Evapotranspiration

Overview

Evapotranspiration ($\text{ET}$) (m/s) is the combined process of water evaporation from soil and open water surfaces, evaporation of water intercepted by the canopy, and transpiration through leaf stomata. These processes remove water from the surface, driving latent heat flux and competing with sensible heat and ground heat fluxes in the surface energy balance.

All evapotranspiration pathways are primarily driven by the vapor pressure difference or specific humidity difference at the surface $\Delta q = q_{\text{sat}}(T_s) - q_a$, where $q_{\text{sat}}(T_s)$ is the saturation specific humidity at surface temperature $T_s$ (kg/kg) and $q_a$ is the atmospheric specific humidity (kg/kg) at a particular reference height.

Each pathway is also modulated by aerodynamic resistance(s) $r_a$ (s/m) (between surface and atmosphere) and possibly stomatal resistance $r_s$ (s/m) (in the case of transpiration).

subtypes(Terrarium.AbstractEvapotranspiration)

2-element Vector{Any}:
 BareGroundEvaporation
 PALADYNCanopyEvapotranspiration

Flux conventions: kinematic humidity vs. liquid water mass

Evapotranspiration computations involve (at least) three different physical domains, each with its own natural units: the atmosphere (humidity fluxes), the surface energy balance (energy fluxes), and soil hydrology (water mass fluxes). To maintain clarity and prevent unit errors, Terrarium distinguishes between kinematic humidity fluxes $Q_h$ used in the calculation of the energy budget and liquid water mass fluxes $E$ that drive the hydrological state of the land surface.

Kinematic humidity flux $Q_h$

The kinematic humidity flux $Q_h$ [m/s] represents the transport of water vapor through the air, scaled by air density:

\[\begin{equation} Q_h = \Delta q \cdot g \quad \text{[m/s]}, \end{equation}\]

where $\Delta q$ (kg/kg) is the specific humidity difference and $g$ (m/s) is the vapor conductance.

Liquid water mass flux $E$

The liquid water mass flux $E$ (m/s) represents the equivalent flux of liquid water that would produce the same mass transfer:

\[\begin{equation} E = Q_h \cdot \frac{\rho_a}{\rho_w} \quad \text{[m/s]}, \end{equation}\]

where $\rho_a$ (kg/m³) is air density and $\rho_w$ (kg/m³) is liquid water density.

Why the distinction matters

The ratio $\rho_a / \rho_w \approx 1.2 / 1000 \approx 1.2 \times 10^{-3}$ means that a kinematic humidity flux of $Q_h = 10^{-5}$ m/s corresponds to a liquid water flux of only $E \approx 1.2 \times 10^{-8}$ m/s. This scaling is critical for:

1. Energy balance consistency: The SEB uses $Q_h$ to compute latent heat ($H_l = \rho_a L Q_h$)
2. Hydrology consistency: Soil moisture budgets use $E$ as a water loss term
3. Mass conservation: The same physical process must remove the same mass from both atmosphere and soil

Summary

Bare ground evaporation

BareGroundEvaporation{NF, GR} <: AbstractEvapotranspiration

variables(BareGroundEvaporation(Float32))

Variables
├─ Prognostic: 
├─ Auxiliary: 
├── ground_evaporation_conductance [m s^-1] on XY{Center, Center}
├── evaporation_ground [m s^-1] on XY{Center, Center}
├─ Inputs: 
├── skin_temperature [°C] on XY{Center, Center}
├─ Namespaces:

Vegetated land evapotranspiration

struct PALADYNCanopyEvapotranspiration{NF, GR<:Terrarium.AbstractGroundEvaporationResistanceFactor} <: Terrarium.AbstractEvapotranspiration{NF}

variables(PALADYNCanopyEvapotranspiration(Float32))

Variables
├─ Prognostic: 
├─ Auxiliary: 
├── ground_evaporation_conductance [m s^-1] on XY{Center, Center}
├── canopy_evaporation_conductance [m s^-1] on XY{Center, Center}
├── transpiration_conductance [m s^-1] on XY{Center, Center}
├── evaporation_canopy [m s^-1] on XY{Center, Center}
├── evaporation_ground [m s^-1] on XY{Center, Center}
├── transpiration [m s^-1] on XY{Center, Center}
├─ Inputs: 
├── skin_temperature [°C] on XY{Center, Center}
├── ground_temperature [°C] on XY{Center, Center}
├─ Namespaces:

Evaporation from the canopy

Canopy evaporation of intercepted water $E_{\text{can}}$ (liquid water flux, m/s) depends on the saturation state of the canopy (fraction of leaves wet). The underlying kinematic humidity flux $Q_{h,\text{can}}$ is:

\[\begin{equation} Q_{h,\text{can}} = f_{\text{can}} \frac{\Delta q}{r_a} \quad \text{[m/s]}, \end{equation}\]

where $f_{\text{can}}$ is the canopy saturation fraction (0 = dry, 1 = saturated) (-) and $\Delta q$ is the specific humidity difference (kg vapor/kg air).

