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

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-revision 1.1 by cnh,
Thu Sep 27 17:45:03 2001 UTC
+revision 1.20 by jmc,
Fri Oct 15 14:44:25 2004 UTC
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+ % $Header$
- %%%% % $Name$
+ % $Name$
- %%%% %\usepackage{oldgerm}
- %%%% % I commented the following because it introduced excessive white space
+ %tci%\documentclass[12pt]{book}
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+ %tci%\usepackage{amsmath}
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+ %tci%\usepackage{html}
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+ %tci%\usepackage{epsfig}
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+ %tci%%TCIDATA{Language=American English}
- %%%% % This allows numbering of subsubsections
- %%%% % This changes the the chapter title
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+ %tci%\input{tcilatex}
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- %%%% \input{tcilatex}
- %%%%
- %%%% \begin{document}
- %%%%
- %%%% \tableofcontents
- \pagebreak
+ %tci%\begin{document}
+ %tci%\tableofcontents
- \part{MITgcm basics}
  % Section: Overview
  % $Header$
  % $Name$
- \section{Introduction}
+ This document provides the reader with the information necessary to
- This documentation provides the reader with the information necessary to
  carry out numerical experiments using MITgcm. It gives a comprehensive
  description of the continuous equations on which the model is based, the
  numerical algorithms the model employs and a description of the associated
-Line 73 
 are available. A number of examples illu
+Line 47 
 are available. A number of examples illu
  both process and general circulation studies of the atmosphere and ocean are
  also presented.
+ \section{Introduction}
+ \begin{rawhtml}
+ <!-- CMIREDIR:innovations: -->
+ \end{rawhtml}
  MITgcm has a number of novel aspects:
  \begin{itemize}
  \item it can be used to study both atmospheric and oceanic phenomena; one
  hydrodynamical kernel is used to drive forward both atmospheric and oceanic
- models - see fig.1%
+ models - see fig \ref{fig:onemodel}
- \marginpar{
- Fig.1 One model}\ref{fig:onemodel}
+ %% CNHbegin
+ \input{part1/one_model_figure}
- \begin{figure}
+ %% CNHend
- \begin{center}
- \resizebox{!}{4in}{
-  \rotatebox{90}{
-  \rotatebox{180}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps}
-  }
-  }
- }
- \end{center}
- \label{fig:onemodel}
- \end{figure}
  \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}
- \begin{figure}
- \begin{center}
- \resizebox{!}{4in}{
-  \rotatebox{90}{
-  \rotatebox{180}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/scales.eps}
-  }
-  }
- }
- \end{center}
- \label{fig:scales}
- \end{figure}
+ %% 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 130 
 computational platforms.
+Line 88 
 computational platforms.
  \end{itemize}
  Key publications reporting on and charting the development of the model are
- listed in an Appendix.
+ \cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99}:
+ \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
  % $Header$
  % $Name$
-Line 143 
 give a feel for the wide range of proble
+Line 144 
 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
- given later. Indeed it is easy to reproduce the results shown here: simply
+ given later. Indeed many of the illustrative examples shown below can be
- download the model (the minimum you need is a PC running linux, together
+ easily reproduced: simply download the model (the minimum you need is a PC
- with a FORTRAN\ 77 compiler) and follow the examples.
+ running Linux, together with a FORTRAN\ 77 compiler) and follow the examples
+ described in detail in the documentation.
  \subsection{Global atmosphere: `Held-Suarez' benchmark}
+ \begin{rawhtml}
+ <!-- CMIREDIR:atmospheric_example: -->
+ \end{rawhtml}
- Fig.E1a.\ref{fig:Held-Suarez} is an instaneous plot of the 500$mb$ height
- field obtained using a 5-level version of the atmospheric pressure isomorph
- run at 2.8$^{\circ }$ resolution. We see fully developed baroclinic eddies
- along the northern hemisphere storm track. There are no mountains or
- land-sea contrast in this calculation, but you can easily put them in. The
- model is driven by relaxation to a radiative-convective equilibrium profile,
- following the description set out in Held and Suarez; 1994 designed to test
- atmospheric hydrodynamical cores - there are no mountains or land-sea
- contrast. As decribed in Adcroft (2001), a `cubed sphere' is used to
- descretize the globe permitting a uniform gridding and obviated the need to
- fourier filter.
- Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
- wind and meridional overturning streamfunction from the 5-level model.
- \begin{figure}
- \begin{center}
- \resizebox{!}{4in}{
-  \rotatebox{90}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps}
-  }
- }
- \end{center}
- \label{fig:hscs}
- \end{figure}
- A regular spherical lat-lon grid can also be used.
- \begin{figure}
- \begin{center}
- \resizebox{!}{4in}{
-  \rotatebox{90}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps}
-  }
- }
- \end{center}
- \label{fig:hslatlon}
- \end{figure}
- \subsection{Ocean gyres}
- \subsection{Global ocean circulation}
+ 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.
- Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
- global ocean model run with 15 vertical levels. The model is driven using
+ Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
- monthly-mean winds with mixed boundary conditions on temperature and
+ temperature field obtained using the atmospheric isomorph of MITgcm run at
- salinity at the surface. Fig.E2b shows the overturning (thermohaline)
+.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
- circulation. Lopped cells are used to represent topography on a regular $%
+ (blue) and warm air along an equatorial band (red). Fully developed
- lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$.
