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revision 1.7 by cnh, Thu Oct 25 12:06:56 2001 UTC revision 1.24 by edhill, Wed Apr 5 02:27:32 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.
# 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 163  warm water northward by the mean flow of Line 222  warm water northward by the mean flow of
222  visible.  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    <!-- CMIREDIR:global_ocean_circulation: -->
232    \end{rawhtml}
233    
234  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 currents at
235  the surface of a 4$^{\circ }$  the surface of a 4$^{\circ }$
# 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
# Line 231  data assimilation studies. Line 303  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 \ref{fig:hf-sensitivity}
305  maps the 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
306  of the overturning streamfunction shown in figure \ref{fig:large-scale-circ}  of the overturning stream-function shown in figure \ref{fig:large-scale-circ}
307  at 60$^{\circ }$N and $  at 60$^{\circ }$N and $
308  \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over  \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
309  a 100 year period. We see that $J$ is  a 100 year period. We see that $J$ is
# 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    <!-- CMIREDIR:ocean_biogeo_cycles: -->
342    \end{rawhtml}
343    
344  MITgcm is being used to study global biogeochemical cycles in the ocean. For  MITgcm is being used to study global biogeochemical cycles in the ocean. For
345  example one can study the effects of interannual changes in meteorological  example one can study the effects of interannual changes in meteorological
# Line 275  telescoping to $\frac{1}{3}^{\circ}\time Line 355  telescoping to $\frac{1}{3}^{\circ}\time
355  %%CNHend  %%CNHend
356    
357  \subsection{Simulations of laboratory experiments}  \subsection{Simulations of laboratory experiments}
358    \begin{rawhtml}
359    <!-- CMIREDIR:classroom_exp: -->
360    \end{rawhtml}
361    
362  Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a  Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
363  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
364  initially homogeneous tank of water ($1m$ in diameter) is driven from its  initially homogeneous tank of water ($1m$ in diameter) is driven from its
365  free surface by a rotating heated disk. The combined action of mechanical  free surface by a rotating heated disk. The combined action of mechanical
366  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 376  stratification of the ACC.
376  % $Name$  % $Name$
377    
378  \section{Continuous equations in `r' coordinates}  \section{Continuous equations in `r' coordinates}
379    \begin{rawhtml}
380    <!-- CMIREDIR:z-p_isomorphism: -->
381    \end{rawhtml}
382    
383  To render atmosphere and ocean models from one dynamical core we exploit  To render atmosphere and ocean models from one dynamical core we exploit
384  `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 387  One system of hydrodynamical equations i
387  and encoded. The model variables have different interpretations depending on  and encoded. The model variables have different interpretations depending on
388  whether the atmosphere or ocean is being studied. Thus, for example, the  whether the atmosphere or ocean is being studied. Thus, for example, the
389  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
390  modeling the atmosphere (left hand side of figure \ref{fig:isomorphic-equations})  modeling the atmosphere (right hand side of figure \ref{fig:isomorphic-equations})
391  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
392  \ref{fig:isomorphic-equations}).  \ref{fig:isomorphic-equations}).
393    
394  %%CNHbegin  %%CNHbegin
# Line 323  see figure \ref{fig:zandp-vert-coord}. Line 409  see figure \ref{fig:zandp-vert-coord}.
409  \input{part1/vertcoord_figure.tex}  \input{part1/vertcoord_figure.tex}
410  %%CNHend  %%CNHend
411    
412  \begin{equation*}  \begin{equation}
413  \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}}
414  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
415  \text{ horizontal mtm}  \text{ horizontal mtm} \label{eq:horizontal_mtm}
416  \end{equation*}  \end{equation}
417    
418  \begin{equation*}  \begin{equation}
419  \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{
420  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
421  vertical mtm}  vertical mtm} \label{eq:vertical_mtm}
422  \end{equation*}  \end{equation}
423    
424  \begin{equation}  \begin{equation}
425  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
426  \partial r}=0\text{ continuity}  \label{eq:continuous}  \partial r}=0\text{ continuity}  \label{eq:continuity}
427  \end{equation}  \end{equation}
428    
429  \begin{equation*}  \begin{equation}
430  b=b(\theta ,S,r)\text{ equation of state}  b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
