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% $Header: /u/u0/gcmpack/mitgcmdoc/part1/appendix_atmos.tex,v 1.5 2001/09/27 01:57:17 cnh Exp $ |
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% $Name: $ |
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\section{Appendix ATMOSPHERE} |
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\subsection{Hydrostatic Primitive Equations for the Atmosphere in pressure |
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coordinates} |
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\label{sect-hpe-p} |
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The hydrostatic primitive equations (HPEs) in p-coordinates are: |
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\begin{eqnarray} |
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\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} |
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_{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}} |
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\label{eq:atmos-mom} \\ |
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\frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\ |
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\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ |
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\partial p} &=&0 \label{eq:atmos-cont} \\ |
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p\alpha &=&RT \label{eq:atmos-eos} \\ |
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c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} |
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\label{eq:atmos-heat} |
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\end{eqnarray} |
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where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure |
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surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot |
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\mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total |
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derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is |
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the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp |
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}{Dt}$ is the vertical velocity in the $p-$coordinate. Equation \ref |
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{eq-p-firstlaw} is the first law of thermodynamics where internal energy $ |
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e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $ |
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p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing. |
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It is convenient to cast the heat equation in terms of potential temperature |
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$\theta $ so that it looks more like a generic conservation law. |
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Differentiating \ref{eq-p-eos-start} we get: |
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\[ |
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p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt} |
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\] |
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which, when added to the heat equation \ref{eq-p-firstlaw} and using $ |
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c_{p}=c_{v}+R$, gives: |
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\begin{equation} |
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c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q} |
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\label{eq-p-heat-interim} |
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\end{equation} |
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Potential temperature is defined: |
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\begin{equation} |
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\theta =T(\frac{p_{c}}{p})^{\kappa } \label{eq:potential-temp} |
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\end{equation} |
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where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience |
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we will make use of the Exner function $\Pi (p)$ which defined by: |
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\begin{equation} |
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\Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa } \label{Exner} |
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\end{equation} |
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The following relations will be useful and are easily expressed in terms of |
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the Exner function: |
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\[ |
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c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi |
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}{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{ |
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\partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p} |
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\frac{Dp}{Dt} |
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\] |
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where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy. |
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The heat equation is obtained by noting that |
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\[ |
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c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta |
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\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt} |
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\] |
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and on substituting into (\ref{eq-p-heat-interim}) gives: |
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\begin{equation} |
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\Pi \frac{D\theta }{Dt}=\mathcal{Q} \label{eq:potential-temperature-equation} |
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\end{equation} |
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which is in conservative form. |
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For convenience in the model we prefer to step forward (\ref{theta-equation} |
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) rather than (\ref{eq-p-firstlaw}). |
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\subsubsection{Boundary conditions} |
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The upper and lower boundary conditions are : |
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\begin{eqnarray} |
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\mbox{at the top:}\;\;p=0 &&\text{, }\omega =\frac{Dp}{Dt}=0 \\ |
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\mbox{at the surface:}\;\;p=p_{s} &&\text{, }\phi =\phi _{topo}=g~Z_{topo} |
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\label{eq:boundary-condition-atmosphere} |
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\end{eqnarray} |
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In $p$-coordinates, the upper boundary acts like a solid boundary ($\omega |
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=0 $); in $z$-coordinates and the lower boundary is analogous to a free |
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surface ($\phi $ is imposed and $\omega \neq 0$). |
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\subsubsection{Splitting the geo-potential} |
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For the purposes of initialization and reducing round-off errors, the model |
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deals with perturbations from reference (or ``standard'') profiles. For |
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example, the hydrostatic geopotential associated with the resting atmosphere |
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is not dynamically relevant and can therefore be subtracted from the |
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equations. The equations written in terms of perturbations are obtained by |
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substituting the following definitions into the previous model equations: |
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\begin{eqnarray} |
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\theta &=&\theta _{o}+\theta ^{\prime } \label{eq:atmos-ref-prof-theta} \\ |
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\alpha &=&\alpha _{o}+\alpha ^{\prime } \label{eq:atmos-ref-prof-alpha}\\ |
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\phi &=&\phi _{o}+\phi ^{\prime } \label{eq:atmos-ref-prof-phi} |
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\end{eqnarray} |
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The reference state (indicated by subscript ``0'') corresponds to |
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horizontally homogeneous atmosphere at rest ($\theta _{o},\alpha _{o},\phi |
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_{o}$) with surface pressure $p_{o}(x,y)$ that satisfies $\phi |
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_{o}(p_{o})=g~Z_{topo}$, defined: |
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\begin{eqnarray*} |
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\theta _{o}(p) &=&f^{n}(p) \\ |
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\alpha _{o}(p) &=&\Pi _{p}\theta _{o} \\ |
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\phi _{o}(p) &=&\phi _{topo}-\int_{p_{0}}^{p}\alpha _{o}dp |
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\end{eqnarray*} |
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%\begin{eqnarray*} |
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%\phi'_\alpha & = & \int^p_{p_o} (\alpha_o -\alpha) dp \\ |
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%\phi'_s(x,y,t) & = & \int_{p_o}^{p_s} \alpha dp |
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%\end{eqnarray*} |
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The final form of the HPE's in p coordinates is then: |
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\begin{eqnarray} |
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\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} |
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_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} |
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\\ |
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\frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 |
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\\ |
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\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ |
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\partial p} &=&0 \\ |
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\frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } |
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\\ |
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\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \label{eq:atmos-prime} |
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\end{eqnarray} |
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