+ baroclinic eddies spawned in the northern hemisphere storm track are
+ evident. There are no mountains or land-sea contrast in this calculation,
+ but you can easily put them in. The model is driven by relaxation to a
- \begin{figure}
+ radiative-convective equilibrium profile, following the description set out
- \begin{center}
+ in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
- \resizebox{!}{4in}{
+ there are no mountains or land-sea contrast.
- % \rotatebox{90}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps}
+ %% CNHbegin
- % }
+ \input{part1/cubic_eddies_figure}
- }
+ %% CNHend
- \end{center}
- \label{fig:horizcirc}
+ As described in Adcroft (2001), a `cubed sphere' is used to discretize the
- \end{figure}
+ globe permitting a uniform griding and obviated the need to Fourier filter.
+ The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
- \begin{figure}
+ grid, of which the cubed sphere is just one of many choices.
- \begin{center}
- \resizebox{!}{4in}{
+ Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
-  \rotatebox{90}{
+ wind from a 20-level configuration of
-  \rotatebox{180}{
+ the model. It compares favorable with more conventional spatial
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps}
+ discretization approaches. The two plots show the field calculated using the
-  }
+ cube-sphere grid and the flow calculated using a regular, spherical polar
-  }
+ latitude-longitude grid. Both grids are supported within the model.
- }
- \end{center}
+ %% CNHbegin
- \label{fig:moc}
+ \input{part1/hs_zave_u_figure}
- \end{figure}
+ %% CNHend
- \subsection{Flow over topography}
- \subsection{Ocean convection}
- Fig.E3 shows convection over a slope using the non-hydrostatic ocean
- isomorph and lopped cells to respresent topography. .....The grid resolution
- is
- \subsection{Boundary forced internal waves}
- \subsection{Carbon outgassing sensitivity}
+ \subsection{Ocean gyres}
+ \begin{rawhtml}
+ <!-- CMIREDIR:oceanic_example: -->
+ \end{rawhtml}
+ \begin{rawhtml}
+ <!-- CMIREDIR:ocean_gyres: -->
+ \end{rawhtml}
+ Baroclinic instability is a ubiquitous process in the ocean, as well as the
+ atmosphere. Ocean eddies play an important role in modifying the
+ hydrographic structure and current systems of the oceans. Coarse resolution
+ models of the oceans cannot resolve the eddy field and yield rather broad,
+ diffusive patterns of ocean currents. But if the resolution of our models is
+ increased until the baroclinic instability process is resolved, numerical
+ solutions of a different and much more realistic kind, can be obtained.
+ Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity
+ field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal
+ resolution on a $lat-lon$
+ grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
+ (to avoid the converging of meridian in northern latitudes). 21 vertical
+ levels are used in the vertical with a `lopped cell' representation of
+ topography. The development and propagation of anomalously warm and cold
+ eddies can be clearly seen in the Gulf Stream region. The transport of
+ warm water northward by the mean flow of the Gulf Stream is also clearly
+ visible.
+ %% CNHbegin
+ \input{part1/atl6_figure}
+ %% CNHend
- Fig.E4 shows....
- \begin{figure}
+ \subsection{Global ocean circulation}
- \begin{center}
+ \begin{rawhtml}
- \resizebox{!}{4in}{
+ <!-- CMIREDIR:global_ocean_circulation: -->
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps}
+ \end{rawhtml}
- }
- \end{center}
+ Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
- \label{fig:co2mrt}
+ the surface of a 4$^{\circ }$
- \end{figure}
+ 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
+ mixed boundary conditions on temperature and salinity at the surface. The
+ transfer properties of ocean eddies, convection and mixing is parameterized
+ in this model.
+ Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
+ circulation of the global ocean in Sverdrups.
+ %%CNHbegin
+ \input{part1/global_circ_figure}
+ %%CNHend
+ \subsection{Convection and mixing over topography}
+ \begin{rawhtml}
+ <!-- CMIREDIR:mixing_over_topography: -->
+ \end{rawhtml}
+ Dense plumes generated by localized cooling on the continental shelf of the
+ ocean may be influenced by rotation when the deformation radius is smaller
+ than the width of the cooling region. Rather than gravity plumes, the
+ mechanism for moving dense fluid down the shelf is then through geostrophic
+ eddies. The simulation shown in the figure \ref{fig:convect-and-topo}
+ (blue is cold dense fluid, red is
+ warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
+ trigger convection by surface cooling. The cold, dense water falls down the
+ slope but is deflected along the slope by rotation. It is found that
+ entrainment in the vertical plane is reduced when rotational control is
+ strong, and replaced by lateral entrainment due to the baroclinic
+ instability of the along-slope current.
+ %%CNHbegin
+ \input{part1/convect_and_topo}
+ %%CNHend
+ \subsection{Boundary forced internal waves}
+ \begin{rawhtml}
+ <!-- CMIREDIR:boundary_forced_internal_waves: -->
+ \end{rawhtml}
+ The unique ability of MITgcm to treat non-hydrostatic dynamics in the
+ presence of complex geometry makes it an ideal tool to study internal wave
+ dynamics and mixing in oceanic canyons and ridges driven by large amplitude
+ barotropic tidal currents imposed through open boundary conditions.
+ Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope
+ topographic variations on
+ internal wave breaking - the cross-slope velocity is in color, the density
+ contoured. The internal waves are excited by application of open boundary
+ conditions on the left. They propagate to the sloping boundary (represented
+ using MITgcm's finite volume spatial discretization) where they break under
+ nonhydrostatic dynamics.
+ %%CNHbegin
+ \input{part1/boundary_forced_waves}
+ %%CNHend
+ \subsection{Parameter sensitivity using the adjoint of MITgcm}
+ \begin{rawhtml}
+ <!-- CMIREDIR:parameter_sensitivity: -->
+ \end{rawhtml}
+ Forward and tangent linear counterparts of MITgcm are supported using an
+ `automatic adjoint compiler'. These can be used in parameter sensitivity and
+ data assimilation studies.
+ As one example of application of the MITgcm adjoint, Figure \ref{fig:hf-sensitivity}
+ maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+ of the overturning stream-function shown in figure \ref{fig:large-scale-circ}
+ at 60$^{\circ }$N and $
+ \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
+ a 100 year period. We see that $J$ is
+ sensitive to heat fluxes over the Labrador Sea, one of the important sources
+ of deep water for the thermohaline circulations. This calculation also
+ yields sensitivities to all other model parameters.
+ %%CNHbegin
+ \input{part1/adj_hf_ocean_figure}
+ %%CNHend
+ \subsection{Global state estimation of the ocean}
+ \begin{rawhtml}
+ <!-- CMIREDIR:global_state_estimation: -->
+ \end{rawhtml}
+ An important application of MITgcm is in state estimation of the global
+ ocean circulation. An appropriately defined `cost function', which measures
+ the departure of the model from observations (both remotely sensed and
+ in-situ) over an interval of time, is minimized by adjusting `control
+ parameters' such as air-sea fluxes, the wind field, the initial conditions
+ etc. Figure \ref{fig:assimilated-globes} shows the large scale planetary
+ circulation and a Hopf-Muller plot of Equatorial sea-surface height.
+ Both are obtained from assimilation bringing the model in to
+ consistency with altimetric and in-situ observations over the period
+-1997.
+ %% CNHbegin
+ \input{part1/assim_figure}
+ %% CNHend
+ \subsection{Ocean biogeochemical cycles}
+ \begin{rawhtml}
+ <!-- CMIREDIR:ocean_biogeo_cycles: -->
+ \end{rawhtml}
+ MITgcm is being used to study global biogeochemical cycles in the ocean. For
+ example one can study the effects of interannual changes in meteorological
+ forcing and upper ocean circulation on the fluxes of carbon dioxide and
+ oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows
+ the annual air-sea flux of oxygen and its relation to density outcrops in
+ the southern oceans from a single year of a global, interannually varying
+ simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution
+ telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).
+ %%CNHbegin
+ \input{part1/biogeo_figure}
+ %%CNHend
+ \subsection{Simulations of laboratory experiments}
+ \begin{rawhtml}
+ <!-- CMIREDIR:classroom_exp: -->
+ \end{rawhtml}
+ Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
+ laboratory experiment inquiring into the dynamics of the Antarctic Circumpolar Current (ACC). An
+ initially homogeneous tank of water ($1m$ in diameter) is driven from its
+ free surface by a rotating heated disk. The combined action of mechanical
+ and thermal forcing creates a lens of fluid which becomes baroclinically
+ unstable. The stratification and depth of penetration of the lens is
+ arrested by its instability in a process analogous to that which sets the
+ stratification of the ACC.
+ %%CNHbegin
+ \input{part1/lab_figure}
+ %%CNHend
  % $Header$
  % $Name$
  \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}
+ %%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 276 
 velocity $\vec{\mathbf{v}}$, active trac
+Line 401 
 velocity $\vec{\mathbf{v}}$, active trac
  depend on $\theta $, $S$, and $p$. The equations that govern the evolution
  of these fields, obtained by applying the laws of classical mechanics and
  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
- a generic vertical coordinate, $r$, see fig.5%
+ a generic vertical coordinate, $r$, so that the appropriate
- \marginpar{
+ kinematic boundary conditions can be applied isomorphically
- Fig.5 The vertical coordinate of model}:
+ see figure \ref{fig:zandp-vert-coord}.
