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

Diff of /manual/s_overview/text/manual.tex

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-revision 1.1 by cnh,
Thu Sep 27 17:45:03 2001 UTC
+revision 1.4 by adcroft,
Thu Oct 11 19:36:56 2001 UTC
 Line 1
- %%%% % $Header$
+ % $Header$
- %%%% % $Name$
+ % $Name$
- %%%% %\usepackage{oldgerm}
- %%%% % I commented the following because it introduced excessive white space
+ %tci%\documentclass[12pt]{book}
- %%%% %\usepackage{palatcm}              % better PDF
+ %tci%\usepackage{amsmath}
- %%%% % page headers and footers
+ %tci%\usepackage{html}
- %%%% %\pagestyle{fancy}
+ %tci%\usepackage{epsfig}
- %%%% % referencing
+ %tci%\usepackage{graphics,subfigure}
- %%%% %% \newcommand{\refequ}[1]{equation (\ref{equ:#1})}
+ %tci%\usepackage{array}
- %%%% %% \newcommand{\refequbig}[1]{Equation (\ref{equ:#1})}
+ %tci%\usepackage{multirow}
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+ %tci%\usepackage{psfrag}
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+ %tci%%TCIDATA{OutputFilter=Latex.dll}
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+ %tci%%TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22}
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+ %tci%%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
- %%%% %% \newcommand{\mathinfigure}[1]{\small\ensuremath{{#1}}}
+ %tci%%TCIDATA{Language=American English}
- %%%% % This allows numbering of subsubsections
- %%%% % This changes the the chapter title
+ %tci%\fancyhead{}
- %%%% %\renewcommand{\chaptername}{Section}
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+ %tci%\fancyhead[RE]{\slshape \leftmark}
+ %tci%\fancyhead[RO,LE]{\thepage}
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+ %tci%\fancyfoot[RO,LE]{ }
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+ %tci%\setcounter{secnumdepth}{3}
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+ %tci%\input{tcilatex}
- %%%% \usepackage{array}
- %%%% \usepackage{multirow}
+ %tci%\begin{document}
- %%%% \usepackage{fancyhdr}
- %%%% \usepackage{psfrag}
+ %tci%\tableofcontents
- %%%%
- %%%% %TCIDATA{OutputFilter=Latex.dll}
- %%%% %TCIDATA{LastRevised=Thursday, September 27, 2001 10:59:02}
- %%%% %TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
- %%%% %TCIDATA{Language=American English}
- %%%%
- %%%% \fancyhead{}
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- %%%% \input{tcilatex}
- %%%%
- %%%% \begin{document}
- %%%%
- %%%% \tableofcontents
- \pagebreak
- \part{MITgcm basics}
+ \part{MIT GCM basics}
  % Section: Overview
-Line 78 
 MITgcm has a number of novel aspects:
+Line 56 
 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%
+ models - see fig
  \marginpar{
  Fig.1 One model}\ref{fig:onemodel}
- \begin{figure}
+ %% CNHbegin
- \begin{center}
+ \input{part1/one_model_figure}
- \resizebox{!}{4in}{
+ %% CNHend
-  \rotatebox{90}{
-  \rotatebox{180}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps}
-  }
-  }
- }
- \end{center}
- \label{fig:onemodel}
- \end{figure}
  \item it has a non-hydrostatic capability and so can be used to study both
- small-scale and large scale processes - see fig.2%
+ small-scale and large scale processes - see fig
  \marginpar{
  Fig.2 All scales}\ref{fig:all-scales}
+ %% CNHbegin
- \begin{figure}
+ \input{part1/all_scales_figure}
- \begin{center}
+ %% CNHend
- \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}
  \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%
+ orthogonal curvilinear grids and shaved cells - see fig
  \marginpar{
- Fig.3 Finite volumes}\ref{fig:Finite volumes}
+ Fig.3 Finite volumes}\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
-Line 147 
 atmospheric winds - see fig.2\ref{fig:al
+Line 109 
 atmospheric winds - see fig.2\ref{fig:al
  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
+ given later. Indeed many of the illustrative examples shown below can be
- download the model (the minimum you need is a PC running linux, together
+ easily reproduced: simply download the model (the minimum you need is a PC
- 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}
- Fig.E1a.\ref{fig:Held-Suarez} is an instaneous plot of the 500$mb$ height
+ A novel feature of MITgcm is its ability to simulate both atmospheric and
- field obtained using a 5-level version of the atmospheric pressure isomorph
+ oceanographic flows at both small and large scales.
