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revision 1.8 by cnh, Thu Oct 25 15:24:01 2001 UTC revision 1.25 by edhill, Sat Apr 8 01:50:49 2006 UTC
# Line 37  Line 37 
37  % $Header$  % $Header$
38  % $Name$  % $Name$
39    
40  \section{Introduction}  This document provides the reader with the information necessary to
   
 This documentation provides the reader with the information necessary to  
41  carry out numerical experiments using MITgcm. It gives a comprehensive  carry out numerical experiments using MITgcm. It gives a comprehensive
42  description of the continuous equations on which the model is based, the  description of the continuous equations on which the model is based, the
43  numerical algorithms the model employs and a description of the associated  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
47  both process and general circulation studies of the atmosphere and ocean are  both process and general circulation studies of the atmosphere and ocean are
48  also presented.  also presented.
49    
50    \section{Introduction}
51    \begin{rawhtml}
52    <!-- CMIREDIR:innovations: -->
53    \end{rawhtml}
54    
55    
56  MITgcm has a number of novel aspects:  MITgcm has a number of novel aspects:
57    
58  \begin{itemize}  \begin{itemize}
# Line 84  computational platforms. Line 88  computational platforms.
88  \end{itemize}  \end{itemize}
89    
90  Key publications reporting on and charting the development of the model are  Key publications reporting on and charting the development of the model are
91  listed in an Appendix.  \cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04}:
92    
93    \begin{verbatim}
94    Hill, C. and J. Marshall, (1995)
95    Application of a Parallel Navier-Stokes Model to Ocean Circulation in
96    Parallel Computational Fluid Dynamics
97    In Proceedings of Parallel Computational Fluid Dynamics: Implementations
98    and Results Using Parallel Computers, 545-552.
99    Elsevier Science B.V.: New York
100    
101    Marshall, J., C. Hill, L. Perelman, and A. Adcroft, (1997)
102    Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling
103    J. Geophysical Res., 102(C3), 5733-5752.
104    
105    Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, (1997)
106    A finite-volume, incompressible Navier Stokes model for studies of the ocean
107    on parallel computers,
108    J. Geophysical Res., 102(C3), 5753-5766.
109    
110    Adcroft, A.J., Hill, C.N. and J. Marshall, (1997)
111    Representation of topography by shaved cells in a height coordinate ocean
112    model
113    Mon Wea Rev, vol 125, 2293-2315
114    
115    Marshall, J., Jones, H. and C. Hill, (1998)
116    Efficient ocean modeling using non-hydrostatic algorithms
117    Journal of Marine Systems, 18, 115-134
118    
119    Adcroft, A., Hill C. and J. Marshall: (1999)
120    A new treatment of the Coriolis terms in C-grid models at both high and low
121    resolutions,
122    Mon. Wea. Rev. Vol 127, pages 1928-1936
123    
124    Hill, C, Adcroft,A., Jamous,D., and J. Marshall, (1999)
125    A Strategy for Terascale Climate Modeling.
126    In Proceedings of the Eighth ECMWF Workshop on the Use of Parallel Processors
127    in Meteorology, pages 406-425
128    World Scientific Publishing Co: UK
129    
130    Marotzke, J, Giering,R., Zhang, K.Q., Stammer,D., Hill,C., and T.Lee, (1999)
131    Construction of the adjoint MIT ocean general circulation model and
132    application to Atlantic heat transport variability
133    J. Geophysical Res., 104(C12), 29,529-29,547.
134    
135    \end{verbatim}
136    
137  We begin by briefly showing some of the results of the model in action to  We begin by briefly showing some of the results of the model in action to
138  give a feel for the wide range of problems that can be addressed using it.  give a feel for the wide range of problems that can be addressed using it.
# Line 94  give a feel for the wide range of proble Line 142  give a feel for the wide range of proble
142    
143  \section{Illustrations of the model in action}  \section{Illustrations of the model in action}
144    
145  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,
146  from convection on the scale of meters in the ocean to the global pattern of  from convection on the scale of meters in the ocean to the global pattern of
147  atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the  atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the
148  kinds of problems the model has been used to study, we briefly describe some  kinds of problems the model has been used to study, we briefly describe some
# Line 102  of them here. A more detailed descriptio Line 150  of them here. A more detailed descriptio
150  numerical algorithm and implementation that lie behind these calculations is  numerical algorithm and implementation that lie behind these calculations is
151  given later. Indeed many of the illustrative examples shown below can be  given later. Indeed many of the illustrative examples shown below can be
152  easily reproduced: simply download the model (the minimum you need is a PC  easily reproduced: simply download the model (the minimum you need is a PC
153  running linux, together with a FORTRAN\ 77 compiler) and follow the examples  running Linux, together with a FORTRAN\ 77 compiler) and follow the examples
154  described in detail in the documentation.  described in detail in the documentation.
