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-revision 1.4 by adcroft,
Thu Oct 11 19:36:56 2001 UTC
+revision 1.30 by jmc,
Wed May 11 18:45:43 2016 UTC
 Line 32
  %tci%\tableofcontents
- \part{MIT GCM basics}
  % Section: Overview
- % $Header$
+ This document provides the reader with the information necessary to
- % $Name$
- \section{Introduction}
- This documentation provides the reader with the information necessary to
  carry out numerical experiments using MITgcm. It gives a comprehensive
  description of the continuous equations on which the model is based, the
  numerical algorithms the model employs and a description of the associated
-Line 51 
 are available. A number of examples illu
+Line 44 
 are available. A number of examples illu
  both process and general circulation studies of the atmosphere and ocean are
  also presented.
+ \section{Introduction}
+ \begin{rawhtml}
+ <!-- CMIREDIR:innovations: -->
+ \end{rawhtml}
  MITgcm has a number of novel aspects:
  \begin{itemize}
  \item it can be used to study both atmospheric and oceanic phenomena; one
  hydrodynamical kernel is used to drive forward both atmospheric and oceanic
- models - see fig
+ models - see fig \ref{fig:onemodel}
- \marginpar{
- Fig.1 One model}\ref{fig:onemodel}
  %% CNHbegin
- \input{part1/one_model_figure}
+ \input{s_overview/text/one_model_figure}
  %% CNHend
  \item it has a non-hydrostatic capability and so can be used to study both
- small-scale and large scale processes - see fig
+ small-scale and large scale processes - see fig \ref{fig:all-scales}
- \marginpar{
- Fig.2 All scales}\ref{fig:all-scales}
  %% CNHbegin
- \input{part1/all_scales_figure}
+ \input{s_overview/text/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
+ 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}
+ \input{s_overview/text/fvol_figure}
  %% CNHend
  \item tangent linear and adjoint counterparts are automatically maintained
-Line 91 
 studies.
+Line 84 
 studies.
  computational platforms.
  \end{itemize}
  Key publications reporting on and charting the development of the model are
- listed in an Appendix.
+ \cite{hill:95,marshall:97a,marshall:97b,adcroft:97,mars-eta:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04}
+ (an overview on the model formulation can also be found in \cite{adcroft:04c}):
+ \begin{verbatim}
+ Hill, C. and J. Marshall, (1995)
+ Application of a Parallel Navier-Stokes Model to Ocean Circulation in
+ Parallel Computational Fluid Dynamics
+ In Proceedings of Parallel Computational Fluid Dynamics: Implementations
+ and Results Using Parallel Computers, 545-552.
+ Elsevier Science B.V.: New York
+ Marshall, J., C. Hill, L. Perelman, and A. Adcroft, (1997)
+ Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling
+ J. Geophysical Res., 102(C3), 5733-5752.
+ Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, (1997)
+ A finite-volume, incompressible Navier Stokes model for studies of the ocean
+ on parallel computers,
+ J. Geophysical Res., 102(C3), 5753-5766.
+ Adcroft, A.J., Hill, C.N. and J. Marshall, (1997)
+ Representation of topography by shaved cells in a height coordinate ocean
+ model
+ Mon Wea Rev, vol 125, 2293-2315
+ Marshall, J., Jones, H. and C. Hill, (1998)
+ Efficient ocean modeling using non-hydrostatic algorithms
+ Journal of Marine Systems, 18, 115-134
+ Adcroft, A., Hill C. and J. Marshall: (1999)
+ A new treatment of the Coriolis terms in C-grid models at both high and low
+ resolutions,
+ Mon. Wea. Rev. Vol 127, pages 1928-1936
+ Hill, C, Adcroft,A., Jamous,D., and J. Marshall, (1999)
+ A Strategy for Terascale Climate Modeling.
