--- manual/s_overview/text/manual.tex	2001/09/27 17:45:03	1.1
+++ manual/s_overview/text/manual.tex	2008/01/17 21:28:22	1.27
@@ -1,69 +1,43 @@
-%%%% % $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 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, September 27, 2001 10:59:02}
-%%%% %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
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.27 2008/01/17 21:28:22 jmc Exp $
+% $Name:  $
+
+%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}
 
-\pagebreak
+%tci%\tableofcontents
 
-\part{MITgcm basics}
 
 % Section: Overview
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.27 2008/01/17 21:28:22 jmc 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
@@ -73,53 +47,37 @@
 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.1%
-\marginpar{
-Fig.1 One model}\ref{fig:onemodel}
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{ 
- \rotatebox{90}{
- \rotatebox{180}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps}
- }
- }
-}
-\end{center}
-\label{fig:onemodel}
-\end{figure}
+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}
-
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{ 
- \rotatebox{90}{
- \rotatebox{180}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/scales.eps}
- }
- }
-}
-\end{center}
-\label{fig:scales}
-\end{figure}
+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
@@ -129,146 +87,317 @@
 computational platforms.
 \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,mars-eta:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04}
+(an overview on the model formulation can also be found in \cite{adcroft:04c}):
+
+\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.
-\pagebreak
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.27 2008/01/17 21:28:22 jmc Exp $
 % $Name:  $
 
 \section{Illustrations of the model in action}
 
-The MITgcm has been designed and used to model a wide range of phenomena,
+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 it is easy to reproduce the results shown here: simply
-download the model (the minimum you need is a PC running linux, together
-with a FORTRAN\ 77 compiler) and follow the examples.
+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
+described in detail in the documentation.
 
 \subsection{Global atmosphere: `Held-Suarez' benchmark}
+\begin{rawhtml}
+<!-- CMIREDIR:atmospheric_example: -->
+\end{rawhtml}
 
-Fig.E1a.\ref{fig:Held-Suarez} is an instaneous plot of the 500$mb$ height
-field obtained using a 5-level version of the atmospheric pressure isomorph
-run at 2.8$^{\circ }$ resolution. We see fully developed baroclinic eddies
-along the northern hemisphere storm track. There are no mountains or
-land-sea contrast in this calculation, but you can easily put them in. The
-model is driven by relaxation to a radiative-convective equilibrium profile,
-following the description set out in Held and Suarez; 1994 designed to test
-atmospheric hydrodynamical cores - there are no mountains or land-sea
-contrast. As decribed in Adcroft (2001), a `cubed sphere' is used to
-descretize the globe permitting a uniform gridding and obviated the need to
-fourier filter.
-
-Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
-wind and meridional overturning streamfunction from the 5-level model.
-
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps}
- }
-}
-\end{center}
-\label{fig:hscs}
-\end{figure}
-
-
-A regular spherical lat-lon grid can also be used.
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps}
- }
-}
-\end{center}
-\label{fig:hslatlon}
-\end{figure}
 
-\subsection{Ocean gyres}
 
-\subsection{Global ocean circulation}
-
-Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
-global ocean model run with 15 vertical levels. The model is driven using
-monthly-mean winds with mixed boundary conditions on temperature and
-salinity at the surface. Fig.E2b shows the overturning (thermohaline)
-circulation. Lopped cells are used to represent topography on a regular $%
-lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$.
-
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
-% \rotatebox{90}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps}
-% }
-}
-\end{center}
-\label{fig:horizcirc}
-\end{figure}
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps}
- }
- }
-}
-\end{center}
-\label{fig:moc}
-\end{figure}
-
-
-\subsection{Flow over topography}
-
-\subsection{Ocean convection}
-
-Fig.E3 shows convection over a slope using the non-hydrostatic ocean
-isomorph and lopped cells to respresent topography. .....The grid resolution
-is
-
-\subsection{Boundary forced internal waves}
+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
+baroclinic eddies spawned in the northern hemisphere storm track are
+evident. There are no mountains or land-sea contrast in this calculation,
+but you can easily put them in. The model is driven by relaxation to a
+radiative-convective equilibrium profile, following the description set out
+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 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.
+
+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. 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{Carbon outgassing sensitivity}
+\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
+hydrographic structure and current systems of the oceans. Coarse resolution
+models of the oceans cannot resolve the eddy field and yield rather broad,
+diffusive patterns of ocean currents. But if the resolution of our models is
+increased until the baroclinic instability process is resolved, numerical
+solutions of a different and much more realistic kind, can be obtained.
+
+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 \textit{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 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/atl6_figure}
+%% CNHend
 
-Fig.E4 shows....
 
