Leaf water potential

Input: ψ_soil (soil water supply)
       E (transpiration rate from energy balance)
       LSC (soil-to-leaf conductance from soil_resistance)
       h (leaf height)
       C (plant capacitance)
       ψ_leaf_old (current leaf water potential)
       dt (time step)
                    │
                    ▼
    ┌──────────────────────────────────────┐
    │  Gravitational head                  │
    │  head = ρ_water · g  (MPa/m)         │
    └──────────────────────────────────────┘
                    │
                    ▼
    ┌──────────────────────────────────────┐
    │  Steady-state target                 │
    │  a = ψ_soil − head·h − E/LSC         │
    │      ↑          ↑         ↑          │
    │    supply    gravity   transpiration │
    │    from      penalty   drawdown      │
    │    soil                              │
    └──────────────────────────────────────┘
                    │
                    ▼
    ┌──────────────────────────────────────┐
    │  Time constant                       │
    │  τ = C / LSC  (seconds)              │
    └──────────────────────────────────────┘
                    │
                    ▼
    ┌──────────────────────────────────────┐
    │  Analytical integration              │
    │  Δψ = (a − ψ₀) · (1 − e^(−dt/τ))     │
    │  ψ_leaf_new = ψ₀ + Δψ                │
    └──────────────────────────────────────┘
                    │
                    ▼
            Return ψ_leaf_new
                    │
          (used by StomataEfficiency
           to check: ψ_leaf > ψ_min ?)

source

leaf_water_potential


def leaf_water_potential(
    physcon:PhysCon, # Physical constants:
- grav : float
    Gravitational acceleration (m/s2).
- denh2o : float
    Density of liquid water (kg/m3).
    leaf:Leaf, # Leaf parameters:
- capac : float
    Plant capacitance (mmol H2O/m2 leaf area/MPa).
    flux:Flux, # Flux variables:
- height : float
    Leaf height (m).
- lsc : float
    Leaf-specific conductance (mmol H2O/m2 leaf/s/MPa).
- psi_leaf : float
    Leaf water potential at beginning of time step (MPa).
- psi_soil : float
    Weighted soil water potential (MPa).
- etflx : float
    Leaf transpiration flux (mol H2O/m2 leaf/s).
- dt : float
    Model time step (s).
)->float: # Leaf water potential at end of time step (MPa).

Calculate leaf water potential for the current transpiration rate.

Solves the plant capacitance equation for the change in leaf water potential over a model time step. The leaf water potential is driven toward a steady-state value set by the balance between soil water supply and transpiration demand, with the rate of approach governed by the plant capacitance.

The steady-state target is:

a = ψ_soil - ρ·g·h - E / LSC

where the three terms represent soil water potential (i.e Pre-dawn), the gravitational head for lifting water to leaf height, and the transpiration-driven drawdown through the soil-to-leaf pathway.

The change in ψ_leaf is integrated analytically as:

Δψ = (a - ψ_leaf) · (1 - exp(-dt / τ))

where τ = C / LSC is the time constant (plant capacitance divided by leaf-specific conductance).

The governing ODE is dy/dt = (a - y) / τ, where y = ψ_leaf, a is the steady-state target, and τ = C / LSC. This is integrated exactly over the time step dt to give:

dy = (a - y) · (1 - exp(-dt / τ))

The factor 1000 in the transpiration term converts flux.etflx from mol H2O/m2/s to mmol H2O/m2/s for consistency with LSC units.

The behaviour:

If dt ≪ τ (short time step relative to the time constant): the exponential ≈ 1 − dt/τ, so Δψ ≈ (a − ψ₀) · dt/τ — a small linear step toward the target
If dt ≫ τ (long time step): the exponential → 0, so Δψ ≈ a − ψ₀ — the leaf jumps all the way to the steady state
If ψ₀ = a already (leaf is at equilibrium): Δψ = 0 — nothing changes

The bathtub analogy applied to dy = (a - y) · (1 - exp(-dt / τ))

A leaf can be viewed as a bathtub where the inflow represents the water supply from the soil and the outflow represents the transpiration. In this analogy the water potential is the bathtub water level. In the absence of capacitance, what comes in immediately comes out (Eq 13.6). However, to represent the effect of capacitance over the water potential, a tau_b term is incorporated (Eq 13.11). tau_b is the ratio between bathtub size and pipe water supply (capacitance / leaf-specific conductance) and represents the time it takes the water level to change. Therefore, if the bathtub is big (large capacitance) and the water supply pipe is small, then the ratio will be large, indicating that it takes a lot of time to change the water level (water potential).

In other words tau_b it is the ratio that determines the respose time for the change in water potential

Parameters:

PhysCon: Physical constants:
- grav: Gravitational acceleration (m/s2).
- denh2o: Density of liquid water (kg/m3).
Leaf: Leaf parameters:
- capac: Plant capacitance (mmol H2O/m2 leaf area/MPa).
Flux: Flux variables:
- height: Leaf height (m).
- lsc: Leaf-specific conductance (mmol H2O/m2 leaf/s/MPa).
- psi_leaf: Leaf water potential at beginning of time step (MPa).
- psi_soil: Weighted soil water potential (MPa).
- etflx: Leaf transpiration flux (mol H2O/m2 leaf/s).
- dt: Model time step (s).

Returns:

leaf_wp: Leaf water potential at end of time step (MPa).

leaf_water_potential

Example leaf_water_potential()