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revision 1.21 by edhill, Sat Oct 16 03:40:12 2004 UTC revision 1.27 by jmc, Thu Jan 17 21:28:22 2008 UTC
# Line 87  studies. Line 87  studies.
87  computational platforms.  computational platforms.
88  \end{itemize}  \end{itemize}
89    
90    
91  Key publications reporting on and charting the development of the model are  Key publications reporting on and charting the development of the model are
92  \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}
93    (an overview on the model formulation can also be found in \cite{adcroft:04c}):
94    
95  \begin{verbatim}  \begin{verbatim}
96  Hill, C. and J. Marshall, (1995)  Hill, C. and J. Marshall, (1995)
# Line 142  give a feel for the wide range of proble Line 144  give a feel for the wide range of proble
144    
145  \section{Illustrations of the model in action}  \section{Illustrations of the model in action}
146    
147  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,
148  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
149  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
150  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 165  both atmospheric and oceanographic flows Line 167  both atmospheric and oceanographic flows
167    
168  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$mb$
169  temperature field obtained using the atmospheric isomorph of MITgcm run at  temperature field obtained using the atmospheric isomorph of MITgcm run at
170  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
171  (blue) and warm air along an equatorial band (red). Fully developed  (blue) and warm air along an equatorial band (red). Fully developed
172  baroclinic eddies spawned in the northern hemisphere storm track are  baroclinic eddies spawned in the northern hemisphere storm track are
173  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 210  diffusive patterns of ocean currents. Bu Line 212  diffusive patterns of ocean currents. Bu
212  increased until the baroclinic instability process is resolved, numerical  increased until the baroclinic instability process is resolved, numerical
213  solutions of a different and much more realistic kind, can be obtained.  solutions of a different and much more realistic kind, can be obtained.
214    
215  Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity  Figure \ref{fig:ocean-gyres} shows the surface temperature and
216  field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal  velocity field obtained from MITgcm run at $\frac{1}{6}^{\circ }$
217  resolution on a $lat-lon$  horizontal resolution on a \textit{lat-lon} grid in which the pole has
218  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
219  (to avoid the converging of meridian in northern latitudes). 21 vertical  converging of meridian in northern latitudes). 21 vertical levels are
220  levels are used in the vertical with a `lopped cell' representation of  used in the vertical with a `lopped cell' representation of
221  topography. The development and propagation of anomalously warm and cold  topography. The development and propagation of anomalously warm and
222  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
223  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
224  visible.  is also clearly visible.
225    
226  %% CNHbegin  %% CNHbegin
227  \input{part1/atl6_figure}  \input{part1/atl6_figure}
# Line 231  visible. Line 233  visible.
233  <!-- CMIREDIR:global_ocean_circulation: -->  <!-- CMIREDIR:global_ocean_circulation: -->
234  \end{rawhtml}  \end{rawhtml}
235    
236  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
237  the surface of a 4$^{\circ }$  currents at the surface of a $4^{\circ }$ global ocean model run with
238  global ocean model run with 15 vertical levels. Lopped cells are used to  15 vertical levels. Lopped cells are used to represent topography on a
239  represent topography on a regular $lat-lon$ grid extending from 70$^{\circ  regular \textit{lat-lon} grid extending from $70^{\circ }N$ to
240  }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
241  mixed boundary conditions on temperature and salinity at the surface. The  mixed boundary conditions on temperature and salinity at the surface.
242  transfer properties of ocean eddies, convection and mixing is parameterized  The transfer properties of ocean eddies, convection and mixing is
243  in this model.  parameterized in this model.
244    
245  Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning  Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
246  circulation of the global ocean in Sverdrups.  circulation of the global ocean in Sverdrups.
# Line 301  Forward and tangent linear counterparts Line 303  Forward and tangent linear counterparts
303  `automatic adjoint compiler'. These can be used in parameter sensitivity and  `automatic adjoint compiler'. These can be used in parameter sensitivity and
304  data assimilation studies.  data assimilation studies.
305    
306  As one example of application of the MITgcm adjoint, Figure \ref{fig:hf-sensitivity}  As one example of application of the MITgcm adjoint, Figure
307  maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude  \ref{fig:hf-sensitivity} maps the gradient $\frac{\partial J}{\partial
308  of the overturning stream-function shown in figure \ref{fig:large-scale-circ}    \mathcal{H}}$where $J$ is the magnitude of the overturning
309  at 60$^{\circ }$N and $  stream-function shown in figure \ref{fig:large-scale-circ} at
310  \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
311  a 100 year period. We see that $J$ is  air-sea heat flux over a 100 year period. We see that $J$ is sensitive
312  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
313  of deep water for the thermohaline circulations. This calculation also  deep water for the thermohaline circulations. This calculation also
314  yields sensitivities to all other model parameters.  yields sensitivities to all other model parameters.
315    
316  %%CNHbegin  %%CNHbegin
# Line 341  consistency with altimetric and in-situ Line 343  consistency with altimetric and in-situ
343  <!-- CMIREDIR:ocean_biogeo_cycles: -->  <!-- CMIREDIR:ocean_biogeo_cycles: -->
344  \end{rawhtml}  \end{rawhtml}
345    
346  MITgcm is being used to study global biogeochemical cycles in the ocean. For  MITgcm is being used to study global biogeochemical cycles in the
347  example one can study the effects of interannual changes in meteorological  ocean. For example one can study the effects of interannual changes in
348  forcing and upper ocean circulation on the fluxes of carbon dioxide and  meteorological forcing and upper ocean circulation on the fluxes of
349  oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows  carbon dioxide and oxygen between the ocean and atmosphere. Figure
350  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
351  the southern oceans from a single year of a global, interannually varying  relation to density outcrops in the southern oceans from a single year
352  simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution  of a global, interannually varying simulation. The simulation is run
353  telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).  at $1^{\circ}\times1^{\circ}$ resolution telescoping to
354    $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not
355    shown).
