--- manual/s_overview/text/manual.tex	2001/10/15 19:34:28	1.5
+++ manual/s_overview/text/manual.tex	2004/03/23 16:47:04	1.19
@@ -1,4 +1,4 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.19 2004/03/23 16:47:04 afe Exp $
 % $Name:  $
 
 %tci%\documentclass[12pt]{book}
@@ -34,12 +34,10 @@
 
 % Section: Overview
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.19 2004/03/23 16:47:04 afe Exp $
 % $Name:  $
 
-\section{Introduction}
-
-This documentation provides the reader with the information necessary to
+This document provides the reader with the information necessary to
 carry out numerical experiments using MITgcm. It gives a comprehensive
 description of the continuous equations on which the model is based, the
 numerical algorithms the model employs and a description of the associated
@@ -49,23 +47,25 @@
 both process and general circulation studies of the atmosphere and ocean are
 also presented.
 
+\section{Introduction}
+\begin{rawhtml}
+<!-- CMIREDIR:innovations: -->
+\end{rawhtml}
+
+
 MITgcm has a number of novel aspects:
 
 \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
-\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 
-\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}
@@ -73,9 +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 
-\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}
@@ -90,33 +88,82 @@
 \end{itemize}
 
 Key publications reporting on and charting the development of the model are
-listed in an Appendix.
+\cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99}:
+
+\begin{verbatim}
+Hill, C. and J. Marshall, (1995)
+Application of a Parallel Navier-Stokes Model to Ocean Circulation in 
+Parallel Computational Fluid Dynamics
+In Proceedings of Parallel Computational Fluid Dynamics: Implementations 
+and Results Using Parallel Computers, 545-552.
+Elsevier Science B.V.: New York
+
+Marshall, J., C. Hill, L. Perelman, and A. Adcroft, (1997)
+Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling
+J. Geophysical Res., 102(C3), 5733-5752.
+
+Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, (1997)
+A finite-volume, incompressible Navier Stokes model for studies of the ocean
+on parallel computers,
+J. Geophysical Res., 102(C3), 5753-5766.
+
+Adcroft, A.J., Hill, C.N. and J. Marshall, (1997)
+Representation of topography by shaved cells in a height coordinate ocean
+model
+Mon Wea Rev, vol 125, 2293-2315
+
+Marshall, J., Jones, H. and C. Hill, (1998)
+Efficient ocean modeling using non-hydrostatic algorithms
+Journal of Marine Systems, 18, 115-134
+
+Adcroft, A., Hill C. and J. Marshall: (1999)
+A new treatment of the Coriolis terms in C-grid models at both high and low
+resolutions,
+Mon. Wea. Rev. Vol 127, pages 1928-1936
+
+Hill, C, Adcroft,A., Jamous,D., and J. Marshall, (1999)
+A Strategy for Terascale Climate Modeling.
+In Proceedings of the Eighth ECMWF Workshop on the Use of Parallel Processors
+in Meteorology, pages 406-425
+World Scientific Publishing Co: UK
+
+Marotzke, J, Giering,R., Zhang, K.Q., Stammer,D., Hill,C., and T.Lee, (1999)
+Construction of the adjoint MIT ocean general circulation model and 
+application to Atlantic heat transport variability
+J. Geophysical Res., 104(C12), 29,529-29,547.
+
+\end{verbatim}
 
 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.
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.19 2004/03/23 16:47:04 afe 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
 given later. Indeed many of the illustrative examples shown below can be
 easily reproduced: simply download the model (the minimum you need is a PC
-running linux, together with a FORTRAN\ 77 compiler) and follow the examples
+running Linux, together with a FORTRAN\ 77 compiler) and follow the examples
 described in detail in the documentation.
 
 \subsection{Global atmosphere: `Held-Suarez' benchmark}
+\begin{rawhtml}
+<!-- CMIREDIR:atmospheric_example: -->
+\end{rawhtml}
+
 
-A novel feature of MITgcm is its ability to simulate both atmospheric and
-oceanographic flows at both small and large scales.
 
