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-revision 1.4 by adcroft,
Thu Oct 11 19:36:56 2001 UTC
+revision 1.8 by cnh,
Thu Oct 25 15:24:01 2001 UTC
 Line 32
  %tci%\tableofcontents
- \part{MIT GCM basics}
  % Section: Overview
  % $Header$
-Line 56 
 MITgcm has a number of novel aspects:
+Line 54 
 MITgcm has a number of novel aspects:
  \begin{itemize}
  \item it can be used to study both atmospheric and oceanic phenomena; one
  hydrodynamical kernel is used to drive forward both atmospheric and oceanic
- models - see fig
+ 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 75 
 Fig.2 All scales}\ref{fig:all-scales}
+Line 69 
 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 96 
 listed in an Appendix.
+Line 88 
 listed in an Appendix.
  We begin by briefly showing some of the results of the model in action to
  give a feel for the wide range of problems that can be addressed using it.
- \pagebreak
  % $Header$
  % $Name$
-Line 105 
 give a feel for the wide range of proble
+Line 96 
 give a feel for the wide range of proble
  The MITgcm has been designed and used to model a wide range of phenomena,
  from convection on the scale of meters in the ocean to the global pattern of
- atmospheric winds - see fig.2\ref{fig:all-scales}. To give a flavor of the
+ atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the
  kinds of problems the model has been used to study, we briefly describe some
  of them here. A more detailed description of the underlying formulation,
  numerical algorithm and implementation that lie behind these calculations is
-Line 116 
 described in detail in the documentation
+Line 107 
 described in detail in the documentation
  \subsection{Global atmosphere: `Held-Suarez' benchmark}
- A novel feature of MITgcm is its ability to simulate both atmospheric and
+ A novel feature of MITgcm is its ability to simulate, using one basic algorithm,
- oceanographic flows at both small and large scales.
+ both atmospheric and oceanographic flows at both small and large scales.
- Fig.E1a.\ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
+ Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
  temperature field obtained using the atmospheric isomorph of MITgcm run at
 .8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
  (blue) and warm air along an equatorial band (red). Fully developed
-Line 135 
 there are no mountains or land-sea contr
+Line 126 
 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 gridding and obviated the need to Fourier filter.
  The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
  grid, of which the cubed sphere is just one of many choices.
- Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
+ Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
- wind and meridional overturning streamfunction from a 20-level version of
+ wind from a 20-level configuration of
  the model. It compares favorable with more conventional spatial
- discretization approaches.
+ discretization approaches. The two plots show the field calculated using the
+ cube-sphere grid and the flow calculated using a regular, spherical polar
- A regular spherical lat-lon grid can also be used.
+ latitude-longitude grid. Both grids are supported within the model.
  %% CNHbegin
  \input{part1/hs_zave_u_figure}
-Line 160 
 diffusive patterns of ocean currents. Bu
+Line 151 
 diffusive patterns of ocean currents. Bu
  increased until the baroclinic instability process is resolved, numerical
  solutions of a different and much more realistic kind, can be obtained.
- Fig. ?.? shows the surface temperature and velocity field obtained from
+ Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity
- MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
+ field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal
+ resolution on a $lat-lon$
  grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
  (to avoid the converging of meridian in northern latitudes). 21 vertical
  levels are used in the vertical with a `lopped cell' representation of
  topography. The development and propagation of anomalously warm and cold
- eddies can be clearly been seen in the Gulf Stream region. The transport of
+ eddies can be clearly seen in the Gulf Stream region. The transport of
  warm water northward by the mean flow of the Gulf Stream is also clearly
  visible.
-Line 177 
 visible.
+Line 169 
 visible.
  \subsection{Global ocean circulation}
- Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
+ Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
+ the surface of a 4$^{\circ }$
  global ocean model run with 15 vertical levels. Lopped cells are used to
  represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
  }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
-Line 185 
 mixed boundary conditions on temperature
+Line 178 
 mixed boundary conditions on temperature
  transfer properties of ocean eddies, convection and mixing is parameterized
  in this model.
- Fig.E2b shows the meridional overturning circulation of the global ocean in
+ Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
- Sverdrups.
+ circulation of the global ocean in Sverdrups.