155    
156  \subsection{Global atmosphere: `Held-Suarez' benchmark}  \subsection{Global atmosphere: `Held-Suarez' benchmark}
157    \begin{rawhtml}
158    <!-- CMIREDIR:atmospheric_example: -->
159    \end{rawhtml}
160    
161    
162    
163  A novel feature of MITgcm is its ability to simulate, using one basic algorithm,  A novel feature of MITgcm is its ability to simulate, using one basic algorithm,
164  both atmospheric and oceanographic flows at both small and large scales.  both atmospheric and oceanographic flows at both small and large scales.
165    
166  Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$  Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
167  temperature field obtained using the atmospheric isomorph of MITgcm run at  temperature field obtained using the atmospheric isomorph of MITgcm run at
168  2.8$^{\circ }$ 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
169  (blue) and warm air along an equatorial band (red). Fully developed  (blue) and warm air along an equatorial band (red). Fully developed
170  baroclinic eddies spawned in the northern hemisphere storm track are  baroclinic eddies spawned in the northern hemisphere storm track are
171  evident. There are no mountains or land-sea contrast in this calculation,  evident. There are no mountains or land-sea contrast in this calculation,
# Line 126  there are no mountains or land-sea contr Line 179  there are no mountains or land-sea contr
179  %% CNHend  %% CNHend
180    
181  As described in Adcroft (2001), a `cubed sphere' is used to discretize the  As described in Adcroft (2001), a `cubed sphere' is used to discretize the
182  globe permitting a uniform gridding and obviated the need to Fourier filter.  globe permitting a uniform griding and obviated the need to Fourier filter.
183  The `vector-invariant' form of MITgcm supports any orthogonal curvilinear  The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
184  grid, of which the cubed sphere is just one of many choices.  grid, of which the cubed sphere is just one of many choices.
185    
# Line 142  latitude-longitude grid. Both grids are Line 195  latitude-longitude grid. Both grids are
195  %% CNHend  %% CNHend
196    
197  \subsection{Ocean gyres}  \subsection{Ocean gyres}
198    \begin{rawhtml}
199    <!-- CMIREDIR:oceanic_example: -->
200    \end{rawhtml}
201    \begin{rawhtml}
202    <!-- CMIREDIR:ocean_gyres: -->
203    \end{rawhtml}
204    
205  Baroclinic instability is a ubiquitous process in the ocean, as well as the  Baroclinic instability is a ubiquitous process in the ocean, as well as the
206  atmosphere. Ocean eddies play an important role in modifying the  atmosphere. Ocean eddies play an important role in modifying the
# Line 151  diffusive patterns of ocean currents. Bu Line 210  diffusive patterns of ocean currents. Bu
210  increased until the baroclinic instability process is resolved, numerical  increased until the baroclinic instability process is resolved, numerical
211  solutions of a different and much more realistic kind, can be obtained.  solutions of a different and much more realistic kind, can be obtained.
212    
213  Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity  Figure \ref{fig:ocean-gyres} shows the surface temperature and
214  field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal  velocity field obtained from MITgcm run at $\frac{1}{6}^{\circ }$
215  resolution on a $lat-lon$  horizontal resolution on a \textit{lat-lon} grid in which the pole has
216  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
217  (to avoid the converging of meridian in northern latitudes). 21 vertical  converging of meridian in northern latitudes). 21 vertical levels are
218  levels are used in the vertical with a `lopped cell' representation of  used in the vertical with a `lopped cell' representation of
219  topography. The development and propagation of anomalously warm and cold  topography. The development and propagation of anomalously warm and
220  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
221  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
222  visible.  is also clearly visible.
