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-revision 1.22 by jmc,
Sun Oct 17 04:15:13 2004 UTC
+revision 1.30 by jmc,
Wed May 11 18:45:43 2016 UTC
 Line 34
  % Section: Overview
- % $Header$
- % $Name$
  This document 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
-Line 61 
 hydrodynamical kernel is used to drive f
+Line 58 
 hydrodynamical kernel is used to drive f
  models - see fig \ref{fig:onemodel}
  %% CNHbegin
- \input{part1/one_model_figure}
+ \input{s_overview/text/one_model_figure}
  %% CNHend
  \item it has a non-hydrostatic capability and so can be used to study both
  small-scale and large scale processes - see fig \ref{fig:all-scales}
  %% CNHbegin
- \input{part1/all_scales_figure}
+ \input{s_overview/text/all_scales_figure}
  %% CNHend
  \item finite volume techniques are employed yielding an intuitive
-Line 76 
 discretization and support for the treat
+Line 73 
 discretization and support for the treat
  orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes}
  %% CNHbegin
- \input{part1/fvol_figure}
+ \input{s_overview/text/fvol_figure}
  %% CNHend
  \item tangent linear and adjoint counterparts are automatically maintained
-Line 87 
 studies.
+Line 84 
 studies.
  computational platforms.
  \end{itemize}
  Key publications reporting on and charting the development of the model are
- \cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99}:
+ \cite{hill:95,marshall:97a,marshall:97b,adcroft:97,mars-eta:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04}
+ (an overview on the model formulation can also be found in \cite{adcroft:04c}):
  \begin{verbatim}
  Hill, C. and J. Marshall, (1995)
-Line 137 
 J. Geophysical Res., 104(C12), 29,529-29
+Line 136 
 J. Geophysical Res., 104(C12), 29,529-29
  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.
- % $Header$
- % $Name$
  \section{Illustrations of the model in action}
- The MITgcm has been designed and used to model a wide range of phenomena,
+ MITgcm has been designed and used to model a wide range of phenomena,
  from convection on the scale of meters in the ocean to the global pattern of
  atmospheric winds - see 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
-Line 165 
 both atmospheric and oceanographic flows
+Line 161 
 both atmospheric and oceanographic flows
  Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
  temperature field obtained using the atmospheric isomorph of MITgcm run at
-.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
+ $2.8^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
  (blue) and warm air along an equatorial band (red). Fully developed
  baroclinic eddies spawned in the northern hemisphere storm track are
  evident. There are no mountains or land-sea contrast in this calculation,
-Line 175 
 in Held and Suarez; 1994 designed to tes
+Line 171 
 in Held and Suarez; 1994 designed to tes
  there are no mountains or land-sea contrast.
  %% CNHbegin
- \input{part1/cubic_eddies_figure}
+ \input{s_overview/text/cubic_eddies_figure}
  %% CNHend
  As described in Adcroft (2001), a `cubed sphere' is used to discretize the
-Line 191 
 cube-sphere grid and the flow calculated
+Line 187 
 cube-sphere grid and the flow calculated
  latitude-longitude grid. Both grids are supported within the model.
  %% CNHbegin
- \input{part1/hs_zave_u_figure}
+ \input{s_overview/text/hs_zave_u_figure}
  %% CNHend
  \subsection{Ocean gyres}
-Line 210 
 diffusive patterns of ocean currents. Bu
+Line 206 
 diffusive patterns of ocean currents. Bu
  increased until the baroclinic instability process is resolved, numerical
  solutions of a different and much more realistic kind, can be obtained.
- Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity
+ Figure \ref{fig:ocean-gyres} shows the surface temperature and
- field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal
+ velocity field obtained from MITgcm run at $\frac{1}{6}^{\circ }$
- resolution on a $lat-lon$
+ horizontal resolution on a \textit{lat-lon} grid in which the pole has
- grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
+ been rotated by $90^{\circ }$ on to the equator (to avoid the
- (to avoid the converging of meridian in northern latitudes). 21 vertical
+ converging of meridian in northern latitudes). 21 vertical levels are
- levels are used in the vertical with a `lopped cell' representation of
+ used in the vertical with a `lopped cell' representation of
- topography. The development and propagation of anomalously warm and cold
+ topography. The development and propagation of anomalously warm and
- eddies can be clearly seen in the Gulf Stream region. The transport of
+ cold eddies can be clearly seen in the Gulf Stream region. The
- warm water northward by the mean flow of the Gulf Stream is also clearly
+ transport of warm water northward by the mean flow of the Gulf Stream
- visible.
