Forest water and energy balance (theory)
Outline
- Preliminary concepts: hydraulics and drought effects
- Forest water balance in
medfate - Transpiration and photosynthesis under the basic model
- Transpiration and photosynthesis under the advanced model
- Plant drought stress and cavitation
- Basic vs. advanced models: a summary of differences
Water potential
The water potential (\(\Psi\)) is the potential energy of water, relative to pure water under reference conditions. It quantifies the tendency of water to move from one area to another.
It has pressure units (e.g. MPa) and can be divided into different components:
But not all components are equally relevant in all contexts
Plant pressure volume curves
The pressure volume curve of a plant tissue or organ is the relationship between relative water content (\(RWC\), in \(kg \, H_2O / kg \, \, H_2O\) at saturation) and the corresponding water potential ( \(\Psi\), in MPa).
The relationship between \(\Psi\) and \(RWC\) is formulating by separating \(\Psi\) into osmotic (solute) potential (\(\Psi_{S}\)) and the turgor pressure potential (\(\Psi_{P}\)):
\[\Psi = \Psi_{S} + \Psi_{P}\] where \[\Psi_{P} = -\pi_0 -\epsilon\cdot (1.0 - RWC)\] and \[ \Psi_{S} = \frac{-\pi_0}{RWC}\]
where \(\pi_0\) (MPa) is the osmotic potential at full turgor (i.e. when \(RWC = 1\)) and \(\epsilon\) is the modulus of elasticity (i.e. the slope of the relationship).
When \(\Psi \leq \Psi_{tlp}\), the water potential at turgor loss point, then \(\Psi_{P} = 0\) and \(\Psi = \Psi_{S}\). If \(\Psi > \Psi_{tlp}\) then the two components are needed.
Hydraulic conductance and vulnerability curves
Hydraulic conductance \(k\) measures how much flux exists along a pathway segment (e.g. soil, stem, leaves, … ) for a given difference in water potential.
Hydraulic conductance decreases when air replaces water in any segment of the pathway.
The vulnerability curve specifies the relationship between water potential (\(\Psi\)) and hydraulic conductance (\(k\)) of a given segment.
Rhizosphere
Conductance is modelled as a van Genuchten (1980) function:
\[k(\Psi) = k_{max} \cdot v^{(n-1)/(2\cdot n)} \cdot ((1-v)^{(n-1)/n}-1)^2\]
Xylem
Conductance is modelled using a Weibull or a Sigmoid:
\[k(\Psi) = k_{max}\cdot e^{-((\Psi/d)^c)}\]
\[k(\Psi) = \frac{k_{max}}{1 + e^{(slope/25) \cdot (\Psi - \Psi_{50})}}\] 
Water potential drop in plants
When stomata are closed (e.g. during night), plant leaf water potential is assumed to be in equilibrium with the water potential in the rhizosphere (neglecting gravity effects).
When stomata are open, a larger transpiration flow (\(E\)) implies a larger drop in water potential along the transpiration pathway due to the negative pressure (suction) that arises 1:
Drought impacts on plants
The decrease in soil water potential caused by drought has multiple effects on plants 1, with some processes ceasing to occur and others becoming important or being promoted, depending on the plant response strategy 2.
Water balance components
The water balance models in available in medfate simulate the following vertical water flows in a given forest stand.
| Component | Symbol | Description |
|---|---|---|
| Infiltration | \(If\) | Water entering the soil from above |
| Capillarity rise | \(Cr\) | Water entering the soil via capillarity from a lower saturated layer |
| Deep drainage | \(Dd\) | Water percolating beyond the root zone |
| Saturation excess | \(Se\) | Excess of water in the soil |
| Soil evaporation | \(Es\) | Evaporation from soil surface |
| Woody transpiration | \(Tr_{woody}\) | Woody plant transpiration |
| Herb transpiration | \(Tr_{herb}\) | Herbaceous plant transpiration |
Variations in soil water content can be summarized as: \[\Delta{V_{soil}} = (If + Cr) - (Dd + Se + Es + Tr_{herb} + Tr_{woody})\]
Soil water inputs
If rainfall occurs during a given day, three processes are simulated to update the water content in soil layers:
Sub-models involved in water inputs
Soil water outputs
Regardless of precipitation, soil moisture can be modified due to the following processes:
Sub-models involved in water outputs
Important
Soil water uptake by plants, hydraulic redistribution and transpiration are modelled differently depending on the water balance model: basic vs advanced.
Soil water fluxes
Three submodels are available to simulate water movement into, out of and within the soil:
A. Multi-bucket model: Water inputs and drainage during a rainy day (and may be the next if over field capacity).
B. Single-domain model: Vertical water movement any day following gravitational and matric potentials (Richards equation). Assumes an homogeneous porous media.
C. Dual permeability model: Flows in the soil matrix following the previous model. Flows in the macropore domain following gravitational forces. The two domains exchange water.
Sub-models of fluxes in the soil
Warning
The three sub-models differ greatly in computational demand (see Exercise 2b).
Maximum canopy transpiration
Maximum canopy transpiration \(Tr_{\max}\) depends on potential evapotranspiration, \(PET\), and the amount of transpirating surface, i.e. the stand leaf area index, thanks to an empirical relationship by Granier 1:
\[\frac{Tr_{\max}}{PET}= -0.006\cdot (LAI^{\phi}_{stand})^2+0.134\cdot LAI^{\phi}_{stand}\]
and therefore:
\[Tr_{\max} = PET \cdot \left( -0.006\cdot (LAI^{\phi}_{stand})^2+0.134\cdot LAI^{\phi}_{stand} \right)\]
Maximum canopy transpiration is divided among plant cohorts according to the amount of light absorbed by each one.