431  \end{equation*}  \end{equation}
432    
433  \begin{equation*}  \begin{equation}
434  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
435  \end{equation*}  \label{eq:potential_temperature}
436    \end{equation}
437    
438  \begin{equation*}  \begin{equation}
439  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
440  \end{equation*}  \label{eq:humidity_salt}
441    \end{equation}
442    
443  Here:  Here:
444    
# Line 424  in later chapters. Line 512  in later chapters.
512  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}):
513    
514  \begin{equation}  \begin{equation}
515  \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)}
516  \label{eq:fixedbc}  \label{eq:fixedbc}
517  \end{equation}  \end{equation}
518    
519  \begin{equation}  \begin{equation}
520  \dot{r}=\frac{Dr}{Dt}atr=R_{moving}\text{ \  \dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
521  (oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}  (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
522  \end{equation}  \end{equation}
523    
524  Here  Here
# Line 523  The boundary conditions at top and botto Line 611  The boundary conditions at top and botto
611  atmosphere)}  \label{eq:moving-bc-atmos}  atmosphere)}  \label{eq:moving-bc-atmos}
612  \end{eqnarray}  \end{eqnarray}
613    
614  Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent  Then the (hydrostatic form of) equations
615  set of atmospheric equations which, for convenience, are written out in $p$  (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yields a consistent
616  coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).  set of atmospheric equations which, for convenience, are written out
617    in $p$ coordinates in Appendix Atmosphere - see
618    eqs(\ref{eq:atmos-prime}).
619    
620  \subsection{Ocean}  \subsection{Ocean}
621    
# Line 560  w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo Line 650  w &=&\frac{D\eta }{Dt}\text{ at }r=R_{mo
650  \end{eqnarray}  \end{eqnarray}
651  where $\eta $ is the elevation of the free surface.  where $\eta $ is the elevation of the free surface.
652    
653  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
654    of oceanic equations
655  which, for convenience, are written out in $z$ coordinates in Appendix Ocean  which, for convenience, are written out in $z$ coordinates in Appendix Ocean
656  - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).  - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
657    
658  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and  \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
659  Non-hydrostatic forms}  Non-hydrostatic forms}
660    \begin{rawhtml}
661    <!-- CMIREDIR:non_hydrostatic: -->
662    \end{rawhtml}
663    
664    
665  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:  Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
666    
# Line 573  Let us separate $\phi $ in to surface, h Line 668  Let us separate $\phi $ in to surface, h
668  \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)
669  \label{eq:phi-split}  \label{eq:phi-split}
670  \end{equation}  \end{equation}
671  and write eq(\ref{incompressible}a,b) in the form:  %and write eq(\ref{eq:incompressible}) in the form:
672    %                  ^- this eq is missing (jmc) ; replaced with:
673    and write eq( \ref{eq:horizontal_mtm}) in the form:
674    
675  \begin{equation}  \begin{equation}
676  \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 666  In the above `${r}$' is the distance fro Line 763  In the above `${r}$' is the distance fro
763    
764  Grad and div operators in spherical coordinates are defined in appendix  Grad and div operators in spherical coordinates are defined in appendix
765  OPERATORS.  OPERATORS.
 \marginpar{  
 Fig.6 Spherical polar coordinate system.}  
766    
767  %%CNHbegin  %%CNHbegin
768  \input{part1/sphere_coord_figure.tex}  \input{part1/sphere_coord_figure.tex}
# Line 675  Fig.6 Spherical polar coordinate system. Line 770  Fig.6 Spherical polar coordinate system.
770    
771  \subsubsection{Shallow atmosphere approximation}  \subsubsection{Shallow atmosphere approximation}
772    
773  Most models are based on the `hydrostatic primitive equations' (HPE's) in  Most models are based on the `hydrostatic primitive equations' (HPE's)
774  which the vertical momentum equation is reduced to a statement of  in which the vertical momentum equation is reduced to a statement of
775  hydrostatic balance and the `traditional approximation' is made in which the  hydrostatic balance and the `traditional approximation' is made in
776  Coriolis force is treated approximately and the shallow atmosphere  which the Coriolis force is treated approximately and the shallow
777  approximation is made.\ The MITgcm need not make the `traditional  atmosphere approximation is made.  MITgcm need not make the
778  approximation'. To be able to support consistent non-hydrostatic forms the  `traditional approximation'. To be able to support consistent
779  shallow atmosphere approximation can be relaxed - when dividing through by $  non-hydrostatic forms the shallow atmosphere approximation can be
780  r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,  relaxed - when dividing through by $ r $ in, for example,
781  the radius of the earth.  (\ref{eq:gu-speherical}), we do not replace $r$ by $a$, the radius of
782    the earth.
