--- manual/s_overview/text/manual.tex	2001/10/10 16:48:45	1.3
+++ manual/s_overview/text/manual.tex	2001/10/15 19:34:28	1.5
@@ -1,63 +1,40 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.3 2001/10/10 16:48:45 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
 % $Name:  $
-%\usepackage{oldgerm}
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-%\usepackage{palatcm}              % better PDF 
-% page headers and footers
-%\pagestyle{fancy}
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-%% \newcommand{\refequ}[1]{equation (\ref{equ:#1})}
-%% \newcommand{\refequbig}[1]{Equation (\ref{equ:#1})}
-%% \newcommand{\reftab}[1]{Tab.~\ref{tab:#1}}
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-%% \newcommand{\reffig}[1]{Fig.~\ref{fig:#1}}
-%% \newcommand{\reffigno}[1]{\ref{fig:#1}}
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-%% \newcommand{\textinfigure}[1]{{\footnotesize\textbf{\textsf{#1}}}}
-%% \newcommand{\mathinfigure}[1]{\small\ensuremath{{#1}}}
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-%\renewcommand{\chaptername}{Section}
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-%%%% \usepackage{amsmath}
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-%%%% \usepackage{graphics,subfigure}
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-%%%% %TCIDATA{OutputFilter=Latex.dll}
-%%%% %TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22}
-%%%% %TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
-%%%% %TCIDATA{Language=American English}
-
-%%%% \fancyhead{}
-%%%% \fancyhead[LO]{\slshape \rightmark}
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-%%%% \input{tcilatex}
-%%%% 
-%%%% \begin{document}
-%%%% 
-%%%% \tableofcontents
-%%%% 
-%%%% \pagebreak
 
-%%%% \part{MIT GCM basics}
+%tci%\documentclass[12pt]{book}
+%tci%\usepackage{amsmath}
+%tci%\usepackage{html}
+%tci%\usepackage{epsfig}
+%tci%\usepackage{graphics,subfigure}
+%tci%\usepackage{array}
+%tci%\usepackage{multirow}
+%tci%\usepackage{fancyhdr}
+%tci%\usepackage{psfrag}
+
+%tci%%TCIDATA{OutputFilter=Latex.dll}
+%tci%%TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22}
+%tci%%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
+%tci%%TCIDATA{Language=American English}
+
+%tci%\fancyhead{}
+%tci%\fancyhead[LO]{\slshape \rightmark}
+%tci%\fancyhead[RE]{\slshape \leftmark}
+%tci%\fancyhead[RO,LE]{\thepage}
+%tci%\fancyfoot[CO,CE]{\today}
+%tci%\fancyfoot[RO,LE]{ }
+%tci%\renewcommand{\headrulewidth}{0.4pt}
+%tci%\renewcommand{\footrulewidth}{0.4pt}
+%tci%\setcounter{secnumdepth}{3}
+%tci%\input{tcilatex}
+
+%tci%\begin{document}
+
+%tci%\tableofcontents
+
 
 % Section: Overview
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.3 2001/10/10 16:48:45 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
 % $Name:  $
 
 \section{Introduction}
@@ -77,7 +54,7 @@
 \begin{itemize}
 \item it can be used to study both atmospheric and oceanic phenomena; one
 hydrodynamical kernel is used to drive forward both atmospheric and oceanic
-models - see fig%
+models - see fig
 \marginpar{
 Fig.1 One model}\ref{fig:onemodel}
 
@@ -86,7 +63,7 @@
 %% CNHend
 
 \item it has a non-hydrostatic capability and so can be used to study both
-small-scale and large scale processes - see fig %
+small-scale and large scale processes - see fig 
 \marginpar{
 Fig.2 All scales}\ref{fig:all-scales}
 
@@ -96,7 +73,7 @@
 
 \item finite volume techniques are employed yielding an intuitive
 discretization and support for the treatment of irregular geometries using
-orthogonal curvilinear grids and shaved cells - see fig %
+orthogonal curvilinear grids and shaved cells - see fig 
 \marginpar{
 Fig.3 Finite volumes}\ref{fig:finite-volumes}
 