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps}
-}
-\end{center}
-\label{fig:co2mrt}
-\end{figure}
+\subsection{Global ocean circulation}
+\begin{rawhtml}
+<!-- CMIREDIR:global_ocean_circulation: -->
+\end{rawhtml}
+
+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 \textit{lat-lon} grid extending from $70^{\circ }N$ to
+$70^{\circ }S$. The model is driven using monthly-mean winds with
+mixed boundary conditions on temperature and salinity at the surface.
+The transfer properties of ocean eddies, convection and mixing is
+parameterized in this model.
+
+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 \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
+entrainment in the vertical plane is reduced when rotational control is
+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}
+\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. \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
+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}
+\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, 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.
+
+%%CNHbegin
+\input{part1/adj_hf_ocean_figure}
+%%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
+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 \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/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. 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 \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 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 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.27 2008/01/17 21:28:22 jmc 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}
+%%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
@@ -276,108 +405,100 @@
 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}:
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{ 
- \rotatebox{90}{
- \rotatebox{180}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/vertcoord.eps}
- }
- }
-}
-\end{center}
-\label{fig:vertcoord}
-\end{figure}
-
-\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}
-\end{equation*}
+a generic vertical coordinate, $r$, so that the appropriate
+kinematic boundary conditions can be applied isomorphically
+see figure \ref{fig:zandp-vert-coord}.
 
-\begin{equation*}
-\frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{%
+%%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}}}
+\text{ horizontal mtm} \label{eq:horizontal_mtm}
+\end{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}
+\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
+\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*}
-\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{  potential temperature}
-\end{equation*}
+\begin{equation}
+\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
+\label{eq:potential_temperature}
+\end{equation}
 
-\begin{equation*}
-\frac{DS}{Dt}=\mathcal{Q}_{S}\text{  humidity/salinity}
-\end{equation*}
+\begin{equation}
+\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
+\label{eq:humidity_salt}
+\end{equation}
 
 Here:
 
 \begin{equation*}
-r\text{ is the vertical coordinate} 
+r\text{ is the vertical coordinate}
 \end{equation*}
 
 \begin{equation*}
 \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{
-is the total derivative} 
+is the total derivative}
 \end{equation*}
 
 \begin{equation*}
-\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%
-\text{ is the `grad' operator} 
+\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
 
 \begin{equation*}
-t\text{ is time} 
+t\text{ is time}
 \end{equation*}
 
 \begin{equation*}
 \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the
-velocity} 
+velocity}
 \end{equation*}
 
 \begin{equation*}
-\phi \text{ is the `pressure'/`geopotential'} 
+\phi \text{ is the `pressure'/`geopotential'}
 \end{equation*}
 
 \begin{equation*}
-\vec{\Omega}\text{ is the Earth's rotation} 
+\vec{\Omega}\text{ is the Earth's rotation}
 \end{equation*}
 
 \begin{equation*}
-b\text{ is the `buoyancy'} 
+b\text{ is the `buoyancy'}
 \end{equation*}
 
 \begin{equation*}
-\theta \text{ is potential temperature} 
+\theta \text{ is potential temperature}
 \end{equation*}
 
 \begin{equation*}
-S\text{ is specific humidity in the atmosphere; salinity in the ocean} 
+S\text{ is specific humidity in the atmosphere; salinity in the ocean}
 \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*}
 
 \begin{equation*}
-\mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }%
-\theta 
+\mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
 \end{equation*}
 
 \begin{equation*}
@@ -385,28 +506,29 @@
 \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
 
 \begin{equation*}
-R_{moving}=R_{o}+\eta 
+R_{moving}=R_{o}+\eta
 \end{equation*}
 where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on
 whether we are in the atmosphere or ocean) of the `moving surface' in the
@@ -417,12 +539,12 @@
 
 \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}
@@ -454,10 +576,10 @@
 
 \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}
@@ -467,19 +589,19 @@
 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.
 