- \begin{figure}
- \begin{center}
- \resizebox{!}{4in}{
-  \rotatebox{90}{
-  \rotatebox{180}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/vertcoord.eps}
-  }
-  }
- }
- \end{center}
- \label{fig:vertcoord}
- \end{figure}
- \begin{equation*}
- \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}%
- \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}%
- \text{ horizontal mtm}
- \end{equation*}
- \begin{equation*}
+ %%CNHbegin
- \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{%
+ \input{part1/vertcoord_figure.tex}
+ %%CNHend
+ \begin{equation}
+ \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}
+ \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
+ \text{ horizontal mtm} \label{eq:horizontal_mtm}
+ \end{equation}
+ \begin{equation}
+ \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
- vertical mtm}
+ vertical mtm} \label{eq:vertical_mtm}
- \end{equation*}
+ \end{equation}
  \begin{equation}
- \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{%
+ \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
- \partial r}=0\text{ continuity}  \label{eq:continuous}
+ \partial r}=0\text{ continuity}  \label{eq:continuity}
  \end{equation}
- \begin{equation*}
+ \begin{equation}
- b=b(\theta ,S,r)\text{ equation of state}
+ b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
- \end{equation*}
+ \end{equation}
- \begin{equation*}
+ \begin{equation}
- \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{  potential temperature}
+ \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}
+ \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
- \end{equation*}
+ \label{eq:humidity_salt}
+ \end{equation}
  Here:
  \begin{equation*}
  r\text{ is the vertical coordinate}
  \end{equation*}
  \begin{equation*}
  \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{
  is the total derivative}
  \end{equation*}
  \begin{equation*}
- \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%
+ \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
  \text{ is the `grad' operator}
  \end{equation*}
- with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%
+ with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}
  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
  is a unit vector in the vertical
  \begin{equation*}
  t\text{ is time}
  \end{equation*}
  \begin{equation*}
  \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the
  velocity}
  \end{equation*}
  \begin{equation*}
  \phi \text{ is the `pressure'/`geopotential'}
  \end{equation*}
  \begin{equation*}
  \vec{\Omega}\text{ is the Earth's rotation}
  \end{equation*}
  \begin{equation*}
  b\text{ is the `buoyancy'}
  \end{equation*}
  \begin{equation*}
  \theta \text{ is potential temperature}
  \end{equation*}
  \begin{equation*}
  S\text{ is specific humidity in the atmosphere; salinity in the ocean}
  \end{equation*}
  \begin{equation*}
- \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{%
+ \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{
  \mathbf{v}}
  \end{equation*}
  \begin{equation*}
- \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }%
+ \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
- \theta
  \end{equation*}
  \begin{equation*}
-Line 385 
 S\text{ is specific humidity in the atmo
+Line 502 
 S\text{ is specific humidity in the atmo
  \end{equation*}
  The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by
- extensive `physics' packages for atmosphere and ocean described in Chapter 6.
+ `physics' and forcing packages for atmosphere and ocean. These are described
+ in later chapters.
  \subsection{Kinematic Boundary conditions}
  \subsubsection{vertical}
- at fixed and moving $r$ surfaces we set (see fig.5):
+ at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
  \begin{equation}
- \dot{r}=0atr=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
+ \dot{r}=0 \text{\ at\ } r=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
  \label{eq:fixedbc}
  \end{equation}
  \begin{equation}
- \dot{r}=\frac{Dr}{Dt}atr=R_{moving}\text{ \
+ \dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
- (oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}
+ (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
  \end{equation}
  Here
  \begin{equation*}
  R_{moving}=R_{o}+\eta
  \end{equation*}
  where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on
  whether we are in the atmosphere or ocean) of the `moving surface' in the
-Line 417 
 of motion.
+Line 535 
 of motion.
  \begin{equation}
  \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0  \label{eq:noflow}
- \end{equation}%
+ \end{equation}
  where $\vec{\mathbf{n}}$ is the normal to a solid boundary.
  \subsection{Atmosphere}
- In the atmosphere, see fig.5, we interpret:
+ In the atmosphere, (see figure \ref{fig:zandp-vert-coord}), we interpret:
  \begin{equation}
  r=p\text{ is the pressure}  \label{eq:atmos-r}
-Line 454 
 where
+Line 572 
 where
  \begin{equation*}
  T\text{ is absolute temperature}
- \end{equation*}%
+ \end{equation*}
  \begin{equation*}
  p\text{ is the pressure}
- \end{equation*}%
+ \end{equation*}
  \begin{eqnarray*}
  &&z\text{ is the height of the pressure surface} \\
  &&g\text{ is the acceleration due to gravity}
-Line 467 
 In the above the ideal gas law, $p=\rho
+Line 585 
 In the above the ideal gas law, $p=\rho
  the Exner function $\Pi (p)$ given by (see Appendix Atmosphere)
  \begin{equation}
  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{eq:exner}
- \end{equation}%
+ \end{equation}
  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas
  constant and $c_{p}$ the specific heat of air at constant pressure.
  At the top of the atmosphere (which is `fixed' in our $r$ coordinate):
  \begin{equation*}
  R_{fixed}=p_{top}=0
  \end{equation*}
  In a resting atmosphere the elevation of the mountains at the bottom is
  given by
  \begin{equation*}
  R_{moving}=R_{o}(x,y)=p_{o}(x,y)
  \end{equation*}
  i.e. the (hydrostatic) pressure at the top of the mountains in a resting
  atmosphere.
-Line 493 
 The boundary conditions at top and botto
+Line 611 
 The boundary conditions at top and botto
  atmosphere)}  \label{eq:moving-bc-atmos}
  \end{eqnarray}
- Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent
+ Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt})
- set of atmospheric equations which, for convenience, are written out in $p$
+ yields a consistent set of atmospheric equations which, for convenience, are written out in $p$
  coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).