+ In Proceedings of the Eighth ECMWF Workshop on the Use of Parallel Processors
+ in Meteorology, pages 406-425
+ World Scientific Publishing Co: UK
+ Marotzke, J, Giering,R., Zhang, K.Q., Stammer,D., Hill,C., and T.Lee, (1999)
+ Construction of the adjoint MIT ocean general circulation model and
+ application to Atlantic heat transport variability
+ J. Geophysical Res., 104(C12), 29,529-29,547.
+ \end{verbatim}
  We begin by briefly showing some of the results of the model in action to
  give a feel for the wide range of problems that can be addressed using it.
- \pagebreak
- % $Header$
- % $Name$
  \section{Illustrations of the model in action}
- The MITgcm has been designed and used to model a wide range of phenomena,
+ MITgcm has been designed and used to model a wide range of phenomena,
  from convection on the scale of meters in the ocean to the global pattern of
- atmospheric winds - see fig.2\ref{fig:all-scales}. To give a flavor of the
+ atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the
  kinds of problems the model has been used to study, we briefly describe some
  of them here. A more detailed description of the underlying formulation,
  numerical algorithm and implementation that lie behind these calculations is
  given later. Indeed many of the illustrative examples shown below can be
  easily reproduced: simply download the model (the minimum you need is a PC
- running linux, together with a FORTRAN\ 77 compiler) and follow the examples
+ running Linux, together with a FORTRAN\ 77 compiler) and follow the examples
  described in detail in the documentation.
  \subsection{Global atmosphere: `Held-Suarez' benchmark}
+ \begin{rawhtml}
+ <!-- CMIREDIR:atmospheric_example: -->
+ \end{rawhtml}
- A novel feature of MITgcm is its ability to simulate both atmospheric and
- oceanographic flows at both small and large scales.
- Fig.E1a.\ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
+ A novel feature of MITgcm is its ability to simulate, using one basic algorithm,
+ both atmospheric and oceanographic flows at both small and large scales.
+ Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
  temperature field obtained using the atmospheric isomorph of MITgcm run at
-.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
+ $2.8^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
  (blue) and warm air along an equatorial band (red). Fully developed
  baroclinic eddies spawned in the northern hemisphere storm track are
  evident. There are no mountains or land-sea contrast in this calculation,
-Line 131 
 in Held and Suarez; 1994 designed to tes
+Line 171 
 in Held and Suarez; 1994 designed to tes
  there are no mountains or land-sea contrast.
  %% CNHbegin
- \input{part1/cubic_eddies_figure}
+ \input{s_overview/text/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 griding and obviated the need to Fourier filter.
  The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
  grid, of which the cubed sphere is just one of many choices.
- Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
+ Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
- wind and meridional overturning streamfunction from a 20-level version of
+ wind from a 20-level configuration of
  the model. It compares favorable with more conventional spatial
- discretization approaches.
+ discretization approaches. The two plots show the field calculated using the
+ cube-sphere grid and the flow calculated using a regular, spherical polar
- A regular spherical lat-lon grid can also be used.
+ latitude-longitude grid. Both grids are supported within the model.
  %% CNHbegin
- \input{part1/hs_zave_u_figure}
+ \input{s_overview/text/hs_zave_u_figure}
  %% CNHend
  \subsection{Ocean gyres}
+ \begin{rawhtml}
+ <!-- CMIREDIR:oceanic_example: -->
+ \end{rawhtml}
+ \begin{rawhtml}
+ <!-- CMIREDIR:ocean_gyres: -->
+ \end{rawhtml}
  Baroclinic instability is a ubiquitous process in the ocean, as well as the
  atmosphere. Ocean eddies play an important role in modifying the
-Line 160 
 diffusive patterns of ocean currents. Bu
+Line 206 
 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
- MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
+ velocity field obtained from MITgcm run at $\frac{1}{6}^{\circ }$
- grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
+ horizontal resolution on a \textit{lat-lon} grid in which the pole has
- (to avoid the converging of meridian in northern latitudes). 21 vertical
+ been rotated by $90^{\circ }$ on to the equator (to avoid the
- levels are used in the vertical with a `lopped cell' representation of
+ converging of meridian in northern latitudes). 21 vertical levels are
- topography. The development and propagation of anomalously warm and cold
+ used in the vertical with a `lopped cell' representation of
- eddies can be clearly been seen in the Gulf Stream region. The transport of
+ topography. The development and propagation of anomalously warm and
- warm water northward by the mean flow of the Gulf Stream is also clearly
+ cold eddies can be clearly seen in the Gulf Stream region. The
- visible.