Note
Granier’s equation is actually species-specific in medfate.
Actual plant transpiration
Actual plant transpiration depends on soil moisture and is calculated for each soil layer \(s\) separately.
A relative whole-plant water conductance, \(k_{rel}\) is defined for any given soil layer \(s\) using:
\[k_{rel}(\Psi_s) = \exp \left \{\ln{(0.5)}\cdot \left[ \frac{\Psi_{s}}{\Psi_{extract}}\right] ^r \right \}\]
where \(\Psi_{extract}\) is the water potential at which transpiration is 50% of maximum, and \(\Psi_s\), the water potential in layer \(s\).
The water extracted by a plant cohort from soil layer \(s\) and transpired, \(Tr_{s}\), is the product:
\[Tr_{s} = Tr_{\max} \cdot k_{rel}(\Psi_{s}) \cdot FRP_{s}\] where \(FRP_{s}\) is the proportion of plant fine roots in layer \(s\).
Important
This transpiration model allows emulating stomatal closure in response to soil water deficit but do not allow modelling stomatal responses to other factors.
Advanced features
The advanced transpiration and photosynthesis model operates at sub-daily time steps.
Temperature and radiation inputs are temporally disaggregated.
- The model is explicit with respect to many additional processes:
| Process | Source |
|---|---|
| Soil & canopy energy balance | Best et al. (2011) Geosci. Mod. Dev. 4, 677-699 |
| Canopy turbulence | Katul et al. (2004) Bound. Lay. Met. 113, 81-109 |
| Sunlit/shade leaf photosynthesis | De pury & Farquhar (1997) Plant, Cell & Env., 20, 537–557 |
| Direct/diffuse short-wave extinction model | Anten & Bastiaans (2016) Canopy photosynthesis: From basics to application |
| Long-wave radiation model | Flerchinger et al. (2009) Wan. J. Life Sci. 57, 5-15 |
| Plant hydraulics & stomatal regulation | [next slides] |
Sperry and Sureau sub-models
Sperry
- Steady-state plant hydraulics 1.
- Optimality-based stomatal regulation 2.
Sureau
- Plant hydraulics of SurEau-ECOS 3, including plant water storage.
- Stomatal regulation based on a semi-empirical model.
Sperry sub-model: Supply function
The supply function describes the steady-state rate of water flow, \(E\), as a function of water potential drop.
The steady-state flow rate \(E_i\) through any element \(i\) is related to the flow-induced drop in water potential across that element, \(\Delta \Psi_i = \Psi_{down} - \Psi_{up}\), by the integral of the vulnerability curve \(k_i(\Psi)\) 1:
\[E_i = \int_{\Psi_{up}}^{\Psi_{down}}{k_i(\Psi) d\Psi}\]
where \(\Psi_{up}\) and \(\Psi_{down}\) are the upstream and downstream water potential values.
The supply function can be integrated across the whole hydraulic network.
\[E(\Psi_{leaf}) = \int_{\Psi_{soil}}^{\Psi_{leaf}}{k(\Psi) d\Psi}\]
Sperry sub-model: Stomatal regulation
1. Cost function
The hydraulic supply function is used to derive a cost function \(\theta(\Psi_{leaf})\), reflecting the increasing damage from cavitation.
\[\theta(\Psi_{leaf}) = \frac{k_{c,max}-k_{c}(\Psi_{leaf})}{k_{c,max}-k_{crit}}\]
where \(k_c(\Psi_{leaf}) = dE/d\Psi(\Psi)\) is the slope of the (whole-plant) supply function.
2. Gain function
The normalized photosynthetic gain function \(\beta(\Psi_{leaf})\) reflects the increase in assimilation rate, with respect to the maximum.
\[\beta(\Psi_{leaf}) = \frac{A(\Psi_{leaf})}{A_{max}}\]
where \(A_{max}\) is the instantaneous maximum (gross) assimilation rate estimated over the full \(\Psi_{leaf}\) range.
3. Profit function
Stomatal regulation can be effectively estimated by determining the maximum of the profit function: \[Profit(\Psi_{leaf}) = \beta(\Psi_{leaf})-\theta(\Psi_{leaf})\]
The maximization is achieved when the slopes of the gain and cost functions are equal: \[\frac{\delta \beta(\Psi_{leaf})}{\delta \Psi_{leaf}} = \frac{\delta \theta(\Psi_{leaf})}{\delta \Psi_{leaf}}\]
Cavitation
If cavitation has occurred in previous steps then the capacity of the plant to transport water is impaired via the estimation of percent loss conductance (PLC).
Basic model
Estimation of PLC:
\[PLC_{stem} = 1 - \exp \left \{ \ln{(0.5)}\cdot \left[ \frac{\Psi_{plant}}{\Psi_{critic}} \right] ^r \right \}\]
Effect on plant transpiration:
\[k_{rel}^{PLC}(\Psi_{s}) = \min \{k_{rel}(\Psi_{s}), 1.0 - PLC_{stem} \}\]
Advanced model
Estimation of PLC:
\(PLC_{stem} = 1 - \frac{k_{stem}(\Psi_{stem})}{k_{max,stem}}\)
Effect on the stem vulnerability curve:
M.C. Escher - Waterfall, 1961