  \subsection{Ocean}
-Line 517 
 At the bottom of the ocean: $R_{fixed}(x
+Line 635 
 At the bottom of the ocean: $R_{fixed}(x
  The surface of the ocean is given by: $R_{moving}=\eta $
- The position of the resting free surface of the ocean is given by $%
+ The position of the resting free surface of the ocean is given by $
  R_{o}=Z_{o}=0$.
  Boundary conditions are:
-Line 525 
 Boundary conditions are:
+Line 643 
 Boundary conditions are:
  \begin{eqnarray}
  w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)}  \label{eq:fixed-bc-ocean}
  \\
- w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) %
+ w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)
  \label{eq:moving-bc-ocean}}
  \end{eqnarray}
  where $\eta $ is the elevation of the free surface.
- Then eq(\ref{eq:continuous}) yields a consistent set of oceanic equations
+ Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yield a consistent set
+ of oceanic equations
  which, for convenience, are written out in $z$ coordinates in Appendix Ocean
  - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
  Non-hydrostatic forms}
+ \begin{rawhtml}
+ <!-- CMIREDIR:non_hydrostatic: -->
+ \end{rawhtml}
  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
  \begin{equation}
  \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
  \label{eq:phi-split}
- \end{equation}%
+ \end{equation}
- and write eq(\ref{incompressible}a,b) in the form:
+ %and write eq(\ref{eq:incompressible}) in the form:
+ %                  ^- this eq is missing (jmc) ; replaced with:
+ and write eq( \ref{eq:horizontal_mtm}) in the form:
  \begin{equation}
  \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
-Line 556 
 _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
+Line 681 
 _{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 576 
 discussion:
+Line 701 
 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}
+ \ \right. \qquad  \label{eq:gu-speherical}
  \end{equation}
  \begin{equation}
  \left.
  \begin{tabular}{l}
  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
- $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\}
+ $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\}
  $ \\
- $-\left\{ -2\Omega u\sin lat\right\} $ \\
+ $-\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}
+ \ \right. \qquad  \label{eq:gv-spherical}
- \end{equation}%
+ \end{equation}
  \qquad \qquad \qquad \qquad \qquad
  \begin{equation}
  \left.
  \begin{tabular}{l}
  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
- ${+}\underline{{2\Omega u\cos lat}}$ \\
+ ${+}\underline{{2\Omega u\cos \varphi}}$ \\
- $\underline{\underline{\mathcal{F}_{\dot{r}}}}$%
+ $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
- \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}
+ \ \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.}
- \begin{figure}
- \begin{center}
- \resizebox{!}{4in}{
-  \rotatebox{90}{
-  \rotatebox{180}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps}
-  }
-  }
- }
- \end{center}
- \label{fig:spcoord}
- \end{figure}
+ %%CNHbegin
+ \input{part1/sphere_coord_figure.tex}
+ %%CNHend
  \subsubsection{Shallow atmosphere approximation}
-Line 661 
 hydrostatic balance and the `traditional
+Line 774 
 hydrostatic balance and the `traditional
  Coriolis force is treated approximately and the shallow atmosphere
  approximation is made.\ The MITgcm need not make the `traditional
  approximation'. To be able to support consistent non-hydrostatic forms the
- shallow atmosphere approximation can be relaxed - when dividing through by $r
+ shallow atmosphere approximation can be relaxed - when dividing through by $
- $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
+ r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
  the radius of the earth.
  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
+ \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
  These are discussed at length in Marshall et al (1997a).
-Line 674 
 terms in Eqs. (\ref{eq:gu-speherical} $\
+Line 788 
 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 688 
 variation of the radial position of a pa
+Line 802 
 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 704 
 only a quasi-non-hydrostatic atmospheric
+Line 818 
 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
  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 717 
 and Bromley, 1995; Marshall et.al.\ 1997
+Line 831 
 and Bromley, 1995; Marshall et.al.\ 1997
  \paragraph{Quasi-nonhydrostatic Atmosphere}
- In the non-hydrostatic version of our atmospheric model we approximate $\dot{%
+ In the non-hydrostatic version of our atmospheric model we approximate $\dot{
  r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical})
  (but only here) by:
  \begin{equation}
  \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt}  \label{eq:quasi-nh-w}
- \end{equation}%
+ \end{equation}
  where $p_{hy}$ is the hydrostatic pressure.
  \subsubsection{Summary of equation sets supported by model}
-Line 751 
 equations in $z-$coordinates are support
+Line 865 
 equations in $z-$coordinates are support
  \subparagraph{Non-hydrostatic}
- Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$%
+ Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
- coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref%
+ coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref
  {eq:ocean-salt}).
  \subsection{Solution strategy}
- The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{%
+ The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
- NH} models is summarized in Fig.7.%
+ NH} models is summarized in Figure \ref{fig:solution-strategy}.
- \marginpar{
+ Under all dynamics, a 2-d elliptic equation is
- Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
  first solved to find the surface pressure and the hydrostatic pressure at
  any level computed from the weight of fluid above. Under \textbf{HPE} and
  \textbf{QH} dynamics, the horizontal momentum equations are then stepped
-Line 769 
 forward and $\dot{r}$ found from continu
+Line 882 
 forward and $\dot{r}$ found from continu
  stepping forward the horizontal momentum equations; $\dot{r}$ is found by
  stepping forward the vertical momentum equation.
