--- manual/s_overview/appendix_ocean.tex	2001/09/11 14:39:38	1.2
+++ manual/s_overview/appendix_ocean.tex	2001/09/27 01:57:17	1.3
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/Attic/appendix_ocean.tex,v 1.2 2001/09/11 14:39:38 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/Attic/appendix_ocean.tex,v 1.3 2001/09/27 01:57:17 cnh Exp $
 % $Name:  $
 
 \section{Appendix OCEAN}
@@ -9,12 +9,12 @@
 HPE's for the ocean written in z-coordinates are obtained. The
 non-Boussinesq equations for oceanic motion 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}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
 \\
 \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
 &=&\epsilon _{nh}\mathcal{F}_{w}  \\
-\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%
+\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
 _{h}+\frac{\partial w}{\partial z} &=&0  \\
 \rho &=&\rho (\theta ,S,p)  \\
 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \\
@@ -41,7 +41,7 @@
 reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
 {eq-zns-cont} gives: 
 \begin{equation}
-\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%
+\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
 v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
 \end{equation}
 where we have used an approximation sign to indicate that we have assumed
@@ -49,12 +49,12 @@
 Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that
 can be explicitly integrated forward: 
 \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}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
 \label{eq-cns-hmom} \\
 \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
 &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\
-\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%
+\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
 v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\
 \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\
 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\
@@ -68,13 +68,13 @@
 `Boussinesq assumption'. The only term that then retains the full variation
 in $\rho $ is the gravitational acceleration: 
 \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}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
 \label{eq-zcb-hmom} \\
-\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%
+\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
 \label{eq-zcb-hydro} \\
-\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{%
+\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{
 \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-zcb-cont} \\
 \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\
 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\
@@ -86,14 +86,14 @@
 term appears as a Helmholtz term in the non-hydrostatic pressure equation).
 These are the \emph{truly} compressible Boussinesq equations. Note that the
 EOS must have the same pressure dependency as the linearized pressure term,
-ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{%
+ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{
 c_{s}^{2}}$, for consistency.
 
 \subsubsection{`Anelastic' z-coordinate equations}
 
 The anelastic approximation filters the acoustic mode by removing the
-time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}%
-). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}%
+time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}
+). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}
 \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
 continuity and EOS. A better solution is to change the dependency on
 pressure in the EOS by splitting the pressure into a reference function of
@@ -104,29 +104,29 @@
 Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
 differentiating the EOS, the continuity equation then becomes: 
 \[
-\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%
-Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%
+\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{
+Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+
 \frac{\partial w}{\partial z}=0 
 \]
 If the time- and space-scales of the motions of interest are longer than
-those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%
+those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},
 \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and 
-$\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{%
+$\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{
 Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
 ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
 _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
 and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the
 anelastic continuity equation: 
 \begin{equation}
-\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-%
+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-
 \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}
 \end{equation}
 A slightly different route leads to the quasi-Boussinesq continuity equation
-where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+%
-\mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%
+where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
+\mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
 _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding: 
 \begin{equation}
-\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%
+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
 \partial \left( \rho _{o}w\right) }{\partial z}=0  \label{eq-za-cont2}
 \end{equation}
 Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same
@@ -135,18 +135,18 @@
 \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}
 \end{equation}
 Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
-and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{%
+and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{
 g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
 full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
 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}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
 \label{eq-zab-hmom} \\
-\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%
+\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
 \label{eq-zab-hydro} \\
-\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%
+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
 \partial \left( \rho _{o}w\right) }{\partial z} &=&0  \label{eq-zab-cont} \\
 \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\
 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\
@@ -159,10 +159,10 @@
 technically, to also remove the dependence of $\rho $ on $p_{o}$. This would
 yield the ``truly'' incompressible Boussinesq equations: 
 \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}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
 \label{eq-ztb-hmom} \\
-\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}%
+\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}
 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
 \label{eq-ztb-hydro} \\
 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
@@ -193,8 +193,8 @@
 This then yields what we can call the semi-compressible Boussinesq
 equations: 
 \begin{eqnarray}
-\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
-_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%
+\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
+_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{
 \mathcal{F}}}  \label{eq:ocean-mom} \\
 \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
 _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}