  %%CNHbegin
  \input{part1/global_circ_figure}
-Line 198 
 Dense plumes generated by localized cool
+Line 191 
 Dense plumes generated by localized cool
  ocean may be influenced by rotation when the deformation radius is smaller
  than the width of the cooling region. Rather than gravity plumes, the
  mechanism for moving dense fluid down the shelf is then through geostrophic
- eddies. The simulation shown in the figure (blue is cold dense fluid, red is
+ eddies. The simulation shown in the figure \ref{fig::convect-and-topo}
+ (blue is cold dense fluid, red is
  warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
  trigger convection by surface cooling. The cold, dense water falls down the
  slope but is deflected along the slope by rotation. It is found that
-Line 217 
 presence of complex geometry makes it an
+Line 211 
 presence of complex geometry makes it an
  dynamics and mixing in oceanic canyons and ridges driven by large amplitude
  barotropic tidal currents imposed through open boundary conditions.
- Fig. ?.? shows the influence of cross-slope topographic variations on
+ Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope
+ topographic variations on
  internal wave breaking - the cross-slope velocity is in color, the density
  contoured. The internal waves are excited by application of open boundary
- conditions on the left.\ They propagate to the sloping boundary (represented
+ conditions on the left. They propagate to the sloping boundary (represented
  using MITgcm's finite volume spatial discretization) where they break under
  nonhydrostatic dynamics.
-Line 234 
 Forward and tangent linear counterparts
+Line 229 
 Forward and tangent linear counterparts
  `automatic adjoint compiler'. These can be used in parameter sensitivity and
  data assimilation studies.
- As one example of application of the MITgcm adjoint, Fig.E4 maps the
+ As one example of application of the MITgcm adjoint, Figure \ref{fig:hf-sensitivity}
- gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+ maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
- of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $
+ of the overturning streamfunction shown in figure \ref{fig:large-scale-circ}
- \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
+ at 60$^{\circ }$N and $
+ \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
+ a 100 year period. We see that $J$ is
  sensitive to heat fluxes over the Labrador Sea, one of the important sources
  of deep water for the thermohaline circulations. This calculation also
  yields sensitivities to all other model parameters.
-Line 253 
 ocean circulation. An appropriately defi
+Line 250 
 ocean circulation. An appropriately defi
  the departure of the model from observations (both remotely sensed and
  insitu) over an interval of time, is minimized by adjusting `control
  parameters' such as air-sea fluxes, the wind field, the initial conditions
- etc. Figure ?.? shows an estimate of the time-mean surface elevation of the
+ etc. Figure \ref{fig:assimilated-globes} shows an estimate of the time-mean
- ocean obtained by bringing the model in to consistency with altimetric and
+ surface elevation of the ocean obtained by bringing the model in to
- in-situ observations over the period 1992-1997.
+ consistency with altimetric and in-situ observations over the period
+-1997. {\bf CHANGE THIS TEXT - FIG FROM PATRICK/CARL/DETLEF}
  %% CNHbegin
  \input{part1/globes_figure}
-Line 266 
 in-situ observations over the period 199
+Line 264 
 in-situ observations over the period 199
  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}
 Line 276 
 from a single year of a global, interann
  \subsection{Simulations of laboratory experiments}
- Figure ?.? shows MITgcm being used to simulate a laboratory experiment
+ Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
- enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
+ laboratory experiment enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
  initially homogeneous tank of water ($1m$ in diameter) is driven from its
  free surface by a rotating heated disk. The combined action of mechanical
  and thermal forcing creates a lens of fluid which becomes baroclinically
  unstable. The stratification and depth of penetration of the lens is
- arrested by its instability in a process analogous to that whic sets the
+ arrested by its instability in a process analogous to that which sets the
  stratification of the ACC.
  %%CNHbegin
 Line 296 
 stratification of the ACC.
  To render atmosphere and ocean models from one dynamical core we exploit
  `isomorphisms' between equation sets that govern the evolution of the
- respective fluids - see fig.4
+ respective fluids - see figure \ref{fig:isomorphic-equations}.
- \marginpar{
+ One system of hydrodynamical equations is written down
- Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down
  and encoded. The model variables have different interpretations depending on
  whether the atmosphere or ocean is being studied. Thus, for example, the
  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
- modeling the atmosphere and height, $z$, if we are modeling the ocean.
+ modeling the atmosphere (left hand side of figure \ref{fig:isomorphic-equations})
+ and height, $z$, if we are modeling the ocean (right hand side of figure
+ \ref{fig:isomorphic-equations}).
