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-revision 1.5 by adcroft,
Mon Oct 15 19:34:28 2001 UTC
+revision 1.18 by afe,
Tue Mar 23 15:29:39 2004 UTC
 Line 37
  % $Header$
  % $Name$
- \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 49 
 are available. A number of examples illu
+Line 47 
 are available. A number of examples illu
  both process and general circulation studies of the atmosphere and ocean are
  also presented.
+ \section{Introduction}
+ \begin{rawhtml}
+ <!-- CMIREDIR:innovations -->
+ \end{rawhtml}
  MITgcm has a number of novel aspects:
  \begin{itemize}
  \item it can be used to study both atmospheric and oceanic phenomena; one
  hydrodynamical kernel is used to drive forward both atmospheric and oceanic
- models - see fig
+ models - see fig \ref{fig:onemodel}
- \marginpar{
- Fig.1 One model}\ref{fig:onemodel}
  %% CNHbegin
  \input{part1/one_model_figure}
  %% CNHend
  \item it has a non-hydrostatic capability and so can be used to study both
- small-scale and large scale processes - see fig
+ small-scale and large scale processes - see fig \ref{fig:all-scales}
- \marginpar{
- Fig.2 All scales}\ref{fig:all-scales}
  %% CNHbegin
  \input{part1/all_scales_figure}
 Line 73 
 Fig.2 All scales}\ref{fig:all-scales}
  \item finite volume techniques are employed yielding an intuitive
  discretization and support for the treatment of irregular geometries using
- orthogonal curvilinear grids and shaved cells - see fig
+ orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes}
- \marginpar{
- Fig.3 Finite volumes}\ref{fig:finite-volumes}
  %% CNHbegin
  \input{part1/fvol_figure}
-Line 90 
 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.
-Line 102 
 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 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
  (blue) and warm air along an equatorial band (red). Fully developed
-Line 132 
 there are no mountains or land-sea contr
+Line 179 
 there are no mountains or land-sea contr
  %% CNHend
  As described in Adcroft (2001), a `cubed sphere' is used to discretize the
- globe permitting a uniform gridding and obviated the need to fourier filter.
+ globe permitting a uniform griding and obviated the need to Fourier filter.
  The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
  grid, of which the cubed sphere is just one of many choices.
- Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
+ Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
- wind and meridional overturning streamfunction from a 20-level version of
+ wind from a 20-level configuration of
  the model. It compares favorable with more conventional spatial
- discretization approaches.
+ discretization approaches. The two plots show the field calculated using the
+ cube-sphere grid and the flow calculated using a regular, spherical polar
- A regular spherical lat-lon grid can also be used.
+ latitude-longitude grid. Both grids are supported within the model.
  %% CNHbegin
  \input{part1/hs_zave_u_figure}
  %% CNHend
  \subsection{Ocean gyres}
+ \begin{rawhtml}
+ <!-- CMIREDIR:oceanic_example -->
+ \end{rawhtml}
+ \begin{rawhtml}
+ <!-- CMIREDIR:ocean_gyres -->
+ \end{rawhtml}
  Baroclinic instability is a ubiquitous process in the ocean, as well as the
  atmosphere. Ocean eddies play an important role in modifying the
-Line 157 
 diffusive patterns of ocean currents. Bu
+Line 210 
 diffusive patterns of ocean currents. Bu
  increased until the baroclinic instability process is resolved, numerical
  solutions of a different and much more realistic kind, can be obtained.
- Fig. ?.? shows the surface temperature and velocity field obtained from
+ Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity
- MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
+ field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal
+ resolution on a $lat-lon$
  grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
  (to avoid the converging of meridian in northern latitudes). 21 vertical
  levels are used in the vertical with a `lopped cell' representation of
  topography. The development and propagation of anomalously warm and cold
- eddies can be clearly been seen in the Gulf Stream region. The transport of
+ eddies can be clearly seen in the Gulf Stream region. The transport of
  warm water northward by the mean flow of the Gulf Stream is also clearly
  visible.