 At the top of the atmosphere (which is `fixed' in our $r$ coordinate):
 
 \begin{equation*}
-R_{fixed}=p_{top}=0 
+R_{fixed}=p_{top}=0
 \end{equation*}
 In a resting atmosphere the elevation of the mountains at the bottom is
 given by 
 \begin{equation*}
-R_{moving}=R_{o}(x,y)=p_{o}(x,y) 
+R_{moving}=R_{o}(x,y)=p_{o}(x,y)
 \end{equation*}
 i.e. the (hydrostatic) pressure at the top of the mountains in a resting
 atmosphere.
@@ -493,9 +615,11 @@
 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 $p$
-coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).
+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}
 
@@ -517,7 +641,7 @@
 
 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:
@@ -525,25 +649,32 @@
 \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.
 
-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:
 
 \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}%
-and write eq(\ref{incompressible}a,b) in the form:
+\end{equation}
+%and write eq(\ref{eq:incompressible}) in the form: 
+%                  ^- this eq is missing (jmc) ; replaced with:
+and write eq( \ref{eq:horizontal_mtm}) in the form:
 
 \begin{equation}
 \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
@@ -556,19 +687,19 @@
 \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:
 
@@ -576,96 +707,86 @@
 \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}%
-\ \right. \qquad   \label{eq:gu-speherical}
+\textit{\ Forcing/Dissipation}
+\end{tabular}
+\ \right. \qquad  \label{eq:gu-speherical}
 \end{equation}
 
 \begin{equation}
 \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}%
-\ \right. \qquad   \label{eq:gv-spherical}
-\end{equation}%
-\qquad \qquad \qquad \qquad \qquad 
+\textit{\ Forcing/Dissipation}
+\end{tabular}
+\ \right. \qquad  \label{eq:gv-spherical}
+\end{equation}
+\qquad \qquad \qquad \qquad \qquad
 
 \begin{equation}
 \left. 
 \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}%
-\ \right.   \label{eq:gw-spherical}
-\end{equation}%
-\qquad \qquad \qquad \qquad \qquad 
+\textit{\ Forcing/Dissipation}
+\end{tabular}
+\ \right.  \label{eq:gw-spherical}
+\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.}
-
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps}
- }
- }
-}
-\end{center}
-\label{fig:spcoord}
-\end{figure}
+OPERATORS.
 
+%%CNHbegin
+\input{part1/sphere_coord_figure.tex}
+%%CNHend
 
 \subsubsection{Shallow atmosphere approximation}
 
-Most models are based on the `hydrostatic primitive equations' (HPE's) in
-which the vertical momentum equation is reduced to a statement of
-hydrostatic balance and the `traditional approximation' is made in which the
-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 $r
-$ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
-the radius of the earth.
+Most models are based on the `hydrostatic primitive equations' (HPE's)
+in which the vertical momentum equation is reduced to a statement of
+hydrostatic balance and the `traditional approximation' is made in
+which the Coriolis force is treated approximately and the shallow
+atmosphere approximation is made.  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 $ 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).
 
@@ -674,13 +795,13 @@
 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
@@ -688,7 +809,7 @@
 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.
 
@@ -699,17 +820,17 @@
 
 \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}
 
-The MIT model presently supports a full non-hydrostatic ocean isomorph, but
+MITgcm presently supports a full non-hydrostatic ocean isomorph, but
 only a quasi-non-hydrostatic atmospheric isomorph.
 
 \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
 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
@@ -717,13 +838,13 @@
 
 \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}
@@ -751,16 +872,15 @@
 
 \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.%
-\marginpar{
-Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
+The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
+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
@@ -769,22 +889,12 @@
 stepping forward the horizontal momentum equations; $\dot{r}$ is found by
 stepping forward the vertical momentum equation.
 
