/[MITgcm]/manual/s_overview/text/manual.tex
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revision 1.19 by afe, Tue Mar 23 16:47:04 2004 UTC revision 1.24 by edhill, Wed Apr 5 02:27:32 2006 UTC
# Line 88  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  \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,marshall:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04}:
92    
93  \begin{verbatim}  \begin{verbatim}
94  Hill, C. and J. Marshall, (1995)  Hill, C. and J. Marshall, (1995)
# Line 142  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 409  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} \label{eq: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{
# Line 611  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) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt})  Then the (hydrostatic form of) equations
615  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
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 666  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{eq:incompressible}) 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 766  OPERATORS. Line 770  OPERATORS.
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 811  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 1096  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  {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
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 1208  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 1237  _{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}}} \label{eq:atmos-prime} \\  _{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 \\
# Line 1283  _{\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{eq-zns-cont} gives:  the reciprocal of the sound speed ($c_{s}$) squared. Substituting into
1296    \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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