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

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-revision 1.2 by cnh,
Tue Oct  9 10:48:03 2001 UTC
+revision 1.7 by cnh,
Thu Oct 25 12:06:56 2001 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
-Line 77 
 MITgcm has a number of novel aspects:
+Line 54 
 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 \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.2%
+ 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}
+ %% CNHend
  \item finite volume techniques are employed yielding an intuitive
  discretization and support for the treatment of irregular geometries using
- orthogonal curvilinear grids and shaved cells - see fig.3%
+ 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}
+ %% CNHend
  \item tangent linear and adjoint counterparts are automatically maintained
  along with the forward model, permitting sensitivity and optimization
-Line 105 
 listed in an Appendix.
+Line 88 
 listed in an Appendix.
  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
  % $Header$
  % $Name$
-Line 114 
 give a feel for the wide range of proble
+Line 96 
 give a feel for the wide range of proble
  The MITgcm has been designed and used to model a wide range of phenomena,
  from convection on the scale of meters in the ocean to the global pattern of
- atmospheric winds - see fig.2\ref{fig:all-scales}. To give a flavor of the
+ atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the
  kinds of problems the model has been used to study, we briefly describe some
  of them here. A more detailed description of the underlying formulation,
  numerical algorithm and implementation that lie behind these calculations is
-Line 125 
 described in detail in the documentation
+Line 107 
 described in detail in the documentation
  \subsection{Global atmosphere: `Held-Suarez' benchmark}
- 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:Held-Suarez} shows an instantaneous plot of the 500$mb$
+ Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
  temperature field obtained using the atmospheric isomorph of MITgcm run at
 .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 139 
 radiative-convective equilibrium profile
+Line 121 
 radiative-convective equilibrium profile
  in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
  there are no mountains or land-sea contrast.
+ %% CNHbegin
+ \input{part1/cubic_eddies_figure}
+ %% CNHend
  As described in Adcroft (2001), a `cubed sphere' is used to discretize the
- globe permitting a uniform gridding and obviated the need to fourier filter.
+ globe permitting a uniform gridding and obviated the need to Fourier filter.
  The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
  grid, of which the cubed sphere is just one of many choices.
- Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
+ 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}
-Line 161 
 diffusive patterns of ocean currents. Bu
+Line 151 
 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}
+ %% CNHend
  \subsection{Global ocean circulation}
- Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
+ Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
+ the surface of a 4$^{\circ }$
  global ocean model run with 15 vertical levels. Lopped cells are used to
  represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
  }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
-Line 181 
 mixed boundary conditions on temperature
+Line 178 
 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}
-Line 190 
 Dense plumes generated by localized cool
+Line 191 
 Dense plumes generated by localized cool
  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 198 
 entrainment in the vertical plane is red
+Line 200 
 entrainment in the vertical plane is red
  strong, and replaced by lateral entrainment due to the baroclinic
  instability of the along-slope current.
+ %%CNHbegin
+ \input{part1/convect_and_topo}
+ %%CNHend
  \subsection{Boundary forced internal waves}
  The unique ability of MITgcm to treat non-hydrostatic dynamics in the
-Line 205 
 presence of complex geometry makes it an
+Line 211 
 presence of complex geometry makes it an
  dynamics and mixing in oceanic canyons and ridges driven by large amplitude
  barotropic tidal currents imposed through open boundary conditions.
- Fig. ?.? shows the influence of cross-slope topographic variations on
+ Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope
+ topographic variations on
  internal wave breaking - the cross-slope velocity is in color, the density
  contoured. The internal waves are excited by application of open boundary
- conditions on the left.\ They propagate to the sloping boundary (represented
+ conditions on the left. They propagate to the sloping boundary (represented
  using MITgcm's finite volume spatial discretization) where they break under
  nonhydrostatic dynamics.
+ %%CNHbegin
+ \input{part1/boundary_forced_waves}
+ %%CNHend
  \subsection{Parameter sensitivity using the adjoint of MITgcm}
  Forward and tangent linear counterparts of MITgcm are supported using an
  `automatic adjoint compiler'. These can be used in parameter sensitivity and
  data assimilation studies.
- As one example of application of the MITgcm adjoint, Fig.E4 maps the
+ 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 $%
+ of the overturning streamfunction shown in figure \ref{fig:large-scale-circ}
- \mathcal{H}$ 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.
+ %%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
-Line 233 
 ocean circulation. An appropriately defi
+Line 250 
 ocean circulation. An appropriately defi
  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
+ etc. Figure \ref{fig:assimilated-globes} shows an estimate of the time-mean
- ocean obtained by bringing the model in to consistency with altimetric and
+ surface elevation of the ocean obtained by bringing the model in to
- in-situ observations over the period 1992-1997.
+ consistency with altimetric and in-situ observations over the period
+-1997. {\bf CHANGE THIS TEXT - FIG FROM PATRICK/CARL/DETLEF}
+ %% CNHbegin
+ \input{part1/globes_figure}
+ %% CNHend
  \subsection{Ocean biogeochemical cycles}
  MITgcm is being used to study global biogeochemical cycles in the ocean. For
  example one can study the effects of interannual changes in meteorological
  forcing and upper ocean circulation on the fluxes of carbon dioxide and
- oxygen between the ocean and atmosphere. The figure shows the annual air-sea
+ 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
- Chris - get figure here: http://puddle.mit.edu/\symbol{126}%
+ telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).
