--- manual/s_overview/text/manual.tex	2001/10/09 10:48:03	1.2
+++ manual/s_overview/text/manual.tex	2001/10/25 12:06:56	1.7
@@ -1,63 +1,40 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
 % $Name:  $
-%\usepackage{oldgerm}
-% I commented the following because it introduced excessive white space
-%\usepackage{palatcm}              % better PDF 
-% page headers and footers
-%\pagestyle{fancy}
-% referencing
-%% \newcommand{\refequ}[1]{equation (\ref{equ:#1})}
-%% \newcommand{\refequbig}[1]{Equation (\ref{equ:#1})}
-%% \newcommand{\reftab}[1]{Tab.~\ref{tab:#1}}
-%% \newcommand{\reftabno}[1]{\ref{tab:#1}}
-%% \newcommand{\reffig}[1]{Fig.~\ref{fig:#1}}
-%% \newcommand{\reffigno}[1]{\ref{fig:#1}}
-% stuff for psfrag
-%% \newcommand{\textinfigure}[1]{{\footnotesize\textbf{\textsf{#1}}}}
-%% \newcommand{\mathinfigure}[1]{\small\ensuremath{{#1}}}
-% This allows numbering of subsubsections
-% This changes the the chapter title
-%\renewcommand{\chaptername}{Section}
-
-
-%%%% \documentclass[12pt]{book}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%% \usepackage{amsmath}
-%%%% \usepackage{html}
-%%%% \usepackage{epsfig}
-%%%% \usepackage{graphics,subfigure}
-%%%% \usepackage{array}
-%%%% \usepackage{multirow}
-%%%% \usepackage{fancyhdr}
-%%%% \usepackage{psfrag}
-
-%%%% %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}
-%%%% \fancyhead[RE]{\slshape \leftmark}
-%%%% \fancyhead[RO,LE]{\thepage}
-%%%% \fancyfoot[CO,CE]{\today}
-%%%% \fancyfoot[RO,LE]{ }
-%%%% \renewcommand{\headrulewidth}{0.4pt}
-%%%% \renewcommand{\footrulewidth}{0.4pt}
-%%%% \setcounter{secnumdepth}{3}
-%%%% \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 1.2 2001/10/09 10:48:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
 % $Name:  $
 
 \section{Introduction}
@@ -77,20 +54,26 @@
 \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.1%
-\marginpar{
-Fig.1 One model}\ref{fig:onemodel}
+models - see fig \ref{fig:onemodel}
+
+%% CNHbegin
+\input{part1/one_model_figure}
+%% CNHend
 
 \item it has a non-hydrostatic capability and so can be used to study both
-small-scale and large scale processes - see fig.2%
-\marginpar{
-Fig.2 All scales}\ref{fig:all-scales}
+small-scale and large scale processes - see fig \ref{fig:all-scales}
+
+%% CNHbegin
+\input{part1/all_scales_figure}
+%% CNHend
 
 \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.3%
-\marginpar{
-Fig.3 Finite volumes}\ref{fig:Finite volumes}
+orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes}
+
+%% CNHbegin
+\input{part1/fvol_figure}
+%% CNHend
 
 \item tangent linear and adjoint counterparts are automatically maintained
 along with the forward model, permitting sensitivity and optimization
@@ -105,16 +88,15 @@
 
 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 1.2 2001/10/09 10:48:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
 % $Name:  $
 
 \section{Illustrations of the model in action}
 
 The MITgcm has been designed and used to model a wide range of phenomena,
 from convection on the scale of meters in the ocean to the global pattern of
-atmospheric winds - see fig.2\ref{fig:all-scales}. To give a flavor of the
+atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the
 kinds of problems the model has been used to study, we briefly describe some
 of them here. A more detailed description of the underlying formulation,
 numerical algorithm and implementation that lie behind these calculations is
@@ -125,10 +107,10 @@
 
 \subsection{Global atmosphere: `Held-Suarez' benchmark}
 
-A novel feature of MITgcm is its ability to simulate both atmospheric and
-oceanographic flows at both small and large scales.
+A novel feature of MITgcm is its ability to simulate, using one basic algorithm, 
+both atmospheric and oceanographic flows at both small and large scales.
 