+ transport of warm water northward by the mean flow of the Gulf Stream
+ is also clearly visible.
  %% CNHbegin
- \input{part1/ocean_gyres_figure}
+ \input{s_overview/text/atl6_figure}
  %% CNHend
  \subsection{Global ocean circulation}
+ \begin{rawhtml}
+ <!-- CMIREDIR:global_ocean_circulation: -->
+ \end{rawhtml}
+ Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean
+ currents at the surface of a $4^{\circ }$ global ocean model run with
+vertical levels. Lopped cells are used to represent topography on a
+ regular \textit{lat-lon} grid extending from $70^{\circ }N$ to
+ $70^{\circ }S$. The model is driven using monthly-mean winds with
+ mixed boundary conditions on temperature and salinity at the surface.
+ The transfer properties of ocean eddies, convection and mixing is
+ parameterized in this model.
- Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
+ Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
- global ocean model run with 15 vertical levels. Lopped cells are used to
+ circulation of the global ocean in Sverdrups.
- 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
- mixed boundary conditions on temperature and salinity at the surface. The
- transfer properties of ocean eddies, convection and mixing is parameterized
- in this model.
- Fig.E2b shows the meridional overturning circulation of the global ocean in
- Sverdrups.
  %%CNHbegin
- \input{part1/global_circ_figure}
+ \input{s_overview/text/global_circ_figure}
  %%CNHend
  \subsection{Convection and mixing over topography}
+ \begin{rawhtml}
+ <!-- CMIREDIR:mixing_over_topography: -->
+ \end{rawhtml}
  Dense plumes generated by localized cooling on the continental shelf of the
  ocean may be influenced by rotation when the deformation radius is smaller
  than the width of the cooling region. Rather than gravity plumes, the
  mechanism for moving dense fluid down the shelf is then through geostrophic
- eddies. The simulation shown in the figure (blue is cold dense fluid, red is
+ eddies. The simulation shown in the figure \ref{fig:convect-and-topo}
+ (blue is cold dense fluid, red is
  warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
  trigger convection by surface cooling. The cold, dense water falls down the
  slope but is deflected along the slope by rotation. It is found that
-Line 207 
 strong, and replaced by lateral entrainm
+Line 263 
 strong, and replaced by lateral entrainm
  instability of the along-slope current.
  %%CNHbegin
- \input{part1/convect_and_topo}
+ \input{s_overview/text/convect_and_topo}
  %%CNHend
  \subsection{Boundary forced internal waves}
+ \begin{rawhtml}
+ <!-- CMIREDIR:boundary_forced_internal_waves: -->
+ \end{rawhtml}
  The unique ability of MITgcm to treat non-hydrostatic dynamics in the
  presence of complex geometry makes it an ideal tool to study internal wave
  dynamics and mixing in oceanic canyons and ridges driven by large amplitude
  barotropic tidal currents imposed through open boundary conditions.
- Fig. ?.? 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}
+ \input{s_overview/text/boundary_forced_waves}
  %%CNHend
  \subsection{Parameter sensitivity using the adjoint of MITgcm}
+ \begin{rawhtml}
+ <!-- CMIREDIR:parameter_sensitivity: -->
+ \end{rawhtml}
  Forward and tangent linear counterparts of MITgcm are supported using an
  `automatic adjoint compiler'. These can be used in parameter sensitivity and
  data assimilation studies.