@@ -117,9 +94,8 @@
 
 We begin by briefly showing some of the results of the model in action to
 give a feel for the wide range of problems that can be addressed using it.
-\pagebreak
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.3 2001/10/10 16:48:45 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
 % $Name:  $
 
 \section{Illustrations of the model in action}
@@ -257,7 +233,7 @@
 
 As one example of application of the MITgcm adjoint, Fig.E4 maps the
 gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
-of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $%
+of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $
 \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
 sensitive to heat fluxes over the Labrador Sea, one of the important sources
 of deep water for the thermohaline circulations. This calculation also
@@ -310,14 +286,14 @@
 \input{part1/lab_figure}
 %%CNHend
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.3 2001/10/10 16:48:45 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
 % $Name:  $
 
 \section{Continuous equations in `r' coordinates}
 
 To render atmosphere and ocean models from one dynamical core we exploit
 `isomorphisms' between equation sets that govern the evolution of the
-respective fluids - see fig.4%
+respective fluids - see fig.4
 \marginpar{
 Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down
 and encoded. The model variables have different interpretations depending on
@@ -335,7 +311,7 @@
 depend on $\theta $, $S$, and $p$. The equations that govern the evolution
 of these fields, obtained by applying the laws of classical mechanics and
 thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
-a generic vertical coordinate, $r$, see fig.5%
+a generic vertical coordinate, $r$, see fig.5
 \marginpar{
 Fig.5 The vertical coordinate of model}:
 
@@ -344,19 +320,19 @@
 %%CNHend
 
 \begin{equation*}
-\frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}%
-\right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}%
+\frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}
+\right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
 \text{ horizontal mtm}
 \end{equation*}
 
 \begin{equation*}
-\frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{%
+\frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
 v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
 vertical mtm}
 \end{equation*}
 
 \begin{equation}
-\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{%
+\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
 \partial r}=0\text{ continuity}  \label{eq:continuous}
 \end{equation}
 
@@ -384,10 +360,10 @@
 \end{equation*}
 
 \begin{equation*}
-\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%
+\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
 \text{ is the `grad' operator}
 \end{equation*}
-with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%
+with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}
 \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
 is a unit vector in the vertical
 
@@ -421,7 +397,7 @@
 \end{equation*}
 
 \begin{equation*}
-\mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{%
+\mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{
 \mathbf{v}}
 \end{equation*}
 
@@ -466,7 +442,7 @@
 
 \begin{equation}
 \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0  \label{eq:noflow}
-\end{equation}%
+\end{equation}
 where $\vec{\mathbf{n}}$ is the normal to a solid boundary.
 
 \subsection{Atmosphere}
@@ -503,10 +479,10 @@
 
 \begin{equation*}
 T\text{ is absolute temperature}
-\end{equation*}%
+\end{equation*}
 \begin{equation*}
 p\text{ is the pressure}
-\end{equation*}%
+\end{equation*}
 \begin{eqnarray*}
 &&z\text{ is the height of the pressure surface} \\
 &&g\text{ is the acceleration due to gravity}
@@ -516,7 +492,7 @@
 the Exner function $\Pi (p)$ given by (see Appendix Atmosphere) 
 \begin{equation}
 \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{eq:exner}
-\end{equation}%
+\end{equation}
 where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas
 constant and $c_{p}$ the specific heat of air at constant pressure.
 
@@ -566,7 +542,7 @@
 
 The surface of the ocean is given by: $R_{moving}=\eta $
 
-The position of the resting free surface of the ocean is given by $%
+The position of the resting free surface of the ocean is given by $
 R_{o}=Z_{o}=0$.
 
 Boundary conditions are:
@@ -574,7 +550,7 @@
 \begin{eqnarray}
 w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)}  \label{eq:fixed-bc-ocean}
 \\
-w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) %
+w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) 
 \label{eq:moving-bc-ocean}}
 \end{eqnarray}
 where $\eta $ is the elevation of the free surface.
@@ -591,7 +567,7 @@
 \begin{equation}
 \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
 \label{eq:phi-split}
-\end{equation}%
+\end{equation}
 and write eq(\ref{incompressible}a,b) in the form:
 
 \begin{equation}
@@ -605,19 +581,19 @@
 \end{equation}
 
 \begin{equation}
-\epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{%
+\epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{
 \partial r}=G_{\dot{r}}  \label{eq:mom-w}
 \end{equation}
 Here $\epsilon _{nh}$ is a non-hydrostatic parameter.
 
-The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref%
+The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref
 {eq:mom-h}) and (\ref{eq:mom-w}) represent advective, metric and Coriolis
-terms in the momentum equations. In spherical coordinates they take the form%
-\footnote{%
+terms in the momentum equations. In spherical coordinates they take the form
+\footnote{
 In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms
-in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref%
+in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref
 {eq:gw-spherical}) are omitted; the singly-underlined terms are included in
-the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (%
+the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (
 \textbf{NH}) includes all terms.} - see Marshall et al 1997a for a full
 discussion:
 
@@ -629,15 +605,15 @@
 \\ 
 $-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\right\} $
 \\ 
-$+\mathcal{F}_{u}$%
-\end{tabular}%
+$+\mathcal{F}_{u}$
+\end{tabular}
 \ \right\} \left\{ 
 \begin{tabular}{l}
 \textit{advection} \\ 
 \textit{metric} \\ 
 \textit{Coriolis} \\ 
-\textit{\ Forcing/Dissipation}%
-\end{tabular}%
+\textit{\ Forcing/Dissipation}
+\end{tabular}
 \ \right. \qquad  \label{eq:gu-speherical}
 \end{equation}
 
@@ -648,17 +624,17 @@
 $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\} 
 $ \\ 
 $-\left\{ -2\Omega u\sin lat\right\} $ \\ 
-$+\mathcal{F}_{v}$%
-\end{tabular}%
+$+\mathcal{F}_{v}$
+\end{tabular}
 \ \right\} \left\{ 
 \begin{tabular}{l}
 \textit{advection} \\ 
 \textit{metric} \\ 
 \textit{Coriolis} \\ 
-\textit{\ Forcing/Dissipation}%
-\end{tabular}%
+\textit{\ Forcing/Dissipation}
+\end{tabular}
 \ \right. \qquad  \label{eq:gv-spherical}
-\end{equation}%
+\end{equation}
 \qquad \qquad \qquad \qquad \qquad
 
 \begin{equation}
@@ -667,24 +643,24 @@
 $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\ 
 $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\ 
 ${+}\underline{{2\Omega u\cos lat}}$ \\ 
-$\underline{\underline{\mathcal{F}_{\dot{r}}}}$%
-\end{tabular}%
+$\underline{\underline{\mathcal{F}_{\dot{r}}}}$
+\end{tabular}
 \ \right\} \left\{ 
 \begin{tabular}{l}
 \textit{advection} \\ 
 \textit{metric} \\ 
 \textit{Coriolis} \\ 
-\textit{\ Forcing/Dissipation}%
-\end{tabular}%
+\textit{\ Forcing/Dissipation}
+\end{tabular}
 \ \right.  \label{eq:gw-spherical}
-\end{equation}%
+\end{equation}
 \qquad \qquad \qquad \qquad \qquad
 
-In the above `${r}$' is the distance from the center of the earth and `$lat$%
+In the above `${r}$' is the distance from the center of the earth and `$lat$
 ' is latitude.
 
 Grad and div operators in spherical coordinates are defined in appendix
-OPERATORS.%
+OPERATORS.
 \marginpar{
 Fig.6 Spherical polar coordinate system.}
 
@@ -700,7 +676,7 @@
 Coriolis force is treated approximately and the shallow atmosphere
 approximation is made.\ The MITgcm need not make the `traditional
 approximation'. To be able to support consistent non-hydrostatic forms the
-shallow atmosphere approximation can be relaxed - when dividing through by $%
+shallow atmosphere approximation can be relaxed - when dividing through by $
 r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
 the radius of the earth.
 
