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| 1 | +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 $g_{s}$. |
| 2 | +## Usage in ClimaLand |
| 3 | +The P-model differs from other canopy component models in two main ways: |
| 4 | +1) The P-model encompasses *both* a photosynthesis model *and* a stomatal conductance model. Therefore, `PModel` must be used for photosynthesis iff `PModelConductance` is used for stomatal conductance. |
| 5 | +2) The P-model requires an extra callback `PModelCallback` which 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 via `SciMLBase`, this callback can be constructed like so: |
| 6 | + |
| 7 | +```julia |
| 8 | +# make the callback |
| 9 | +pmodel_cb = ClimaLand.make_PModel_callback(FT, start_date, t0, dt, canopy) |
| 10 | + |
| 11 | +# add this callback to the CallbackSet with driver, diag |
| 12 | +cb = SciMLBase.CallbackSet(driver_cb, diag_cb, pmodel_cb); |
| 13 | +``` |
| 14 | + |
| 15 | +**To be implemented**: when using the P-model within a LandSimulation, automatically detect this and use the P-model callback. |
| 16 | +## Theory |
| 17 | +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. |
| 18 | +$$ \mathrm{Cost} = a \frac{E}{A} + b \frac{V_{c\max}}{A} $$ |
| 19 | +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 |
| 20 | +$$ a \frac{\partial (E/A)}{\partial \chi} = -b \frac{\partial (V_{c\max} / A)}{ \partial \chi} $$ |
| 21 | +The second assumption, called the "coordination hypothesis" (see Chen et al., 1993), states that $V_{c\max}$ |
| 22 | +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. |
| 23 | +## Implementation (under construction) |
| 24 | +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: |
| 25 | +$$ |
| 26 | +\tau \dfrac{ d\bar{x}}{dt} = \bar{x} - x \implies \bar{x}_{t+1} = \alpha \bar{x}_{t} + (1 - \alpha) x_{t+1} |
| 27 | +$$ |
| 28 | +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). |
| 29 | + |
| 30 | +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 |
| 31 | +$$ |
| 32 | +V_{c\max} = V_{c\max 25}^{\mathrm{opt}} \cdot \underbrace{ \exp \left( \dfrac{\Delta H_{a}(T - T_{0})}{T_{0} TR} \right) }_{ \text{activation (Arrhenius)} } \cdot \underbrace{ \dfrac{1 + e^{(T_{0}\Delta S - \Delta H_{d})/RT_{0}}}{1 + e^{(T\Delta S - \Delta H_{d})/RT}} }_{ \text{deactivation} } |
| 33 | +$$ |
| 34 | +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 $c_{i}$ |
| 35 | +$$ |
| 36 | +c_{i} =\dfrac{ \xi^\mathrm{opt} c_{a} + \Gamma^* \sqrt{ \max(D,0) }}{\xi^\mathrm{opt} + \sqrt{ \max(D,0) }} |
| 37 | +$$ |
| 38 | +where $D$ is the vapor pressure deficit (VPD). Finally, carboxylation- and light-limited assimilation is computed as |
| 39 | +$$ |
| 40 | +\begin{align*} |
| 41 | +A_{c} &= V_{c\max} \dfrac{c_{i} - \Gamma^*}{c_{i} + K} \\ |
| 42 | +A_{j} &= \dfrac{1}{4} \underbrace{ \dfrac{4\phi_{0} I_{abs}}{\sqrt{ 1 + \left(\dfrac{4\phi_{0}I_{abs}}{J_{\max}}\right)^2 }} }_{ J } \cdot \dfrac{c_{i} - \Gamma^*}{c_{i} + 2 \Gamma^*} |
| 43 | +\end{align*} |
| 44 | +$$ |
| 45 | +The remaining steps for computing $A_{n}$ and GPP are identical to the Farquhar model. |
| 46 | + |
| 47 | +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 |
| 48 | +$$ |
| 49 | +g_{s} = \dfrac{D_{H_{2}O}}{D_{CO_{2}}} \dfrac{A_{n}}{c_{a}- c_{i}} |
| 50 | +$$ |
| 51 | +where $\dfrac{D_{H_{2}O}}{D_{CO_{2}}} \approx 1.6$ is treated as a constant (`Drel`) in our model. |
| 52 | +## Citations |
| 53 | +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) |
| 54 | + |
| 55 | +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](https://doi.org/10.1029/2021MS002767) |
| 56 | + |
| 57 | +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](https://doi.org/10.5194/gmd-13-1545-2020) |
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