- 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}
+ Fig.E1a.\ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
- \begin{center}
+ temperature field obtained using the atmospheric isomorph of MITgcm run at
- \resizebox{!}{4in}{
+.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
-  \rotatebox{90}{
+ (blue) and warm air along an equatorial band (red). Fully developed
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps}
+ 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
- \end{center}
+ radiative-convective equilibrium profile, following the description set out
- \label{fig:hscs}
+ in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
- \end{figure}
+ there are no mountains or land-sea contrast.
+ %% CNHbegin
+ \input{part1/cubic_eddies_figure}
+ %% CNHend
+ As described in Adcroft (2001), a `cubed sphere' is used to discretize the
+ globe permitting a uniform gridding and obviated the need to fourier filter.
+ The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
+ grid, of which the cubed sphere is just one of many choices.
+ Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
+ wind and meridional overturning streamfunction from a 20-level version of
+ the model. It compares favorable with more conventional spatial
+ discretization approaches.
  A regular spherical lat-lon grid can also be used.
- \begin{figure}
+ %% CNHbegin
- \begin{center}
+ \input{part1/hs_zave_u_figure}
- \resizebox{!}{4in}{
+ %% CNHend
-  \rotatebox{90}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps}
-  }
- }
- \end{center}
- \label{fig:hslatlon}
- \end{figure}
  \subsection{Ocean gyres}
+ 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.
+ Fig. ?.? shows the surface temperature and velocity field obtained from
+ MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
+ grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
+ (to avoid the converging of meridian in northern latitudes). 21 vertical
+ levels are used in the vertical with a `lopped cell' representation of
+ topography. The development and propagation of anomalously warm and cold
+ eddies can be clearly been seen in the Gulf Stream region. The transport of
+ warm water northward by the mean flow of the Gulf Stream is also clearly
+ visible.
+ %% CNHbegin
+ \input{part1/ocean_gyres_figure}
+ %% CNHend
  \subsection{Global ocean circulation}
  Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
- global ocean model run with 15 vertical levels. The model is driven using
+ global ocean model run with 15 vertical levels. Lopped cells are used to
- monthly-mean winds with mixed boundary conditions on temperature and
+ represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
- salinity at the surface. Fig.E2b shows the overturning (thermohaline)
+ }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
- circulation. Lopped cells are used to represent topography on a regular $%
+ mixed boundary conditions on temperature and salinity at the surface. The
- lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$.
+ transfer properties of ocean eddies, convection and mixing is parameterized
+ in this model.
- \begin{figure}
+ Fig.E2b shows the meridional overturning circulation of the global ocean in
- \begin{center}
+ Sverdrups.
- \resizebox{!}{4in}{
- % \rotatebox{90}{
+ %%CNHbegin
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps}
+ \input{part1/global_circ_figure}
- % }
+ %%CNHend
- }
- \end{center}
+ \subsection{Convection and mixing over topography}
- \label{fig:horizcirc}
- \end{figure}
+ Dense plumes generated by localized cooling on the continental shelf of the
+ ocean may be influenced by rotation when the deformation radius is smaller
- \begin{figure}
+ than the width of the cooling region. Rather than gravity plumes, the
- \begin{center}
+ mechanism for moving dense fluid down the shelf is then through geostrophic
- \resizebox{!}{4in}{
+ eddies. The simulation shown in the figure (blue is cold dense fluid, red is
-  \rotatebox{90}{
+ warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
-  \rotatebox{180}{
+ trigger convection by surface cooling. The cold, dense water falls down the
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps}
+ 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.
- \end{center}
- \label{fig:moc}
+ %%CNHbegin
- \end{figure}
+ \input{part1/convect_and_topo}
+ %%CNHend
- \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}
- \subsection{Carbon outgassing sensitivity}
+ 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
- Fig.E4 shows....
+ dynamics and mixing in oceanic canyons and ridges driven by large amplitude
+ barotropic tidal currents imposed through open boundary conditions.
- \begin{figure}
- \begin{center}
+ Fig. ?.? shows the influence of cross-slope topographic variations on
- \resizebox{!}{4in}{
+ internal wave breaking - the cross-slope velocity is in color, the density
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps}
+ contoured. The internal waves are excited by application of open boundary
- }
+ conditions on the left.\ They propagate to the sloping boundary (represented
- \end{center}
+ using MITgcm's finite volume spatial discretization) where they break under
- \label{fig:co2mrt}
+ nonhydrostatic dynamics.