783    
784  \subsubsection{Hydrostatic and quasi-hydrostatic forms}  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
785  \label{sec:hydrostatic_and_quasi-hydrostatic_forms}  \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
# Line 720  et.al., 1997a. As in \textbf{HPE }only a Line 816  et.al., 1997a. As in \textbf{HPE }only a
816    
817  \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}  \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}
818    
819  The MIT model presently supports a full non-hydrostatic ocean isomorph, but  MITgcm presently supports a full non-hydrostatic ocean isomorph, but
820  only a quasi-non-hydrostatic atmospheric isomorph.  only a quasi-non-hydrostatic atmospheric isomorph.
821    
822  \paragraph{Non-hydrostatic Ocean}  \paragraph{Non-hydrostatic Ocean}
# Line 730  In the non-hydrostatic ocean model all t Line 826  In the non-hydrostatic ocean model all t
826  three dimensional elliptic equation must be solved subject to Neumann  three dimensional elliptic equation must be solved subject to Neumann
827  boundary conditions (see below). It is important to note that use of the  boundary conditions (see below). It is important to note that use of the
828  full \textbf{NH} does not admit any new `fast' waves in to the system - the  full \textbf{NH} does not admit any new `fast' waves in to the system - the
829  incompressible condition eq(\ref{eq:continuous})c has already filtered out  incompressible condition eq(\ref{eq:continuity}) has already filtered out
830  acoustic modes. It does, however, ensure that the gravity waves are treated  acoustic modes. It does, however, ensure that the gravity waves are treated
831  accurately with an exact dispersion relation. The \textbf{NH} set has a  accurately with an exact dispersion relation. The \textbf{NH} set has a
832  complete angular momentum principle and consistent energetics - see White  complete angular momentum principle and consistent energetics - see White
# Line 779  coordinates are supported - see eqs(\ref Line 875  coordinates are supported - see eqs(\ref
875  \subsection{Solution strategy}  \subsection{Solution strategy}
876    
877  The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{  The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
878  NH} models is summarized in Fig.7.  NH} models is summarized in Figure \ref{fig:solution-strategy}.
879  \marginpar{  Under all dynamics, a 2-d elliptic equation is
 Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is  
880  first solved to find the surface pressure and the hydrostatic pressure at  first solved to find the surface pressure and the hydrostatic pressure at
881  any level computed from the weight of fluid above. Under \textbf{HPE} and  any level computed from the weight of fluid above. Under \textbf{HPE} and
882  \textbf{QH} dynamics, the horizontal momentum equations are then stepped  \textbf{QH} dynamics, the horizontal momentum equations are then stepped
# Line 838  atmospheric pressure pushing down on the Line 933  atmospheric pressure pushing down on the
933    
934  \subsubsection{Surface pressure}  \subsubsection{Surface pressure}
935    
936  The surface pressure equation can be obtained by integrating continuity, (  The surface pressure equation can be obtained by integrating continuity,
937  \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$  (\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
938    
939  \begin{equation*}  \begin{equation*}
940  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
# Line 864  r $. The above can be rearranged to yiel Line 959  r $. The above can be rearranged to yiel
959  where we have incorporated a source term.  where we have incorporated a source term.
960    
961  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
962  (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can  (atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
963  be written  be written
964  \begin{equation}  \begin{equation}
965  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
# Line 872  be written Line 967  be written
967  \end{equation}  \end{equation}
968  where $b_{s}$ is the buoyancy at the surface.  where $b_{s}$ is the buoyancy at the surface.
969    
970  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
971  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
972  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
973  surface' and `rigid lid' approaches are available.  surface' and `rigid lid' approaches are available.
974    
975  \subsubsection{Non-hydrostatic pressure}  \subsubsection{Non-hydrostatic pressure}
976    
977  Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{  Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
978  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation  $\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
979  (\ref{incompressible}), we deduce that:  (\ref{eq:continuity}), we deduce that:
980    
981  \begin{equation}  \begin{equation}
982  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
# Line 911  tangential component of velocity, $v_{T} Line 1006  tangential component of velocity, $v_{T}
1006  depending on the form chosen for the dissipative terms in the momentum  depending on the form chosen for the dissipative terms in the momentum
1007  equations - see below.  equations - see below.