+ is also clearly visible.
  %% CNHbegin
- \input{part1/atl6_figure}
+ \input{s_overview/text/atl6_figure}
  %% CNHend
-Line 231 
 visible.
+Line 227 
 visible.
  <!-- CMIREDIR:global_ocean_circulation: -->
  \end{rawhtml}
- Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
+ Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean
- the surface of a 4$^{\circ }$
+ currents at the surface of a $4^{\circ }$ global ocean model run with
- global ocean model run with 15 vertical levels. Lopped cells are used to
+vertical levels. Lopped cells are used to represent topography on a
- represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
+ regular \textit{lat-lon} grid extending from $70^{\circ }N$ to
- }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
+ $70^{\circ }S$. The model is driven using monthly-mean winds with
- mixed boundary conditions on temperature and salinity at the surface. The
+ mixed boundary conditions on temperature and salinity at the surface.
- transfer properties of ocean eddies, convection and mixing is parameterized
+ The transfer properties of ocean eddies, convection and mixing is
- in this model.
+ 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}
+ \input{s_overview/text/global_circ_figure}
  %%CNHend
  \subsection{Convection and mixing over topography}
-Line 267 
 strong, and replaced by lateral entrainm
+Line 263 
 strong, and replaced by lateral entrainm
  instability of the along-slope current.
  %%CNHbegin
- \input{part1/convect_and_topo}
+ \input{s_overview/text/convect_and_topo}
  %%CNHend
  \subsection{Boundary forced internal waves}
-Line 289 
 using MITgcm's finite volume spatial dis
+Line 285 
 using MITgcm's finite volume spatial dis
  nonhydrostatic dynamics.
  %%CNHbegin
- \input{part1/boundary_forced_waves}
+ \input{s_overview/text/boundary_forced_waves}
  %%CNHend
  \subsection{Parameter sensitivity using the adjoint of MITgcm}
-Line 301 
 Forward and tangent linear counterparts
+Line 297 
 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, Figure \ref{fig:hf-sensitivity}
+ As one example of application of the MITgcm adjoint, Figure
- maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
+ \ref{fig:hf-sensitivity} maps the gradient $\frac{\partial J}{\partial
- of the overturning stream-function shown in figure \ref{fig:large-scale-circ}
+   \mathcal{H}}$where $J$ is the magnitude of the overturning
- at 60$^{\circ }$N and $
+ stream-function shown in figure \ref{fig:large-scale-circ} at
- \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
+ $60^{\circ }N$ and $ \mathcal{H}(\lambda,\varphi)$ is the mean, local
- a 100 year period. We see that $J$ is
+ air-sea heat flux over a 100 year period. We see that $J$ is sensitive
- sensitive to heat fluxes over the Labrador Sea, one of the important sources
+ to heat fluxes over the Labrador Sea, one of the important sources of
- of deep water for the thermohaline circulations. This calculation also
+ deep water for the thermohaline circulations. This calculation also
  yields sensitivities to all other model parameters.
  %%CNHbegin
- \input{part1/adj_hf_ocean_figure}
+ \input{s_overview/text/adj_hf_ocean_figure}
  %%CNHend
  \subsection{Global state estimation of the ocean}
-Line 333 
 consistency with altimetric and in-situ
+Line 329 
 consistency with altimetric and in-situ
 -1997.
