# Equations

## Introduction

A dynamic core ASAM (All Scale Atmopsheric Model)of the moist compressible Euler equation in conservative form is presented.

The thermodynamic equations applied here differ slightly from those used in most numerical models. Traditionally, the specific heats of water vapor and liquid water are ignored in numerical models, so that $R_m \approx R$, $c_{pml} \approx c_p$, and $c_{vml} \approx c_v$, yielding the traditional potential temperature equation. Unlike the complex bulk microphysical models typically employed in a majority of mesoscale models, a simple and differentiable parameterization that converts water vapor into total cloud substance was utilized in the model so far. The microphysical model is completed by the addition of the traditional bulk parameterization of the falling of rain. The orography is incorporated in the model through a special grid system, where the orography is represented by cut cells in a Cartesian grid. The time integration is accomplished by a linear implicit method of Rosenbrock type. Because the method is fully implicit, the approach is able to employ time steps that result in Courant-Friedrichs-Lewy (CFL) numbers greater than one for advection, gravity, and sound waves; however, the dynamical time scale of the problem will be respected for accuracy by a dynamic time step procedure.

## The dry compressible Euler equation

$\begin{matrix} \frac{\partial \rho}{\partial t} +\nabla (\rho \mathbf{v}) & = & 0 \\[0,2cm] \frac{\partial \rho \mathbf{v}}{\partial t}+\nabla (\rho \mathbf{v} \cdot \mathbf{v}) & = & - \nabla p - \rho\mathbf{g} - 2 \Omega\times(\rho \mathbf{v}) \\[0,2cm] \frac{\partial \rho \theta}{\partial t}+\nabla (\rho\mathbf{v} \theta) & = & Q_{\theta} \\[0,2cm] \quad p & = & \rho R \theta (p/p_{0})^\kappa \quad \mbox{or} \\[0,2cm] \quad p & = & \left(\frac{R\rho \theta}{p_0^{\kappa}}\right)^{1/(1-\kappa)} \end{matrix}$

## The moist compressible Euler equation

Some thermodynamic definitions

• Density $\rho\,$ is now the density of the moist air.
• New densities for water vapor $\rho_v\,$ and cloud water $\rho_l\,$.
• Define mixing ratios $r_v=\frac{\rho_v}{\rho_d}$ and $r_l=\frac{\rho_l}{\rho_d}$.
• Specific heat of moist air at constant pressure $c_{pml}=c_p+c_{pv}r_v+c_{pl}r_l\,$.
• Specific heat of moist air at constant volume $c_{vml}=c_v+c_{vv}r_v+c_{pl}r_l\,$.
• Gas constant of moist air $R_m=R+R_vr_v\,$.
• Latent heat of vaporization $L_v=L_{v0}-(c_{pl}-c_{pv})(T-T_0)\,$.

Redefined dry potential temperature

$\tilde{\pi} =\left(\frac{p}{p_0}\right)^{R_m/c_{pml}},\quad \theta=\frac{T}{\tilde{ \pi}}$

After Bryan and Fritsch the following equation for the new defined dry potential temperature $\theta\,$ can be derived in the moist case

$\frac{D \ln \theta}{Dt}=-\frac{L_v}{c_{pml}T}\frac{Dr_v}{Dt}-\ln \tilde {\pi} \left(\left(\frac{R_v}{R_m}-\frac{c_{pv}}{c_{pml}}\right)\frac{Dr_v}{Dt}-\frac{c_{pl}}{c_{pml}}\frac{Dr_l}{Dt}\right)$

together with equations for the mixing ratios $r_v\,$ and $r_l\,$

$\frac{D r_v}{Dt}=- \frac{D r_l}{Dt}=r_{Cond}$ ,

where $r_{Cond}\,$ is the transfer rate of condensation. With this definition the dry potential temperature is conserved if no phase changes occur. This is not the case for the classical definition of the Exner pressure, where $\pi =\left(\frac{p}{p_0}\right)^{R/c_{p}}$.

Density potential temperature

Reformulated equation of state

$p=\rho RT\frac{1+r_v/\epsilon}{1+r_v+r_l} \quad \mbox{with} \quad \epsilon=R_d/R_v$.

Density potential temperature

$\theta_\rho=\theta \frac{1+r_v/\epsilon}{1+r_v+r_l}$.

With this definition we have

$p=\rho R\theta_\rho\left(\frac{p}{p_0}\right)^{R_m/c_{pml}} \quad \mbox{or} \quad p=\left(\frac{R\Theta_\rho}{p_0^{\kappa_m}}\right)^{1/(1-\kappa_m)}$

with $\kappa_m=R_m/c_{pml}\,$ and $\Theta_\rho=\rho\theta_\rho\,$. Use the product rule to derive an equation for $\theta_\rho\,$.