1008    
1009  Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that:  Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
1010    
1011  \begin{equation}  \begin{equation}
1012  \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}  \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
# Line 951  If the flow is `close' to hydrostatic ba Line 1046  If the flow is `close' to hydrostatic ba
1046  converges rapidly because $\phi _{nh}\ $is then only a small correction to  converges rapidly because $\phi _{nh}\ $is then only a small correction to
1047  the hydrostatic pressure field (see the discussion in Marshall et al, a,b).  the hydrostatic pressure field (see the discussion in Marshall et al, a,b).
1048    
1049  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})
1050  does not vanish at $r=R_{moving}$, and so refines the pressure there.  does not vanish at $r=R_{moving}$, and so refines the pressure there.
1051    
1052  \subsection{Forcing/dissipation}  \subsection{Forcing/dissipation}
# Line 959  does not vanish at $r=R_{moving}$, and s Line 1054  does not vanish at $r=R_{moving}$, and s
1054  \subsubsection{Forcing}  \subsubsection{Forcing}
1055    
1056  The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by  The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by
1057  `physics packages' described in detail in chapter ??.  `physics packages' and forcing packages. These are described later on.
1058    
1059  \subsubsection{Dissipation}  \subsubsection{Dissipation}
1060    
# Line 1006  salinity ... ). Line 1101  salinity ... ).
1101    
1102  \subsection{Vector invariant form}  \subsection{Vector invariant form}
1103    
1104  For some purposes it is advantageous to write momentum advection in eq(\ref  For some purposes it is advantageous to write momentum advection in
1105  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:  eq(\ref {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the
1106    (so-called) `vector invariant' form:
1107    
1108  \begin{equation}  \begin{equation}
1109  \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 1025  to discretize the model. Line 1121  to discretize the model.
1121    
1122  \subsection{Adjoint}  \subsection{Adjoint}
1123    
1124  Tangent linear and adjoint counterparts of the forward model and described  Tangent linear and adjoint counterparts of the forward model are described
1125  in Chapter 5.  in Chapter 5.
1126    
1127  % $Header$  % $Header$
# Line 1118  In $p$-coordinates, the upper boundary a Line 1214  In $p$-coordinates, the upper boundary a
1214  surface ($\phi $ is imposed and $\omega \neq 0$).  surface ($\phi $ is imposed and $\omega \neq 0$).
1215    
1216  \subsubsection{Splitting the geo-potential}  \subsubsection{Splitting the geo-potential}
1217    \label{sec:hpe-p-geo-potential-split}
1218    
1219  For the purposes of initialization and reducing round-off errors, the model  For the purposes of initialization and reducing round-off errors, the model
1220  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}$, defined: Line 1244  _{o}(p_{o})=g~Z_{topo}$, defined:
1244  The final form of the HPE's in p coordinates is then:  The final form of the HPE's in p coordinates is then:
1245  \begin{eqnarray}  \begin{eqnarray}
1246  \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}}
1247  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}}
1248    \label{eq:atmos-prime} \\
1249  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
1250  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1251  \partial p} &=&0 \\  \partial p} &=&0 \\
1252  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
1253  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }
1254  \end{eqnarray}  \end{eqnarray}
1255    
1256  % $Header$  % $Header$
# Line 1171  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z} Line 1269  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}
1269  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
1270  &=&\epsilon _{nh}\mathcal{F}_{w} \\  &=&\epsilon _{nh}\mathcal{F}_{w} \\
1271  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
1272  _{h}+\frac{\partial w}{\partial z} &=&0 \\  _{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
1273  \rho &=&\rho (\theta ,S,p) \\  \rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
1274  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
1275  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zns-salt}
1276    \label{eq:non-boussinesq}
1277  \end{eqnarray}  \end{eqnarray}
1278  These equations permit acoustics modes, inertia-gravity waves,  These equations permit acoustics modes, inertia-gravity waves,
1279  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
1280  mode. As written, they cannot be integrated forward consistently - if we  mode. As written, they cannot be integrated forward consistently - if we
1281  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
1282  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 1192  _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\ Line 1291  _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
1291  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}
1292  \end{equation}  \end{equation}
1293    
1294  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
1295  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref  the reciprocal of the sound speed ($c_{s}$) squared. Substituting into
1296  {eq-zns-cont} gives:  \ref{eq-zns-cont} gives:
1297  \begin{equation}  \begin{equation}
1298  \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{
1299  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
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