223    
224  %% CNHbegin  %% CNHbegin
225  \input{part1/ocean_gyres_figure}  \input{part1/atl6_figure}
226  %% CNHend  %% CNHend
227    
228    
229  \subsection{Global ocean circulation}  \subsection{Global ocean circulation}
230    \begin{rawhtml}
231  Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at  <!-- CMIREDIR:global_ocean_circulation: -->
232  the surface of a 4$^{\circ }$  \end{rawhtml}
233  global ocean model run with 15 vertical levels. Lopped cells are used to  
234  represent topography on a regular $lat-lon$ grid extending from 70$^{\circ  Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean
235  }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with  currents at the surface of a $4^{\circ }$ global ocean model run with
236  mixed boundary conditions on temperature and salinity at the surface. The  15 vertical levels. Lopped cells are used to represent topography on a
237  transfer properties of ocean eddies, convection and mixing is parameterized  regular \textit{lat-lon} grid extending from $70^{\circ }N$ to
238  in this model.  $70^{\circ }S$. The model is driven using monthly-mean winds with
239    mixed boundary conditions on temperature and salinity at the surface.
240    The transfer properties of ocean eddies, convection and mixing is
241    parameterized in this model.
242    
243  Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning  Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
244  circulation of the global ocean in Sverdrups.  circulation of the global ocean in Sverdrups.
# Line 186  circulation of the global ocean in Sverd Line 248  circulation of the global ocean in Sverd
248  %%CNHend  %%CNHend
249    
250  \subsection{Convection and mixing over topography}  \subsection{Convection and mixing over topography}
251    \begin{rawhtml}
252    <!-- CMIREDIR:mixing_over_topography: -->
253    \end{rawhtml}
254    
255    
256  Dense plumes generated by localized cooling on the continental shelf of the  Dense plumes generated by localized cooling on the continental shelf of the
257  ocean may be influenced by rotation when the deformation radius is smaller  ocean may be influenced by rotation when the deformation radius is smaller
258  than the width of the cooling region. Rather than gravity plumes, the  than the width of the cooling region. Rather than gravity plumes, the
259  mechanism for moving dense fluid down the shelf is then through geostrophic  mechanism for moving dense fluid down the shelf is then through geostrophic
260  eddies. The simulation shown in the figure \ref{fig::convect-and-topo}  eddies. The simulation shown in the figure \ref{fig:convect-and-topo}
261  (blue is cold dense fluid, red is  (blue is cold dense fluid, red is
262  warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to  warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
263  trigger convection by surface cooling. The cold, dense water falls down the  trigger convection by surface cooling. The cold, dense water falls down the
# Line 205  instability of the along-slope current. Line 271  instability of the along-slope current.
271  %%CNHend  %%CNHend
272    
273  \subsection{Boundary forced internal waves}  \subsection{Boundary forced internal waves}
274    \begin{rawhtml}
275    <!-- CMIREDIR:boundary_forced_internal_waves: -->
276    \end{rawhtml}
277    
278  The unique ability of MITgcm to treat non-hydrostatic dynamics in the  The unique ability of MITgcm to treat non-hydrostatic dynamics in the
279  presence of complex geometry makes it an ideal tool to study internal wave  presence of complex geometry makes it an ideal tool to study internal wave
# Line 224  nonhydrostatic dynamics. Line 293  nonhydrostatic dynamics.
293  %%CNHend  %%CNHend
294    
295  \subsection{Parameter sensitivity using the adjoint of MITgcm}  \subsection{Parameter sensitivity using the adjoint of MITgcm}
296    \begin{rawhtml}
297    <!-- CMIREDIR:parameter_sensitivity: -->
298    \end{rawhtml}
299    
300  Forward and tangent linear counterparts of MITgcm are supported using an  Forward and tangent linear counterparts of MITgcm are supported using an
301  `automatic adjoint compiler'. These can be used in parameter sensitivity and  `automatic adjoint compiler'. These can be used in parameter sensitivity and
302  data assimilation studies.  data assimilation studies.
303    
304  As one example of application of the MITgcm adjoint, Figure \ref{fig:hf-sensitivity}  As one example of application of the MITgcm adjoint, Figure
305  maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude  \ref{fig:hf-sensitivity} maps the gradient $\frac{\partial J}{\partial
306  of the overturning streamfunction shown in figure \ref{fig:large-scale-circ}    \mathcal{H}}$where $J$ is the magnitude of the overturning
307  at 60$^{\circ }$N and $  stream-function shown in figure \ref{fig:large-scale-circ} at
308  \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
309  a 100 year period. We see that $J$ is  air-sea heat flux over a 100 year period. We see that $J$ is sensitive
310  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
311  of deep water for the thermohaline circulations. This calculation also  deep water for the thermohaline circulations. This calculation also
312  yields sensitivities to all other model parameters.  yields sensitivities to all other model parameters.