- As one example of application of the MITgcm adjoint, Fig.E4 maps the
+ As one example of application of the MITgcm adjoint, Figure
- gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+ \ref{fig:hf-sensitivity} maps the gradient $\frac{\partial J}{\partial
- of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $
+   \mathcal{H}}$where $J$ is the magnitude of the overturning
- \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
+ stream-function shown in figure \ref{fig:large-scale-circ} at
- sensitive to heat fluxes over the Labrador Sea, one of the important sources
+ $60^{\circ }N$ and $ \mathcal{H}(\lambda,\varphi)$ is the mean, local
- of deep water for the thermohaline circulations. This calculation also
+ 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}
+ \input{s_overview/text/adj_hf_ocean_figure}
  %%CNHend
  \subsection{Global state estimation of the ocean}
+ \begin{rawhtml}
+ <!-- CMIREDIR:global_state_estimation: -->
+ \end{rawhtml}
  An important application of MITgcm is in state estimation of the global
  ocean circulation. An appropriately defined `cost function', which measures
  the departure of the model from observations (both remotely sensed and
- insitu) over an interval of time, is minimized by adjusting `control
+ in-situ) over an interval of time, is minimized by adjusting `control
  parameters' such as air-sea fluxes, the wind field, the initial conditions
- etc. Figure ?.? shows an estimate of the time-mean surface elevation of the
+ etc. Figure \ref{fig:assimilated-globes} shows the large scale planetary
- ocean obtained by bringing the model in to consistency with altimetric and
+ circulation and a Hopf-Muller plot of Equatorial sea-surface height.
- in-situ observations over the period 1992-1997.
+ Both are obtained from assimilation bringing the model in to
+ consistency with altimetric and in-situ observations over the period
+-1997.
  %% CNHbegin
- \input{part1/globes_figure}
+ \input{s_overview/text/assim_figure}
  %% CNHend
  \subsection{Ocean biogeochemical cycles}
+ \begin{rawhtml}
- MITgcm is being used to study global biogeochemical cycles in the ocean. For
+ <!-- CMIREDIR:ocean_biogeo_cycles: -->
- example one can study the effects of interannual changes in meteorological
+ \end{rawhtml}
- 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
+ MITgcm is being used to study global biogeochemical cycles in the
- flux of oxygen and its relation to density outcrops in the southern oceans
+ ocean. For example one can study the effects of interannual changes in
- from a single year of a global, interannually varying simulation.
+ meteorological forcing and upper ocean circulation on the fluxes of
+ carbon dioxide and oxygen between the ocean and atmosphere. Figure
+ \ref{fig:biogeo} shows the annual air-sea flux of oxygen and its
+ relation to density outcrops in the southern oceans from a single year
+ of a global, interannually varying simulation. The simulation is run
+ at $1^{\circ}\times1^{\circ}$ resolution telescoping to
+ $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not
+ shown).
  %%CNHbegin
- \input{part1/biogeo_figure}
+ \input{s_overview/text/biogeo_figure}
  %%CNHend
  \subsection{Simulations of laboratory experiments}
+ \begin{rawhtml}
+ <!-- CMIREDIR:classroom_exp: -->
+ \end{rawhtml}
- Figure ?.? shows MITgcm being used to simulate a laboratory experiment
+ Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
- enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
+ laboratory experiment inquiring into the dynamics of the Antarctic Circumpolar Current (ACC). An
  initially homogeneous tank of water ($1m$ in diameter) is driven from its
  free surface by a rotating heated disk. The combined action of mechanical
  and thermal forcing creates a lens of fluid which becomes baroclinically
  unstable. The stratification and depth of penetration of the lens is
- arrested by its instability in a process analogous to that whic sets the
+ arrested by its instability in a process analogous to that which sets the
  stratification of the ACC.
  %%CNHbegin
- \input{part1/lab_figure}
+ \input{s_overview/text/lab_figure}
  %%CNHend
- % $Header$
- % $Name$
  \section{Continuous equations in `r' coordinates}
+ \begin{rawhtml}
+ <!-- CMIREDIR:z-p_isomorphism: -->
+ \end{rawhtml}
  To render atmosphere and ocean models from one dynamical core we exploit
  `isomorphisms' between equation sets that govern the evolution of the
- respective fluids - see fig.4
+ respective fluids - see figure \ref{fig:isomorphic-equations}.