-Fig.E1a.\ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
+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.
+
+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
@@ -132,22 +179,28 @@
 %% 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 griding 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}
+\begin{rawhtml}
+<!-- CMIREDIR:oceanic_example: -->
+\end{rawhtml}
+\begin{rawhtml}
+<!-- CMIREDIR:ocean_gyres: -->
+\end{rawhtml}
 
 Baroclinic instability is a ubiquitous process in the ocean, as well as the
 atmosphere. Ocean eddies play an important role in modifying the
@@ -157,24 +210,29 @@
 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}
+\input{part1/atl6_figure}
 %% CNHend
 
 
 \subsection{Global ocean circulation}
+\begin{rawhtml}
+<!-- CMIREDIR:global_ocean_circulation: -->
+\end{rawhtml}
 
-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
@@ -182,20 +240,25 @@
 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}
+\begin{rawhtml}
+<!-- CMIREDIR:mixing_over_topography: -->
+\end{rawhtml}
+
 
 Dense plumes generated by localized cooling on the continental shelf of the
 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
@@ -208,16 +271,20 @@
 %%CNHend
 
 \subsection{Boundary forced internal waves}
+\begin{rawhtml}
+<!-- CMIREDIR:boundary_forced_internal_waves: -->
+\end{rawhtml}
 
 The unique ability of MITgcm to treat non-hydrostatic dynamics in the
 presence of complex geometry makes it an ideal tool to study internal wave
 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.
 
@@ -226,15 +293,20 @@
 %%CNHend
 
 \subsection{Parameter sensitivity using the adjoint of MITgcm}
+\begin{rawhtml}
+<!-- CMIREDIR:parameter_sensitivity: -->
+\end{rawhtml}
 
 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 stream-function 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.
@@ -244,62 +316,80 @@
 %%CNHend
 
 \subsection{Global state estimation of the ocean}
+\begin{rawhtml}
+<!-- CMIREDIR:global_state_estimation: -->
+\end{rawhtml}
+
 
 An important application of MITgcm is in state estimation of the global
 ocean circulation. An appropriately defined `cost function', which measures
 the departure of the model from observations (both remotely sensed and
-insitu) over an interval of time, is minimized by adjusting `control
+in-situ) 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 the large scale planetary
+circulation and a Hopf-Muller plot of Equatorial sea-surface height.
+Both are obtained from assimilation bringing the model in to
+consistency with altimetric and in-situ observations over the period
+1992-1997.
 
 %% CNHbegin
-\input{part1/globes_figure}
+\input{part1/assim_figure}
 %% CNHend
 
 \subsection{Ocean biogeochemical cycles}
+\begin{rawhtml}
+<!-- CMIREDIR:ocean_biogeo_cycles: -->
+\end{rawhtml}
 
 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.
+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}
+\begin{rawhtml}
+<!-- CMIREDIR:classroom_exp: -->
+\end{rawhtml}
 
-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 inquiring into 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.
 
 %%CNHbegin
 \input{part1/lab_figure}
 %%CNHend
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.19 2004/03/23 16:47:04 afe Exp $
 % $Name:  $
 
 \section{Continuous equations in `r' coordinates}
+\begin{rawhtml}
+<!-- CMIREDIR:z-p_isomorphism: -->
+\end{rawhtml}
 
 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 (right hand side of figure \ref{fig:isomorphic-equations})
+and height, $z$, if we are modeling the ocean (left hand side of figure
+\ref{fig:isomorphic-equations}).
 