- \begin{figure}
+ %%CNHbegin
- \begin{center}
+ \input{part1/solution_strategy_figure.tex}
- \resizebox{!}{4in}{
+ %%CNHend
-  \rotatebox{90}{
-  \rotatebox{180}{
-   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps}
-  }
-  }
- }
- \end{center}
- \label{fig:solnstart}
- \end{figure}
  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
- course, some complication that goes with the inclusion of $\cos \phi \ $%
+ course, some complication that goes with the inclusion of $\cos \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 794 
 Marshall et al, 1997) resulting in a non
+Line 897 
 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 808 
 Hydrostatic pressure is obtained by inte
+Line 912 
 Hydrostatic pressure is obtained by inte
  vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:
  \begin{equation*}
- \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}%
+ \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}
  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
  \end{equation*}
  and so
-Line 826 
 atmospheric pressure pushing down on the
+Line 930 
 atmospheric pressure pushing down on the
  \subsubsection{Surface pressure}
- The surface pressure equation can be obtained by integrating continuity, (%
+ The surface pressure equation can be obtained by integrating continuity,
- \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
+ (\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
  \begin{equation*}
- \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}%
+ \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
  }_{h}+\partial _{r}\dot{r}\right) dr=0
  \end{equation*}
  Thus:
  \begin{equation*}
  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
- +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%
+ +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}
  _{h}dr=0
  \end{equation*}
- where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $%
+ where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $
  r $. The above can be rearranged to yield, using Leibnitz's theorem:
  \begin{equation}
  \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot
  \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}
  \label{eq:free-surface}
- \end{equation}%
+ \end{equation}
  where we have incorporated a source term.
  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
- (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
+ (atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
  be written
  \begin{equation}
  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
  \label{eq:phi-surf}
- \end{equation}%
+ \end{equation}
  where $b_{s}$ is the buoyancy at the surface.
- In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref%
+ In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref
  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
  surface' and `rigid lid' approaches are available.
  \subsubsection{Non-hydrostatic pressure}
- Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{%
+ Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
- \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
+ $\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
- (\ref{incompressible}), we deduce that:
+ (\ref{eq:continuity}), we deduce that:
  \begin{equation}
- \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{%
+ \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
- \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .%
+ \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .
  \vec{\mathbf{F}}  \label{eq:3d-invert}
  \end{equation}
-Line 893 
 coasts (in the ocean) and the bottom:
+Line 997 
 coasts (in the ocean) and the bottom:
  \end{equation}
  where $\widehat{n}$ is a vector of unit length normal to the boundary. The
  kinematic condition (\ref{nonormalflow}) is also applied to the vertical
- velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $%
+ velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $
  \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the
  tangential component of velocity, $v_{T}$, at all solid boundaries,
  depending on the form chosen for the dissipative terms in the momentum
  equations - see below.
- Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that:
+ Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
  \begin{equation}
  \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
-Line 910 
 where
+Line 1014 
 where
  \begin{equation*}
  \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi
  _{s}+\mathbf{\nabla }\phi _{hyd}\right)
- \end{equation*}%
+ \end{equation*}
  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
  (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
  exploit classical 3D potential theory and, by introducing an appropriately
- chosen $\delta $-function sheet of `source-charge', replace the inhomogenous
+ chosen $\delta $-function sheet of `source-charge', replace the
- boundary condition on pressure by a homogeneous one. The source term $rhs$
+ inhomogeneous boundary condition on pressure by a homogeneous one. The
- in (\ref{eq:3d-invert}) is the divergence of the vector $\vec{\mathbf{F}}.$
+ source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $
- By simultaneously setting $%
+ \vec{\mathbf{F}}.$ By simultaneously setting $
  \begin{array}{l}
- \widehat{n}.\vec{\mathbf{F}}%
+ \widehat{n}.\vec{\mathbf{F}}
- \end{array}%
+ \end{array}
  =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
- self-consistent but simpler homogenised Elliptic problem is obtained:
+ self-consistent but simpler homogenized Elliptic problem is obtained:
  \begin{equation*}
  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
- \end{equation*}%
+ \end{equation*}
  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
- that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref%
+ that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref
  {eq:inhom-neumann-nh}) the modified boundary condition becomes:
  \begin{equation}
-Line 939 
 If the flow is `close' to hydrostatic ba
+Line 1043 
 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 947 
 does not vanish at $r=R_{moving}$, and s
+Line 1051 
 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 957 
 Many forms of momentum dissipation are a
+Line 1061 
 Many forms of momentum dissipation are a
  biharmonic frictions are commonly used:
  \begin{equation}
- D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}%
+ D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}
  +A_{4}\nabla _{h}^{4}v  \label{eq:dissipation}
  \end{equation}
  where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity
-Line 968 
 friction. These coefficients are the sam
+Line 1072 
 friction. These coefficients are the sam
  The mixing terms for the temperature and salinity equations have a similar
  form to that of momentum except that the diffusion tensor can be
- non-diagonal and have varying coefficients. $\qquad $%
+ non-diagonal and have varying coefficients. $\qquad $
  \begin{equation}
  D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
  _{h}^{4}(T,S)  \label{eq:diffusion}
  \end{equation}
- where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%
+ where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $
  horizontal coefficient for biharmonic diffusion. In the simplest case where
  the subgrid-scale fluxes of heat and salt are parameterized with constant
  horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,
-Line 984 
 reduces to a diagonal matrix with consta
+Line 1088 
 reduces to a diagonal matrix with consta
  \begin{array}{ccc}
  K_{h} & 0 & 0 \\
 & K_{h} & 0 \\
-& 0 & K_{v}%
+& 0 & K_{v}
  \end{array}
  \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}
  \end{equation}
-Line 994 
 salinity ... ).
