\section{Pressure coordinate} The primitive equations in the $(\lambda , \mu, p)$ -coordinates without scaling. That means $D$ and $zeta$ in Appendix A and B have the units: $s^-1$, $T$ is in $K$, $p$ in $Pa$, $\phi$ in $m^2 s^{-2}$ and $\vec{\nu}$ in $m s^{-1}$.\\ Conservation of momentum (vorticity and divergence equation) \begin{equation} {\displaystyle \frac{\partial \zeta}{\partial t} = - \vec{\nu} \cdot \bigtriangledown (\zeta + f) - \omega \frac{\partial \zeta}{\partial p} - (\zeta + f) \bigtriangledown \cdot \vec{\nu} + \vec{k} \cdot (\frac{\partial \vec{\nu}}{\partial p} \times \bigtriangledown \omega) + P_\zeta} \end{equation} \begin{equation} {\displaystyle \frac{\partial D}{\partial t} = \vec{k} \cdot \bigtriangledown \times (\zeta + f) \vec{\nu} - \bigtriangledown \cdot (\omega \frac{\partial \vec{\nu}}{\partial p}) - \bigtriangledown^2 (\phi + \frac{\vec{\nu}^2}{2}) + P_D} \end{equation} Hydrostatic approximation (using the equation of state) \begin{equation} {\displaystyle \frac{\partial \phi}{\partial p} = - \frac{1}{\rho} = - \frac{RT}{p}} \end{equation} Conservation of mass (continuity equation) \begin{equation} {\displaystyle \bigtriangledown \cdot \vec{\nu} + \frac{ \partial \omega}{\partial p} = 0} \end{equation} Thermodynamic equation ( J= diabatic heating per unit mass) \begin{equation} {\displaystyle \frac{d T}{d t} = \frac{\omega}{c_p \rho} + \frac{J}{c_p} + P_T} \end{equation} \section{Sigma-system} $\sigma=p/p_s$ ranges monotonically from zero at the top of the atmosphere to unity at the ground. For $\xi=x,y $ or $t$ \begin{equation} {\displaystyle (\frac{\partial }{\partial \xi})_p =\frac{\partial }{\partial \xi} -\sigma \frac{\partial \ln p_s}{\partial \xi} \frac{\partial }{\partial \sigma}} \end{equation} \begin{equation} \frac{\partial }{\partial p} =\frac{\partial \sigma }{\partial p} \frac{\partial }{\partial \sigma} =\frac{1 }{p_s} \frac{\partial }{\partial \sigma} \end{equation} The vertical velocity in the p-coordinate system $\omega$ and in the new $\sigma$-coordinate system $\dot {\sigma}$ are given by \cite{phillips1957} \begin{equation} \omega= \frac{p}{p_s} [\vec{V} \cdot \nabla p_s - \int\limits_{0}^{\sigma} \nabla \cdot p_s \vec{V} d \sigma] = p [\vec{V} \cdot \nabla \ln p_s] - p_s \int\limits_{0}^{\sigma} A d \sigma \end{equation} \begin{equation} \dot {\sigma}= \sigma \int\limits_{0}^{1} A d \sigma - \int\limits_{0}^{\sigma} A d \sigma \end{equation} with $A=D+\vec{V} \cdot \nabla \ln p_s = \frac{1}{p_s} \nabla \cdot p_s \vec{V}$. The primitive equations in the $(\lambda , \mu, \sigma)$ -coordinates without scaling Conservation of momentum (vorticity and divergence equation) \begin{equation} {\displaystyle \frac{\partial \zeta}{\partial t} = \frac{1}{a(1 - \mu^2)} \frac{\partial F_\nu}{\partial \lambda} - \frac{1}{a} \frac{\partial F_u}{\partial \mu} + P_\zeta} \end{equation} \begin{equation} {\displaystyle \frac{\partial D}{\partial t} = \frac{1}{a (1 - \mu^2)} \frac{\partial F_u}{\partial \lambda} + \frac{1}{a} \frac{ \partial F_\nu}{\partial \mu} - \bigtriangledown^2 (E + \phi +T_0 \ln p_s) + P_D} \end{equation} Hydrostatic approximation (using the equation of state) \begin{equation} {\displaystyle \frac{\partial \phi}{\partial \ln \sigma} = - TR} \end{equation} Conservation of mass (continuity equation) \begin{equation} {\displaystyle \frac{\partial \ln p_s}{\partial t} = - \frac{U}{a (1 - \mu^2)} \frac{\partial \ln p_s}{\partial \lambda} - \frac{V}{a} \frac{\partial \ln p_s}{\partial \mu} - D - \frac{\partial \dot{\sigma}}{\partial \sigma} = - \int\limits_{0}^{1} (D+\vec{V} \cdot \nabla \ln p_s) d \sigma} \end{equation} Thermodynamic equation ( J= diabatic heating per unit mass) \begin{equation} {\displaystyle \frac{\partial T}{\partial t} = F_T - \dot{\sigma} \frac{\partial T}{\partial \sigma} + \kappa T [\vec{V} \cdot \nabla \ln p_s - \frac{1}{\sigma}\int\limits_{0}^{\sigma} A d \sigma] +\frac{J}{c_p} + P_T} \end{equation} ${\displaystyle E = \frac{U^2 + V^2}{2(1 - \mu^2)} }$ ${\displaystyle F_u = ( \zeta + f ) V - \dot{\sigma} \frac{\partial U}{\partial \sigma} - \frac{RT}{a} \frac{\partial \ln p_s}{\partial \lambda}} $ ${\displaystyle F_\nu = - (\zeta + f)U - \dot{\sigma} \frac{\partial V}{\partial\sigma} - (1 - \mu^2) \frac{RT}{a} \frac{\partial \ln p_s}{\partial \mu}} $ ${\displaystyle F_T = - \frac{U}{a(1-\mu^2)} \frac{\partial T}{\partial \lambda} - \frac{V}{a} \frac{\partial T}{\partial \mu} } $ $A=D+\vec{V} \cdot \nabla \ln p_s = \frac{1}{p_s} \nabla \cdot p_s \vec{V}$. \section{Matrix {\em B}} For the implicit scheme, fast (linear) gravity modes and the slower non-linear terms are separated. \\ ${\displaystyle\frac{ \partial D }{\partial t}= { N_D} - \bigtriangledown^2 (\phi + T_0 \ln p_s)} $\\ ${\displaystyle \frac{\partial \ln p_s}{\partial t} = N_p - \int\limits_{0}^{1} D d \sigma}$\\ ${\displaystyle \frac{\partial T'}{\partial t} = N_T- [ \sigma \int\limits_{0}^{1} D d \sigma - \int\limits_{0}^{\sigma} D d \sigma ] \frac{\partial T_0}{\partial \sigma} + \kappa T_0 [- \int\limits_{0}^{\sigma} D d \ln \sigma] }$\\ ${\displaystyle \frac{\partial \phi}{\partial \ln \sigma} = - T} $\\ The set of differential equations are approximated by its finite difference analogues using the notation (for each variable $D$, $T$, $\ln p_s$, and $\phi$)\\ ${\displaystyle \overline{Q}^t = 0.5 (Q^{t + \Delta t} + Q^{t - \Delta t}) =Q^{t - \Delta t} + \Delta t \delta_t Q} $ and ${\displaystyle \delta_t Q = \frac{Q^{t + \Delta t} - Q^{t - \Delta t}}{2 \Delta t}}$\\ The hydrostatic approximation using an angular momentum conserving finite-difference scheme is solved at half levels\\ ${\displaystyle \phi_{r+0.5}-\phi_{r-0.5}= T_r \cdot \ln \frac{\sigma_{r+0.5}}{\sigma_{r-0.5}}}$\\ Full level values of geopotential are given by\\ ${\displaystyle \phi_{r}=\phi_{r+0.5}+\alpha_r T_r }$ with ${\displaystyle \alpha_r=1-\frac{\sigma_{r-0.5}}{\Delta \sigma_r} \ln \frac{\sigma_{r+0.5}}{\sigma_{r-0.5}}}$ and $ \Delta \sigma_r=\sigma_{r+0.5} - \sigma_{r-0.5}$\\ Now, the implicit formulation for the divergence is derived using the conservation of mass, the hydrostatic approximation and the thermodynamic equation at discrete time steps\\ ${\displaystyle \delta_t { D} = { N_D} - \bigtriangledown^2 (\overline{\phi}^t + T_0 [\ln p_s^{t - \Delta t} + \Delta t \delta_t \ln p_s])} $\\ ${\displaystyle \delta_t \ln p_s = N_p - L_p [D^{t - \Delta t} + \Delta t \delta_t D]}$\\ ${\displaystyle \overline{ \phi - \phi_s}^t = L_{\phi} [T^{t - \Delta t} + \Delta t \delta_t T}]$\\ ${\displaystyle \delta_t T' = N_T - L_T [D^{t - \Delta t} + \Delta t \delta_t D]} $\\ The set of differential equations for each level $ k (k=1,..,n)$ written in vector form leads to the matrix $ {\cal B}$ with n rows and n columns. The matrix $ {\cal B} = {\cal L}_{\phi} {\cal L}_T + \vec{T}_0 \vec{L}_p = {\cal B}(\sigma , \kappa , \vec{T}_0)$ is constant in time. The variables ${\vec{D},\vec{T},\vec{T}',\vec{\phi}-\vec{\phi}_s}$ $\vec{N}_D$ and $\vec{N}_T$ are represented by column vectors with values at each level. $L_p$, $L_T$ and $L_{\phi}$ contain the effect of the divergence (or the gravity waves) on the surface pressure tendency, the temperature tendency and the geopotential.\\ $\vec{L}_p =(\Delta \sigma_1, ..., \Delta \sigma_n)$ is a row vector with $ \Delta \sigma_n=\sigma_{n+0.5} - \sigma_{n-0.5}$.\\ ${\cal L}_{\phi}= {\left(\begin{array}{*{5}{c}} 1 & \alpha_{21} &\alpha_{31}& \cdots & \alpha_{n1} \\ 0 &\alpha_{22} &\alpha_{32}& \ddots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & \alpha_{nn} \\ \end{array} \right)} $\\ For $i=j: {\displaystyle \alpha_{jj}=1- [ \frac{\sigma_{j-0.5}}{\sigma_{j+0.5}-\sigma_{j-0.5}} (\ln \sigma_{j+0.5} - \ln \sigma_{j-0.5})]}$\\ $i>j: \alpha_{ij}=\ln \sigma_{j+0.5} - \ln \sigma_{j-0.5}$\\ $ij$: $ \gamma_{ij}= \frac{1}{2} \Delta \sigma_{i} [ \Delta T_{0.5} \sigma_{1} ] $ \\ for $j>1$ and\\ $i=j$: $ \gamma_{jj}= \frac{1}{2} [\Delta T_{j-0.5} \sigma_{j-0.5} + \Delta T_{j+0.5} (\sigma_{j+0.5}-1) ] $ \\ $ij$: $ \gamma_{ij}= \frac{\Delta \sigma_{i}}{2 \Delta \sigma_{j} } [\Delta T_{j-0.5} \sigma_{j-0.5} + \Delta T_{j+0.5} \sigma_{j+0.5} ] $ \\ %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: