/[MITgcm]/manual/s_overview/text/manual.tex

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-revision 1.3 by cnh,
Wed Oct 10 16:48:45 2001 UTC
+revision 1.24 by edhill,
Wed Apr  5 02:27:32 2006 UTC
 Line 1
  % $Header$
  % $Name$
- %\usepackage{oldgerm}
- % I commented the following because it introduced excessive white space
- %\usepackage{palatcm}              % better PDF
- % page headers and footers
- %\pagestyle{fancy}
- % referencing
- %% \newcommand{\refequ}[1]{equation (\ref{equ:#1})}
- %% \newcommand{\refequbig}[1]{Equation (\ref{equ:#1})}
- %% \newcommand{\reftab}[1]{Tab.~\ref{tab:#1}}
- %% \newcommand{\reftabno}[1]{\ref{tab:#1}}
- %% \newcommand{\reffig}[1]{Fig.~\ref{fig:#1}}
- %% \newcommand{\reffigno}[1]{\ref{fig:#1}}
- % stuff for psfrag
- %% \newcommand{\textinfigure}[1]{{\footnotesize\textbf{\textsf{#1}}}}
- %% \newcommand{\mathinfigure}[1]{\small\ensuremath{{#1}}}
- % This allows numbering of subsubsections
- % This changes the the chapter title
- %\renewcommand{\chaptername}{Section}
- %%%% \documentclass[12pt]{book}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- %%%% \usepackage{amsmath}
- %%%% \usepackage{html}
- %%%% \usepackage{epsfig}
- %%%% \usepackage{graphics,subfigure}
- %%%% \usepackage{array}
- %%%% \usepackage{multirow}
- %%%% \usepackage{fancyhdr}
- %%%% \usepackage{psfrag}
- %%%% %TCIDATA{OutputFilter=Latex.dll}
- %%%% %TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22}
- %%%% %TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
- %%%% %TCIDATA{Language=American English}
- %%%% \fancyhead{}
- %%%% \fancyhead[LO]{\slshape \rightmark}
- %%%% \fancyhead[RE]{\slshape \leftmark}
- %%%% \fancyhead[RO,LE]{\thepage}
- %%%% \fancyfoot[CO,CE]{\today}
- %%%% \fancyfoot[RO,LE]{ }
- %%%% \renewcommand{\headrulewidth}{0.4pt}
- %%%% \renewcommand{\footrulewidth}{0.4pt}
- %%%% \setcounter{secnumdepth}{3}
- %%%% \input{tcilatex}
- %%%%
- %%%% \begin{document}
- %%%%
- %%%% \tableofcontents
- %%%%
- %%%% \pagebreak
- %%%% \part{MIT GCM basics}
+ %tci%\documentclass[12pt]{book}
+ %tci%\usepackage{amsmath}
+ %tci%\usepackage{html}
+ %tci%\usepackage{epsfig}
+ %tci%\usepackage{graphics,subfigure}
+ %tci%\usepackage{array}
+ %tci%\usepackage{multirow}
+ %tci%\usepackage{fancyhdr}
+ %tci%\usepackage{psfrag}
+ %tci%%TCIDATA{OutputFilter=Latex.dll}
+ %tci%%TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22}
+ %tci%%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
+ %tci%%TCIDATA{Language=American English}
+ %tci%\fancyhead{}
+ %tci%\fancyhead[LO]{\slshape \rightmark}
+ %tci%\fancyhead[RE]{\slshape \leftmark}
+ %tci%\fancyhead[RO,LE]{\thepage}
+ %tci%\fancyfoot[CO,CE]{\today}
+ %tci%\fancyfoot[RO,LE]{ }
+ %tci%\renewcommand{\headrulewidth}{0.4pt}
+ %tci%\renewcommand{\footrulewidth}{0.4pt}
+ %tci%\setcounter{secnumdepth}{3}
+ %tci%\input{tcilatex}
+ %tci%\begin{document}
+ %tci%\tableofcontents
  % Section: Overview
  % $Header$
  % $Name$
- \section{Introduction}
+ This document provides the reader with the information necessary to
- This documentation 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
-Line 72 
 are available. A number of examples illu
+Line 47 
 are available. A number of examples illu
  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%
+ models - see fig \ref{fig:onemodel}
- \marginpar{
- Fig.1 One model}\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 %
+ small-scale and large scale processes - see fig \ref{fig:all-scales}
- \marginpar{
- Fig.2 All scales}\ref{fig:all-scales}
  %% CNHbegin
  \input{part1/all_scales_figure}
-Line 96 
 Fig.2 All scales}\ref{fig:all-scales}
+Line 73 
 Fig.2 All scales}\ref{fig:all-scales}
  \item finite volume techniques are employed yielding an intuitive
  discretization and support for the treatment of irregular geometries using
- orthogonal curvilinear grids and shaved cells - see fig %
+ orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes}
- \marginpar{
- Fig.3 Finite volumes}\ref{fig:finite-volumes}
  %% CNHbegin
  \input{part1/fvol_figure}
-Line 113 
 computational platforms.
+Line 88 
 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,marshall:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04}:
+ \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
  % $Header$
  % $Name$
  \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 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
+ A novel feature of MITgcm is its ability to simulate, using one basic algorithm,
- oceanographic flows at both small and large scales.
+ both atmospheric and oceanographic flows at both small and large scales.
- Fig.E1a.\ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
+ Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
  temperature field obtained using the atmospheric isomorph of MITgcm run at
 .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
-Line 156 
 there are no mountains or land-sea contr
+Line 179 
 there are no mountains or land-sea contr
  %% 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
+ Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
- wind and meridional overturning streamfunction from a 20-level version of
+ wind from a 20-level configuration of
  the model. It compares favorable with more conventional spatial
- discretization approaches.
+ discretization approaches. The two plots show the field calculated using the
+ cube-sphere grid and the flow calculated using a regular, spherical polar
- A regular spherical lat-lon grid can also be used.
+ 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
-Line 181 
 diffusive patterns of ocean currents. Bu
+Line 210 
 diffusive patterns of ocean currents. Bu
  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
+ Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity
- MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
+ 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
-Line 206 
 mixed boundary conditions on temperature
+Line 240 
 mixed boundary conditions on temperature
  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
+ Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
- Sverdrups.
+ 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
-Line 232 
 instability of the along-slope current.
+Line 271 
 instability of the along-slope current.
  %%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.
-Line 250 
 nonhydrostatic dynamics.
+Line 293 
 nonhydrostatic dynamics.
  %%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
+ As one example of application of the MITgcm adjoint, Figure \ref{fig:hf-sensitivity}
- gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+ maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
- of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $%
+ of the overturning stream-function shown in figure \ref{fig:large-scale-circ}
- \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
+ 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.
-Line 268 
 yields sensitivities to all other model
+Line 316 
 yields sensitivities to all other model
  %%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
+ etc. Figure \ref{fig:assimilated-globes} shows the large scale planetary
- ocean obtained by bringing the model in to consistency with altimetric and
+ circulation and a Hopf-Muller plot of Equatorial sea-surface height.
- in-situ observations over the period 1992-1997.
+ Both are obtained from assimilation bringing the model in to
+ consistency with altimetric and in-situ observations over the period
+-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
+ oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows
- flux of oxygen and its relation to density outcrops in the southern oceans
+ the annual air-sea flux of oxygen and its relation to density outcrops in
- from a single year of a global, interannually varying simulation.
+ 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
+ Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
- enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
+ 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
-Line 314 
 stratification of the ACC.
+Line 376 
 stratification of the ACC.
  % $Name$
  \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%
+ respective fluids - see figure \ref{fig:isomorphic-equations}.
- \marginpar{
+ One system of hydrodynamical equations is written down
- Fig.4. Isomorphisms}. 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}
-Line 335 
 velocity $\vec{\mathbf{v}}$, active trac
+Line 401 
 velocity $\vec{\mathbf{v}}$, active trac
  depend on $\theta $, $S$, and $p$. The equations that govern the evolution
  of these fields, obtained by applying the laws of classical mechanics and
  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
- a generic vertical coordinate, $r$, see fig.5%
+ a generic vertical coordinate, $r$, so that the appropriate
- \marginpar{
+ kinematic boundary conditions can be applied isomorphically
- Fig.5 The vertical coordinate of model}:
+ see figure \ref{fig:zandp-vert-coord}.