@@ -713,13 +689,13 @@
 are neglected and `${r}$' is replaced by `$a$', the mean radius of the
 earth. Once the pressure is found at one level - e.g. by inverting a 2-d
 Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be
-computed at all other levels by integration of the hydrostatic relation, eq(%
+computed at all other levels by integration of the hydrostatic relation, eq(
 \ref{eq:hydrostatic}).
 
 In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
 gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
 \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
-contribution to the pressure field: only the terms underlined twice in Eqs. (%
+contribution to the pressure field: only the terms underlined twice in Eqs. (
 \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero
 and, simultaneously, the shallow atmosphere approximation is relaxed. In 
 \textbf{QH}\ \textit{all} the metric terms are retained and the full
@@ -743,7 +719,7 @@
 
 \paragraph{Non-hydrostatic Ocean}
 
-In the non-hydrostatic ocean model all terms in equations Eqs.(\ref%
+In the non-hydrostatic ocean model all terms in equations Eqs.(\ref
 {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A
 three dimensional elliptic equation must be solved subject to Neumann
 boundary conditions (see below). It is important to note that use of the
@@ -756,13 +732,13 @@
 
 \paragraph{Quasi-nonhydrostatic Atmosphere}
 
-In the non-hydrostatic version of our atmospheric model we approximate $\dot{%
+In the non-hydrostatic version of our atmospheric model we approximate $\dot{
 r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical})
 (but only here) by:
 
 \begin{equation}
 \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt}  \label{eq:quasi-nh-w}
-\end{equation}%
+\end{equation}
 where $p_{hy}$ is the hydrostatic pressure.
 
 \subsubsection{Summary of equation sets supported by model}
@@ -790,14 +766,14 @@
 
 \subparagraph{Non-hydrostatic}
 
-Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$%
-coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref%
+Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
+coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref
 {eq:ocean-salt}).
 
 \subsection{Solution strategy}
 
-The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{%
-NH} models is summarized in Fig.7.%
+The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
+NH} models is summarized in Fig.7.
 \marginpar{
 Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
 first solved to find the surface pressure and the hydrostatic pressure at
@@ -813,7 +789,7 @@
 %%CNHend
 
 There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
-course, some complication that goes with the inclusion of $\cos \phi \ $%
+course, some complication that goes with the inclusion of $\cos \phi \ $
 Coriolis terms and the relaxation of the shallow atmosphere approximation.
 But this leads to negligible increase in computation. In \textbf{NH}, in
 contrast, one additional elliptic equation - a three-dimensional one - must
@@ -837,7 +813,7 @@
 vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:
 
 \begin{equation*}
-\int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}%
+\int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}
 \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
 \end{equation*}
 and so
@@ -855,11 +831,11 @@
 
 \subsubsection{Surface pressure}
 
-The surface pressure equation can be obtained by integrating continuity, (%
+The surface pressure equation can be obtained by integrating continuity, (
 \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
 
 \begin{equation*}
-\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}%
+\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
 }_{h}+\partial _{r}\dot{r}\right) dr=0
 \end{equation*}
 
@@ -867,17 +843,17 @@
 
 \begin{equation*}
 \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
-+\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%
++\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}
 _{h}dr=0
 \end{equation*}
-where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $%
+where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $
 r $. The above can be rearranged to yield, using Leibnitz's theorem:
 
 \begin{equation}
 \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot
 \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}
 \label{eq:free-surface}
-\end{equation}%
+\end{equation}
 where we have incorporated a source term.
 
 Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
@@ -886,23 +862,23 @@
 \begin{equation}
 \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
 \label{eq:phi-surf}
-\end{equation}%
+\end{equation}
 where $b_{s}$ is the buoyancy at the surface.
 
-In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref%
+In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref
 {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
 elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
 surface' and `rigid lid' approaches are available.
 