-Fig.E1a.\ref{fig:Held-Suarez} shows an instantaneous plot of the 500$mb$
+Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
 temperature field obtained using the atmospheric isomorph of MITgcm run at
 2.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
 (blue) and warm air along an equatorial band (red). Fully developed
@@ -139,17 +121,25 @@
 in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
 there are no mountains or land-sea contrast.
 
+%% CNHbegin
+\input{part1/cubic_eddies_figure}
+%% CNHend
+
 As described in Adcroft (2001), a `cubed sphere' is used to discretize the
-globe permitting a uniform gridding and obviated the need to fourier filter.
+globe permitting a uniform gridding and obviated the need to Fourier filter.
 The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
 grid, of which the cubed sphere is just one of many choices.
 
-Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
-wind and meridional overturning streamfunction from a 20-level version of
+Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
+wind from a 20-level configuration of
 the model. It compares favorable with more conventional spatial
-discretization approaches.
-
-A regular spherical lat-lon grid can also be used.
+discretization approaches. The two plots show the field calculated using the
+cube-sphere grid and the flow calculated using a regular, spherical polar
+latitude-longitude grid. Both grids are supported within the model.
+
+%% CNHbegin
+\input{part1/hs_zave_u_figure}
+%% CNHend
 
 \subsection{Ocean gyres}
 
@@ -161,19 +151,26 @@
 increased until the baroclinic instability process is resolved, numerical
 solutions of a different and much more realistic kind, can be obtained.
 
-Fig. ?.? shows the surface temperature and velocity field obtained from
-MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
+Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity 
+field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal 
+resolution on a $lat-lon$
 grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
 (to avoid the converging of meridian in northern latitudes). 21 vertical
 levels are used in the vertical with a `lopped cell' representation of
 topography. The development and propagation of anomalously warm and cold
-eddies can be clearly been seen in the Gulf Stream region. The transport of
+eddies can be clearly seen in the Gulf Stream region. The transport of
 warm water northward by the mean flow of the Gulf Stream is also clearly
 visible.
 
+%% CNHbegin
+\input{part1/ocean_gyres_figure}
+%% CNHend
+
+
 \subsection{Global ocean circulation}
 
-Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
+Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at 
+the surface of a 4$^{\circ }$
 global ocean model run with 15 vertical levels. Lopped cells are used to
 represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
 }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
@@ -181,8 +178,12 @@
 transfer properties of ocean eddies, convection and mixing is parameterized
 in this model.
 
-Fig.E2b shows the meridional overturning circulation of the global ocean in
-Sverdrups.
+Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning 
+circulation of the global ocean in Sverdrups.
+
+%%CNHbegin
+\input{part1/global_circ_figure}
+%%CNHend
 
 \subsection{Convection and mixing over topography}
 
@@ -190,7 +191,8 @@
 ocean may be influenced by rotation when the deformation radius is smaller
 than the width of the cooling region. Rather than gravity plumes, the
 mechanism for moving dense fluid down the shelf is then through geostrophic
-eddies. The simulation shown in the figure (blue is cold dense fluid, red is
+eddies. The simulation shown in the figure \ref{fig::convect-and-topo}
+(blue is cold dense fluid, red is
 warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
 trigger convection by surface cooling. The cold, dense water falls down the
 slope but is deflected along the slope by rotation. It is found that
@@ -198,6 +200,10 @@
 strong, and replaced by lateral entrainment due to the baroclinic
 instability of the along-slope current.
 
+%%CNHbegin
+\input{part1/convect_and_topo}
+%%CNHend
+
 \subsection{Boundary forced internal waves}
 
 The unique ability of MITgcm to treat non-hydrostatic dynamics in the
@@ -205,27 +211,38 @@
 dynamics and mixing in oceanic canyons and ridges driven by large amplitude
 barotropic tidal currents imposed through open boundary conditions.
 
