Thermodynamics and the Working Fluid

The thermodynamic formulation underpins the model's conservation properties. ClimaAtmos prognoses the specific total energy of moist air, so the definitions of internal energy, enthalpy, and latent heat determine whether energy budgets close. This page sets out that formulation, following [1].

The formulation is implemented in Thermodynamics.jl, shared with the other CliMA components, and the constants are defined in ClimaParams.jl. This page explains the assumptions and the reasoning; consult the Thermodynamics.jl documentation for the function-level interface.

The working fluid

The working fluid is moist air: a mixture of dry air and water vapor, both treated as ideal gases, together with condensed water — suspended in clouds and falling as precipitation.

All constituents are in thermal equilibrium, at the same temperature. Condensates may sediment or fall, but they carry the air's temperature. The model therefore neglects temperature differences between condensate and ambient air, such as those from evaporative cooling of raindrops, when computing fluid energetics. Microphysical process rates are unaffected.

Constituents need not be in full thermodynamic equilibrium. Out-of-equilibrium phases such as supercooled liquid are therefore permitted, and the condensed phases can carry their own prognostic equations with microphysics schemes in which non-equilibrium phases coexist. Suspended condensate and precipitation are part of the working fluid, which suits bulk microphysics schemes that represent these components through moments of a particle size distribution.

Composition is described by mass fractions: dry air $q_d$, water vapor $q_v$, liquid $q_l$, and ice $q_i$. The condensate fraction is $q_c = q_l + q_i$, and the total water specific humidity is $q_t = q_v + q_c$. Since these account for every component, $q_t + q_d = 1$.

Bulk microphysics schemes often split the condensate further into suspended cloud condensate and precipitation, $q_l = q_l^{cl} + q_l^{pr}$ and $q_i = q_i^{cl} + q_i^{pr}$. The equations of motion allow for that distinction; see Microphysics.

Equation of state

The pressure of the working fluid is the sum of the partial pressures of dry air and water vapor. The volume — but not the mass — of the condensed phases is neglected, because the specific volume of condensate is smaller than that of the gas phases by a factor of about $10^3$. The equation of state is then

\[p = \rho R_m T,\]

with the specific mixture gas constant of moist air

\[R_m(q) = R_d (1 - q_t) + R_v q_v = R_d \left[ 1 + (\epsilon_{dv} - 1) q_t - \epsilon_{dv} q_c \right],\]

which depends on the specific humidities through the ratio of the gas constants of water vapor and dry air, $\epsilon_{dv} = R_v / R_d \approx 1.61$.

A calorically perfect fluid

The primary thermodynamic assumption is that the fluid is calorically perfect: the isochoric specific heat capacities of the constituents are constants. The isobaric capacities are then constant too, and the capacities of moist air are the mass-weighted sums of those of its constituents,

\[c_{vm}(q) = c_{vd} + (c_{vv} - c_{vd}) q_t + (c_{vl} - c_{vv}) q_l + (c_{vi} - c_{vv}) q_i,\]

with the standard relation $c_{pm}(q) = c_{vm}(q) + R_m(q)$ carrying over from the constituents to the mixture. Real specific heat capacities vary slightly with temperature; treating them as constant introduces an error below 1% for dry air and at most a few percent for the water phases.

The assumption has a consequence for latent heats [3]. Kirchhoff's relation, $dL/dT = \Delta c_p$, integrates under constant specific heats to a latent heat that is linear in temperature,

\[L(T) = L_0 + \Delta c_p (T - T_0),\]

for a reference temperature $T_0$ and the latent heat $L_0$ at that temperature. This applies to vaporization, fusion, and sublimation, and the expected relation $L_s(T) = L_v(T) + L_f(T)$ follows.

Energies and enthalpies

Specific internal energies of the constituents are referenced to $T_0$ [4]. Two reference values may be chosen independently, because dry air and water cannot be converted into one another: the reference specific internal energy of liquid water is set to zero and that of dry air to $-R_d T_0$, which simplifies the later enthalpy and enthalpy-flux expressions. The internal energy of moist air is the mass-weighted sum,

\[I(T, q) = c_{vm}(q) (T - T_0) + (q_t - q_c) I_{v,0} - q_i I_{i,0} - (1 - q_t) R_d T_0 ,\]

which can be inverted for temperature. That inversion is how the model recovers temperature when specific internal energy and the specific humidities are the primary thermodynamic state variables.

Specific enthalpies follow by adding $p_\mu / \rho_\mu$ to each constituent's internal energy, neglecting the specific volumes of the condensed phases:

\[h(T, q) = c_{pm}(q) (T - T_0) + (q_t - q_c) L_{v,0} - q_i L_{f,0} = I(T, q) + R_m T .\]

Enthalpy is central to describing transport, appearing in both the grid-scale and the subgrid-scale flux terms of the governing equations.

Saturation vapor pressure

The Clausius–Clapeyron relation, combined with the linear latent heat above, integrates to a closed-form expression for the saturation vapor pressure — the Rankine–Kirchhoff approximation [5, 6]. For a mixture of liquid and ice out of thermodynamic equilibrium, as in mixed-phase clouds, a thermodynamically consistent saturation vapor pressure uses the liquid-fraction-weighted average [7], $L = \lambda_f L_v + (1 - \lambda_f) L_s$ with $\lambda_f = q_l / q_c$.

The expression is invariant to the choice of reference temperature, provided the reference latent heats are shifted with it. With the triple point of water as the reference, the saturation vapor pressure over liquid between 248 and 325 K lies within 0.4% of measured values, and the ratio of saturation vapor pressure over ice to that over liquid between 233 and 273 K within 0.6% [1].

Optional local thermodynamic equilibrium

The model can optionally assume local thermodynamic equilibrium for cloudy air — excluding falling precipitation — as many atmosphere models do. Gibbs' phase rule then implies that three thermodynamic state variables suffice to determine the partitioning into water phases, so the suspended condensate specific humidities are obtained by saturation adjustment from the cloudy-air total humidity, density, and internal energy, while the precipitation humidities remain prognostic.

Strict equilibrium requires the liquid fraction to be a Heaviside function of temperature. Supercooled liquid is out of equilibrium and exists between the homogeneous ice nucleation temperature and the freezing temperature; a ramp function between the two is often used to represent it [8]. That ramp is available in the code, and it remains an approximation, since out-of-equilibrium phases depend on the history of the air mass as well as on its thermodynamic state. By default the model gives the condensed phases their own prognostic equations instead. See Microphysics for the available schemes.

Reference temperature invariance

The reference temperature $T_0$ is arbitrary, and the model's physics must not depend on it. The formulation is constructed so that it does not: a shift $T_0 \to T_0 + \delta T_0$, with the reference latent heats shifted consistently by Kirchhoff's relation, offsets each constituent's specific internal energy and enthalpy by a constant.

The offset for dry air differs from that for water, but the offsets for all three water phases are identical. The extra terms in the total energy equation therefore sum to zero through the total water conservation law, leaving the dynamics unchanged. The same holds for the surface enthalpy flux boundary condition. What remains of the dependence on $T_0$ is the accuracy of the linear approximation to the latent heats, which is why the triple point of water, a value typical for the atmosphere, is the choice made here.

This invariance gives a check on any new energy flux added to the model: a flux that breaks it is not thermodynamically consistent, and will not conserve energy (see Conservation Properties). The subgrid-scale enthalpy flux decomposition in Hyperdiffusion and in the vertical diffusion scheme follows from this requirement.

Where this is implemented

The choice between equilibrium and non-equilibrium condensate is a configuration option; see Configuration Options and Microphysics.