  %% CNHbegin
- \input{part1/assim_figure}
+ \input{s_overview/text/assim_figure}
  %% CNHend
  \subsection{Ocean biogeochemical cycles}
-Line 341 
 consistency with altimetric and in-situ
+Line 337 
 consistency with altimetric and in-situ
  <!-- CMIREDIR:ocean_biogeo_cycles: -->
  \end{rawhtml}
- MITgcm is being used to study global biogeochemical cycles in the ocean. For
+ MITgcm is being used to study global biogeochemical cycles in the
- example one can study the effects of interannual changes in meteorological
+ ocean. For example one can study the effects of interannual changes in
- forcing and upper ocean circulation on the fluxes of carbon dioxide and
+ meteorological forcing and upper ocean circulation on the fluxes of
- oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows
+ carbon dioxide and oxygen between the ocean and atmosphere. Figure
- the annual air-sea flux of oxygen and its relation to density outcrops in
+ \ref{fig:biogeo} shows the annual air-sea flux of oxygen and its
- the southern oceans from a single year of a global, interannually varying
+ relation to density outcrops in the southern oceans from a single year
- simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution
+ of a global, interannually varying simulation. The simulation is run
- telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).
+ at $1^{\circ}\times1^{\circ}$ resolution telescoping to
+ $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not
+ shown).
  %%CNHbegin
- \input{part1/biogeo_figure}
+ \input{s_overview/text/biogeo_figure}
  %%CNHend
  \subsection{Simulations of laboratory experiments}
-Line 369 
 arrested by its instability in a process
+Line 367 
 arrested by its instability in a process
  stratification of the ACC.
  %%CNHbegin
- \input{part1/lab_figure}
+ \input{s_overview/text/lab_figure}
  %%CNHend
- % $Header$
- % $Name$
  \section{Continuous equations in `r' coordinates}
  \begin{rawhtml}
  <!-- CMIREDIR:z-p_isomorphism: -->
-Line 392 
 and height, $z$, if we are modeling the
+Line 387 
 and height, $z$, if we are modeling the
  \ref{fig:isomorphic-equations}).
  %%CNHbegin
- \input{part1/zandpcoord_figure.tex}
+ \input{s_overview/text/zandpcoord_figure.tex}
  %%CNHend
  The state of the fluid at any time is characterized by the distribution of
-Line 406 
 kinematic boundary conditions can be app
+Line 401 
 kinematic boundary conditions can be app
  see figure \ref{fig:zandp-vert-coord}.
  %%CNHbegin
- \input{part1/vertcoord_figure.tex}
+ \input{s_overview/text/vertcoord_figure.tex}
  %%CNHend
  \begin{equation}
-Line 657 
 which, for convenience, are written out
+Line 652 
 which, for convenience, are written out
  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
  Non-hydrostatic forms}
+ \label{sec:all_hydrostatic_forms}
  \begin{rawhtml}
  <!-- CMIREDIR:non_hydrostatic: -->
  \end{rawhtml}
-Line 765 
 Grad and div operators in spherical coor
+Line 761 
 Grad and div operators in spherical coor
  OPERATORS.
  %%CNHbegin
- \input{part1/sphere_coord_figure.tex}
+ \input{s_overview/text/sphere_coord_figure.tex}
  %%CNHend
  \subsubsection{Shallow atmosphere approximation}
- Most models are based on the `hydrostatic primitive equations' (HPE's) in
+ Most models are based on the `hydrostatic primitive equations' (HPE's)
- which the vertical momentum equation is reduced to a statement of
+ in which the vertical momentum equation is reduced to a statement of
- hydrostatic balance and the `traditional approximation' is made in which the
+ hydrostatic balance and the `traditional approximation' is made in
- Coriolis force is treated approximately and the shallow atmosphere
+ which the Coriolis force is treated approximately and the shallow
- approximation is made.\ The MITgcm need not make the `traditional
+ atmosphere approximation is made.  MITgcm need not make the
- approximation'. To be able to support consistent non-hydrostatic forms the
+ `traditional approximation'. To be able to support consistent
- shallow atmosphere approximation can be relaxed - when dividing through by $
+ non-hydrostatic forms the shallow atmosphere approximation can be
- r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
+ relaxed - when dividing through by $ r $ in, for example,
- the radius of the earth.
+ (\ref{eq:gu-speherical}), we do not replace $r$ by $a$, the radius of
+ the earth.