  %% CNHbegin
- \input{part1/ocean_gyres_figure}
+ \input{part1/atl6_figure}
  %% CNHend
  \subsection{Global ocean circulation}
+ \begin{rawhtml}
+ <!-- CMIREDIR:global_ocean_circulation -->
+ \end{rawhtml}
- Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
+ Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
+ the surface of a 4$^{\circ }$
  global ocean model run with 15 vertical levels. Lopped cells are used to
  represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
  }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
-Line 182 
 mixed boundary conditions on temperature
+Line 240 
 mixed boundary conditions on temperature
  transfer properties of ocean eddies, convection and mixing is parameterized
  in this model.
- Fig.E2b shows the meridional overturning circulation of the global ocean in
+ Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
- Sverdrups.
+ circulation of the global ocean in Sverdrups.
  %%CNHbegin
  \input{part1/global_circ_figure}
  %%CNHend
  \subsection{Convection and mixing over topography}
+ \begin{rawhtml}
+ <!-- CMIREDIR:mixing_over_topography -->
+ \end{rawhtml}
  Dense plumes generated by localized cooling on the continental shelf of the
  ocean may be influenced by rotation when the deformation radius is smaller
  than the width of the cooling region. Rather than gravity plumes, the
  mechanism for moving dense fluid down the shelf is then through geostrophic
- eddies. The simulation shown in the figure (blue is cold dense fluid, red is
+ eddies. The simulation shown in the figure \ref{fig:convect-and-topo}
+ (blue is cold dense fluid, red is
  warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
  trigger convection by surface cooling. The cold, dense water falls down the
  slope but is deflected along the slope by rotation. It is found that
-Line 208 
 instability of the along-slope current.
+Line 271 
 instability of the along-slope current.
  %%CNHend
  \subsection{Boundary forced internal waves}
+ \begin{rawhtml}
+ <!-- CMIREDIR:boundary_forced_internal_waves -->
+ \end{rawhtml}
  The unique ability of MITgcm to treat non-hydrostatic dynamics in the
  presence of complex geometry makes it an ideal tool to study internal wave
  dynamics and mixing in oceanic canyons and ridges driven by large amplitude
  barotropic tidal currents imposed through open boundary conditions.
- Fig. ?.? shows the influence of cross-slope topographic variations on
+ Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope
+ topographic variations on
  internal wave breaking - the cross-slope velocity is in color, the density
  contoured. The internal waves are excited by application of open boundary
- conditions on the left.\ They propagate to the sloping boundary (represented
+ conditions on the left. They propagate to the sloping boundary (represented
  using MITgcm's finite volume spatial discretization) where they break under
  nonhydrostatic dynamics.
-Line 226 
 nonhydrostatic dynamics.
+Line 293 
 nonhydrostatic dynamics.
  %%CNHend
  \subsection{Parameter sensitivity using the adjoint of MITgcm}
+ \begin{rawhtml}
+ <!-- CMIREDIR:parameter_sensitivity -->
+ \end{rawhtml}
  Forward and tangent linear counterparts of MITgcm are supported using an
  `automatic adjoint compiler'. These can be used in parameter sensitivity and
  data assimilation studies.
- As one example of application of the MITgcm adjoint, Fig.E4 maps the
+ As one example of application of the MITgcm adjoint, Figure \ref{fig:hf-sensitivity}
- gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+ maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
- of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $
+ of the overturning stream-function shown in figure \ref{fig:large-scale-circ}
- \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
+ at 60$^{\circ }$N and $
+ \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
+ a 100 year period. We see that $J$ is
  sensitive to heat fluxes over the Labrador Sea, one of the important sources
  of deep water for the thermohaline circulations. This calculation also
  yields sensitivities to all other model parameters.
-Line 244 
 yields sensitivities to all other model
+Line 316 
 yields sensitivities to all other model
  %%CNHend
  \subsection{Global state estimation of the ocean}
+ \begin{rawhtml}
+ <!-- CMIREDIR:global_state_estimation -->
+ \end{rawhtml}
  An important application of MITgcm is in state estimation of the global
  ocean circulation. An appropriately defined `cost function', which measures
  the departure of the model from observations (both remotely sensed and
- insitu) over an interval of time, is minimized by adjusting `control
+ in-situ) over an interval of time, is minimized by adjusting `control
  parameters' such as air-sea fluxes, the wind field, the initial conditions
- etc. Figure ?.? shows an estimate of the time-mean surface elevation of the
+ etc. Figure \ref{fig:assimilated-globes} shows the large scale planetary
- ocean obtained by bringing the model in to consistency with altimetric and
+ circulation and a Hopf-Muller plot of Equatorial sea-surface height.