-\begin{figure}
-\begin{center}
-\resizebox{!}{4in}{
- \rotatebox{90}{
- \rotatebox{180}{
-  \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps}
- }
- }
-}
-\end{center}
-\label{fig:solnstart}
-\end{figure}
-
+%%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
@@ -794,6 +904,7 @@
 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
@@ -808,8 +919,8 @@
 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}%
-\right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr 
+\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
 
@@ -826,54 +937,54 @@
 
 \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}%
-}_{h}+\partial _{r}\dot{r}\right) dr=0 
+\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
+}_{h}+\partial _{r}\dot{r}\right) dr=0
 \end{equation*}
 
 Thus:
 
 \begin{equation*}
 \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
-+\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%
-_{h}dr=0 
++\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
-(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) 
+\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 ($\epsilon _{nh}=0$), 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{%
-\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}
 
@@ -893,13 +1004,13 @@
 \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
 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}}
@@ -910,25 +1021,25 @@
 \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 inhomogenous
-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 $%
+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 $
 \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 homogenised Elliptic problem is obtained:
+self-consistent but simpler homogenized Elliptic problem is obtained:
 
 \begin{equation*}
-\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad 
-\end{equation*}%
+\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
+\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}
@@ -939,7 +1050,7 @@
 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}
@@ -947,7 +1058,7 @@
 \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}
 
@@ -957,7 +1068,7 @@
 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
@@ -968,12 +1079,12 @@
 
 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.
 \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, $\underline{\underline{K}}$,
@@ -984,7 +1095,7 @@
 \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}
@@ -994,29 +1105,30 @@
 
 \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:
+For some purposes it is advantageous to write momentum advection in
+eq(\ref {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}%
-+\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %
-\left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right] 
+\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.
 
 \subsection{Adjoint}
 
-Tangent linear and adoint counterparts of the forward model and described in
-Chapter 5.
+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 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.27 2008/01/17 21:28:22 jmc Exp $
 % $Name:  $
 
 \section{Appendix ATMOSPHERE}
@@ -1028,23 +1140,23 @@
 
 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}}%
-_{h}+\mathbf{\nabla }_{p}\phi  &=&\vec{\mathbf{\mathcal{F}}}
+\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 }{%
+\frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
+\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
 \partial p} &=&0  \label{eq:atmos-cont} \\
-p\alpha  &=&RT  \label{eq:atmos-eos} \\
+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 
@@ -1052,36 +1164,36 @@
 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 
 \begin{equation*}
 c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta 
-\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt} 
+\frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
 \end{equation*}
 and on substituting into (\ref{eq-p-heat-interim}) gives: 
 \begin{equation}
@@ -1090,7 +1202,7 @@
 \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}
@@ -1106,6 +1218,7 @@
 surface ($\phi $ is imposed and $\omega \neq 0$).
 
 \subsubsection{Splitting the geo-potential}
+\label{sec:hpe-p-geo-potential-split}
 
 For the purposes of initialization and reducing round-off errors, the model
 deals with perturbations from reference (or ``standard'') profiles. For
@@ -1134,16 +1247,17 @@
 
 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}}} \\
+\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
+_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} 
+\label{eq:atmos-prime} \\
 \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}
+\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } 
 \end{eqnarray}
 
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.27 2008/01/17 21:28:22 jmc Exp $
 % $Name:  $
 
 \section{Appendix OCEAN}
@@ -1154,21 +1268,22 @@
 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}}%
-_{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}%
+\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
+_{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%
+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:
@@ -1180,11 +1295,11 @@
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}
 \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: 
+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:
 \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
@@ -1192,12 +1307,12 @@
 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} \\
@@ -1211,65 +1326,65 @@
 `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
 height and a perturbation: 
 \begin{equation*}
-\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime }) 
+\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
 \end{equation*}
 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{\partial w}{\partial z}=0 
+\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
@@ -1278,18 +1393,18 @@
 \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} \\
@@ -1302,10 +1417,10 @@
 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}
@@ -1324,20 +1439,20 @@
 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}
@@ -1348,7 +1463,7 @@
 \\
 \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.
@@ -1359,7 +1474,7 @@
 _{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 1.1 2001/09/27 17:45:03 cnh Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/text/manual.tex,v 1.27 2008/01/17 21:28:22 jmc Exp $
 % $Name:  $
 
 \section{Appendix:OPERATORS}
@@ -1372,35 +1487,34 @@
 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}
 \end{equation*}
-$\qquad \qquad \qquad \qquad $
 
 \begin{equation*}
-\dot{r}=\frac{Dr}{Dt} 
+\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.
 
-The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in
+The `grad' ($\nabla $) and `div' ($\nabla\cdot$) operators are defined by, in
 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}%
-\right) 
+\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\}
-+\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r} 
+\nabla\cdot 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}