- \end{figure}
+ %%CNHbegin
+ \input{part1/boundary_forced_waves}
+ %%CNHend
+ \subsection{Parameter sensitivity using the adjoint of MITgcm}
+ Forward and tangent linear counterparts of MITgcm are supported using an
+ `automatic adjoint compiler'. These can be used in parameter sensitivity and
+ data assimilation studies.
+ As one example of application of the MITgcm adjoint, Fig.E4 maps the
+ gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+ of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $
+ \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
+ sensitive to heat fluxes over the Labrador Sea, one of the important sources
+ of deep water for the thermohaline circulations. This calculation also
+ yields sensitivities to all other model parameters.
+ %%CNHbegin
+ \input{part1/adj_hf_ocean_figure}
+ %%CNHend
+ \subsection{Global state estimation of the ocean}
+ An important application of MITgcm is in state estimation of the global
+ 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
+ parameters' such as air-sea fluxes, the wind field, the initial conditions
+ etc. Figure ?.? shows an estimate of the time-mean surface elevation of the
+ ocean obtained by bringing the model in to consistency with altimetric and
+ in-situ observations over the period 1992-1997.
+ %% CNHbegin
+ \input{part1/globes_figure}
+ %% CNHend
+ \subsection{Ocean biogeochemical cycles}
+ MITgcm is being used to study global biogeochemical cycles in the ocean. For
+ example one can study the effects of interannual changes in meteorological
+ forcing and upper ocean circulation on the fluxes of carbon dioxide and
+ oxygen between the ocean and atmosphere. The figure shows the annual air-sea
+ flux of oxygen and its relation to density outcrops in the southern oceans
+ from a single year of a global, interannually varying simulation.
+ %%CNHbegin
+ \input{part1/biogeo_figure}
+ %%CNHend
+ \subsection{Simulations of laboratory experiments}
+ Figure ?.? shows MITgcm being used to simulate a laboratory experiment
+ enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
+ 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
+ stratification of the ACC.
+ %%CNHbegin
+ \input{part1/lab_figure}
+ %%CNHend
  % $Header$
  % $Name$
-Line 262 
 Fig.E4 shows....
+Line 296 
 Fig.E4 shows....
  To render atmosphere and ocean models from one dynamical core we exploit
  `isomorphisms' between equation sets that govern the evolution of the
- respective fluids - see fig.4%
+ respective fluids - see fig.4
  \marginpar{
  Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down
  and encoded. The model variables have different interpretations depending on
-Line 270 
 whether the atmosphere or ocean is being
+Line 304 
 whether the atmosphere or ocean is being
  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
  modeling the atmosphere and height, $z$, if we are modeling the ocean.
+ %%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
  `geopotential' $\phi $ and density $\rho =\rho (\theta ,S,p)$ which may
  depend on $\theta $, $S$, and $p$. The equations that govern the evolution
  of these fields, obtained by applying the laws of classical mechanics and
  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
- a generic vertical coordinate, $r$, see fig.5%
+ a generic vertical coordinate, $r$, see fig.5
  \marginpar{
  Fig.5 The vertical coordinate of model}:
- \begin{figure}
+ %%CNHbegin
- \begin{center}
+ \input{part1/vertcoord_figure.tex}
- \resizebox{!}{4in}{
+ %%CNHend
-  \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}}%
+ \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}
  \end{equation*}
  \begin{equation*}
- \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{%
+ \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
  vertical mtm}
  \end{equation*}
  \begin{equation}
- \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{%
+ \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
  \partial r}=0\text{ continuity}  \label{eq:continuous}
  \end{equation}
  \begin{equation*}
  b=b(\theta ,S,r)\text{ equation of state}
  \end{equation*}
  \begin{equation*}
- \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{  potential temperature}
+ \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
  \end{equation*}
  \begin{equation*}
- \frac{DS}{Dt}=\mathcal{Q}_{S}\text{  humidity/salinity}
+ \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
  \end{equation*}
  Here:
  \begin{equation*}
  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}
  \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
  \begin{equation*}
  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}
  \end{equation*}
  \begin{equation*}
  \phi \text{ is the `pressure'/`geopotential'}
  \end{equation*}
  \begin{equation*}
  \vec{\Omega}\text{ is the Earth's rotation}
  \end{equation*}
  \begin{equation*}
  b\text{ is the `buoyancy'}
  \end{equation*}
  \begin{equation*}
  \theta \text{ is potential temperature}
  \end{equation*}
  \begin{equation*}
  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 }%
+ \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
- \theta
  \end{equation*}
  \begin{equation*}
-Line 406 
 at fixed and moving $r$ surfaces we set
+Line 434 
 at fixed and moving $r$ surfaces we set
  Here
  \begin{equation*}
  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
-Line 417 
 of motion.