  %%CNHbegin
  \input{part1/zandpcoord_figure.tex}
-Line 314 
 velocity $\vec{\mathbf{v}}$, active trac
+Line 315 
 velocity $\vec{\mathbf{v}}$, active trac
  depend on $\theta $, $S$, and $p$. The equations that govern the evolution
  of these fields, obtained by applying the laws of classical mechanics and
  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
- a generic vertical coordinate, $r$, see fig.5
+ a generic vertical coordinate, $r$, so that the appropriate
- \marginpar{
+ kinematic boundary conditions can be applied isomorphically
- Fig.5 The vertical coordinate of model}:
+ see figure \ref{fig:zandp-vert-coord}.
  %%CNHbegin
  \input{part1/vertcoord_figure.tex}
-Line 325 
 Fig.5 The vertical coordinate of model}:
+Line 326 
 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:humidtity_salt}
+ \end{equation}
  Here:
-Line 413 
 S\text{ is specific humidity in the atmo
+Line 416 
 S\text{ is specific humidity in the atmo
  \end{equation*}
  The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by
- extensive `physics' packages for atmosphere and ocean described in Chapter 6.
+ `physics' and forcing packages for atmosphere and ocean. These are described
+ in later chapters.
  \subsection{Kinematic Boundary conditions}
  \subsubsection{vertical}
- at fixed and moving $r$ surfaces we set (see fig.5):
+ at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
  \begin{equation}
  \dot{r}=0atr=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
-Line 450 
 where $\vec{\mathbf{n}}$ is the normal t
+Line 454 
 where $\vec{\mathbf{n}}$ is the normal t
  \subsection{Atmosphere}
- In the atmosphere, see fig.5, we interpret:
+ In the atmosphere, (see figure \ref{fig:zandp-vert-coord}), we interpret:
  \begin{equation}
  r=p\text{ is the pressure}  \label{eq:atmos-r}
-Line 521 
 The boundary conditions at top and botto
+Line 525 
 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_slainty})
- 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 558 
 w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo
+Line 562 
 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_slainty}) 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}).
-Line 571 
 Let us separate $\phi $ in to surface, h
+Line 576 
 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 604 
 discussion:
+Line 609 
 discussion:
  \left.
  \begin{tabular}{l}
  $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\
- $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan lat}{{r}}\right\} $
+ $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} $
  \\
- $-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\right\} $
+ $-\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\right\} $
  \\
  $+\mathcal{F}_{u}$
  \end{tabular}
-Line 624 
 $+\mathcal{F}_{u}$
+Line 629 
 $+\mathcal{F}_{u}$
  \left.
  \begin{tabular}{l}
  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
- $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\}
+ $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\}
  $ \\
- $-\left\{ -2\Omega u\sin lat\right\} $ \\
+ $-\left\{ -2\Omega u\sin \varphi \right\} $ \\
  $+\mathcal{F}_{v}$
  \end{tabular}
  \ \right\} \left\{
-Line 645 
 $+\mathcal{F}_{v}$
+Line 650 
 $+\mathcal{F}_{v}$
  \begin{tabular}{l}
  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
- ${+}\underline{{2\Omega u\cos lat}}$ \\
+ ${+}\underline{{2\Omega u\cos \varphi}}$ \\
  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
  \end{tabular}
  \ \right\} \left\{
-Line 659 
 $\underline{\underline{\mathcal{F}_{\dot
+Line 664 
 $\underline{\underline{\mathcal{F}_{\dot
  \end{equation}
  \qquad \qquad \qquad \qquad \qquad
- In the above `${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 684 
 r $ in, for example, (\ref{eq:gu-speheri
+Line 687 
 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 697 
 computed at all other levels by integrat
+Line 701 
 computed at all other levels by integrat
  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
- \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
+ \varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
  contribution to the pressure field: only the terms underlined twice in Eqs. (
  \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero
  and, simultaneously, the shallow atmosphere approximation is relaxed. In
-Line 706 
 variation of the radial position of a pa
+Line 710 
 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 727 
 In the non-hydrostatic ocean model all t
+Line 731 
 In the non-hydrostatic ocean model all t
  three dimensional elliptic equation must be solved subject to Neumann
  boundary conditions (see below). It is important to note that use of the
  full \textbf{NH} does not admit any new `fast' waves in to the system - the
- incompressible condition eq(\ref{eq:continuous})c has already filtered out
+ incompressible condition eq(\ref{eq:continuity}) has already filtered out
  acoustic modes. It does, however, ensure that the gravity waves are treated
  accurately with an exact dispersion relation. The \textbf{NH} set has a
  complete angular momentum principle and consistent energetics - see White
-Line 776 
 coordinates are supported - see eqs(\ref
+Line 780 
 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 792 
 stepping forward the vertical momentum e
+Line 795 
 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 802 
 Marshall et al, 1997) resulting in a non
+Line 805 
 Marshall et al, 1997) resulting in a non
  hydrostatic limit, is as computationally economic as the \textbf{HPEs}.