  %%CNHbegin
  \input{part1/vertcoord_figure.tex}
  %%CNHend
- \begin{equation*}
+ \begin{equation}
- \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}%
+ \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}}}%
+ \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
- \text{ horizontal mtm}
+ \text{ horizontal mtm} \label{eq:horizontal_mtm}
- \end{equation*}
+ \end{equation}
- \begin{equation*}
+ \begin{equation}
- \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{%
+ \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
- vertical mtm}
+ vertical mtm} \label{eq:vertical_mtm}
- \end{equation*}
+ \end{equation}
  \begin{equation}
- \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{%
+ \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
- \partial r}=0\text{ continuity}  \label{eq:continuous}
+ \partial r}=0\text{ continuity}  \label{eq:continuity}
  \end{equation}
- \begin{equation*}
+ \begin{equation}
- b=b(\theta ,S,r)\text{ equation of state}
+ b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
- \end{equation*}
+ \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:
-Line 384 
 is the total derivative}
+Line 452 
 is the total derivative}
  \end{equation*}
  \begin{equation*}
- \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%
+ \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
  \text{ is the `grad' operator}
  \end{equation*}
- with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%
+ with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}
  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
  is a unit vector in the vertical
-Line 421 
 S\text{ is specific humidity in the atmo
+Line 489 
 S\text{ is specific humidity in the atmo
  \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*}
-Line 434 
 S\text{ is specific humidity in the atmo
+Line 502 
 S\text{ is specific humidity in the atmo
  \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{ \
+ \dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
- (oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}
+ (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
  \end{equation}
  Here
-Line 466 
 of motion.
+Line 535 
 of motion.
  \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}
-Line 503 
 where
+Line 572 
 where
  \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}
-Line 516 
 In the above the ideal gas law, $p=\rho
+Line 585 
 In the above the ideal gas law, $p=\rho
  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.
-Line 542 
 The boundary conditions at top and botto
+Line 611 
 The boundary conditions at top and botto
  atmosphere)}  \label{eq:moving-bc-atmos}
  \end{eqnarray}
- Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent
+ Then the (hydrostatic form of) equations
- set of atmospheric equations which, for convenience, are written out in $p$
+ (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yields a consistent
- coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).
+ set of atmospheric equations which, for convenience, are written out
+ in $p$ coordinates in Appendix Atmosphere - see
+ eqs(\ref{eq:atmos-prime}).
  \subsection{Ocean}
-Line 566 
 At the bottom of the ocean: $R_{fixed}(x
+Line 637 
 At the bottom of the ocean: $R_{fixed}(x
  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:
-Line 574 
 Boundary conditions are:
+Line 645 
 Boundary conditions are:
  \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}%
+ \end{equation}
- and write eq(\ref{incompressible}a,b) in the form:
+ %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
-Line 605 
 _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
+Line 683 
 _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
  \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%
+ terms in the momentum equations. In spherical coordinates they take the form
- \footnote{%
+ \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:
-Line 625 
 discussion:
+Line 703 
 discussion:
  \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\} $
  \\
- $+\mathcal{F}_{u}$%
+ $+\mathcal{F}_{u}$
- \end{tabular}%
+ \end{tabular}
  \ \right\} \left\{
  \begin{tabular}{l}
  \textit{advection} \\
  \textit{metric} \\
  \textit{Coriolis} \\
- \textit{\ Forcing/Dissipation}%
+ \textit{\ Forcing/Dissipation}
- \end{tabular}%
+ \end{tabular}
  \ \right. \qquad  \label{eq:gu-speherical}
