--- manual/s_overview/appendix_atmos.tex 2001/08/08 16:16:18 1.1.1.1 +++ manual/s_overview/appendix_atmos.tex 2001/09/27 01:57:17 1.5 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/Attic/appendix_atmos.tex,v 1.1.1.1 2001/08/08 16:16:18 adcroft Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/Attic/appendix_atmos.tex,v 1.5 2001/09/27 01:57:17 cnh Exp $ % $Name: $ \section{Appendix ATMOSPHERE} @@ -10,23 +10,24 @@ The hydrostatic primitive equations (HPEs) in p-coordinates are: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}} -\label{eq-p-hmom-start} \\ +\label{eq:atmos-mom} \\ \frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\ -\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{% -\partial p} &=&0 \label{eq-p-cont-start} \\ -p\alpha &=&RT \label{eq-p-eos-start} \\ -c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} \label{eq-p-firstlaw} +\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ +\partial p} &=&0 \label{eq:atmos-cont} \\ +p\alpha &=&RT \label{eq:atmos-eos} \\ +c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} +\label{eq:atmos-heat} \end{eqnarray} where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is -the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp% +the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation \ref -{eq-p-firstlaw} is the first law of thermodynamics where internal energy $% -e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $% +{eq-p-firstlaw} is the first law of thermodynamics where internal energy $ +e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $ p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing. It is convenient to cast the heat equation in terms of potential temperature @@ -35,7 +36,7 @@ \[ p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt} \] -which, when added to the heat equation \ref{eq-p-firstlaw} and using $% +which, when added to the heat equation \ref{eq-p-firstlaw} and using $ c_{p}=c_{v}+R$, gives: \begin{equation} c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q} @@ -43,7 +44,7 @@ \end{equation} Potential temperature is defined: \begin{equation} -\theta =T(\frac{p_{c}}{p})^{\kappa } \label{eq-potential-temp} +\theta =T(\frac{p_{c}}{p})^{\kappa } \label{eq:potential-temp} \end{equation} where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience we will make use of the Exner function $\Pi (p)$ which defined by: @@ -54,8 +55,8 @@ the Exner function: \[ c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi -}{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{% -\partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}% +}{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{ +\partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p} \frac{Dp}{Dt} \] where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy. @@ -67,20 +68,21 @@ \] and on substituting into (\ref{eq-p-heat-interim}) gives: \begin{equation} -\Pi \frac{D\theta }{Dt}=\mathcal{Q} \label{theta-equation} +\Pi \frac{D\theta }{Dt}=\mathcal{Q} \label{eq:potential-temperature-equation} \end{equation} which is in conservative form. -For convenience in the model we prefer to step forward (\ref{theta-equation}% +For convenience in the model we prefer to step forward (\ref{theta-equation} ) rather than (\ref{eq-p-firstlaw}). \subsubsection{Boundary conditions} The upper and lower boundary conditions are : -\begin{eqnarray*} +\begin{eqnarray} \mbox{at the top:}\;\;p=0 &&\text{, }\omega =\frac{Dp}{Dt}=0 \\ \mbox{at the surface:}\;\;p=p_{s} &&\text{, }\phi =\phi _{topo}=g~Z_{topo} -\end{eqnarray*} +\label{eq:boundary-condition-atmosphere} +\end{eqnarray} In $p$-coordinates, the upper boundary acts like a solid boundary ($\omega =0 $); in $z$-coordinates and the lower boundary is analogous to a free surface ($\phi $ is imposed and $\omega \neq 0$). @@ -93,11 +95,11 @@ is not dynamically relevant and can therefore be subtracted from the equations. The equations written in terms of perturbations are obtained by substituting the following definitions into the previous model equations: -\begin{eqnarray*} -\theta &=&\theta _{o}+\theta ^{\prime } \\ -\alpha &=&\alpha _{o}+\alpha ^{\prime } \\ -\phi &=&\phi _{o}+\phi ^{\prime } -\end{eqnarray*} +\begin{eqnarray} +\theta &=&\theta _{o}+\theta ^{\prime } \label{eq:atmos-ref-prof-theta} \\ +\alpha &=&\alpha _{o}+\alpha ^{\prime } \label{eq:atmos-ref-prof-alpha}\\ +\phi &=&\phi _{o}+\phi ^{\prime } \label{eq:atmos-ref-prof-phi} +\end{eqnarray} The reference state (indicated by subscript ``0'') corresponds to horizontally homogeneous atmosphere at rest ($\theta _{o},\alpha _{o},\phi _{o}$) with surface pressure $p_{o}(x,y)$ that satisfies $\phi @@ -114,14 +116,14 @@ The final form of the HPE's in p coordinates is then: \begin{eqnarray} -\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% +\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} -\label{eq-p-hmom} \\ + \\ \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 -\label{eq-p-hydro} \\ -\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{% -\partial p} &=&0 \label{eq-p-cont} \\ + \\ +\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ +\partial p} &=&0 \\ \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } -\label{eq-p-eos} \\ -\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \label{eq-p-heat} + \\ +\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \label{eq:atmos-prime} \end{eqnarray}