+Line 445 
 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}
-Line 454 
 where
+Line 482 
 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 467 
 In the above the ideal gas law, $p=\rho
+Line 495 
 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.
  At the top of the atmosphere (which is `fixed' in our $r$ coordinate):
  \begin{equation*}
  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)
  \end{equation*}
  i.e. the (hydrostatic) pressure at the top of the mountains in a resting
  atmosphere.
-Line 517 
 At the bottom of the ocean: $R_{fixed}(x
+Line 545 
 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 525 
 Boundary conditions are:
+Line 553 
 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.
-Line 542 
 Let us separate $\phi $ in to surface, h
+Line 570 
 Let us separate $\phi $ in to surface, h
  \begin{equation}
  \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
  \label{eq:phi-split}
- \end{equation}%
+ \end{equation}
  and write eq(\ref{incompressible}a,b) in the form:
  \begin{equation}
-Line 556 
 _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
+Line 584 
 _{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 580 
 $-\left\{ \underline{\frac{u\dot{r}}{{r}
+Line 608 
 $-\left\{ \underline{\frac{u\dot{r}}{{r}
  \\
  $-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\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}
+ \ \right. \qquad  \label{eq:gu-speherical}
  \end{equation}
  \begin{equation}
-Line 599 
 $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
+Line 627 
 $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
  $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\}
  $ \\
  $-\left\{ -2\Omega u\sin lat\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}
+ \ \right. \qquad  \label{eq:gv-spherical}
- \end{equation}%
+ \end{equation}
  \qquad \qquad \qquad \qquad \qquad
  \begin{equation}
  \left.
-Line 618 
 $+\mathcal{F}_{v}$%
+Line 646 
 $+\mathcal{F}_{v}$%
  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
  ${+}\underline{{2\Omega u\cos lat}}$ \\
- $\underline{\underline{\mathcal{F}_{\dot{r}}}}$%
+ $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
- \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}
+ \ \right.  \label{eq:gw-spherical}
- \end{equation}%
+ \end{equation}
  \qquad \qquad \qquad \qquad \qquad
- In the above `${r}$' is the distance from the center of the earth and `$lat$%
+ In the above `${r}$' is the distance from the center of the earth and `$lat$
  ' is latitude.
  Grad and div operators in spherical coordinates are defined in appendix
- OPERATORS.%
+ OPERATORS.
  \marginpar{
  Fig.6 Spherical polar coordinate system.}
- \begin{figure}
+ %%CNHbegin
- \begin{center}
+ \input{part1/sphere_coord_figure.tex}
- \resizebox{!}{4in}{
+ %%CNHend
-  \rotatebox{90}{
-  \rotatebox{180}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps}
-  }
-  }
- }
- \end{center}
- \label{fig:spcoord}
- \end{figure}
  \subsubsection{Shallow atmosphere approximation}
-Line 661 
 hydrostatic balance and the `traditional
+Line 679 
 hydrostatic balance and the `traditional
  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
+ shallow atmosphere approximation can be relaxed - when dividing through by $
- $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
+ 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}
-Line 674 
 terms in Eqs. (\ref{eq:gu-speherical} $\
+Line 692 
 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
- 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 688 
 variation of the radial position of a pa
+Line 706 
 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
  \end{equation*}
  making a small correction to the hydrostatic pressure.
-Line 704 
 only a quasi-non-hydrostatic atmospheric
+Line 722 
 only a quasi-non-hydrostatic atmospheric
  \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
-Line 717 
 and Bromley, 1995; Marshall et.al.\ 1997
+Line 735 
 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 751 
 equations in $z-$coordinates are support
+Line 769 
 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 Fig.7.
  \marginpar{
  Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
  first solved to find the surface pressure and the hydrostatic pressure at
-Line 769 
 forward and $\dot{r}$ found from continu
+Line 787 
 forward and $\dot{r}$ found from continu
  stepping forward the horizontal momentum equations; $\dot{r}$ is found by
  stepping forward the vertical momentum equation.
- \begin{figure}
+ %%CNHbegin
- \begin{center}
+ \input{part1/solution_strategy_figure.tex}
- \resizebox{!}{4in}{
+ %%CNHend
-  \rotatebox{90}{
-  \rotatebox{180}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps}
-  }
-  }
- }
- \end{center}
- \label{fig:solnstart}
- \end{figure}
  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
- course, some complication that goes with the inclusion of $\cos \phi \ $%
+ course, some complication that goes with the inclusion of $\cos \phi \ $
  Coriolis terms and the relaxation of the shallow atmosphere approximation.
