# 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.