- \marginpar{
+ One system of hydrodynamical equations is written down
- Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down
  and encoded. The model variables have different interpretations depending on
  whether the atmosphere or ocean is being studied. Thus, for example, the
  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
- modeling the atmosphere and height, $z$, if we are modeling the ocean.
+ modeling the atmosphere (right hand side of figure \ref{fig:isomorphic-equations})
+ and height, $z$, if we are modeling the ocean (left hand side of figure
+ \ref{fig:isomorphic-equations}).
  %%CNHbegin
- \input{part1/zandpcoord_figure.tex}
+ \input{s_overview/text/zandpcoord_figure.tex}
  %%CNHend
  The state of the fluid at any time is characterized by the distribution of
-Line 314 
 velocity $\vec{\mathbf{v}}$, active trac
+Line 396 
 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}
+ \input{s_overview/text/vertcoord_figure.tex}
  %%CNHend
- \begin{equation*}
+ \begin{equation}
  \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}
  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
- \text{ horizontal mtm}
+ \text{ horizontal mtm} \label{eq:horizontal_mtm}
- \end{equation*}
+ \end{equation}
- \begin{equation*}
+ \begin{equation}
  \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
- vertical mtm}
+ vertical mtm} \label{eq:vertical_mtm}
- \end{equation*}
+ \end{equation}
  \begin{equation}
  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
- \partial r}=0\text{ continuity}  \label{eq:continuous}
+ \partial r}=0\text{ continuity}  \label{eq:continuity}
  \end{equation}
- \begin{equation*}
+ \begin{equation}
- b=b(\theta ,S,r)\text{ equation of state}
+ b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
- \end{equation*}
+ \end{equation}
- \begin{equation*}
+ \begin{equation}
  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
- \end{equation*}
+ \label{eq:potential_temperature}
+ \end{equation}
- \begin{equation*}
+ \begin{equation}
  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
- \end{equation*}
+ \label{eq:humidity_salt}
+ \end{equation}
  Here:
-Line 413 
 S\text{ is specific humidity in the atmo
+Line 497 
 S\text{ is specific humidity in the atmo
  \end{equation*}
  The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by
- extensive `physics' packages for atmosphere and ocean described in Chapter 6.
+ `physics' and forcing packages for atmosphere and ocean. These are described
+ in later chapters.
  \subsection{Kinematic Boundary conditions}
  \subsubsection{vertical}
- at fixed and moving $r$ surfaces we set (see fig.5):
+ at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
  \begin{equation}
- \dot{r}=0atr=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
+ \dot{r}=0 \text{\ at\ } r=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
  \label{eq:fixedbc}
  \end{equation}
  \begin{equation}
- \dot{r}=\frac{Dr}{Dt}atr=R_{moving}\text{ \
+ \dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
- (oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}
+ (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
  \end{equation}
  Here
-Line 450 
 where $\vec{\mathbf{n}}$ is the normal t
+Line 535 
 where $\vec{\mathbf{n}}$ is the normal t
  \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 521 
 The boundary conditions at top and botto
+Line 606 
 The boundary conditions at top and botto
  atmosphere)}  \label{eq:moving-bc-atmos}
  \end{eqnarray}
- Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent
+ Then the (hydrostatic form of) equations
- set of atmospheric equations which, for convenience, are written out in $p$
+ (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yields a consistent
- coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).
+ set of atmospheric equations which, for convenience, are written out
+ in $p$ coordinates in Appendix Atmosphere - see
+ eqs(\ref{eq:atmos-prime}).
  \subsection{Ocean}
-Line 558 
 w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo
+Line 645 
 w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo
  \end{eqnarray}
  where $\eta $ is the elevation of the free surface.
- Then eq(\ref{eq:continuous}) yields a consistent set of oceanic equations
+ Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yield a consistent set
+ of oceanic equations
  which, for convenience, are written out in $z$ coordinates in Appendix Ocean
  - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
  Non-hydrostatic forms}
+ \label{sec:all_hydrostatic_forms}
+ \begin{rawhtml}
+ <!-- CMIREDIR:non_hydrostatic: -->
+ \end{rawhtml}
  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
-Line 571 
 Let us separate $\phi $ in to surface, h
+Line 664 
 Let us separate $\phi $ in to surface, h
  \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
  \label{eq:phi-split}
  \end{equation}
- and write eq(\ref{incompressible}a,b) in the form:
+ %and write eq(\ref{eq:incompressible}) in the form:
+ %                  ^- this eq is missing (jmc) ; replaced with:
+ and write eq( \ref{eq:horizontal_mtm}) in the form:
  \begin{equation}
  \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
-Line 604 
 discussion:
+Line 699 
 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\} $
  \\
  $+\mathcal{F}_{u}$
  \end{tabular}
-Line 624 
 $+\mathcal{F}_{u}$
+Line 719 
 $+\mathcal{F}_{u}$
  \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\} $ \\
  $+\mathcal{F}_{v}$
  \end{tabular}
  \ \right\} \left\{
-Line 645 
 $+\mathcal{F}_{v}$
+Line 740 
 $+\mathcal{F}_{v}$
  \begin{tabular}{l}
  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
- ${+}\underline{{2\Omega u\cos lat}}$ \\
+ ${+}\underline{{2\Omega u\cos \varphi}}$ \\
  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
  \end{tabular}
  \ \right\} \left\{
-Line 659 
 $\underline{\underline{\mathcal{F}_{\dot
+Line 754 
 $\underline{\underline{\mathcal{F}_{\dot
  \end{equation}
  \qquad \qquad \qquad \qquad \qquad
- In the above `${r}$' is the distance from the center of the earth and `$lat$
+ In the above `${r}$' is the distance from the center of the earth and `$\varphi$
  ' is latitude.
  Grad and div operators in spherical coordinates are defined in appendix
  OPERATORS.
- \marginpar{
- Fig.6 Spherical polar coordinate system.}
  %%CNHbegin
- \input{part1/sphere_coord_figure.tex}
+ \input{s_overview/text/sphere_coord_figure.tex}
  %%CNHend
  \subsubsection{Shallow atmosphere approximation}
- Most models are based on the `hydrostatic primitive equations' (HPE's) in
+ Most models are based on the `hydrostatic primitive equations' (HPE's)
- which the vertical momentum equation is reduced to a statement of
+ in which the vertical momentum equation is reduced to a statement of
- hydrostatic balance and the `traditional approximation' is made in which the
+ hydrostatic balance and the `traditional approximation' is made in
- Coriolis force is treated approximately and the shallow atmosphere
+ which the Coriolis force is treated approximately and the shallow
- approximation is made.\ The MITgcm need not make the `traditional
+ atmosphere approximation is made.  MITgcm need not make the
- approximation'. To be able to support consistent non-hydrostatic forms the
+ `traditional approximation'. To be able to support consistent
- shallow atmosphere approximation can be relaxed - when dividing through by $
+ non-hydrostatic forms the shallow atmosphere approximation can be
- r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
+ relaxed - when dividing through by $ r $ in, for example,
- the radius of the earth.
+ (\ref{eq:gu-speherical}), we do not replace $r$ by $a$, the radius of
+ the earth.
  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
+ \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
  These are discussed at length in Marshall et al (1997a).
-Line 697 
 computed at all other levels by integrat
+Line 792 
 computed at all other levels by integrat
  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
- \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
+ \varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
  contribution to the pressure field: only the terms underlined twice in Eqs. (
  \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero
  and, simultaneously, the shallow atmosphere approximation is relaxed. In
-Line 706 
 variation of the radial position of a pa
+Line 801 
 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 717 
 et.al., 1997a. As in \textbf{HPE }only a
+Line 812 
 et.al., 1997a. As in \textbf{HPE }only a
  \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}
- The MIT model presently supports a full non-hydrostatic ocean isomorph, but
+ MITgcm presently supports a full non-hydrostatic ocean isomorph, but
  only a quasi-non-hydrostatic atmospheric isomorph.
