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# 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 106  running Linux, together with a FORTRAN\ Line 154  running Linux, together with a FORTRAN\
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 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
# 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 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  in-situ) 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 inquiring 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} \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 426  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  (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}  (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
522  \end{equation}  \end{equation}
523    
# Line 525  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 569  which, for convenience, are written out Line 657  which, for convenience, are written out
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 576  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 676  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 721  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 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  {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 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}}} \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 1193  _{\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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