree_share(surface_space). The canopy respiration is that of the pools (PoolBasedAutotrophicRespirationModel), and the cveg diagnostic reports their sum, which the vegetation leaderboard compares with the live woody carbon of Xu et al. [31].

Parameters

The wood allocation, turnover time and its temperature factor, and the fine-root respiration rate are calibrated against the live woody carbon of Xu et al. [31] and the ecosystem respiration of the carbon leaderboard (CarbonTracker NEE plus GOSIF GPP): the wood turnover time is an effective value, which includes the disturbances that keep the observed biomass below its potential. The TOML names start with carbon_; see toml/default_parameters.toml.