Slab Lake Model

The slab lake model represents inland water bodies (lakes, reservoirs) as a fixed-depth, fully mixed water column that exchanges energy with the atmosphere at its surface and with the underlying sediment at its base. The lake evolves energy prognostically and can undergo freezing and thawing, while maintaining a constant depth. In ClimaLand, the slab lake replaces the soil surface at grid points identified as inland water by a user-provided mask.

Model Overview

The slab lake is a single-layer energy balance model. At each timestep:

1. Surface fluxes (radiation, sensible heat, latent heat, precipitation enthalpy) drive the lake energy budget
2. Heat is exchanged with the underlying soil sediment through thermal conduction
3. Runoff adjusts to maintain constant lake depth (the fixed-depth constraint)
4. Lake temperature and liquid fraction are diagnosed from internal energy

The key simplification is that the lake is fully mixed: there is no vertical temperature stratification within the water column.

Governing Equation

The prognostic variable is the lake internal energy per unit lake area, $U$ (J m$^{-2}$):

\[\frac{dU}{dt} = -Q_{\rm sfc} - Q_{\rm runoff} + Q_{\rm sed}\]

where we have:

• the net surface energy flux into the atmosphere, $Q_{\rm sfc}$, per unit lake area (W m$^{-2}$)
• the energy carried away by runoff, $Q_{\rm runoff}$, per unit lake area (W m$^{-2}$)
• the heat flux from the lake to the underlying sediment, per unit lake area, $Q_{\rm sed}$ (W m$^{-2}$)

Surface Energy Flux

The surface energy balance includes radiative, turbulent, and precipitation terms.

\[Q_{\rm sfc} = R_n + \text{LHF} + \text{SHF} + P_{\rm liq} \, \rho_l \, c_{p,l} \, (T_{\rm air} - T_0)\]

where $R_n$ is the net radiation (adjusted for lake albedo), LHF and SHF are the latent and sensible heat fluxes, and the last term is the enthalpy flux of liquid precipitation arriving at the air temperature $T_{\rm air}$.

Lake-Sediment Heat Exchange

Heat is exchanged between the lake mixed layer and the top soil layer through a series-conductance model:

\[Q_{\rm sed} = -G_{\rm eff} \left(T_{\rm lake} - T_{\rm soil}\right)\]

where the effective conductance $G_{\rm eff}$ combines the lake conductance $G$ with the soil thermal conductance:

\[G_{\rm eff} = \frac{1}{\frac{1}{G} + \frac{\Delta z_{\rm soil}}{2 \, \kappa_{\rm soil}}}\]

• the lake conductance, $G$ (W m$^{-2}$ K$^{-1}$), a single tunable parameter
• the top soil layer thermal conductivity, $\kappa_{\rm soil}$ (W m$^{-1}$ K$^{-1}$)
• the top soil layer thickness, $\Delta z_{\rm soil}$ (m)
• the lake and top soil temperatures, $T_{\rm lake}$ and $T_{\rm soil}$ (K)

The conductance $G$ encapsulates the lake-side thermal conductance (combining lake depth, conductivity, and mixing effects) into a single parameter that can be calibrated.

Fixed-Depth Constraint and Runoff

The lake maintains a constant depth. The net surface water flux is:

\[F_{\rm water} = P_{\rm liq} + E_{\rm liq} + E_{\rm ice}\]

where $P_{\rm liq}$ is the liquid precipitation rate, $E_{\rm liq}$ is the liquid evaporation rate, and $E_{\rm ice}$ is the ice sublimation rate. To enforce constant depth, the runoff is defined as:

\[R_{\rm lake} = -F_{\rm water}\]

Positive runoff corresponds to net precipitation (water drains away), while negative runoff corresponds to net evaporation (water must be supplied to maintain depth). The associated energy flux is:

\[Q_{\rm runoff} = R_{\rm lake} \cdot \rho \widehat{e}(T_{\rm lake}, q_l)\]

where $\rho \widehat{e}$ is the volumetric internal energy (blending ice and liquid properties according to $q_l$) evaluated at the lake temperature. Both runoff (effluent) and supplied water are assumed to carry energy at the lake temperature $T_{\rm lake}$. For runoff this is physically motivated: precipitation equilibrates with the lake before draining away as effluent. For evaporation-driven supply (negative runoff), the same assumption is adopted for simplicity, though in reality the source water temperature may differ.