-Fig. ?.? shows the influence of cross-slope topographic variations on
+Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope 
+topographic variations on
 internal wave breaking - the cross-slope velocity is in color, the density
 contoured. The internal waves are excited by application of open boundary
-conditions on the left.\ They propagate to the sloping boundary (represented
+conditions on the left. They propagate to the sloping boundary (represented
 using MITgcm's finite volume spatial discretization) where they break under
 nonhydrostatic dynamics.
 
+%%CNHbegin
+\input{part1/boundary_forced_waves}
+%%CNHend
+
 \subsection{Parameter sensitivity using the adjoint of MITgcm}
 
 Forward and tangent linear counterparts of MITgcm are supported using an
 `automatic adjoint compiler'. These can be used in parameter sensitivity and
 data assimilation studies.
 
-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 $%
-\mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
+As one example of application of the MITgcm adjoint, Figure \ref{fig:hf-sensitivity}
+maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+of the overturning streamfunction shown in figure \ref{fig:large-scale-circ}
+at 60$^{\circ }$N and $
+\mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
+a 100 year period. 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
 yields sensitivities to all other model parameters.
 
+%%CNHbegin
+\input{part1/adj_hf_ocean_figure}
+%%CNHend
+
 \subsection{Global state estimation of the ocean}
 
 An important application of MITgcm is in state estimation of the global
@@ -233,47 +250,64 @@
 the departure of the model from observations (both remotely sensed and
 insitu) over an interval of time, is minimized by adjusting `control
 parameters' such as air-sea fluxes, the wind field, the initial conditions
-etc. Figure ?.? shows an estimate of the time-mean surface elevation of the
-ocean obtained by bringing the model in to consistency with altimetric and
-in-situ observations over the period 1992-1997.
+etc. Figure \ref{fig:assimilated-globes} shows an estimate of the time-mean
+surface elevation of the ocean obtained by bringing the model in to
+consistency with altimetric and in-situ observations over the period
+1992-1997. {\bf CHANGE THIS TEXT - FIG FROM PATRICK/CARL/DETLEF}
+
+%% CNHbegin
+\input{part1/globes_figure}
+%% CNHend
 
 \subsection{Ocean biogeochemical cycles}
 
 MITgcm is being used to study global biogeochemical cycles in the ocean. For
 example one can study the effects of interannual changes in meteorological
 forcing and upper ocean circulation on the fluxes of carbon dioxide and
-oxygen between the ocean and atmosphere. The figure shows the annual air-sea
-flux of oxygen and its relation to density outcrops in the southern oceans
-from a single year of a global, interannually varying simulation.
-
-Chris - get figure here: http://puddle.mit.edu/\symbol{126}%
-mick/biogeochem.html
+oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows 
+the annual air-sea flux of oxygen and its relation to density outcrops in 
+the southern oceans from a single year of a global, interannually varying 
+simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution
+telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).
+
+%%CNHbegin
+\input{part1/biogeo_figure}
+%%CNHend
 
 \subsection{Simulations of laboratory experiments}
 
-Figure ?.? shows MITgcm being used to simulate a laboratory experiment
-enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
+Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a 
+laboratory experiment enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
 initially homogeneous tank of water ($1m$ in diameter) is driven from its
 free surface by a rotating heated disk. The combined action of mechanical
 and thermal forcing creates a lens of fluid which becomes baroclinically
 unstable. The stratification and depth of penetration of the lens is
-arrested by its instability in a process analogous to that whic sets the
+arrested by its instability in a process analogous to that which sets the
 stratification of the ACC.
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.2 2001/10/09 10:48:03 cnh Exp $
+%%CNHbegin
+\input{part1/lab_figure}
+%%CNHend
+
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh 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%
-\marginpar{
-Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down
+respective fluids - see figure \ref{fig:isomorphic-equations}. 
+One system of hydrodynamical equations is written down
 and encoded. The model variables have different interpretations depending on
 whether the atmosphere or ocean is being studied. Thus, for example, the
 vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
-modeling the atmosphere and height, $z$, if we are modeling the ocean.
+modeling the atmosphere (left hand side of figure \ref{fig:isomorphic-equations})
+and height, $z$, if we are modeling the ocean (right hand side of figure
+\ref{fig:isomorphic-equations}).
+
+%%CNHbegin
+\input{part1/zandpcoord_figure.tex}
+%%CNHend
 