  \subsection{Finding the pressure field}
+ \label{sec:finding_the_pressure_field}
  Unlike the prognostic variables $u$, $v$, $w$, $\theta $ and $S$, the
  pressure field must be obtained diagnostically. We proceed, as before, by
-Line 834 
 atmospheric pressure pushing down on the
+Line 838 
 atmospheric pressure pushing down on the
  \subsubsection{Surface pressure}
- The surface pressure equation can be obtained by integrating continuity, (
+ The surface pressure equation can be obtained by integrating continuity,
- \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
+ (\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
  \begin{equation*}
  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
-Line 860 
 r $. The above can be rearranged to yiel
+Line 864 
 r $. The above can be rearranged to yiel
  where we have incorporated a source term.
  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
- (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
+ (atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
  be written
  \begin{equation}
  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
-Line 868 
 be written
+Line 872 
 be written
  \end{equation}
  where $b_{s}$ is the buoyancy at the surface.
- In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref
+ In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref
  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
  surface' and `rigid lid' approaches are available.
  \subsubsection{Non-hydrostatic pressure}
- Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
+ Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
- \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
+ $\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
- (\ref{incompressible}), we deduce that:
+ (\ref{eq:continuity}), we deduce that:
  \begin{equation}
  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
-Line 907 
 tangential component of velocity, $v_{T}
+Line 911 
 tangential component of velocity, $v_{T}
  depending on the form chosen for the dissipative terms in the momentum
  equations - see below.
- Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that:
+ Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
  \begin{equation}
  \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
-Line 947 
 If the flow is `close' to hydrostatic ba
+Line 951 
 If the flow is `close' to hydrostatic ba
  converges rapidly because $\phi _{nh}\ $is then only a small correction to
  the hydrostatic pressure field (see the discussion in Marshall et al, a,b).
- The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{homneuman})
+ The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh})
  does not vanish at $r=R_{moving}$, and so refines the pressure there.
  \subsection{Forcing/dissipation}
-Line 955 
 does not vanish at $r=R_{moving}$, and s
+Line 959 
 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 1003 
 salinity ... ).
+Line 1007 
 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 1021 
 to discretize the model.
+Line 1025 
 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 1048 
 c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt}
+Line 1052 
 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 1143 
 _{o}(p_{o})=g~Z_{topo}$, defined:
+Line 1147 
 _{o}(p_{o})=g~Z_{topo}$, 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 1167 
 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}
+Line 1171 
 _{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
-Line 1189 
 _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
+Line 1194 
 _{\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 1380 
 In spherical coordinates, the velocity c
+Line 1384 
 In spherical coordinates, the velocity c
  and vertical direction respectively, are given by (see Fig.2) :
  \begin{equation*}
- u=r\cos \phi \frac{D\lambda }{Dt}
+ u=r\cos \varphi \frac{D\lambda }{Dt}
  \end{equation*}
  \begin{equation*}
- v=r\frac{D\phi }{Dt}\qquad
+ v=r\frac{D\varphi }{Dt}\qquad
  \end{equation*}
  $\qquad \qquad \qquad \qquad $
-Line 1392 
 $\qquad \qquad \qquad \qquad $
+Line 1396 
 $\qquad \qquad \qquad \qquad $
  \dot{r}=\frac{Dr}{Dt}
  \end{equation*}
- Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial
+ Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
  distance of the particle from the center of the earth, $\Omega $ is the
  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
-Line 1400 
 The `grad' ($\nabla $) and `div' ($\nabl
+Line 1404 
 The `grad' ($\nabla $) and `div' ($\nabl
  spherical coordinates:
  \begin{equation*}
- \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }
+ \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda }
- ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}
+ ,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r}
  \right)
  \end{equation*}
  \begin{equation*}
- \nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial
+ \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
- \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}
+ \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
  +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
  \end{equation*}

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