# Prognostic Equations (Dry)¶

The following partial differential equations governing dry compressible flow are solved in ERF for mass, momentum, potential temperature, and scalars:

\begin{align}\begin{aligned}\frac{\partial \rho}{\partial t} &= - \nabla \cdot (\rho \mathbf{u}),\\\frac{\partial (\rho \mathbf{u})}{\partial t} &= - \nabla \cdot (\rho \mathbf{u} \mathbf{u}) - \nabla p^\prime + \delta_{i,3}\mathbf{B} - \nabla \cdot \tau + \mathbf{F},\\\frac{\partial (\rho \theta)}{\partial t} &= - \nabla \cdot (\rho \mathbf{u} \theta) + \nabla \cdot ( \rho \alpha_{T}\ \nabla \theta) + F_{\rho \theta},\\\frac{\partial (\rho C)}{\partial t} &= - \nabla \cdot (\rho \mathbf{u} C) + \nabla \cdot (\rho \alpha_{C}\ \nabla C)\end{aligned}\end{align}

where

• $$\tau$$ is the viscous stress tensor,

$\tau_{ij} = -2\mu \sigma_{ij},$

with $$\sigma_{ij} = S_{ij} -D_{ij}$$ being the deviatoric part of the strain rate, and

$S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right), \hspace{24pt} D_{ij} = \frac{1}{3} S_{kk} \delta_{ij} = \frac{1}{3} (\nabla \cdot \mathbf{u}) \delta_{ij},$
• $$\mathbf{F}$$ and $$F_{\rho \theta}$$ are the forcing terms described in Physical Forcings,

• $$\mathbf{g} = (0,0,-g)$$ is the gravity vector,

• the potential temperature $$\theta$$ is defined from temperature $$T$$ and pressure $$p$$ as

$\theta = T \left( \frac{p_0}{p} \right)^{R_d / c_p}.$
• pressure and density are defined as perturbations from a hydrostatically stratified background state, i.e.

$p = \overline{p}(z) + p^\prime \hspace{24pt} \rho = \overline{\rho}(z) + \rho^\prime$

with

$\frac{d \overline{p}}{d z} = - \overline{\rho} g$

## Assumptions¶

The assumptions involved in deriving these equations from first principles are:

• Continuum behavior

• Ideal gas behavior ($$p = \rho R_d T$$) with constant specific heats ($$c_p,c_v$$)

• Constant mixture molecular weight (therefore constant $$R_d$$)

• Viscous heating is negligible

• No chemical reactions, second order diffusive processes or radiative heat transfer

• Newtonian viscous stress with no bulk viscosity contribution (i.e., $$\kappa S_{kk} \delta_{ij}$$)

• Depending on the simulation mode, the transport coefficients $$\mu$$, $$\rho\alpha_C$$, and $$\rho\alpha_T$$ may correspond to the molecular transport coefficients, turbulent transport coefficients computed from an LES or PBL model, or a combination. See the sections on DNS vs. LES modes and PBL schemes for more details.

## Diagnostic Relationships¶

In order to close the above prognostic equations, a relationship between the pressure and the other state variables must be specified. This is obtained by re-expressing the ideal gas equation of state in terms of $$\theta$$:

$p = \left( \frac{\rho R_d \theta}{p_0^{R_d / c_p}} \right)^\gamma = p_0 \left( \frac{\rho R_d \theta}{p_0} \right)^\gamma$

## Nomenclature¶

Here $$\rho, T, \theta$$, and $$p$$ are the density, temperature, potential temperature and pressure, respectively; these variables are all defined at cell centers. $$C$$ is an advected quantity, i.e., a tracer, also defined at cell centers. $$\mathbf{u}$$ and $$(\rho \mathbf{u})$$ are the velocity and momentum, respectively, and are defined on faces.

$$R_d$$ and $$c_p$$ are the gas constant and specific heat capacity for dry air respectively, and $$\gamma = c_p / (c_p - R_d)$$ . $$p_0$$ is a reference value for pressure.