 %%CNHbegin
 \input{part1/zandpcoord_figure.tex}
@@ -311,9 +401,9 @@
 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}
@@ -322,31 +412,33 @@
 \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}}}
-\text{ horizontal mtm}
+\text{ horizontal mtm} \label{eq:horizontal_mtm}
 \end{equation*}
 
-\begin{equation*}
+\begin{equation}
 \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*}
+vertical mtm} \label{eq:vertical_mtm}
+\end{equation}
 
 \begin{equation}
 \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
-\partial r}=0\text{ continuity}  \label{eq:continuous}
+\partial r}=0\text{ continuity}  \label{eq:continuity}
 \end{equation}
 
-\begin{equation*}
-b=b(\theta ,S,r)\text{ equation of state}
-\end{equation*}
+\begin{equation}
+b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
+\end{equation}
 
-\begin{equation*}
+\begin{equation}
 \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
-\end{equation*}
+\label{eq:potential_temperature}
+\end{equation}
 
-\begin{equation*}
+\begin{equation}
 \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
-\end{equation*}
+\label{eq:humidity_salt}
+\end{equation}
 
 Here:
 
@@ -410,22 +502,23 @@
 \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)}
+\dot{r}=0 \text{\ at\ } r=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
 \label{eq:fixedbc}
 \end{equation}
 
 \begin{equation}
-\dot{r}=\frac{Dr}{Dt}atr=R_{moving}\text{ \
-(oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}
+\dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
+(ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
 \end{equation}
 
 Here
@@ -447,7 +540,7 @@
 
 \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}
@@ -518,8 +611,8 @@
 atmosphere)}  \label{eq:moving-bc-atmos}
 \end{eqnarray}
 
-Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent
-set of atmospheric equations which, for convenience, are written out in $p$
+Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) 
+yields a consistent set of atmospheric equations which, for convenience, are written out in $p$
 coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).
 
 \subsection{Ocean}
@@ -555,12 +648,17 @@
 \end{eqnarray}
 where $\eta $ is the elevation of the free surface.
 
-Then eq(\ref{eq:continuous}) yields a consistent set of oceanic equations
+Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yield a consistent set 
+of oceanic equations
 which, for convenience, are written out in $z$ coordinates in Appendix Ocean
 - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
 
 \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
 Non-hydrostatic forms}
+\begin{rawhtml}
+<!-- CMIREDIR:non_hydrostatic: -->
+\end{rawhtml}
+
 
 Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
 
@@ -568,7 +666,7 @@
 \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
 \label{eq:phi-split}
 \end{equation}
-and write eq(\ref{incompressible}a,b) in the form:
+and write eq(\ref{eq:incompressible}) in the form:
 
 \begin{equation}
 \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
@@ -601,9 +699,9 @@
 \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}
@@ -621,9 +719,9 @@
 \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\} $ \\ 
+$-\left\{ -2\Omega u\sin \varphi \right\} $ \\ 
 $+\mathcal{F}_{v}$
 \end{tabular}
 \ \right\} \left\{ 
@@ -642,7 +740,7 @@
 \begin{tabular}{l}
 $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\ 
 $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\ 
-${+}\underline{{2\Omega u\cos lat}}$ \\ 
+${+}\underline{{2\Omega u\cos \varphi}}$ \\ 
 $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
 \end{tabular}
 \ \right\} \left\{ 
@@ -656,13 +754,11 @@
 \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.
-\marginpar{
-Fig.6 Spherical polar coordinate system.}
 
 %%CNHbegin
 \input{part1/sphere_coord_figure.tex}
@@ -681,6 +777,7 @@
 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).
 
@@ -694,7 +791,7 @@
 
 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
+\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 
@@ -703,7 +800,7 @@
 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.
 