 \subsubsection{Non-hydrostatic pressure}
 
-Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{%
+Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
 \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
 (\ref{incompressible}), we deduce that:
 
 \begin{equation}
-\nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{%
-\nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .%
+\nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
+\nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .
 \vec{\mathbf{F}}  \label{eq:3d-invert}
 \end{equation}
 
@@ -922,7 +898,7 @@
 \end{equation}
 where $\widehat{n}$ is a vector of unit length normal to the boundary. The
 kinematic condition (\ref{nonormalflow}) is also applied to the vertical
-velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $%
+velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $
 \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the
 tangential component of velocity, $v_{T}$, at all solid boundaries,
 depending on the form chosen for the dissipative terms in the momentum
@@ -939,25 +915,25 @@
 \begin{equation*}
 \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi
 _{s}+\mathbf{\nabla }\phi _{hyd}\right) 
-\end{equation*}%
+\end{equation*}
 presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
 (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
 exploit classical 3D potential theory and, by introducing an appropriately
 chosen $\delta $-function sheet of `source-charge', replace the
 inhomogeneous boundary condition on pressure by a homogeneous one. The
-source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $%
-\vec{\mathbf{F}}.$ By simultaneously setting $%
+source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $
+\vec{\mathbf{F}}.$ By simultaneously setting $
 \begin{array}{l}
-\widehat{n}.\vec{\mathbf{F}}%
-\end{array}%
+\widehat{n}.\vec{\mathbf{F}}
+\end{array}
 =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
 self-consistent but simpler homogenized Elliptic problem is obtained:
 
 \begin{equation*}
 \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
-\end{equation*}%
+\end{equation*}
 where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
-that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref%
+that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref
 {eq:inhom-neumann-nh}) the modified boundary condition becomes:
 
 \begin{equation}
@@ -986,7 +962,7 @@
 biharmonic frictions are commonly used:
 
 \begin{equation}
-D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}%
+D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}
 +A_{4}\nabla _{h}^{4}v  \label{eq:dissipation}
 \end{equation}
 where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity
@@ -997,12 +973,12 @@
 
 The mixing terms for the temperature and salinity equations have a similar
 form to that of momentum except that the diffusion tensor can be
-non-diagonal and have varying coefficients. $\qquad $%
+non-diagonal and have varying coefficients. $\qquad $
 \begin{equation}
 D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
 _{h}^{4}(T,S)  \label{eq:diffusion}
 \end{equation}
-where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%
+where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $
 horizontal coefficient for biharmonic diffusion. In the simplest case where
 the subgrid-scale fluxes of heat and salt are parameterized with constant
 horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,
@@ -1013,7 +989,7 @@
 \begin{array}{ccc}
 K_{h} & 0 & 0 \\ 
 0 & K_{h} & 0 \\ 
-0 & 0 & K_{v}%
+0 & 0 & K_{v}
 \end{array}
 \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}
 \end{equation}
@@ -1023,19 +999,19 @@
 
 \subsection{Vector invariant form}
 
-For some purposes it is advantageous to write momentum advection in eq(\ref%
+For some purposes it is advantageous to write momentum advection in eq(\ref
 {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
 
 \begin{equation}
-\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}%
-+\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %
+\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
++\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla 
 \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
 \label{eq:vi-identity}
-\end{equation}%
+\end{equation}
 This permits alternative numerical treatments of the non-linear terms based
 on their representation as a vorticity flux. Because gradients of coordinate
 vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit
-representation of the metric terms in (\ref{eq:gu-speherical}), (\ref%
+representation of the metric terms in (\ref{eq:gu-speherical}), (\ref
 {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information
 about the geometry is contained in the areas and lengths of the volumes used
 to discretize the model.
@@ -1045,7 +1021,7 @@
 Tangent linear and adjoint counterparts of the forward model and described
 in Chapter 5.
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.3 2001/10/10 16:48:45 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
 % $Name:  $
 