  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
  \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
-Line 815 
 et.al., 1997a. As in \textbf{HPE }only a
+Line 812 
 et.al., 1997a. As in \textbf{HPE }only a
  \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}
- The MIT model presently supports a full non-hydrostatic ocean isomorph, but
+ MITgcm presently supports a full non-hydrostatic ocean isomorph, but
  only a quasi-non-hydrostatic atmospheric isomorph.
  \paragraph{Non-hydrostatic Ocean}
-Line 885 
 stepping forward the horizontal momentum
+Line 882 
 stepping forward the horizontal momentum
  stepping forward the vertical momentum equation.
  %%CNHbegin
- \input{part1/solution_strategy_figure.tex}
+ \input{s_overview/text/solution_strategy_figure.tex}
  %%CNHend
  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
-Line 1074 
 friction. These coefficients are the sam
+Line 1071 
 friction. These coefficients are the sam
  The mixing terms for the temperature and salinity equations have a similar
  form to that of momentum except that the diffusion tensor can be
- non-diagonal and have varying coefficients. $\qquad $
+ non-diagonal and have varying coefficients.
  \begin{equation}
  D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
  _{h}^{4}(T,S)  \label{eq:diffusion}
-Line 1123 
 to discretize the model.
+Line 1120 
 to discretize the model.
  Tangent linear and adjoint counterparts of the forward model are described
  in Chapter 5.
- % $Header$
- % $Name$
  \section{Appendix ATMOSPHERE}
  \subsection{Hydrostatic Primitive Equations for the Atmosphere in pressure
-Line 1145 
 p\alpha &=&RT  \label{eq:atmos-eos} \\
+Line 1139 
 p\alpha &=&RT  \label{eq:atmos-eos} \\
  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}
  \end{eqnarray}
  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
- surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
+ surfaces) component of velocity, $\frac{D}{Dt}=\frac{\partial}{\partial t}
- \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
+ +\vec{\mathbf{v}}_{h}\cdot \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$
- derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter, $\phi =gz$ is
+ is the total derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter,
- the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
+ $\phi =gz$ is the geopotential, $\alpha =1/\rho $ is the specific volume,
- }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
+ $\omega =\frac{Dp }{Dt}$ is the vertical velocity in the $p-$coordinate.
- {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
+ Equation(\ref {eq:atmos-heat}) is the first law of thermodynamics where internal
- e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $
+ energy $e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass
- p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.
+ and $p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.
  It is convenient to cast the heat equation in terms of potential temperature
  $\theta $ so that it looks more like a generic conservation law.
-Line 1252 
 _{h}+\mathbf{\nabla }_{p}\phi ^{\prime }
+Line 1246 
 _{h}+\mathbf{\nabla }_{p}\phi ^{\prime }
  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }
  \end{eqnarray}
- % $Header$
- % $Name$
  \section{Appendix OCEAN}
  \subsection{Equations of motion for the ocean}
-Line 1469 
 the perturbation density. Nevertheless,
+Line 1460 
 the perturbation density. Nevertheless,
  _{nh}=0$ form of these equations that are used throughout the ocean modeling
  community and referred to as the primitive equations (HPE).
- % $Header$
- % $Name$
  \section{Appendix:OPERATORS}
  \subsection{Coordinate systems}
-Line 1486 
 u=r\cos \varphi \frac{D\lambda }{Dt}
+Line 1474 
 u=r\cos \varphi \frac{D\lambda }{Dt}
  \end{equation*}
  \begin{equation*}
- v=r\frac{D\varphi }{Dt}\qquad
+ v=r\frac{D\varphi }{Dt}
  \end{equation*}
- $\qquad \qquad \qquad \qquad $
  \begin{equation*}
  \dot{r}=\frac{Dr}{Dt}
-Line 1498 
 Here $\varphi $ is the latitude, $\lambd
+Line 1485 
 Here $\varphi $ is the latitude, $\lambd
  distance of the particle from the center of the earth, $\Omega $ is the
  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
- The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in
+ The `grad' ($\nabla $) and `div' ($\nabla\cdot$) operators are defined by, in
  spherical coordinates:
  \begin{equation*}
-Line 1508 
 spherical coordinates:
+Line 1495 
 spherical coordinates:
  \end{equation*}
  \begin{equation*}
- \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
+ \nabla\cdot v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
  \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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