313    
314  %%CNHbegin  %%CNHbegin
# Line 244  yields sensitivities to all other model Line 316  yields sensitivities to all other model
316  %%CNHend  %%CNHend
317    
318  \subsection{Global state estimation of the ocean}  \subsection{Global state estimation of the ocean}
319    \begin{rawhtml}
320    <!-- CMIREDIR:global_state_estimation: -->
321    \end{rawhtml}
322    
323    
324  An important application of MITgcm is in state estimation of the global  An important application of MITgcm is in state estimation of the global
325  ocean circulation. An appropriately defined `cost function', which measures  ocean circulation. An appropriately defined `cost function', which measures
326  the departure of the model from observations (both remotely sensed and  the departure of the model from observations (both remotely sensed and
327  insitu) over an interval of time, is minimized by adjusting `control  in-situ) over an interval of time, is minimized by adjusting `control
328  parameters' such as air-sea fluxes, the wind field, the initial conditions  parameters' such as air-sea fluxes, the wind field, the initial conditions
329  etc. Figure \ref{fig:assimilated-globes} shows an estimate of the time-mean  etc. Figure \ref{fig:assimilated-globes} shows the large scale planetary
330  surface elevation of the ocean obtained by bringing the model in to  circulation and a Hopf-Muller plot of Equatorial sea-surface height.
331    Both are obtained from assimilation bringing the model in to
332  consistency with altimetric and in-situ observations over the period  consistency with altimetric and in-situ observations over the period
333  1992-1997. {\bf CHANGE THIS TEXT - FIG FROM PATRICK/CARL/DETLEF}  1992-1997.
334    
335  %% CNHbegin  %% CNHbegin
336  \input{part1/globes_figure}  \input{part1/assim_figure}
337  %% CNHend  %% CNHend
338    
339  \subsection{Ocean biogeochemical cycles}  \subsection{Ocean biogeochemical cycles}
340    \begin{rawhtml}
341  MITgcm is being used to study global biogeochemical cycles in the ocean. For  <!-- CMIREDIR:ocean_biogeo_cycles: -->
342  example one can study the effects of interannual changes in meteorological  \end{rawhtml}
343  forcing and upper ocean circulation on the fluxes of carbon dioxide and  
344  oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows  MITgcm is being used to study global biogeochemical cycles in the
345  the annual air-sea flux of oxygen and its relation to density outcrops in  ocean. For example one can study the effects of interannual changes in
346  the southern oceans from a single year of a global, interannually varying  meteorological forcing and upper ocean circulation on the fluxes of
347  simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution  carbon dioxide and oxygen between the ocean and atmosphere. Figure
348  telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).  \ref{fig:biogeo} shows the annual air-sea flux of oxygen and its
349    relation to density outcrops in the southern oceans from a single year
350    of a global, interannually varying simulation. The simulation is run
351    at $1^{\circ}\times1^{\circ}$ resolution telescoping to
352    $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not
353    shown).
354    
355  %%CNHbegin  %%CNHbegin
356  \input{part1/biogeo_figure}  \input{part1/biogeo_figure}
357  %%CNHend  %%CNHend
358    
359  \subsection{Simulations of laboratory experiments}  \subsection{Simulations of laboratory experiments}
360    \begin{rawhtml}
361    <!-- CMIREDIR:classroom_exp: -->
362    \end{rawhtml}
363    
364  Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a  Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
365  laboratory experiment 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
366  initially homogeneous tank of water ($1m$ in diameter) is driven from its  initially homogeneous tank of water ($1m$ in diameter) is driven from its
367  free surface by a rotating heated disk. The combined action of mechanical  free surface by a rotating heated disk. The combined action of mechanical
368  and thermal forcing creates a lens of fluid which becomes baroclinically  and thermal forcing creates a lens of fluid which becomes baroclinically
# Line 293  stratification of the ACC. Line 378  stratification of the ACC.
378  % $Name$  % $Name$
379    
380  \section{Continuous equations in `r' coordinates}  \section{Continuous equations in `r' coordinates}
381    \begin{rawhtml}
382    <!-- CMIREDIR:z-p_isomorphism: -->
383    \end{rawhtml}
384    
385  To render atmosphere and ocean models from one dynamical core we exploit  To render atmosphere and ocean models from one dynamical core we exploit
386  `isomorphisms' between equation sets that govern the evolution of the  `isomorphisms' between equation sets that govern the evolution of the
# Line 301  One system of hydrodynamical equations i Line 389  One system of hydrodynamical equations i
389  and encoded. The model variables have different interpretations depending on  and encoded. The model variables have different interpretations depending on
390  whether the atmosphere or ocean is being studied. Thus, for example, the  whether the atmosphere or ocean is being studied. Thus, for example, the
391  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
392  modeling the atmosphere (left hand side of figure \ref{fig:isomorphic-equations})  modeling the atmosphere (right hand side of figure \ref{fig:isomorphic-equations})
393  and height, $z$, if we are modeling the ocean (right hand side of figure  and height, $z$, if we are modeling the ocean (left hand side of figure
394  \ref{fig:isomorphic-equations}).  \ref{fig:isomorphic-equations}).
