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revision 1.18 by afe, Tue Mar 23 15:29:39 2004 UTC revision 1.26 by edhill, Wed Jun 28 15:22:13 2006 UTC
# Line 49  also presented. Line 49  also presented.
49    
50  \section{Introduction}  \section{Introduction}
51  \begin{rawhtml}  \begin{rawhtml}
52  <!-- CMIREDIR:innovations -->  <!-- CMIREDIR:innovations: -->
53  \end{rawhtml}  \end{rawhtml}
54    
55    
# 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 155  described in detail in the documentation Line 155  described in detail in the documentation
155    
156  \subsection{Global atmosphere: `Held-Suarez' benchmark}  \subsection{Global atmosphere: `Held-Suarez' benchmark}
157  \begin{rawhtml}  \begin{rawhtml}
158  <!-- CMIREDIR:atmospheric_example -->  <!-- CMIREDIR:atmospheric_example: -->
159  \end{rawhtml}  \end{rawhtml}
160    
161    
# Line 165  both atmospheric and oceanographic flows Line 165  both atmospheric and oceanographic flows
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 196  latitude-longitude grid. Both grids are Line 196  latitude-longitude grid. Both grids are
196    
197  \subsection{Ocean gyres}  \subsection{Ocean gyres}
198  \begin{rawhtml}  \begin{rawhtml}
199  <!-- CMIREDIR:oceanic_example -->  <!-- CMIREDIR:oceanic_example: -->
200  \end{rawhtml}  \end{rawhtml}
201  \begin{rawhtml}  \begin{rawhtml}
202  <!-- CMIREDIR:ocean_gyres -->  <!-- CMIREDIR:ocean_gyres: -->
203  \end{rawhtml}  \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
# Line 210  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/atl6_figure}  \input{part1/atl6_figure}
# Line 228  visible. Line 228  visible.
228    
229  \subsection{Global ocean circulation}  \subsection{Global ocean circulation}
230  \begin{rawhtml}  \begin{rawhtml}
231  <!-- CMIREDIR:global_ocean_circulation -->  <!-- CMIREDIR:global_ocean_circulation: -->
232  \end{rawhtml}  \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
235  the surface of a 4$^{\circ }$  currents at the surface of a $4^{\circ }$ global ocean model run with
236  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
237  represent topography on a regular $lat-lon$ grid extending from 70$^{\circ  regular \textit{lat-lon} grid extending from $70^{\circ }N$ to
238  }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
239  mixed boundary conditions on temperature and salinity at the surface. The  mixed boundary conditions on temperature and salinity at the surface.
240  transfer properties of ocean eddies, convection and mixing is parameterized  The transfer properties of ocean eddies, convection and mixing is
241  in this model.  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 249  circulation of the global ocean in Sverd Line 249  circulation of the global ocean in Sverd
249    
250  \subsection{Convection and mixing over topography}  \subsection{Convection and mixing over topography}
251  \begin{rawhtml}  \begin{rawhtml}
252  <!-- CMIREDIR:mixing_over_topography -->  <!-- CMIREDIR:mixing_over_topography: -->
253  \end{rawhtml}  \end{rawhtml}
254    
255    
# Line 272  instability of the along-slope current. Line 272  instability of the along-slope current.
272    
273  \subsection{Boundary forced internal waves}  \subsection{Boundary forced internal waves}
274  \begin{rawhtml}  \begin{rawhtml}
275  <!-- CMIREDIR:boundary_forced_internal_waves -->  <!-- CMIREDIR:boundary_forced_internal_waves: -->