- in-situ observations over the period 1992-1997.
+ Both are obtained from assimilation bringing the model in to
+ consistency with altimetric and in-situ observations over the period
+-1997.
  %% CNHbegin
- \input{part1/globes_figure}
+ \input{part1/assim_figure}
  %% CNHend
  \subsection{Ocean biogeochemical cycles}
+ \begin{rawhtml}
+ <!-- CMIREDIR:ocean_biogeo_cycles -->
+ \end{rawhtml}
  MITgcm is being used to study global biogeochemical cycles in the ocean. For
  example one can study the effects of interannual changes in meteorological
  forcing and upper ocean circulation on the fluxes of carbon dioxide and
- oxygen between the ocean and atmosphere. The figure shows the annual air-sea
+ oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows
- flux of oxygen and its relation to density outcrops in the southern oceans
+ the annual air-sea flux of oxygen and its relation to density outcrops in
- from a single year of a global, interannually varying simulation.
+ the southern oceans from a single year of a global, interannually varying
+ simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution
+ telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).
  %%CNHbegin
  \input{part1/biogeo_figure}
  %%CNHend
  \subsection{Simulations of laboratory experiments}
+ \begin{rawhtml}
+ <!-- CMIREDIR:classroom_exp -->
+ \end{rawhtml}
- Figure ?.? shows MITgcm being used to simulate a laboratory experiment
+ Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
- enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
+ laboratory experiment inquiring into the dynamics of the Antarctic Circumpolar Current (ACC). An
  initially homogeneous tank of water ($1m$ in diameter) is driven from its
  free surface by a rotating heated disk. The combined action of mechanical
  and thermal forcing creates a lens of fluid which becomes baroclinically
  unstable. The stratification and depth of penetration of the lens is
- arrested by its instability in a process analogous to that whic sets the
+ arrested by its instability in a process analogous to that which sets the
  stratification of the ACC.
  %%CNHbegin
-Line 290 
 stratification of the ACC.
+Line 376 
 stratification of the ACC.
  % $Name$
  \section{Continuous equations in `r' coordinates}
+ \begin{rawhtml}
+ <!-- CMIREDIR:z-p_isomorphism -->
+ \end{rawhtml}
  To render atmosphere and ocean models from one dynamical core we exploit
  `isomorphisms' between equation sets that govern the evolution of the
- respective fluids - see fig.4
+ respective fluids - see figure \ref{fig:isomorphic-equations}.
- \marginpar{
+ One system of hydrodynamical equations is written down
- Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down
  and encoded. The model variables have different interpretations depending on
  whether the atmosphere or ocean is being studied. Thus, for example, the
  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
- modeling the atmosphere and height, $z$, if we are modeling the ocean.
+ modeling the atmosphere (right hand side of figure \ref{fig:isomorphic-equations})
+ and height, $z$, if we are modeling the ocean (left hand side of figure
+ \ref{fig:isomorphic-equations}).
  %%CNHbegin
  \input{part1/zandpcoord_figure.tex}
-Line 311 
 velocity $\vec{\mathbf{v}}$, active trac
+Line 401 
 velocity $\vec{\mathbf{v}}$, active trac
  depend on $\theta $, $S$, and $p$. The equations that govern the evolution
  of these fields, obtained by applying the laws of classical mechanics and
  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
- a generic vertical coordinate, $r$, see fig.5
+ a generic vertical coordinate, $r$, so that the appropriate
- \marginpar{
+ kinematic boundary conditions can be applied isomorphically
- Fig.5 The vertical coordinate of model}:
+ see figure \ref{fig:zandp-vert-coord}.
  %%CNHbegin
  \input{part1/vertcoord_figure.tex}
-Line 322 
 Fig.5 The vertical coordinate of model}:
+Line 412 
 Fig.5 The vertical coordinate of model}:
  \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*}
- \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 410 
 S\text{ is specific humidity in the atmo
+Line 502 
 S\text{ is specific humidity in the atmo
  \end{equation*}
  The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by
- extensive `physics' packages for atmosphere and ocean described in Chapter 6.
+ `physics' and forcing packages for atmosphere and ocean. These are described
+ in later chapters.