395    
396  %%CNHbegin  %%CNHbegin
# Line 323  see figure \ref{fig:zandp-vert-coord}. Line 411  see figure \ref{fig:zandp-vert-coord}.
411  \input{part1/vertcoord_figure.tex}  \input{part1/vertcoord_figure.tex}
412  %%CNHend  %%CNHend
413    
414  \begin{equation*}  \begin{equation}
415  \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}  \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}
416  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
417  \text{ horizontal mtm} \label{eq:horizontal_mtm}  \text{ horizontal mtm} \label{eq:horizontal_mtm}
418  \end{equation*}  \end{equation}
419    
420  \begin{equation}  \begin{equation}
421  \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{  \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
# Line 351  b=b(\theta ,S,r)\text{ equation of state Line 439  b=b(\theta ,S,r)\text{ equation of state
439    
440  \begin{equation}  \begin{equation}
441  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
442  \label{eq:humidtity_salt}  \label{eq:humidity_salt}
443  \end{equation}  \end{equation}
444    
445  Here:  Here:
# Line 426  in later chapters. Line 514  in later chapters.
514  at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):  at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
515    
516  \begin{equation}  \begin{equation}
517  \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)}
518  \label{eq:fixedbc}  \label{eq:fixedbc}
519  \end{equation}  \end{equation}
520    
521  \begin{equation}  \begin{equation}
522  \dot{r}=\frac{Dr}{Dt}atr=R_{moving}\text{ \  \dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
523  (oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}  (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
524  \end{equation}  \end{equation}
525    
526  Here  Here
# Line 525  The boundary conditions at top and botto Line 613  The boundary conditions at top and botto
613  atmosphere)}  \label{eq:moving-bc-atmos}  atmosphere)}  \label{eq:moving-bc-atmos}
614  \end{eqnarray}  \end{eqnarray}
615    
616  Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_slainty})  Then the (hydrostatic form of) equations
617  yields a consistent set of atmospheric equations which, for convenience, are written out in $p$  (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yields a consistent
618  coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).  set of atmospheric equations which, for convenience, are written out
619    in $p$ coordinates in Appendix Atmosphere - see
620    eqs(\ref{eq:atmos-prime}).
621    
622  \subsection{Ocean}  \subsection{Ocean}
623    
# Line 562  w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo Line 652  w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo
652  \end{eqnarray}  \end{eqnarray}
653  where $\eta $ is the elevation of the free surface.  where $\eta $ is the elevation of the free surface.
654    
655  Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_slainty}) yield a consistent set  Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yield a consistent set
656  of oceanic equations  of oceanic equations
657  which, for convenience, are written out in $z$ coordinates in Appendix Ocean  which, for convenience, are written out in $z$ coordinates in Appendix Ocean
658  - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).  - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
659    
660  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
661  Non-hydrostatic forms}  Non-hydrostatic forms}
662    \begin{rawhtml}
663    <!-- CMIREDIR:non_hydrostatic: -->
664    \end{rawhtml}
665    
666    
667  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
668    
# Line 576  Let us separate $\phi $ in to surface, h Line 670  Let us separate $\phi $ in to surface, h
670  \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)  \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
671  \label{eq:phi-split}  \label{eq:phi-split}
672  \end{equation}  \end{equation}
673  and write eq(\ref{eq:incompressible}) in the form:  %and write eq(\ref{eq:incompressible}) in the form:
674    %                  ^- this eq is missing (jmc) ; replaced with:
675    and write eq( \ref{eq:horizontal_mtm}) in the form:
676    
677  \begin{equation}  \begin{equation}
678  \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi  \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