  \paragraph{Non-hydrostatic Ocean}
-Line 727 
 In the non-hydrostatic ocean model all t
+Line 822 
 In the non-hydrostatic ocean model all t
  three dimensional elliptic equation must be solved subject to Neumann
  boundary conditions (see below). It is important to note that use of the
  full \textbf{NH} does not admit any new `fast' waves in to the system - the
- incompressible condition eq(\ref{eq:continuous})c has already filtered out
+ incompressible condition eq(\ref{eq:continuity}) has already filtered out
  acoustic modes. It does, however, ensure that the gravity waves are treated
  accurately with an exact dispersion relation. The \textbf{NH} set has a
  complete angular momentum principle and consistent energetics - see White
-Line 776 
 coordinates are supported - see eqs(\ref
+Line 871 
 coordinates are supported - see eqs(\ref
  \subsection{Solution strategy}
  The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
- NH} models is summarized in Fig.7.
+ NH} models is summarized in Figure \ref{fig:solution-strategy}.
- \marginpar{
+ Under all dynamics, a 2-d elliptic equation is
- Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
  first solved to find the surface pressure and the hydrostatic pressure at
  any level computed from the weight of fluid above. Under \textbf{HPE} and
  \textbf{QH} dynamics, the horizontal momentum equations are then stepped
-Line 788 
 stepping forward the horizontal momentum
+Line 882 
 stepping forward the horizontal momentum
  stepping forward the vertical momentum equation.
  %%CNHbegin
- \input{part1/solution_strategy_figure.tex}
+ \input{s_overview/text/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 802 
 Marshall et al, 1997) resulting in a non
+Line 896 
 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 834 
 atmospheric pressure pushing down on the
+Line 929 
 atmospheric pressure pushing down on the
  \subsubsection{Surface pressure}
- The surface pressure equation can be obtained by integrating continuity, (
+ The surface pressure equation can be obtained by integrating continuity,
- \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
+ (\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
  \begin{equation*}
  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
-Line 860 
 r $. The above can be rearranged to yiel
+Line 955 
 r $. The above can be rearranged to yiel
  where we have incorporated a source term.
  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
- (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
+ (atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
  be written
  \begin{equation}
  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
-Line 868 
 be written
+Line 963 
 be written
  \end{equation}
  where $b_{s}$ is the buoyancy at the surface.
- In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref
+ In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref
  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
  surface' and `rigid lid' approaches are available.
  \subsubsection{Non-hydrostatic pressure}
- Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
+ Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
- \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
+ $\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
- (\ref{incompressible}), we deduce that:
+ (\ref{eq:continuity}), we deduce that:
  \begin{equation}
  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
-Line 907 
 tangential component of velocity, $v_{T}
+Line 1002 
 tangential component of velocity, $v_{T}
  depending on the form chosen for the dissipative terms in the momentum
  equations - see below.
- Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that:
+ Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
  \begin{equation}
  \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
-Line 947 
 If the flow is `close' to hydrostatic ba
+Line 1042 
 If the flow is `close' to hydrostatic ba
  converges rapidly because $\phi _{nh}\ $is then only a small correction to
  the hydrostatic pressure field (see the discussion in Marshall et al, a,b).
- The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{homneuman})
+ The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh})
  does not vanish at $r=R_{moving}$, and so refines the pressure there.
  \subsection{Forcing/dissipation}
-Line 955 
 does not vanish at $r=R_{moving}$, and s
+Line 1050 
 does not vanish at $r=R_{moving}$, and s
  \subsubsection{Forcing}
  The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by
- `physics packages' described in detail in chapter ??.
+ `physics packages' and forcing packages. These are described later on.
  \subsubsection{Dissipation}
-Line 976 
 friction. These coefficients are the sam
+Line 1071 
 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.
  \begin{equation}
  D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
  _{h}^{4}(T,S)  \label{eq:diffusion}
-Line 1002 
 salinity ... ).
+Line 1097 
 salinity ... ).