  But this leads to negligible increase in computation. In \textbf{NH}, in
  contrast, one additional elliptic equation - a three-dimensional one - must
-Line 808 
 Hydrostatic pressure is obtained by inte
+Line 816 
 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 826 
 atmospheric pressure pushing down on the
+Line 834 
 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 $r=R_{moving}$
  \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*}
  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
  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 ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref
  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
  surface' and `rigid lid' approaches are available.
  \subsubsection{Non-hydrostatic pressure}
- Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{%
+ Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
  (\ref{incompressible}), we deduce that:
  \begin{equation}
- \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{%
+ \nabla _{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 893 
 coasts (in the ocean) and the bottom:
+Line 901 
 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
-Line 910 
 where
+Line 918 
 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 inhomogenous
+ chosen $\delta $-function sheet of `source-charge', replace the
- boundary condition on pressure by a homogeneous one. The source term $rhs$
+ inhomogeneous boundary condition on pressure by a homogeneous one. The
- in (\ref{eq:3d-invert}) is the divergence of the vector $\vec{\mathbf{F}}.$
+ source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $
- 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 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*}%
+ \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 957 
 Many forms of momentum dissipation are a
+Line 965 
 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 968 
 friction. These coefficients are the sam
+Line 976 
 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 984 
 reduces to a diagonal matrix with consta
+Line 992 
 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 994 
 salinity ... ).
+Line 1002 
 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 eq(\ref
  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
  \begin{equation}
- \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}%
+ \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 adoint counterparts of the forward model and described in
+ Tangent linear and adjoint counterparts of the forward model and described
- Chapter 5.
+ in Chapter 5.
  % $Header$
  % $Name$
-Line 1028 
 coordinates}
+Line 1036 
 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}}}
+ _{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} \\
+ \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} \\
+ 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%
+ 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 1052 
 $\theta $ so that it looks more like a g
+Line 1060 
 $\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
  \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}
  \end{equation*}
  and on substituting into (\ref{eq-p-heat-interim}) gives:
  \begin{equation}
-Line 1090 
 and on substituting into (\ref{eq-p-heat
+Line 1098 
 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 1134 
 _{o}(p_{o})=g~Z_{topo}$, defined:
+Line 1142 
 _{o}(p_{o})=g~Z_{topo}$, 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}}} \\
  \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}
-Line 1154 
 We review here the method by which the s
+Line 1162 
 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 \\
- \rho  &=&\rho (\theta ,S,p) \\
+ \rho &=&\rho (\theta ,S,p) \\
  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
- \end{eqnarray}%
+ \end{eqnarray}
  These equations permit acoustics modes, inertia-gravity waves,
  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
  mode. As written, they cannot be integrated forward consistently - if we
  step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
- consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref%
+ consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
  {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
  therefore necessary to manipulate the system as follows. Differentiating the
  EOS (equation of state) gives:
-Line 1181 
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
+Line 1189 
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
  \end{equation}
  Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
- reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref%
+ reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
  {eq-zns-cont} gives:
  \begin{equation}
- \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%
+ \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
  \end{equation}
  where we have used an approximation sign to indicate that we have assumed
-Line 1192 
 adiabatic motion, dropping the $\frac{D\
+Line 1200 
 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 1211 
 wherever it appears in a product (ie. no
+Line 1219 
 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
  height and a perturbation:
  \begin{equation*}
  \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{%
+ \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 1278 
 equation if:
+Line 1286 
 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 1302 
 Here, the objective is to drop the depth
+Line 1310 
 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 1324 
 retain compressibility effects in the de
+Line 1332 
 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 1348 
 _{c}}\frac{\partial p^{\prime }}{\partia
+Line 1356 
 _{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 1372 
 In spherical coordinates, the velocity c
+Line 1380 
 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}
  \end{equation*}
  \begin{equation*}
  v=r\frac{D\phi }{Dt}\qquad
  \end{equation*}
  $\qquad \qquad \qquad \qquad $
  \begin{equation*}
  \dot{r}=\frac{Dr}{Dt}
  \end{equation*}
  Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial
-Line 1392 
 The `grad' ($\nabla $) and `div' ($\nabl
+Line 1400 
 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 \phi }\frac{\partial }{\partial \lambda }
- ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%
+ ,\frac{1}{r}\frac{\partial }{\partial \phi },\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}
  \end{equation*}
- %%%% \end{document}
+ %tci%\end{document}

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