# Line 676  OPERATORS. Line 772  OPERATORS.
772    
773  \subsubsection{Shallow atmosphere approximation}  \subsubsection{Shallow atmosphere approximation}
774    
775  Most models are based on the `hydrostatic primitive equations' (HPE's) in  Most models are based on the `hydrostatic primitive equations' (HPE's)
776  which the vertical momentum equation is reduced to a statement of  in which the vertical momentum equation is reduced to a statement of
777  hydrostatic balance and the `traditional approximation' is made in which the  hydrostatic balance and the `traditional approximation' is made in
778  Coriolis force is treated approximately and the shallow atmosphere  which the Coriolis force is treated approximately and the shallow
779  approximation is made.\ The MITgcm need not make the `traditional  atmosphere approximation is made.  MITgcm need not make the
780  approximation'. To be able to support consistent non-hydrostatic forms the  `traditional approximation'. To be able to support consistent
781  shallow atmosphere approximation can be relaxed - when dividing through by $  non-hydrostatic forms the shallow atmosphere approximation can be
782  r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,  relaxed - when dividing through by $ r $ in, for example,
783  the radius of the earth.  (\ref{eq:gu-speherical}), we do not replace $r$ by $a$, the radius of
784    the earth.
785    
786  \subsubsection{Hydrostatic and quasi-hydrostatic forms}  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
787  \label{sec:hydrostatic_and_quasi-hydrostatic_forms}  \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
# Line 721  et.al., 1997a. As in \textbf{HPE }only a Line 818  et.al., 1997a. As in \textbf{HPE }only a
818    
819  \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}  \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}
820    
821  The MIT model presently supports a full non-hydrostatic ocean isomorph, but  MITgcm presently supports a full non-hydrostatic ocean isomorph, but
822  only a quasi-non-hydrostatic atmospheric isomorph.  only a quasi-non-hydrostatic atmospheric isomorph.
823    
824  \paragraph{Non-hydrostatic Ocean}  \paragraph{Non-hydrostatic Ocean}
# Line 1006  salinity ... ). Line 1103  salinity ... ).
1103    
1104  \subsection{Vector invariant form}  \subsection{Vector invariant form}
1105    
1106  For some purposes it is advantageous to write momentum advection in eq(\ref  For some purposes it is advantageous to write momentum advection in
1107  {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the (so-called) `vector invariant' form:  eq(\ref {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the
1108    (so-called) `vector invariant' form:
1109    
1110  \begin{equation}  \begin{equation}
1111  \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}
# Line 1118  In $p$-coordinates, the upper boundary a Line 1216  In $p$-coordinates, the upper boundary a
1216  surface ($\phi $ is imposed and $\omega \neq 0$).  surface ($\phi $ is imposed and $\omega \neq 0$).
1217    
1218  \subsubsection{Splitting the geo-potential}  \subsubsection{Splitting the geo-potential}
1219    \label{sec:hpe-p-geo-potential-split}
1220    
1221  For the purposes of initialization and reducing round-off errors, the model  For the purposes of initialization and reducing round-off errors, the model
1222  deals with perturbations from reference (or ``standard'') profiles. For  deals with perturbations from reference (or ``standard'') profiles. For
# Line 1147  _{o}(p_{o})=g~Z_{topo}$, defined: Line 1246  _{o}(p_{o})=g~Z_{topo}$, defined:
1246  The final form of the HPE's in p coordinates is then:  The final form of the HPE's in p coordinates is then:
1247  \begin{eqnarray}  \begin{eqnarray}
1248  \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}}
1249  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-prime} \\  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}}
1250    \label{eq:atmos-prime} \\
1251  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
1252  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1253  \partial p} &=&0 \\  \partial p} &=&0 \\
# Line 1178  _{h}+\frac{\partial w}{\partial z} &=&0 Line 1278  _{h}+\frac{\partial w}{\partial z} &=&0
1278  \label{eq:non-boussinesq}  \label{eq:non-boussinesq}
1279  \end{eqnarray}  \end{eqnarray}
1280  These equations permit acoustics modes, inertia-gravity waves,  These equations permit acoustics modes, inertia-gravity waves,
1281  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
1282  mode. As written, they cannot be integrated forward consistently - if we  mode. As written, they cannot be integrated forward consistently - if we
1283  step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be  step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
1284  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
# Line 1193  _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\ Line 1293  _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
1293  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}
1294  \end{equation}  \end{equation}
1295    
1296  Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the  Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is
1297  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref{eq-zns-cont} gives:  the reciprocal of the sound speed ($c_{s}$) squared. Substituting into
1298    \ref{eq-zns-cont} gives:
1299  \begin{equation}  \begin{equation}
1300  \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{
1301  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}

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