  \end{equation}
-Line 645 
 $+\mathcal{F}_{u}$%
+Line 723 
 $+\mathcal{F}_{u}$%
  \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\} $ \\
- $+\mathcal{F}_{v}$%
+ $+\mathcal{F}_{v}$
- \end{tabular}%
+ \end{tabular}
  \ \right\} \left\{
  \begin{tabular}{l}
  \textit{advection} \\
  \textit{metric} \\
  \textit{Coriolis} \\
- \textit{\ Forcing/Dissipation}%
+ \textit{\ Forcing/Dissipation}
- \end{tabular}%
+ \end{tabular}
  \ \right. \qquad  \label{eq:gv-spherical}
- \end{equation}%
+ \end{equation}
  \qquad \qquad \qquad \qquad \qquad
  \begin{equation}
-Line 666 
 $+\mathcal{F}_{v}$%
+Line 744 
 $+\mathcal{F}_{v}$%
  \begin{tabular}{l}
  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
- ${+}\underline{{2\Omega u\cos lat}}$ \\
+ ${+}\underline{{2\Omega u\cos \varphi}}$ \\
- $\underline{\underline{\mathcal{F}_{\dot{r}}}}$%
+ $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
- \end{tabular}%
+ \end{tabular}
  \ \right\} \left\{
  \begin{tabular}{l}
  \textit{advection} \\
  \textit{metric} \\
  \textit{Coriolis} \\
- \textit{\ Forcing/Dissipation}%
+ \textit{\ Forcing/Dissipation}
- \end{tabular}%
+ \end{tabular}
  \ \right.  \label{eq:gw-spherical}
- \end{equation}%
+ \end{equation}
  \qquad \qquad \qquad \qquad \qquad
- In the above `${r}$' is the distance from the center of the earth and `$lat$%
+ In the above `${r}$' is the distance from the center of the earth and `$\varphi$
  ' is latitude.
  Grad and div operators in spherical coordinates are defined in appendix
- OPERATORS.%
+ OPERATORS.
- \marginpar{
- Fig.6 Spherical polar coordinate system.}
  %%CNHbegin
  \input{part1/sphere_coord_figure.tex}
-Line 694 
 Fig.6 Spherical polar coordinate system.
+Line 770 
 Fig.6 Spherical polar coordinate system.
  \subsubsection{Shallow atmosphere approximation}
- Most models are based on the `hydrostatic primitive equations' (HPE's) in
+ Most models are based on the `hydrostatic primitive equations' (HPE's)
- which the vertical momentum equation is reduced to a statement of
+ in which the vertical momentum equation is reduced to a statement of
- hydrostatic balance and the `traditional approximation' is made in which the
+ hydrostatic balance and the `traditional approximation' is made in
- Coriolis force is treated approximately and the shallow atmosphere
+ which the Coriolis force is treated approximately and the shallow
- approximation is made.\ The MITgcm need not make the `traditional
+ atmosphere approximation is made.  MITgcm need not make the
- approximation'. To be able to support consistent non-hydrostatic forms the
+ `traditional approximation'. To be able to support consistent
- shallow atmosphere approximation can be relaxed - when dividing through by $%
+ non-hydrostatic forms the shallow atmosphere approximation can be
- r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
+ relaxed - when dividing through by $ r $ in, for example,
- the radius of the earth.
+ (\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).
-Line 713 
 terms in Eqs. (\ref{eq:gu-speherical} $\
+Line 791 
 terms in Eqs. (\ref{eq:gu-speherical} $\
  are neglected and `${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
+ \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. (%
+ 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
-Line 727 
 variation of the radial position of a pa
+Line 805 
 variation of the radial position of a pa
  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.
-Line 738 
 et.al., 1997a. As in \textbf{HPE }only a
+Line 816 
 et.al., 1997a. As in \textbf{HPE }only a
  \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
-Line 756 
 and Bromley, 1995; Marshall et.al.\ 1997
+Line 834 
 and Bromley, 1995; Marshall et.al.\ 1997
  \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}
-Line 790 
 equations in $z-$coordinates are support
+Line 868 
 equations in $z-$coordinates are support
  \subparagraph{Non-hydrostatic}
- Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$%
+ Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
- coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref%
+ 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{%
+ The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
- NH} models is summarized in Fig.7.%
+ NH} models is summarized in Figure \ref{fig:solution-strategy}.