Thermodynamics and Phase Change

The lake temperature and liquid fraction are diagnosed from the internal energy. The model distinguishes three regimes.

Fully frozen ($q_l \leq 0$)

\[T_{\rm lake} = T_0 + \frac{U + \rho_l \cdot d \cdot L_{f,0}}{\rho_l \cdot d \cdot c_{p,i}}\]

Mixed phase ($0 < q_l < 1$)

\[T_{\rm lake} = T_{\rm freeze}\]

Fully liquid ($q_l \geq 1$)

\[T_{\rm lake} = T_0 + \frac{U}{\rho_l \cdot d \cdot c_{p,l}}\]

where we have:

• the reference temperature, $T_0$ (K)
• the density of liquid water, $\rho_l$ (kg m$^{-3}$)
• the lake depth, $d$ (m)
• the specific latent heat of fusion, $L_{f,0}$ (J kg$^{-1}$)
• the specific heat capacities of ice and liquid water, $c_{p,i}$ and $c_{p,l}$ (J kg$^{-1}$ K$^{-1}$)
• the freezing temperature, $T_{\rm freeze}$ (K)

Liquid Fraction

The liquid fraction $q_l$ is determined by linear interpolation between the fully frozen and fully liquid energy states:

\[q_l = \frac{U - U_{\rm ice}}{U_{\rm liq} - U_{\rm ice}}\]

where the bounding energies are:

\[\begin{align} U_{\rm ice} &= \rho_l \cdot d \cdot c_{p,i} \cdot (T_{\rm freeze} - T_0) - \rho_l \cdot d \cdot L_{f,0} \\ U_{\rm liq} &= \rho_l \cdot d \cdot c_{p,l} \cdot (T_{\rm freeze} - T_0) \end{align}\]

\[U_{\rm ice}\]

is the internal energy when the lake is fully frozen at $T_{\rm freeze}$, and $U_{\rm liq}$ is the energy when fully liquid at $T_{\rm freeze}$.

Surface Albedo

The lake surface albedo varies with phase state, interpolating linearly between open-water and ice-covered values:

\[\alpha_{\rm lake} = q_l \cdot \alpha_{\rm liquid} + (1 - q_l) \cdot \alpha_{\rm ice}\]

Both the PAR and NIR albedo bands use this same value. Default values are $\alpha_{\rm liquid} = 0.08$ and $\alpha_{\rm ice} = 0.6$.

Model Assumptions

1. Well-mixed column: The lake has no vertical temperature gradient. This is a reasonable approximation for shallow or wind-mixed lakes.
2. Fixed depth: Lake volume does not change. Excess water is removed as runoff; water deficits are implicitly supplied.
3. No horizontal transport: Each lake grid point is independent.
4. Phase-dependent albedo: Albedo transitions smoothly between open water and ice.
5. No vegetation: Canopy processes are not active at lake points.
6. Isolated sediment: The soil beneath the lake has zero water and energy flux at the top boundary (except for the lake-sediment heat flux), so it is hydrologically isolated regardless of its saturation state.

Parameters

Prognostic Variables

Diagnostic Variables

Inland Water Mask

The inland water mask is a gridded fractional inland water coverage field. In the convenience LandModel constructors it is derived internally from the default IMERG-based dataset; for manual SlabLakeModel construction, a custom mask field can be provided directly. This is used to weight the lake fluxes, total energy, and total water per unit lake area and make them per unit grid area.