  \subsection{Vector invariant form}
- For some purposes it is advantageous to write momentum advection in eq(\ref
+ For some purposes it is advantageous to write momentum advection in
- {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
+ eq(\ref {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the
+ (so-called) `vector invariant' form:
  \begin{equation}
  \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
-Line 1021 
 to discretize the model.
+Line 1117 
 to discretize the model.
  \subsection{Adjoint}
- Tangent linear and adjoint counterparts of the forward model and described
+ Tangent linear and adjoint counterparts of the forward model are described
  in Chapter 5.
- % $Header$
- % $Name$
  \section{Appendix ATMOSPHERE}
  \subsection{Hydrostatic Primitive Equations for the Atmosphere in pressure
-Line 1046 
 p\alpha &=&RT  \label{eq:atmos-eos} \\
+Line 1139 
 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}
  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
+ surfaces) component of velocity, $\frac{D}{Dt}=\frac{\partial}{\partial t}
- \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
+ +\vec{\mathbf{v}}_{h}\cdot \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$
- derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is
+ is the total derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter,
- the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
+ $\phi =gz$ is the geopotential, $\alpha =1/\rho $ is the specific volume,
- }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
+ $\omega =\frac{Dp }{Dt}$ is the vertical velocity in the $p-$coordinate.
- {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
+ Equation(\ref {eq:atmos-heat}) is the first law of thermodynamics where internal
- e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $
+ energy $e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass
- p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.
+ 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
  $\theta $ so that it looks more like a generic conservation law.
-Line 1114 
 In $p$-coordinates, the upper boundary a
+Line 1207 
 In $p$-coordinates, the upper boundary a
  surface ($\phi $ is imposed and $\omega \neq 0$).
  \subsubsection{Splitting the geo-potential}
+ \label{sec:hpe-p-geo-potential-split}
  For the purposes of initialization and reducing round-off errors, the model
  deals with perturbations from reference (or ``standard'') profiles. For
-Line 1143 
 _{o}(p_{o})=g~Z_{topo}$, defined:
+Line 1237 
 _{o}(p_{o})=g~Z_{topo}$, 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}}
- _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\
+ _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}}
+ \label{eq:atmos-prime} \\
  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
  \partial p} &=&0 \\
  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
- \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}
+ \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }
  \end{eqnarray}
- % $Header$
- % $Name$
  \section{Appendix OCEAN}
  \subsection{Equations of motion for the ocean}
-Line 1167 
 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}
+Line 1259 
 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}
  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
  &=&\epsilon _{nh}\mathcal{F}_{w} \\
  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
- _{h}+\frac{\partial w}{\partial z} &=&0 \\
+ _{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
- \rho &=&\rho (\theta ,S,p) \\
+ \rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
- \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
+ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
- \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
+ \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zns-salt}
+ \label{eq:non-boussinesq}
  \end{eqnarray}
  These equations permit acoustics modes, inertia-gravity waves,
- non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
+ non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
  mode. As written, they cannot be integrated forward consistently - if we
  step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
-Line 1188 
 _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
+Line 1281 
 _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}
  \end{equation}
- Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
+ Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is
- reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
+ the reciprocal of the sound speed ($c_{s}$) squared. Substituting into
- {eq-zns-cont} gives:
+ \ref{eq-zns-cont} gives:
  \begin{equation}
  \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
-Line 1367 
 the perturbation density. Nevertheless,
+Line 1460 
 the perturbation density. Nevertheless,
  _{nh}=0$ form of these equations that are used throughout the ocean modeling
  community and referred to as the primitive equations (HPE).
- % $Header$
- % $Name$
  \section{Appendix:OPERATORS}
  \subsection{Coordinate systems}
-Line 1380 
 In spherical coordinates, the velocity c
+Line 1470 
 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}
  \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
+ Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
  distance of the particle from the center of the earth, $\Omega $ is the
  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
- The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in
+ The `grad' ($\nabla $) and `div' ($\nabla\cdot$) operators are defined by, in
  spherical coordinates:
  \begin{equation*}
- \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }
+ \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\cdot 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*}

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