When $f_{\text{can}} = 0$ (completely dry canopy), $Q_{h,\text{can}} = 0$. When $f_{\text{can}} = 1$ (wet canopy), evaporation proceeds at the potential rate.

The canopy evaporation vapor conductance is computed as

\[\begin{equation} g_{\text{can}} = \frac{f_{\text{can}}}{r_a} \quad \text{[m/s]}, \end{equation}\]

so that $Q_{h,\text{can}} = g_{\text{can}} \cdot \Delta q$.

The liquid water flux $E_{\text{can}}$ is then obtained by rescaling:

\[\begin{equation} E_{\text{can}} = Q_{h,\text{can}} \cdot \frac{\rho_a}{\rho_w} \quad \text{[m/s]}. \end{equation}\]

Ground evaporation

Evaporation from exposed soil or under-canopy surfaces $E_{\text{ground}}$ (liquid water flux, m/s) is limited by soil water availability. The underlying kinematic humidity flux $Q_{h,\text{ground}}$ is:

\[\begin{equation} Q_{h,\text{ground}} = \beta \frac{\Delta q}{r_a + r_e} \quad \text{[m/s]}, \end{equation}\]

where $\beta$ is the ground evaporation resistance factor (0 to 1) (-), $r_a$ is aerodynamic resistance (s/m), and $r_e$ is the aerodynamic resistance between ground and canopy (s/m).

The ground evaporation vapor conductance is computed as

\[\begin{equation} g_{\text{ground}} = \frac{\beta}{r_a + r_e} \quad \text{[m/s]}, \end{equation}\]

so that $Q_{h,\text{ground}} = g_{\text{ground}} \cdot \Delta q$.

The liquid water flux $E_{\text{ground}}$ is then:

\[\begin{equation} E_{\text{ground}} = Q_{h,\text{ground}} \cdot \frac{\rho_a}{\rho_w} \quad \text{[m/s]}. \end{equation}\]

The resistance factor $\beta$ is computed from soil moisture in the upper layer: $\beta = 1$ when soil is wet (at field capacity) and $\beta \to 0$ as soil dries.

Transpiration

Plant transpiration occurs through stomata and is controlled by stomatal conductance. The kinematic humidity flux for transpiration $Q_{h,\text{trp}}$ is:

\[\begin{equation} Q_{h,\text{trp}} = \frac{\Delta q}{r_a + r_s} \quad \text{[m/s]}, \end{equation}\]

where $r_s = 1 / g_w$ is the stomatal resistance (s/m) and $g_w$ is the stomatal conductance (m/s) (computed from photosynthesis; see Stomatal conductance).

The transpiration vapor conductance is computed as

\[\begin{equation} g_{\text{trp}} = \frac{1}{r_a + r_s} \quad \text{[m/s]}, \end{equation}\]

so that $Q_{h,\text{trp}} = g_{\text{trp}} \cdot \Delta q$.

The liquid water flux $E_{\text{trp}}$ is then:

\[\begin{equation} E_{\text{trp}} = Q_{h,\text{trp}} \cdot \frac{\rho_a}{\rho_w} \quad \text{[m/s]}. \end{equation}\]

High stomatal conductance (when photosynthetically active) leads to low stomatal resistance and high transpiration. This creates a strong coupling between carbon uptake (photosynthesis) and water loss (transpiration).

Total evapotranspiration

The PALADYN approach combines all three pathways in parallel. The total kinematic humidity flux is:

\[\begin{equation} Q_{h,\text{total}} = Q_{h,\text{can}} + Q_{h,\text{ground}} + Q_{h,\text{trp}} \quad \text{[m/s]}, \end{equation}\]

which is converted to latent heat flux for the surface energy balance:

\[\begin{equation} H_l = \rho_a \, L_v \, Q_{h,\text{total}} \quad \text{[W/m²]}, \end{equation}\]

where $L_v$ is the latent heat of vaporization (J/kg).