$\frac{D\theta_\rho}{Dt} = \frac{D\theta}{Dt}\frac{1+r_v/\epsilon}{1+r_v+r_l} + \frac{\theta}{1+r_v+r_l}\frac{1}{\epsilon}\frac{Dr_v}{Dt} - \frac{1+r_v/\epsilon}{(1+r_v+r_l)^2} \left( \frac{Dr_v}{Dt}+\frac{Dr_l}{Dt} \right)$

and convert to a flux form representation with respect to the full density.

$\begin{matrix} \frac{\partial \rho \theta_{\rho}}{\partial t}+\nabla (\rho {\mathbf{v}} \theta_{\rho}) & = & Q_{\theta_{\rho}} \\[0,2cm] \frac{\partial \rho q_v}{\partial t}+\nabla (\rho {\mathbf{v}} q_v ) & = & Q_{q_l} \\[0,2cm] \frac{\partial \rho q_l}{\partial t}+\nabla (\rho {\mathbf{v}} q_l) & = & Q_{q_v} \\[0,2cm] \end{matrix}$

Representation of $\dot{q}_v=-\dot{q}_l\,$ in the absence of rain.

$\dot{q}_v=-\Phi_c(q_v-qvs)q_{lmax}\tanh\left(\frac{q_l}{q_{lmax}}\right)$

where $\Phi_c\,$ is a relaxation factor whose value depends on the grid size, $qvs\,$ is the saturation ratio and $q_{lmax}\,$ is a constant (1.e − 5).

Changes in the numerics

• More complicated source term in the equation for the "potential temperature".
• Additional equations for the water substances.
• Inclusion of the fall velocity in the advection routine.
• Dependency of the pressure from $q_v\,$ and $q_l\,$ is not taken in to account in the Jacobian part of the momentum equation.

## Transformations in spherical and cartesian systems

The spherical coordinates (λ,φ) used are defined in the ranges as follows:

$\lambda = \left[ 0; 2\pi \right]$
$\phi = \left[ -\pi/2 ; \pi/2 \right]$

spherical coordinates $\rightarrow$ cartesian coordinates

$x = r \cdot \cos{\phi} \cdot \cos{\lambda}$
$y = r \cdot \sin{\phi} \cdot \cos{\lambda}$
$z = r \cdot \sin{\phi}$

cartesian coordinates $\rightarrow$ spherical coordinates

$\lambda = \operatorname{atan2} \left( y,x \right)$
$\phi = \arcsin{\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}}$
$r = \sqrt{x^{2}+y^{2}+z^{2}}$

unit vectors in curvilinear system

$\begin{pmatrix} \mathbf{\hat{e}_{r}} \\ \mathbf{\hat{e}_{\lambda}} \\ \mathbf{\hat{e}_{\phi}} \end{pmatrix} = \begin{pmatrix} -\sin{\lambda} & \cos{\lambda} & 0 \\ -\cos{\lambda}\sin{\phi} & -\sin{\lambda} \sin{\phi} & \cos{\phi} \\ \cos{\lambda} \cos{\phi} & \sin{\lambda} \cos{\phi} & \sin{\phi} \\ \end{pmatrix}^{\mathbf{T}} \begin{pmatrix} \mathbf{e_{x}} \\ \mathbf{e_{y}} \\ \mathbf{e_{z}} \end{pmatrix}$

$\mathbf{\hat{e}_{r}} = \begin{pmatrix} -\sin{\lambda} \\ -\cos{\lambda}\sin{\phi} \\ \cos{\lambda} \cos{\phi} \end{pmatrix} , \qquad \mathbf{\hat{e}_{\lambda}} = \begin{pmatrix} \cos{\lambda} \\ -\sin{\lambda} \sin{\phi} \\ \sin{\lambda} \cos{\phi} \end{pmatrix} , \qquad \mathbf{\hat{e}_{\phi}} = \begin{pmatrix} 0 \\ \cos{\phi} \\ \sin{\phi} \end{pmatrix}$

unit vectors in cartesian system

$\begin{pmatrix} \mathbf{e_{x}} \\ \mathbf{e_{y}} \\ \mathbf{e_{z}} \end{pmatrix} = \begin{pmatrix} -\sin{\lambda} & \cos{\lambda} & 0 \\ -\cos{\lambda}\sin{\phi} & -\sin{\lambda} \sin{\phi} & \cos{\phi} \\ \cos{\lambda} \cos{\phi} & \sin{\lambda} \cos{\phi} & \sin{\phi} \\ \end{pmatrix} \begin{pmatrix} \mathbf{\hat{e}_{r}} \\ \mathbf{\hat{e}_{\lambda}} \\ \mathbf{\hat{e}_{\phi}} \end{pmatrix}$

$\mathbf{e_{x}} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} , \qquad \mathbf{e_{y}} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} , \qquad \mathbf{e_{z}} = \begin{pmatrix} 0 \\0 \\ 1\end{pmatrix}$