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.