# Prognostic Equations (Dry)

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

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

\(\mathbf{F}\) 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

pressure and density are defined as perturbations from a hydrostatically stratified background state, i.e.

with

## 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 \(\theta\):

## 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.