+Line 1098 
 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:
+ {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the (so-called) `vector invariant' form:
  \begin{equation}
- \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}%
+ \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
- +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %
+ +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla
  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
  \label{eq:vi-identity}
- \end{equation}%
+ \end{equation}
  This permits alternative numerical treatments of the non-linear terms based
  on their representation as a vorticity flux. Because gradients of coordinate
  vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit
- representation of the metric terms in (\ref{eq:gu-speherical}), (\ref%
+ representation of the metric terms in (\ref{eq:gu-speherical}), (\ref
  {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information
  about the geometry is contained in the areas and lengths of the volumes used
  to discretize the model.
  \subsection{Adjoint}
- Tangent linear and adoint counterparts of the forward model and described in
+ Tangent linear and adjoint counterparts of the forward model are described
- Chapter 5.
+ in Chapter 5.
  % $Header$
  % $Name$
-Line 1028 
 coordinates}
+Line 1132 
 coordinates}
  The hydrostatic primitive equations (HPEs) in p-coordinates are:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
- _{h}+\mathbf{\nabla }_{p}\phi  &=&\vec{\mathbf{\mathcal{F}}}
+ _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
  \label{eq:atmos-mom} \\
- \frac{\partial \phi }{\partial p}+\alpha  &=&0  \label{eq-p-hydro-start} \\
+ \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
- \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%
+ \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
  \partial p} &=&0  \label{eq:atmos-cont} \\
- p\alpha  &=&RT  \label{eq:atmos-eos} \\
+ p\alpha &=&RT  \label{eq:atmos-eos} \\
  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}
- \end{eqnarray}%
+ \end{eqnarray}
  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
- derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is
+ 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 1052 
 $\theta $ so that it looks more like a g
+Line 1156 
 $\theta $ so that it looks more like a g
  Differentiating (\ref{eq:atmos-eos}) we get:
  \begin{equation*}
  p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
- \end{equation*}%
+ \end{equation*}
- which, when added to the heat equation (\ref{eq:atmos-heat}) and using $%
+ which, when added to the heat equation (\ref{eq:atmos-heat}) and using $
  c_{p}=c_{v}+R$, gives:
  \begin{equation}
  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
  \label{eq-p-heat-interim}
- \end{equation}%
+ \end{equation}
  Potential temperature is defined:
  \begin{equation}
  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}
- \end{equation}%
+ \end{equation}
  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience
  we will make use of the Exner function $\Pi (p)$ which defined by:
  \begin{equation}
  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{Exner}
- \end{equation}%
+ \end{equation}
  The following relations will be useful and are easily expressed in terms of
  the Exner function:
  \begin{equation*}
  c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi
- }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{%
+ }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
- \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}%
+ \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}
  \frac{Dp}{Dt}
- \end{equation*}%
+ \end{equation*}
  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.
  The heat equation is obtained by noting that
  \begin{equation*}
  c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta
  \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
  \end{equation*}
  and on substituting into (\ref{eq-p-heat-interim}) gives:
  \begin{equation}
-Line 1090 
 and on substituting into (\ref{eq-p-heat
+Line 1194 
 and on substituting into (\ref{eq-p-heat
  \end{equation}
  which is in conservative form.
- For convenience in the model we prefer to step forward (\ref%
+ For convenience in the model we prefer to step forward (\ref
  {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).