The total liquid water flux (sum of component liquid water fluxes) is used as a source/sink in soil hydrology:

\[\begin{equation} E_{\text{total}} = E_{\text{can}} + E_{\text{ground}} + E_{\text{trp}} \quad \text{[m/s]}. \end{equation}\]

Evaporation flux computation

All evapotranspiration pathways share a unified functional form for the kinematic humidity flux:

\[\begin{equation} Q_h = \Delta q \cdot g \quad \text{[m/s]}, \end{equation}\]

where $\Delta q$ [kg/kg] is the specific humidity difference and $g$ [m/s] is the vapor conductance specific to each pathway. The unified function humidity_flux handles all three pathways:

humidity_flux(
    _::Terrarium.AbstractEvapotranspiration,
    Δq,
    g
) -> Any

Note: The function returns $Q_h$ (kinematic humidity flux). To obtain the liquid water mass flux $E$ for soil hydrology, multiply by the density ratio $\rho_a/\rho_w$ as shown in the equations above.

Conductance functions

The vapor conductances for each pathway are computed separately and stored as auxiliary fields during the compute_auxiliary! pass. These conductances are skin-temperature-independent (held fixed during the surface energy balance solve).

Transpiration conductance

transpiration_conductance(
    _::PALADYNCanopyEvapotranspiration{NF},
    rₐ,
    g_stm
) -> Any

Canopy evaporation conductance

canopy_evaporation_conductance(
    _::PALADYNCanopyEvapotranspiration,
    f_can,
    rₐ
) -> Any

Ground evaporation conductance

ground_evaporation_conductance(
    _::BareGroundEvaporation,
    β,
    rₐ
) -> Any

ground_evaporation_conductance(
    _::PALADYNCanopyEvapotranspiration,
    β,
    rₐ,
    rₐ_can
) -> Any

Ground resistance parameterizations

The ground evaporation resistance factor $\beta$ is computed from soil moisture using parameterizations of AbstractGroundEvaporationResistanceFactor:

struct ConstantEvaporationResistanceFactor{NF} <: Terrarium.AbstractGroundEvaporationResistanceFactor

struct SoilMoistureResistanceFactor{NF} <: Terrarium.AbstractGroundEvaporationResistanceFactor

ground_evaporation_resistance_factor(i, j, grid, fields, :AbstractGroundEvaporationResistanceFactor, args...)

Process interface

compute_auxiliary!(
    state,
    grid,
    evaporation::BareGroundEvaporation,
    ::NoCanopyInterception,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere
)
compute_auxiliary!(
    state,
    grid,
    evaporation::BareGroundEvaporation,
    ::NoCanopyInterception,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere,
    soil::Union{Nothing, Terrarium.AbstractSoil}
)
compute_auxiliary!(
    state,
    grid,
    evaporation::BareGroundEvaporation,
    ::NoCanopyInterception,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere,
    soil::Union{Nothing, Terrarium.AbstractSoil},
    snow::Union{Nothing, Terrarium.AbstractSnow}
)

compute_auxiliary!(
    state,
    grid,
    evapotranspiration::PALADYNCanopyEvapotranspiration,
    interception::Terrarium.AbstractCanopyInterception,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere,
    soil::Terrarium.AbstractSoil,
    vegetation::Terrarium.AbstractVegetation,
    args...
)

Coupling to soil hydrology

Subtypes of AbstractEvapotranpsiration automatically inherit a default implementation of the forcing interface for SoilHydrology, which computes the contribution of ET to the soil moisture tendency of the uppermost soil layer using ground_evapotranspiration_flux.

forcing(
    i,
    j,
    k,
    grid,
    clock,
    fields,
    evapotranspiration::Terrarium.AbstractEvapotranspiration,
    _::Terrarium.AbstractSoilHydrology,
    args...
) -> Any

Kernel functions

ground_evapotranspiration_flux(i, j, grid, fields, ::AbstractEvapotranspiration)

compute_surface_humidity_flux(
    i,
    j,
    grid,
    fields,
    evaporation::BareGroundEvaporation,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere,
    args...
) -> Any

compute_surface_humidity_flux(
    i,
    j,
    grid,
    fields,
    evtr::PALADYNCanopyEvapotranspiration,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere,
    args...
) -> Any

compute_evapotranspiration_conductances!(
    out,
    i,
    j,
    grid,
    fields,
    evaporation::BareGroundEvaporation,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere
) -> Any
compute_evapotranspiration_conductances!(
    out,
    i,
    j,
    grid,
    fields,
    evaporation::BareGroundEvaporation,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere,
    soil::Union{Nothing, Terrarium.AbstractSoil}
) -> Any

compute_evapotranspiration_conductances!(
    out,
    i,
    j,
    grid,
    fields,
    evapotranspiration::PALADYNCanopyEvapotranspiration,
    interception::Terrarium.AbstractCanopyInterception,
    constants::PhysicalConstants,
    atmos::Terrarium.AbstractAtmosphere,
    soil::Terrarium.AbstractSoil,
    vegetation::Terrarium.AbstractVegetation,
    args...
) -> Any