@@ -724,7 +821,7 @@
 three dimensional elliptic equation must be solved subject to Neumann
 boundary conditions (see below). It is important to note that use of the
 full \textbf{NH} does not admit any new `fast' waves in to the system - the
-incompressible condition eq(\ref{eq:continuous})c has already filtered out
+incompressible condition eq(\ref{eq:continuity}) has already filtered out
 acoustic modes. It does, however, ensure that the gravity waves are treated
 accurately with an exact dispersion relation. The \textbf{NH} set has a
 complete angular momentum principle and consistent energetics - see White
@@ -773,9 +870,8 @@
 \subsection{Solution strategy}
 
 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
+NH} models is summarized in Figure \ref{fig:solution-strategy}.
+Under all dynamics, a 2-d elliptic equation is
 first solved to find the surface pressure and the hydrostatic pressure at
 any level computed from the weight of fluid above. Under \textbf{HPE} and 
 \textbf{QH} dynamics, the horizontal momentum equations are then stepped
@@ -789,7 +885,7 @@
 %%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 \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
@@ -799,6 +895,7 @@
 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
@@ -831,8 +928,8 @@
 
 \subsubsection{Surface pressure}
 
-The surface pressure equation can be obtained by integrating continuity, (
-\ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
+The surface pressure equation can be obtained by integrating continuity,
+(\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
 
 \begin{equation*}
 \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
@@ -857,7 +954,7 @@
 where we have incorporated a source term.
 
 Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
-(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
+(atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
 be written 
 \begin{equation}
 \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
@@ -865,16 +962,16 @@
 \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 ($\epsilon _{nh}=0$), equations (\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 $\frac{
-\partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
-(\ref{incompressible}), we deduce that:
+Taking the horizontal divergence of (\ref{eq:mom-h}) and adding 
+$\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
+(\ref{eq:continuity}), we deduce that:
 
 \begin{equation}
 \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
@@ -904,7 +1001,7 @@
 depending on the form chosen for the dissipative terms in the momentum
 equations - see below.
 
-Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that:
+Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
 
 \begin{equation}
 \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
@@ -944,7 +1041,7 @@
 converges rapidly because $\phi _{nh}\ $is then only a small correction to
 the hydrostatic pressure field (see the discussion in Marshall et al, a,b).
 
-The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{homneuman})
+The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh})
 does not vanish at $r=R_{moving}$, and so refines the pressure there.
 
 \subsection{Forcing/dissipation}
@@ -952,7 +1049,7 @@
 \subsubsection{Forcing}
 
 The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by
-`physics packages' described in detail in chapter ??.
+`physics packages' and forcing packages. These are described later on.
 
 \subsubsection{Dissipation}
 
@@ -1000,7 +1097,7 @@
 \subsection{Vector invariant form}
 
 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:
+{eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the (so-called) `vector invariant' form:
 
 \begin{equation}
 \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
@@ -1018,10 +1115,10 @@
 
 \subsection{Adjoint}
 
-Tangent linear and adjoint counterparts of the forward model and described
+Tangent linear and adjoint counterparts of the forward model are described
 in Chapter 5.
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.19 2004/03/23 16:47:04 afe Exp $
 % $Name:  $
 
 \section{Appendix ATMOSPHERE}
@@ -1045,7 +1142,7 @@
 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
+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 $
@@ -1140,15 +1237,15 @@
 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}}
-_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\
+_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-prime} \\
 \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
 \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}
+\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } 
 \end{eqnarray}
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.5 2001/10/15 19:34:28 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.19 2004/03/23 16:47:04 afe Exp $
 % $Name:  $
 
 \section{Appendix OCEAN}
@@ -1164,13 +1261,14 @@
 \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}}
-_{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}
+_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
+\rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
+\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
+\frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zns-salt}
+\label{eq:non-boussinesq}
 \end{eqnarray}
 These equations permit acoustics modes, inertia-gravity waves,
-non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
+non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
 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
@@ -1186,8 +1284,7 @@
 \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
-{eq-zns-cont} gives: 
+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{
 v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
@@ -1364,7 +1461,7 @@
 _{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 1.5 2001/10/15 19:34:28 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.19 2004/03/23 16:47:04 afe Exp $
 % $Name:  $
 
 \section{Appendix:OPERATORS}
@@ -1377,11 +1474,11 @@
 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 $
 
@@ -1389,7 +1486,7 @@
 \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.
 
@@ -1397,14 +1494,14 @@
 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*}
 
