The P-model is model for photosynthesis and stomatal conductance that extends the Farquhar model in two main ways: 1) it assumes that all plants adjust their internal CO2 concentration such that carbon assimilation is maximized relative to the combination of two costs—the cost of supporting RuBisCo-limited photosynthesis and the cost of transpiration through stomata; 2) it assumes that RuBisCo-limited and light-limited assimilation rates are equal, i.e., that $A_c = A_j$ (coordination hypothesis). These two additional constraints allow for the prediction of photosynthetic parameters such as $V_{c\max}$, $J_{\max}$, and stomatal conductance $g_{s}$.
Usage in ClimaLand
The P-model differs from other canopy component models in two main ways:
- The P-model encompasses both a photosynthesis model and a stomatal conductance model. Therefore,
PModelmust be used for photosynthesis iffPModelConductanceis used for stomatal conductance. - The P-model requires an extra callback
PModelCallbackwhich checks if it is local noon every model timestep and updates the optimal parameters according to local noon (see Implementation for scientific background). When you are constructing a simulation viaSciMLBase, this callback can be constructed like so:
# make the callback
pmodel_cb = ClimaLand.make_PModel_callback(FT, start_date, t0, dt, canopy)
# add this callback to the CallbackSet with driver, diag
cb = SciMLBase.CallbackSet(driver_cb, diag_cb, pmodel_cb);To be implemented: when using the P-model within a LandSimulation, automatically detect this and use the P-model callback.
Theory
The first assumption, which we'll call the "cost minimization principle", can be mathematically written as follows. We define a cost which is dependent on the transpiration $E$, maximum rate of Rubisco limited photosynthesis $V_{c\max}$, normalized to the rate of assimilation $A$. $a$ and $b$ are dimensionless numbers that represent the relative weight of each cost. $ \mathrm{Cost} = a \frac{E}{A} + b \frac{V{c\max}}{A} $ Minimization of this cost represents the general principle that plants seek to maximize assimilation while keeping water loss low ($E$ term) and not overly investing in synthesizing Rubisco protein. The parameter which is adjusted is \chi = \dfrac{c{i}}{c{a}}$, the ratio of internal to ambient (external) CO2 concentration. Thus, the first constraint is mathematically written as $ a \frac{\partial (E/A)}{\partial \chi} = -b \frac{\partial (V{c\max} / A)}{ \partial \chi} $ The second assumption, called the "coordination hypothesis" (see Chen et al., 1993), states that $V_{c\max}$ varies according to APAR such that it is neither in excess or in deficit of what is required for full utilization of the light. Mathematically, this means that $A_{c} = A_{j}$. These two constraints, when added to the Farquhar model, allows for the computation of $V_{c\max}$ from environmental conditions.
Implementation (under construction)
We follow the scheme of Mengoli et al., (2022) who propose updating optimal parameters $V_{c\max 25}$, $J_{\max25}$, and $\xi$ according to conditions at local noon (near maximal APAR). These are then timestepped via the forward Euler discretization of the following ODE: $ \tau \dfrac{ d\bar{x}}{dt} = \bar{x} - x \implies \bar{x}{t+1} = \alpha \bar{x}{t} + (1 - \alpha) x_{t+1} $ where $\bar{x}$ is the acclimated parameter, $x$ is the parameter computed at local noon, $\alpha = 1 - \dfrac{\Delta t}{\tau}$ and $\tau$ is the timescale of acclimation. Since we update this equation every local noon, $\Delta t$ is 1 day and $\tau = 15$ days corresponds to $\alpha = \dfrac{14}{15} \approx 0.933$ (the default timescale used in Mengoli 2022).
At every model timestep, the latest acclimated values $V_{c\max 25}^{\mathrm{opt}}$, $J_{\max 25}^{\mathrm{opt}}$, are then adjusted to instantaneous values via modified-Arrhenius type functions of form $ V{c\max} = V{c\max 25}^{\mathrm{opt}} \cdot \underbrace{ \exp \left( \dfrac{\Delta H{a}(T - T{0})}{T{0} TR} \right) }{ \text{activation (Arrhenius)} } \cdot \underbrace{ \dfrac{1 + e^{(T{0}\Delta S - \Delta H{d})/RT{0}}}{1 + e^{(T\Delta S - \Delta H{d})/RT}} }{ \text{deactivation} } $ where \Delta H{a}$, $\Delta H_{d}$ are standard enthalpies of activation and deactivation, $T_{0} = 298.15$ K is the reference temperature, and $\Delta S$ is the entropy change in deactivation. The acclimated $\xi^\mathrm{opt}$ is used to compute the instantaneous intercellular CO2 concentration $c_{i}$ $ c{i} =\dfrac{ \xi^\mathrm{opt} c{a} + \Gamma^* \sqrt{ \max(D,0) }}{\xi^\mathrm{opt} + \sqrt{ \max(D,0) }} $ where $D$ is the vapor pressure deficit (VPD). Finally, carboxylation- and light-limited assimilation is computed as $ \begin{align} A{c} &= V{c\max} \dfrac{c_{i} - \Gamma^}{c{i} + K} \ A{j} &= \dfrac{1}{4} \underbrace{ \dfrac{4\phi{0} I{abs}}{\sqrt{ 1 + \left(\dfrac{4\phi{0}I{abs}}{J{\max}}\right)^2 }} }{ J } \cdot \dfrac{c{i} - \Gamma^*}{c{i} + 2 \Gamma^} \end{align} $ The remaining steps for computing $A_{n}$ and GPP are identical to the Farquhar model.
The P-model also gives a stomatal conductance model because it predicts $\chi = \dfrac{c_{a}}{c_{i}}$. From Fick's law, we have that $ g{s} = \dfrac{D{H{2}O}}{D{CO{2}}} \dfrac{A{n}}{c{a}- c{i}} $ where $\dfrac{D_{H_{2}O}}{D_{CO_{2}}} \approx 1.6$ is treated as a constant (Drel) in our model.
Citations
Chen, J.-L., Reynolds, J. F., Harley, P. C. & Tenhunen, J. D. Coordination theory of leaf nitrogen distribution in a canopy. Oecologia 93, 63–69 (1993)
Mengoli, G., Agustí-Panareda, A., Boussetta, S., Harrison, S. P., Trotta, C., & Prentice, I. C. (2022). Ecosystem photosynthesis in land-surface models: A first-principles approach incorporating acclimation. Journal of Advances in Modeling Earth Systems, 14, e2021MS002767. https://doi.org/10.1029/2021MS002767
Stocker, B. D., Wang, H., Smith, N. G., Harrison, S. P., Keenan, T. F., Sandoval, D., Davis, T., & Prentice, I. C. (2020). P-model v1.0: An optimality-based light use efficiency model for simulating ecosystem gross primary production. Geoscientific Model Development, 13(3), 1545–1581. https://doi.org/10.5194/gmd-13-1545-2020