356    
357  %%CNHbegin  %%CNHbegin
358  \input{part1/biogeo_figure}  \input{part1/biogeo_figure}
# Line 770  OPERATORS. Line 774  OPERATORS.
774    
775  \subsubsection{Shallow atmosphere approximation}  \subsubsection{Shallow atmosphere approximation}
776    
777  Most models are based on the `hydrostatic primitive equations' (HPE's) in  Most models are based on the `hydrostatic primitive equations' (HPE's)
778  which the vertical momentum equation is reduced to a statement of  in which the vertical momentum equation is reduced to a statement of
779  hydrostatic balance and the `traditional approximation' is made in which the  hydrostatic balance and the `traditional approximation' is made in
780  Coriolis force is treated approximately and the shallow atmosphere  which the Coriolis force is treated approximately and the shallow
781  approximation is made.\ The MITgcm need not make the `traditional  atmosphere approximation is made.  MITgcm need not make the
782  approximation'. To be able to support consistent non-hydrostatic forms the  `traditional approximation'. To be able to support consistent
783  shallow atmosphere approximation can be relaxed - when dividing through by $  non-hydrostatic forms the shallow atmosphere approximation can be
784  r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,  relaxed - when dividing through by $ r $ in, for example,
785  the radius of the earth.  (\ref{eq:gu-speherical}), we do not replace $r$ by $a$, the radius of
786    the earth.
787    
788  \subsubsection{Hydrostatic and quasi-hydrostatic forms}  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
789  \label{sec:hydrostatic_and_quasi-hydrostatic_forms}  \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
# Line 815  et.al., 1997a. As in \textbf{HPE }only a Line 820  et.al., 1997a. As in \textbf{HPE }only a
820    
821  \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}  \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}
822    
823  The MIT model presently supports a full non-hydrostatic ocean isomorph, but  MITgcm presently supports a full non-hydrostatic ocean isomorph, but
824  only a quasi-non-hydrostatic atmospheric isomorph.  only a quasi-non-hydrostatic atmospheric isomorph.
825    
826  \paragraph{Non-hydrostatic Ocean}  \paragraph{Non-hydrostatic Ocean}
# Line 1074  friction. These coefficients are the sam Line 1079  friction. These coefficients are the sam
1079    
1080  The mixing terms for the temperature and salinity equations have a similar  The mixing terms for the temperature and salinity equations have a similar
1081  form to that of momentum except that the diffusion tensor can be  form to that of momentum except that the diffusion tensor can be
1082  non-diagonal and have varying coefficients. $\qquad $  non-diagonal and have varying coefficients.
1083  \begin{equation}  \begin{equation}
1084  D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla  D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
1085  _{h}^{4}(T,S)  \label{eq:diffusion}  _{h}^{4}(T,S)  \label{eq:diffusion}
# Line 1213  In $p$-coordinates, the upper boundary a Line 1218  In $p$-coordinates, the upper boundary a
1218  surface ($\phi $ is imposed and $\omega \neq 0$).  surface ($\phi $ is imposed and $\omega \neq 0$).
1219    
1220  \subsubsection{Splitting the geo-potential}  \subsubsection{Splitting the geo-potential}
1221    \label{sec:hpe-p-geo-potential-split}
1222    
1223  For the purposes of initialization and reducing round-off errors, the model  For the purposes of initialization and reducing round-off errors, the model
1224  deals with perturbations from reference (or ``standard'') profiles. For  deals with perturbations from reference (or ``standard'') profiles. For
# Line 1485  u=r\cos \varphi \frac{D\lambda }{Dt} Line 1491  u=r\cos \varphi \frac{D\lambda }{Dt}
1491  \end{equation*}  \end{equation*}
1492    
1493  \begin{equation*}  \begin{equation*}
1494  v=r\frac{D\varphi }{Dt}\qquad  v=r\frac{D\varphi }{Dt}
1495  \end{equation*}  \end{equation*}
 $\qquad \qquad \qquad \qquad $  
1496    
1497  \begin{equation*}  \begin{equation*}
1498  \dot{r}=\frac{Dr}{Dt}  \dot{r}=\frac{Dr}{Dt}
# Line 1497  Here $\varphi $ is the latitude, $\lambd Line 1502  Here $\varphi $ is the latitude, $\lambd
1502  distance of the particle from the center of the earth, $\Omega $ is the  distance of the particle from the center of the earth, $\Omega $ is the
1503  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.  angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
1504    
1505  The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in  The `grad' ($\nabla $) and `div' ($\nabla\cdot$) operators are defined by, in
1506  spherical coordinates:  spherical coordinates:
1507    
1508  \begin{equation*}  \begin{equation*}
# Line 1507  spherical coordinates: Line 1512  spherical coordinates:
1512  \end{equation*}  \end{equation*}
1513    
1514  \begin{equation*}  \begin{equation*}
1515  \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
1516  \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}  \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
1517  +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}  +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
1518  \end{equation*}  \end{equation*}
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