 The state of the fluid at any time is characterized by the distribution of
 velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a
@@ -281,24 +315,28 @@
 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%
-\marginpar{
-Fig.5 The vertical coordinate of model}:
+a generic vertical coordinate, $r$, so that the appropriate
+kinematic boundary conditions can be applied isomorphically
+see figure \ref{fig:zandp-vert-coord}.
+
+%%CNHbegin
+\input{part1/vertcoord_figure.tex}
+%%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}
 
@@ -326,10 +364,10 @@
 \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
 
@@ -363,7 +401,7 @@
 \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*}
 
@@ -376,13 +414,14 @@
 \end{equation*}
 
 The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by
-extensive `physics' packages for atmosphere and ocean described in Chapter 6.
+`physics' and forcing packages for atmosphere and ocean. These are described
+in later chapters.
 
 \subsection{Kinematic Boundary conditions}
 
 \subsubsection{vertical}
 
-at fixed and moving $r$ surfaces we set (see fig.5):
+at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
 
 \begin{equation}
 \dot{r}=0atr=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
@@ -408,12 +447,12 @@
 
 \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}
 
-In the atmosphere, see fig.5, we interpret:
+In the atmosphere, (see figure \ref{fig:zandp-vert-coord}), we interpret:
 
 \begin{equation}
 r=p\text{ is the pressure}  \label{eq:atmos-r}
@@ -445,10 +484,10 @@
 
 \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}
@@ -458,7 +497,7 @@
 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.
 
@@ -508,7 +547,7 @@
 
 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:
@@ -516,7 +555,7 @@
 \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.
@@ -533,7 +572,7 @@
 \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}
@@ -547,19 +586,19 @@
 \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:
 
@@ -567,19 +606,19 @@
 \left. 
 \begin{tabular}{l}
 $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\ 
-$-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan lat}{{r}}\right\} $
+$-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} $
 \\ 
-$-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\right\} $
+$-\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\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}
 
@@ -587,20 +626,20 @@
 \left. 
 \begin{tabular}{l}
 $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\ 
-$-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\} 
+$-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\} 
 $ \\ 
-$-\left\{ -2\Omega u\sin lat\right\} $ \\ 
-$+\mathcal{F}_{v}$%
-\end{tabular}%
+$-\left\{ -2\Omega u\sin \varphi \right\} $ \\ 
+$+\mathcal{F}_{v}$
+\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}
@@ -608,28 +647,32 @@
 \begin{tabular}{l}
 $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\ 
 $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\ 
-${+}\underline{{2\Omega u\cos lat}}$ \\ 
-$\underline{\underline{\mathcal{F}_{\dot{r}}}}$%
-\end{tabular}%
+${+}\underline{{2\Omega u\cos \varphi}}$ \\ 
+$\underline{\underline{\mathcal{F}_{\dot{r}}}}$
+\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 `$\varphi$
 ' is latitude.
 
 Grad and div operators in spherical coordinates are defined in appendix
-OPERATORS.%
+OPERATORS.
 \marginpar{
 Fig.6 Spherical polar coordinate system.}
 
+%%CNHbegin
+\input{part1/sphere_coord_figure.tex}
+%%CNHend
+
 \subsubsection{Shallow atmosphere approximation}
 
 Most models are based on the `hydrostatic primitive equations' (HPE's) in
@@ -638,11 +681,12 @@
 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.
 
 \subsubsection{Hydrostatic and quasi-hydrostatic forms}
+\label{sec:hydrostatic_and_quasi-hydrostatic_forms}
 
 These are discussed at length in Marshall et al (1997a).
 