 \section{Appendix ATMOSPHERE}
@@ -1057,23 +1033,23 @@
 
 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: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 }{%
+\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}%
+\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, $\omega =\frac{Dp%
-}{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref%
-{eq:atmos-heat}) 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 $%
+the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
+}{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
+{eq:atmos-heat}) 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 
@@ -1081,30 +1057,30 @@
 Differentiating (\ref{eq:atmos-eos}) we get: 
 \begin{equation*}
 p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
-\end{equation*}%
-which, when added to the heat equation (\ref{eq:atmos-heat}) and using $%
+\end{equation*}
+which, when added to the heat equation (\ref{eq:atmos-heat}) and using $
 c_{p}=c_{v}+R$, gives: 
 \begin{equation}
 c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
 \label{eq-p-heat-interim}
-\end{equation}%
+\end{equation}
 Potential temperature is defined: 
 \begin{equation}
 \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}
-\end{equation}%
+\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: 
 \begin{equation}
 \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{Exner}
-\end{equation}%
+\end{equation}
 The following relations will be useful and are easily expressed in terms of
 the Exner function: 
 \begin{equation*}
 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}
-\end{equation*}%
+\end{equation*}
 where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.
 
 The heat equation is obtained by noting that 
@@ -1119,7 +1095,7 @@
 \end{equation}
 which is in conservative form.
 
-For convenience in the model we prefer to step forward (\ref%
+For convenience in the model we prefer to step forward (\ref
 {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).
 
 \subsubsection{Boundary conditions}
@@ -1163,16 +1139,16 @@
 
 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}}} \\
 \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
-\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%
+\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
 \partial p} &=&0 \\
 \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
 \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}
 \end{eqnarray}
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.3 2001/10/10 16:48:45 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
 % $Name:  $
 
 \section{Appendix OCEAN}
@@ -1183,21 +1159,21 @@
 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 } \\
 \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
-\end{eqnarray}%
+\end{eqnarray}
 These equations permit acoustics modes, inertia-gravity waves,
 non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
 mode. As written, they cannot be integrated forward consistently - if we
 step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
-consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref%
+consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
 {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
 therefore necessary to manipulate the system as follows. Differentiating the
 EOS (equation of state) gives:
@@ -1210,10 +1186,10 @@
 \end{equation}
 
 Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
-reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref%
+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
@@ -1221,12 +1197,12 @@
 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} \\
@@ -1240,32 +1216,32 @@
 `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} \\
 \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}
 \end{eqnarray}
 These equations still retain acoustic modes. But, because the
-``compressible'' terms are linearized, the pressure equation \ref%
+``compressible'' terms are linearized, the pressure equation \ref
 {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
 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
@@ -1276,29 +1252,29 @@
 Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
 differentiating the EOS, the continuity equation then becomes: 
 \begin{equation*}
-\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
 \end{equation*}
 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
@@ -1307,18 +1283,18 @@
 \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} \\
@@ -1331,10 +1307,10 @@
 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}
@@ -1353,20 +1329,20 @@
 density thus: 
 \begin{equation*}
 \rho =\rho _{o}+\rho ^{\prime }
-\end{equation*}%
+\end{equation*}
 We then assert that variations with depth of $\rho _{o}$ are unimportant
 while the compressible effects in $\rho ^{\prime }$ are: 
 \begin{equation*}
 \rho _{o}=\rho _{c}
-\end{equation*}%
+\end{equation*}
 \begin{equation*}
 \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}
-\end{equation*}%
+\end{equation*}
 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}
@@ -1377,7 +1353,7 @@
 \\
 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\
 \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}
-\end{eqnarray}%
+\end{eqnarray}
 Note that the hydrostatic pressure of the resting fluid, including that
 associated with $\rho _{c}$, is subtracted out since it has no effect on the
 dynamics.
@@ -1388,7 +1364,7 @@
 _{nh}=0$ form of these equations that are used throughout the ocean modeling
 community and referred to as the primitive equations (HPE).
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.3 2001/10/10 16:48:45 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
 % $Name:  $
 
 \section{Appendix:OPERATORS}
@@ -1421,8 +1397,8 @@
 spherical coordinates:
 
 \begin{equation*}
-\nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%
-,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%
+\nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }
+,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}
 \right)
 \end{equation*}
 
@@ -1432,4 +1408,4 @@
 +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
 \end{equation*}
 
-%%%% \end{document}
+%tci%\end{document}