  \subsubsection{Boundary conditions}
-Line 1134 
 _{o}(p_{o})=g~Z_{topo}$, defined:
+Line 1238 
 _{o}(p_{o})=g~Z_{topo}$, defined:
  The final form of the HPE's in p coordinates is then:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
- _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\
+ _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-prime} \\
  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
- \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%
+ \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
  \partial p} &=&0 \\
  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
- \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}
+ \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }
  \end{eqnarray}
  % $Header$
-Line 1154 
 We review here the method by which the s
+Line 1258 
 We review here the method by which the s
  HPE's for the ocean written in z-coordinates are obtained. The
  non-Boussinesq equations for oceanic motion are:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\
  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
  &=&\epsilon _{nh}\mathcal{F}_{w} \\
- \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%
+ \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
- _{h}+\frac{\partial w}{\partial z} &=&0 \\
+ _{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
- \rho  &=&\rho (\theta ,S,p) \\
+ \rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
- \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
+ \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
- \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
+ \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zns-salt}
- \end{eqnarray}%
+ \label{eq:non-boussinesq}
+ \end{eqnarray}
  These equations permit acoustics modes, inertia-gravity waves,
- non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
+ non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
  mode. As written, they cannot be integrated forward consistently - if we
  step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
- consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref%
+ consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
  {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
  therefore necessary to manipulate the system as follows. Differentiating the
  EOS (equation of state) gives:
-Line 1181 
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
+Line 1286 
 _{\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:
- {eq-zns-cont} gives:
  \begin{equation}
- \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%
+ \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
  \end{equation}
  where we have used an approximation sign to indicate that we have assumed
-Line 1192 
 adiabatic motion, dropping the $\frac{D\
+Line 1296 
 adiabatic motion, dropping the $\frac{D\
  Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that
  can be explicitly integrated forward:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
  \label{eq-cns-hmom} \\
  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\
- \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{%
+ \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\
  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\
  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\
-Line 1211 
 wherever it appears in a product (ie. no
+Line 1315 
 wherever it appears in a product (ie. no
  `Boussinesq assumption'. The only term that then retains the full variation
  in $\rho $ is the gravitational acceleration:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
  \label{eq-zcb-hmom} \\
- \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%
+ \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
  \label{eq-zcb-hydro} \\
- \frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{%
+ \frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{
  \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-zcb-cont} \\
  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\
  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\
  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}
  \end{eqnarray}
  These equations still retain acoustic modes. But, because the
- ``compressible'' terms are linearized, the pressure equation \ref%
+ ``compressible'' terms are linearized, the pressure equation \ref
  {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
  term appears as a Helmholtz term in the non-hydrostatic pressure equation).
  These are the \emph{truly} compressible Boussinesq equations. Note that the
  EOS must have the same pressure dependency as the linearized pressure term,
- ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{%
+ ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{
  c_{s}^{2}}$, for consistency.
  \subsubsection{`Anelastic' z-coordinate equations}
  The anelastic approximation filters the acoustic mode by removing the
- time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}%
+ time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}
- ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}%
+ ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}
  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
  continuity and EOS. A better solution is to change the dependency on
  pressure in the EOS by splitting the pressure into a reference function of
  height and a perturbation:
  \begin{equation*}
  \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
  \end{equation*}
  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
  differentiating the EOS, the continuity equation then becomes:
  \begin{equation*}
- \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%
+ \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{
- Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%
+ Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+
  \frac{\partial w}{\partial z}=0
  \end{equation*}
  If the time- and space-scales of the motions of interest are longer than
- those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%
+ those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},
  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and
- $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{%
+ $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{
  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
  and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the
  anelastic continuity equation:
  \begin{equation}
- \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-%
+ \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-
  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}
  \end{equation}
  A slightly different route leads to the quasi-Boussinesq continuity equation
- where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+%
+ where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
- \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%
+ \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:
  \begin{equation}
- \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%
+ \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
  \partial \left( \rho _{o}w\right) }{\partial z}=0  \label{eq-za-cont2}
  \end{equation}
  Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same
-Line 1278 
 equation if:
+Line 1382 
 equation if:
  \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}
  \end{equation}
  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
- and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{%
+ and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{
  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
  then:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
  \label{eq-zab-hmom} \\
- \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%
+ \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
  \label{eq-zab-hydro} \\
- \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%
+ \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
  \partial \left( \rho _{o}w\right) }{\partial z} &=&0  \label{eq-zab-cont} \\
  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\
  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\
-Line 1302 
 Here, the objective is to drop the depth
+Line 1406 
 Here, the objective is to drop the depth
  technically, to also remove the dependence of $\rho $ on $p_{o}$. This would
  yield the ``truly'' incompressible Boussinesq equations:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
  \label{eq-ztb-hmom} \\
- \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}%
+ \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}
  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
  \label{eq-ztb-hydro} \\
  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
-Line 1324 
 retain compressibility effects in the de
+Line 1428 
 retain compressibility effects in the de
  density thus:
  \begin{equation*}
  \rho =\rho _{o}+\rho ^{\prime }
- \end{equation*}%
+ \end{equation*}
  We then assert that variations with depth of $\rho _{o}$ are unimportant
  while the compressible effects in $\rho ^{\prime }$ are:
  \begin{equation*}
  \rho _{o}=\rho _{c}
- \end{equation*}%
+ \end{equation*}
  \begin{equation*}
  \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}
- \end{equation*}%
+ \end{equation*}
  This then yields what we can call the semi-compressible Boussinesq
  equations:
  \begin{eqnarray}
- \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
- _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%
+ _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{
  \mathcal{F}}}  \label{eq:ocean-mom} \\
  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
-Line 1348 
 _{c}}\frac{\partial p^{\prime }}{\partia
+Line 1452 
 _{c}}\frac{\partial p^{\prime }}{\partia
  \\
  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\
  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}
- \end{eqnarray}%
+ \end{eqnarray}
  Note that the hydrostatic pressure of the resting fluid, including that
  associated with $\rho _{c}$, is subtracted out since it has no effect on the
  dynamics.
-Line 1372 
 In spherical coordinates, the velocity c
+Line 1476 
 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 $
  \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.
-Line 1392 
 The `grad' ($\nabla $) and `div' ($\nabl
+Line 1496 
 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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