@@ -651,13 +695,13 @@
 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. (%
+\varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
+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
@@ -665,7 +709,7 @@
 vertical momentum equation (\ref{eq:mom-w}) becomes:
 
 \begin{equation*}
-\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat
+\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi
 \end{equation*}
 making a small correction to the hydrostatic pressure.
 
@@ -681,7 +725,7 @@
 
 \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
@@ -694,13 +738,13 @@
 
 \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}
@@ -728,14 +772,14 @@
 
 \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
@@ -746,8 +790,12 @@
 stepping forward the horizontal momentum equations; $\dot{r}$ is found by
 stepping forward the vertical momentum equation.
 
+%%CNHbegin
+\input{part1/solution_strategy_figure.tex}
+%%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 \varphi \ $
 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
@@ -757,6 +805,7 @@
 hydrostatic limit, is as computationally economic as the \textbf{HPEs}.
 
 \subsection{Finding the pressure field}
+\label{sec:finding_the_pressure_field}
 
 Unlike the prognostic variables $u$, $v$, $w$, $\theta $ and $S$, the
 pressure field must be obtained diagnostically. We proceed, as before, by
@@ -771,7 +820,7 @@
 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
@@ -789,11 +838,11 @@
 
 \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*}
 
@@ -801,17 +850,17 @@
 
 \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
@@ -820,23 +869,23 @@
 \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}
 
@@ -856,7 +905,7 @@
 \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
@@ -873,25 +922,25 @@
 \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}
@@ -920,7 +969,7 @@
 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
@@ -931,12 +980,12 @@
 
 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, $\underline{\underline{K}}$,
@@ -947,7 +996,7 @@
 \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}
@@ -957,19 +1006,19 @@
 
 \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.
@@ -979,7 +1028,7 @@
 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 1.2 2001/10/09 10:48:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
 % $Name:  $
 
 \section{Appendix ATMOSPHERE}
@@ -991,23 +1040,23 @@
 
 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 $%
+derivative, $f=2\Omega \sin \varphi$ 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 $
 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 
@@ -1015,30 +1064,30 @@
 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 
@@ -1053,7 +1102,7 @@
 \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}
@@ -1097,16 +1146,16 @@
 
 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 1.2 2001/10/09 10:48:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
 % $Name:  $
 
 \section{Appendix OCEAN}
@@ -1117,21 +1166,21 @@
 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:
@@ -1144,10 +1193,10 @@
 \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
@@ -1155,12 +1204,12 @@
 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} \\
@@ -1174,32 +1223,32 @@
 `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
@@ -1210,29 +1259,29 @@
 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
@@ -1241,18 +1290,18 @@
 \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} \\
@@ -1265,10 +1314,10 @@
 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}
@@ -1287,20 +1336,20 @@
 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}
@@ -1311,7 +1360,7 @@
 \\
 \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.
@@ -1322,7 +1371,7 @@
 _{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 1.2 2001/10/09 10:48:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.7 2001/10/25 12:06:56 cnh Exp $
 % $Name:  $
 
 \section{Appendix:OPERATORS}
@@ -1335,11 +1384,11 @@
 and vertical direction respectively, are given by (see Fig.2) :
 
 \begin{equation*}
-u=r\cos \phi \frac{D\lambda }{Dt}
+u=r\cos \varphi \frac{D\lambda }{Dt}
 \end{equation*}
 
 \begin{equation*}
-v=r\frac{D\phi }{Dt}\qquad
+v=r\frac{D\varphi }{Dt}\qquad
 \end{equation*}
 $\qquad \qquad \qquad \qquad $
 
@@ -1347,7 +1396,7 @@
 \dot{r}=\frac{Dr}{Dt}
 \end{equation*}
 
-Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial
+Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
 distance of the particle from the center of the earth, $\Omega $ is the
 angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
 
@@ -1355,15 +1404,15 @@
 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 \varphi }\frac{\partial }{\partial \lambda }
+,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r}
 \right)
 \end{equation*}
 
 \begin{equation*}
-\nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial
-\lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}
+\nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
+\lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
 +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
 \end{equation*}
 
-%%%% \end{document}
+%tci%\end{document}
