Prognostic Equations (Moist)

Model 1: Warm Moisture with no Precipitation

With this model, which is analogous to that in FASTEddy, we consider a mixture of dry air \(\rho_d\) and nonprecipitating water vapor \(\rho_v\), assumed to be a perfect ideal gas with constant heat capacities \(C_{vd}\), \(C_{vv}\), \(C_{pd}\), \(C_{pv}\), and (non-precipitating) cloud water \(\rho_c\).

Neglecting the volume occupied by all water not in vapor form, we have

\[p = p_d + p_v = \rho_d R_d T + \rho_v R_v T\]

where \(p_d\) and \(p_v\) are the partial pressures of dry air and water vapor, respectively, and \(R_d\) and \(R_v\) are the gas constants for dry air and water vapor, respectively.

We define the mixing ratio of each moist component, \(q_s\), as the mass density of species \(s\) relative to the density of dry air, i.e. \(q_s = \frac{\rho_s}{\rho_d}\).

Governing Equations

The governing equations for this model are

\[ \begin{align}\begin{aligned}\frac{\partial \rho_d}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u})\\\frac{\partial (\rho_d \mathbf{u})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \mathbf{u}) - \frac{1}{1 + q_v + q_c} \nabla p^\prime - \nabla \cdot \tau + \mathbf{F} + \delta_{i,3}\mathbf{B}\\\frac{\partial (\rho_d \theta_d)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \theta_d) + \nabla \cdot ( \rho_d \alpha_{T}\ \nabla \theta_d) + \frac{\theta_d L_v}{T_d C_{pd}} f_{cond}\\\frac{\partial (\rho_d q_v)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} q_v + \nabla \cdot (\rho_d \alpha \nabla q_v) - f_{cond}\\\frac{\partial (\rho_d q_c)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} q_c + \nabla \cdot (\rho_d \alpha \nabla q_c) + f_{cond}\end{aligned}\end{align} \]

Here \(L_v\) is the latent heat of vaporization, \(\theta_d\) is the (dry) potential temperature \(\mathbf{B}\) is the buoyancy force, which is defined in Buoyancy.

The pressure perturbation is computed as

\[p^\prime = p_0 \left( \frac{R_d \rho_d \theta_m}{p_0} \right)^\gamma - p_0\]

where \(\gamma = C_{pd} / C_{vd}\) and

\[\theta_m = \theta_d (1 + \frac{R_v}{R_d} q_v)\]

is the moist potential temperature. We note that this is the only place \(\theta_m\) is used; we evolve \(\theta_d\) above.

Model 2: Full Moisture Including Precipitation

With this model, in addition to dry air \(\rho_d\) and nonprecipitating water vapor \(\rho_v\), assumed to be a perfect ideal gas with constant heat capacities \(C_{vd}\), \(C_{vv}\), \(C_{pd}\), \(C_{pv}\), we include non-precipitating condensates \(\rho_c + \rho_i\), and precipitating condensates \(\rho_p = \rho_{rain} + \rho_{snow} + \rho_{graupel}\). Here \(\rho_c\) is the density of cloud water and \(\rho_i\) is the density of cloud ice, and we define the sum of all non-precipitating moist quantities to be \(\rho_T = \rho_v + \rho_c + \rho_i\). All condensates are treated as incompressible; cloud water and ice have constant heat capacities \(C_p\) and \(C_i\), respectively.

Neglecting the volume occupied by all water not in vapor form, we have

\[p = p_d + p_v = \rho_d R_d T + \rho_v R_v T\]

where \(p_d\) and \(p_v\) are the partial pressures of dry air and water vapor, respectively, and \(R_d\) and \(R_v\) are the gas constants for dry air and water vapor, respectively.

We define the mixing ratio of each moist component, \(q_s\), as the mass density of species \(s\) relative to the density of dry air, i.e. \(q_s = \frac{\rho_s}{\rho_d}\).

We define the total potential temperature

\[\theta = \frac{\sum_s \rho_s \theta_s}{\sum_s \rho_s} \approx (\theta_d + q_v \theta_v + q_i \theta_i + q_c \theta_c).\]

and write the EOS as

\[T = \theta (\frac{p}{p_0})^\frac{R^\star}{C_p^\star}\]

or

\[p = p_0 (\frac{\Pi}{C_p^\star})^{\frac{C_p^\star}{R^\star}}\]

where \(p_0\) is the reference pressure. and

\[\Pi = C_p^\star (\frac{p}{\alpha p_0})^\frac{R^\star}{C_p^\star}\]

with \(\alpha = \frac{R^\star}{p}(\frac{p}{p_0})^\frac{R^\star}{c_p^\star} \theta\)

here, \(R^\star = R_{d} + q_v R_{v} + q_i R_{i} + q_p R_{p}\), and \(C_p^\star = C_{pd} + q_v C_{pv} + q_i C_{pi} + q_p C_{pp}\).

\(R_d\), \(R_v\), \(R_i\), and \(R_p\) are the gas constants for dry air, water vapor, cloud ice, precipitating condensates, respectively. \(C_{pd}\), \(C_{pv}\), \(C_{pi}\), and \(C_{pp}\) are the specific heats for dry air, water vapor, cloud ice, and precipitating condensates, respectively.

Governing Equations

We assume that all species have same average speed, Then the governing equations become

\[ \begin{align}\begin{aligned}\frac{\partial \rho_d}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} + \mathbf{F}_\rho)\\\frac{\partial (\rho_d \mathbf{u})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \mathbf{u} + \mathbf{F}_u) - \frac{1}{1 + q_T + q_p} \nabla p^\prime - \nabla \cdot \tau + \mathbf{F} + \delta_{i,3}\mathbf{B}\\\frac{\partial (\rho_d \theta)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \theta + F_{\theta}) + \nabla \cdot ( \rho_d \alpha_{T}\ \nabla \theta) + F_Q\\\frac{\partial (\rho_d q_T)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} q_T +F_{q_{T}}) - Q\\\frac{\partial (\rho_d q_p)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} q_p + F_{q_{p}}) + Q\end{aligned}\end{align} \]

In this set of equations, the subgrid turbulent parameterization effects are included with fluxes \(F_\rho\), \(F_u\), \(F_C\), \(F_{\theta}\), \(F_{q_{T}}\), \(F_{q_{r}}\). \(\mathbf{F}\) stands for the external force, and \(Q\) and \(F_Q\) represent the mass and energy transformation of water vapor to/from water through condensation/evaporation, which is determined by the microphysics parameterization processes. \(\mathbf{B}\) is the buoyancy force, which is defined in Buoyancy.