  \subsection{Kinematic Boundary conditions}
  \subsubsection{vertical}
- at fixed and moving $r$ surfaces we set (see fig.5):
+ at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
  \begin{equation}
- \dot{r}=0atr=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
+ \dot{r}=0 \text{\ at\ } r=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
  \label{eq:fixedbc}
  \end{equation}
  \begin{equation}
- \dot{r}=\frac{Dr}{Dt}atr=R_{moving}\text{ \
+ \dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
- (oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}
+ (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
  \end{equation}
  Here
-Line 447 
 where $\vec{\mathbf{n}}$ is the normal t
+Line 540 
 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 518 
 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 555 
 w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo
+Line 648 
 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}
+ \begin{rawhtml}
+ <!-- CMIREDIR:non_hydrostatic -->
+ \end{rawhtml}
  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
-Line 568 
 Let us separate $\phi $ in to surface, h
+Line 666 
 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:
  \begin{equation}
  \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
-Line 601 
 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 621 
 $+\mathcal{F}_{u}$
+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 642 
 $+\mathcal{F}_{v}$
+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 656 
 $\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}
-Line 681 
 r $ in, for example, (\ref{eq:gu-speheri
+Line 777 
 r $ in, for example, (\ref{eq:gu-speheri
  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 694 
 computed at all other levels by integrat
+Line 791 
 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 703 
 variation of the radial position of a pa
+Line 800 
 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 724 
 In the non-hydrostatic ocean model all t
+Line 821 
 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 773 
 coordinates are supported - see eqs(\ref
+Line 870 
 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 789 
 stepping forward the vertical momentum e
+Line 885 
 stepping forward the vertical momentum e
  %%CNHend
  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
- course, some complication that goes with the inclusion of $\cos \phi \ $
+ course, some complication that goes with the inclusion of $\cos \varphi \ $
  Coriolis terms and the relaxation of the shallow atmosphere approximation.
  But this leads to negligible increase in computation. In \textbf{NH}, in
  contrast, one additional elliptic equation - a three-dimensional one - must
-Line 799 
 Marshall et al, 1997) resulting in a non
+Line 895 
 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 831 
 atmospheric pressure pushing down on the
+Line 928 
 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 857 
 r $. The above can be rearranged to yiel
+Line 954 
 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 865 
 be written
+Line 962 
 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 904 
 tangential component of velocity, $v_{T}
+Line 1001 
 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 944 
 If the flow is `close' to hydrostatic ba
+Line 1041 
 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 952 
 does not vanish at $r=R_{moving}$, and s
+Line 1049 
 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 1000 
 salinity ... ).
+Line 1097 
 salinity ... ).
  \subsection{Vector invariant form}
  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}
-Line 1018 
 to discretize the model.
+Line 1115 
 to discretize the model.
  \subsection{Adjoint}
- Tangent linear and adjoint counterparts of the forward model and described
+ Tangent linear and adjoint counterparts of the forward model are described
  in Chapter 5.
  % $Header$
-Line 1045 
 c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt}
+Line 1142 
 c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt}
  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, $\omega =\frac{Dp
  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
-Line 1140 
 _{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}
  % $Header$
-Line 1164 
 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}
+Line 1261 
 _{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 1186 
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
+Line 1284 
 _{\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{
  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
-Line 1377 
 In spherical coordinates, the velocity c
+Line 1474 
 In spherical coordinates, the velocity c
  and vertical direction respectively, are given by (see Fig.2) :
  \begin{equation*}
- u=r\cos \phi \frac{D\lambda }{Dt}
+ u=r\cos \varphi \frac{D\lambda }{Dt}
  \end{equation*}
  \begin{equation*}
- v=r\frac{D\phi }{Dt}\qquad
+ v=r\frac{D\varphi }{Dt}\qquad
  \end{equation*}
  $\qquad \qquad \qquad \qquad $
-Line 1389 
 $\qquad \qquad \qquad \qquad $
+Line 1486 
 $\qquad \qquad \qquad \qquad $
  \dot{r}=\frac{Dr}{Dt}
  \end{equation*}
- Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial
+ Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
  distance of the particle from the center of the earth, $\Omega $ is the
  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
-Line 1397 
 The `grad' ($\nabla $) and `div' ($\nabl
+Line 1494 
 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*}

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