- \marginpar{
+ Under all dynamics, a 2-d elliptic equation is
- Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
  first solved to find the surface pressure and the hydrostatic pressure at
  any level computed from the weight of fluid above. Under \textbf{HPE} and
  \textbf{QH} dynamics, the horizontal momentum equations are then stepped
-Line 813 
 stepping forward the vertical momentum e
+Line 890 
 stepping forward the vertical momentum e
  %%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
-Line 823 
 Marshall et al, 1997) resulting in a non
+Line 900 
 Marshall et al, 1997) resulting in a non
  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
-Line 837 
 Hydrostatic pressure is obtained by inte
+Line 915 
 Hydrostatic pressure is obtained by inte
  vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:
  \begin{equation*}
- \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}%
+ \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}
  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
  \end{equation*}
  and so
-Line 855 
 atmospheric pressure pushing down on the
+Line 933 
 atmospheric pressure pushing down on the
  \subsubsection{Surface pressure}
- The surface pressure equation can be obtained by integrating continuity, (%
+ The surface pressure equation can be obtained by integrating continuity,
- \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
+ (\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}%
+ \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
  }_{h}+\partial _{r}\dot{r}\right) dr=0
  \end{equation*}
-Line 867 
 Thus:
+Line 945 
 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}}%
+ +\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)
  \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 $\frac{%
+ Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
- \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
+ $\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
- (\ref{incompressible}), we deduce that:
+ (\ref{eq:continuity}), we deduce that:
  \begin{equation}
- \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{%
+ \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 }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .
  \vec{\mathbf{F}}  \label{eq:3d-invert}
  \end{equation}
-Line 922 
 coasts (in the ocean) and the bottom:
+Line 1000 
 coasts (in the ocean) and the bottom:
  \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}}
-Line 939 
 where
+Line 1017 
 where
  \begin{equation*}
  \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi
  _{s}+\mathbf{\nabla }\phi _{hyd}\right)
- \end{equation*}%
+ \end{equation*}
  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
  (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
  exploit classical 3D potential theory and, by introducing an appropriately
  chosen $\delta $-function sheet of `source-charge', replace the
  inhomogeneous boundary condition on pressure by a homogeneous one. The
- source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $%
+ source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $
- \vec{\mathbf{F}}.$ By simultaneously setting $%
+ \vec{\mathbf{F}}.$ By simultaneously setting $
  \begin{array}{l}
- \widehat{n}.\vec{\mathbf{F}}%
+ \widehat{n}.\vec{\mathbf{F}}
- \end{array}%
+ \end{array}
  =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
  self-consistent but simpler homogenized Elliptic problem is obtained:
  \begin{equation*}
  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
- \end{equation*}%
+ \end{equation*}
  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
- that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref%
+ that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref
  {eq:inhom-neumann-nh}) the modified boundary condition becomes:
  \begin{equation}
-Line 968 
 If the flow is `close' to hydrostatic ba
+Line 1046 
 If the flow is `close' to hydrostatic ba
  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}
-Line 976 
 does not vanish at $r=R_{moving}$, and s
+Line 1054 
 does not vanish at $r=R_{moving}$, and s
  \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}
-Line 986 
 Many forms of momentum dissipation are a
+Line 1064 
 Many forms of momentum dissipation are a
  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
-Line 997 
 friction. These coefficients are the sam
+Line 1075 
 friction. These coefficients are the sam
  The mixing terms for the temperature and salinity equations have a similar
  form to that of momentum except that the diffusion tensor can be
- non-diagonal and have varying coefficients. $\qquad $%
+ non-diagonal and have varying coefficients. $\qquad $
  \begin{equation}
  D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
  _{h}^{4}(T,S)  \label{eq:diffusion}
  \end{equation}
- where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%
+ where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $
  horizontal coefficient for biharmonic diffusion. In the simplest case where
  the subgrid-scale fluxes of heat and salt are parameterized with constant
  horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,
-Line 1013 
 reduces to a diagonal matrix with consta
+Line 1091 
 reduces to a diagonal matrix with consta
  \begin{array}{ccc}
  K_{h} & 0 & 0 \\
 & K_{h} & 0 \\
-& 0 & K_{v}%
+& 0 & K_{v}
  \end{array}
  \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}
  \end{equation}
-Line 1023 
 salinity ... ).
+Line 1101 
 salinity ... ).
  \subsection{Vector invariant form}
- For some purposes it is advantageous to write momentum advection in eq(\ref%
+ For some purposes it is advantageous to write momentum advection in
- {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
+ 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}%
+ \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( \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 adjoint counterparts of the forward model and described
+ Tangent linear and adjoint counterparts of the forward model are described
  in Chapter 5.