- mick/biogeochem.html
+ %%CNHbegin
+ \input{part1/biogeo_figure}
+ %%CNHend
  \subsection{Simulations of laboratory experiments}
- 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 enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
  initially homogeneous tank of water ($1m$ in diameter) is driven from its
  free surface by a rotating heated disk. The combined action of mechanical
  and thermal forcing creates a lens of fluid which becomes baroclinically
  unstable. The stratification and depth of penetration of the lens is
- arrested by its instability in a process analogous to that whic sets the
+ arrested by its instability in a process analogous to that which sets the
  stratification of the ACC.
+ %%CNHbegin
+ \input{part1/lab_figure}
+ %%CNHend
  % $Header$
  % $Name$
-Line 267 
 stratification of the ACC.
+Line 296 
 stratification of the ACC.
  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 (left hand side of figure \ref{fig:isomorphic-equations})
+ and height, $z$, if we are modeling the ocean (right hand side of figure
+ \ref{fig:isomorphic-equations}).
+ %%CNHbegin
+ \input{part1/zandpcoord_figure.tex}
+ %%CNHend
  The state of the fluid at any time is characterized by the distribution of
  velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a
-Line 281 
 velocity $\vec{\mathbf{v}}$, active trac
+Line 315 
 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*}
- \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}
-Line 326 
 is the total derivative}
+Line 364 
 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 363 
 S\text{ is specific humidity in the atmo
+Line 401 
 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 376 
 S\text{ is specific humidity in the atmo
+Line 414 
 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)}
-Line 408 
 of motion.
+Line 447 
 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 445 
 where
+Line 484 
 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 458 
 In the above the ideal gas law, $p=\rho
+Line 497 
 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 508 
 At the bottom of the ocean: $R_{fixed}(x
+Line 547 
 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 516 
 Boundary conditions are:
+Line 555 
 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 533 
 Let us separate $\phi $ in to surface, h
+Line 572 
 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 547 
 _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
+Line 586 
 _{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 567 
 discussion:
+Line 606 
 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 587 
 $+\mathcal{F}_{u}$%
+Line 626 
 $+\mathcal{F}_{u}$%
  \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 608 
 $+\mathcal{F}_{v}$%
+Line 647 
 $+\mathcal{F}_{v}$%
  \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}
+ %%CNHend
  \subsubsection{Shallow atmosphere approximation}
  Most models are based on the `hydrostatic primitive equations' (HPE's) in
-Line 638 
 hydrostatic balance and the `traditional
+Line 681 
 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 $%
+ shallow atmosphere approximation can be relaxed - when dividing through by $
  r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
  the radius of the earth.
  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
+ \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
  These are discussed at length in Marshall et al (1997a).
-Line 651 
 terms in Eqs. (\ref{eq:gu-speherical} $\
+Line 695 
 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 665 
 variation of the radial position of a pa
+Line 709 
 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 681 
 only a quasi-non-hydrostatic atmospheric
+Line 725 
 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 694 
 and Bromley, 1995; Marshall et.al.\ 1997
+Line 738 
 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 728 
 equations in $z-$coordinates are support
+Line 772 
 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 746 
 forward and $\dot{r}$ found from continu
+Line 790 
 forward and $\dot{r}$ found from continu
  stepping forward the horizontal momentum equations; $\dot{r}$ is found by
  stepping forward the vertical momentum equation.
+ %%CNHbegin
+ \input{part1/solution_strategy_figure.tex}
+ %%CNHend
  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
- course, some complication that goes with the inclusion of $\cos \phi \ $%
+ course, some complication that goes with the inclusion of $\cos \varphi \ $
  Coriolis terms and the relaxation of the shallow atmosphere approximation.
  But this leads to negligible increase in computation. In \textbf{NH}, in
  contrast, one additional elliptic equation - a three-dimensional one - must
-Line 757 
 Marshall et al, 1997) resulting in a non
+Line 805 
 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 771 
 Hydrostatic pressure is obtained by inte
+Line 820 
 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 789 
 atmospheric pressure pushing down on the
+Line 838 
 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*}
-Line 801 
 Thus:
+Line 850 
 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
-Line 820 
 be written
+Line 869 
 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 856 
 coasts (in the ocean) and the bottom:
+Line 905 
 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 873 
 where
+Line 922 
 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 920 
 Many forms of momentum dissipation are a
+Line 969 
 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 931 
 friction. These coefficients are the sam
+Line 980 
 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 947 
 reduces to a diagonal matrix with consta
+Line 996 
 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 957 
 salinity ... ).
+Line 1006 
 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.
-Line 991 
 coordinates}
+Line 1040 
 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 1015 
 $\theta $ so that it looks more like a g
+Line 1064 
 $\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 1053 
 and on substituting into (\ref{eq-p-heat
+Line 1102 
 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 1097 
 _{o}(p_{o})=g~Z_{topo}$, defined:
+Line 1146 
 _{o}(p_{o})=g~Z_{topo}$, 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 1117 
 We review here the method by which the s
+Line 1166 
 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) \\
  \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 1144 
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
+Line 1193 
 _{\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 1155 
 adiabatic motion, dropping the $\frac{D\
+Line 1204 
 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 1174 
 wherever it appears in a product (ie. no
+Line 1223 
 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 1210 
 height and a perturbation:
+Line 1259 
 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 1241 
 equation if:
+Line 1290 
 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 1265 
 Here, the objective is to drop the depth
+Line 1314 
 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 1287 
 retain compressibility effects in the de
+Line 1336 
 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 1311 
 _{c}}\frac{\partial p^{\prime }}{\partia
+Line 1360 
 _{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 1335 
 In spherical coordinates, the velocity c
+Line 1384 
 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 1347 
 $\qquad \qquad \qquad \qquad $
+Line 1396 
 $\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 1355 
 The `grad' ($\nabla $) and `div' ($\nabl
+Line 1404 
 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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