276  \end{rawhtml}  \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
# Line 294  nonhydrostatic dynamics. Line 294  nonhydrostatic dynamics.
294    
295  \subsection{Parameter sensitivity using the adjoint of MITgcm}  \subsection{Parameter sensitivity using the adjoint of MITgcm}
296  \begin{rawhtml}  \begin{rawhtml}
297  <!-- CMIREDIR:parameter_sensitivity -->  <!-- CMIREDIR:parameter_sensitivity: -->
298  \end{rawhtml}  \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 $\frac{\partial J}{\partial
306  of the overturning stream-function shown in figure \ref{fig:large-scale-circ}    \mathcal{H}}$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 317  yields sensitivities to all other model Line 317  yields sensitivities to all other model
317    
318  \subsection{Global state estimation of the ocean}  \subsection{Global state estimation of the ocean}
319  \begin{rawhtml}  \begin{rawhtml}
320  <!-- CMIREDIR:global_state_estimation -->  <!-- CMIREDIR:global_state_estimation: -->
321  \end{rawhtml}  \end{rawhtml}
322    
323    
# Line 338  consistency with altimetric and in-situ Line 338  consistency with altimetric and in-situ
338    
339  \subsection{Ocean biogeochemical cycles}  \subsection{Ocean biogeochemical cycles}
340  \begin{rawhtml}  \begin{rawhtml}
341  <!-- CMIREDIR:ocean_biogeo_cycles -->  <!-- CMIREDIR:ocean_biogeo_cycles: -->
342  \end{rawhtml}  \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
345  example one can study the effects of interannual changes in meteorological  ocean. For example one can study the effects of interannual changes in
346  forcing and upper ocean circulation on the fluxes of carbon dioxide and  meteorological forcing and upper ocean circulation on the fluxes of
347  oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows  carbon dioxide and oxygen between the ocean and atmosphere. Figure
348  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
349  the southern oceans from a single year of a global, interannually varying  relation to density outcrops in the southern oceans from a single year
350  simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution  of a global, interannually varying simulation. The simulation is run
351  telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).  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}
# Line 356  telescoping to $\frac{1}{3}^{\circ}\time Line 358  telescoping to $\frac{1}{3}^{\circ}\time
358    
359  \subsection{Simulations of laboratory experiments}  \subsection{Simulations of laboratory experiments}
360  \begin{rawhtml}  \begin{rawhtml}
361  <!-- CMIREDIR:classroom_exp -->  <!-- CMIREDIR:classroom_exp: -->
362  \end{rawhtml}  \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
# Line 377  stratification of the ACC. Line 379  stratification of the ACC.
379    
380  \section{Continuous equations in `r' coordinates}  \section{Continuous equations in `r' coordinates}
381  \begin{rawhtml}  \begin{rawhtml}
382  <!-- CMIREDIR:z-p_isomorphism -->  <!-- CMIREDIR:z-p_isomorphism: -->
383  \end{rawhtml}  \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
# Line 409  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 611  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_salt})  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 656  which, for convenience, are written out Line 660  which, for convenience, are written out
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}  \begin{rawhtml}
663  <!-- CMIREDIR:non_hydrostatic -->  <!-- CMIREDIR:non_hydrostatic: -->
664  \end{rawhtml}  \end{rawhtml}
665    
666    
# Line 666  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 766  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 811  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 1070  friction. These coefficients are the sam Line 1077  friction. These coefficients are the sam
1077    
1078  The mixing terms for the temperature and salinity equations have a similar  The mixing terms for the temperature and salinity equations have a similar
1079  form to that of momentum except that the diffusion tensor can be  form to that of momentum except that the diffusion tensor can be
1080  non-diagonal and have varying coefficients. $\qquad $  non-diagonal and have varying coefficients.
1081  \begin{equation}  \begin{equation}
1082  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
1083  _{h}^{4}(T,S)  \label{eq:diffusion}  _{h}^{4}(T,S)  \label{eq:diffusion}
# Line 1096  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 1208  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 1237  _{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 1283  _{\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}
# Line 1478  u=r\cos \varphi \frac{D\lambda }{Dt} Line 1489  u=r\cos \varphi \frac{D\lambda }{Dt}
1489  \end{equation*}  \end{equation*}
1490    
1491  \begin{equation*}  \begin{equation*}
1492  v=r\frac{D\varphi }{Dt}\qquad  v=r\frac{D\varphi }{Dt}
1493  \end{equation*}  \end{equation*}
 $\qquad \qquad \qquad \qquad $  
1494    
1495  \begin{equation*}  \begin{equation*}
1496  \dot{r}=\frac{Dr}{Dt}  \dot{r}=\frac{Dr}{Dt}
# Line 1490  Here $\varphi $ is the latitude, $\lambd Line 1500  Here $\varphi $ is the latitude, $\lambd
1500  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
1501  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.
1502    
1503  The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in  The `grad' ($\nabla $) and `div' ($\nabla\cdot$) operators are defined by, in
1504  spherical coordinates:  spherical coordinates:
1505    
1506  \begin{equation*}  \begin{equation*}
# Line 1500  spherical coordinates: Line 1510  spherical coordinates:
1510  \end{equation*}  \end{equation*}
1511    
1512  \begin{equation*}  \begin{equation*}
1513  \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
1514  \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}  \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
1515  +\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}
1516  \end{equation*}  \end{equation*}
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