  % $Header$
-Line 1057 
 coordinates}
+Line 1136 
 coordinates}
  The hydrostatic primitive equations (HPEs) in p-coordinates are:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
  \label{eq:atmos-mom} \\
  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
- \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%
+ \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
  \partial p} &=&0  \label{eq:atmos-cont} \\
  p\alpha &=&RT  \label{eq:atmos-eos} \\
  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}
- \end{eqnarray}%
+ \end{eqnarray}
  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
- derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is
+ 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%
+ the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
- }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref%
+ }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
- {eq:atmos-heat}) is the first law of thermodynamics where internal energy $%
+ {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 $%
+ e=c_{v}T$, $T$ 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
-Line 1081 
 $\theta $ so that it looks more like a g
+Line 1160 
 $\theta $ so that it looks more like a g
  Differentiating (\ref{eq:atmos-eos}) we get:
  \begin{equation*}
  p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
- \end{equation*}%
+ \end{equation*}
- which, when added to the heat equation (\ref{eq:atmos-heat}) and using $%
+ 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 }{%
+ }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
- \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}%
+ \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
-Line 1119 
 and on substituting into (\ref{eq-p-heat
+Line 1198 
 and on substituting into (\ref{eq-p-heat
  \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}
-Line 1135 
 In $p$-coordinates, the upper boundary a
+Line 1214 
 In $p$-coordinates, the upper boundary a
  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
-Line 1163 
 _{o}(p_{o})=g~Z_{topo}$, defined:
+Line 1243 
 _{o}(p_{o})=g~Z_{topo}$, defined:
  The final form of the HPE's in p coordinates is then:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
- _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\
+ _{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}
  % $Header$
-Line 1183 
 We review here the method by which the s
+Line 1264 
 We review here the method by which the s
  HPE's for the ocean written in z-coordinates are obtained. The
  non-Boussinesq equations for oceanic motion are:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\
  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
  &=&\epsilon _{nh}\mathcal{F}_{w} \\
- \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%
+ \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
- _{h}+\frac{\partial w}{\partial z} &=&0 \\
+ _{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
- \rho &=&\rho (\theta ,S,p) \\
+ \rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
- \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
+ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
- \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
+ \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zns-salt}
- \end{eqnarray}%
+ \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:
-Line 1209 
 _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
+Line 1291 
 _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}
  \end{equation}
- Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
+ Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is
- reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref%
+ the reciprocal of the sound speed ($c_{s}$) squared. Substituting into
- {eq-zns-cont} gives:
+ \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
-Line 1221 
 adiabatic motion, dropping the $\frac{D\
+Line 1303 
 adiabatic motion, dropping the $\frac{D\
  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} \\
-Line 1240 
 wherever it appears in a product (ie. no
+Line 1322 
 wherever it appears in a product (ie. no
  `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}%
+ 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}%
+ ). 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
-Line 1276 
 height and a perturbation:
+Line 1358 
 height and a perturbation:
  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{%
+ \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}+%
+ 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}+%
+ where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
- \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%
+ \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ 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
-Line 1307 
 equation if:
+Line 1389 
 equation if:
  \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} \\
-Line 1331 
 Here, the objective is to drop the depth
+Line 1413 
 Here, the objective is to drop the depth
  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}
-Line 1353 
 retain compressibility effects in the de
+Line 1435 
 retain compressibility effects in the de
  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}}%
+ \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{%
+ _{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}
-Line 1377 
 _{c}}\frac{\partial p^{\prime }}{\partia
+Line 1459 
 _{c}}\frac{\partial p^{\prime }}{\partia
  \\
  \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.
-Line 1401 
 In spherical coordinates, the velocity c
+Line 1483 
 In spherical coordinates, the velocity c
  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 $
-Line 1413 
 $\qquad \qquad \qquad \qquad $
+Line 1495 
 $\qquad \qquad \qquad \qquad $
  \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.
-Line 1421 
 The `grad' ($\nabla $) and `div' ($\nabl
+Line 1503 
 The `grad' ($\nabla $) and `div' ($\nabl
  spherical coordinates:
  \begin{equation*}
- \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%
+ \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda }
- ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%
+ ,\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
+ \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
- \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}
+ \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}

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