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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}:
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 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
# 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} \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  (oceansurface,bottomoftheatmosphere)}  \label{eq:movingbc}  (ocean surface,bottom of the atmosphere)}  \label{eq:movingbc}
522  \end{equation}  \end{equation}
523    
524  Here  Here
# 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 1006  salinity ... ). Line 1100  salinity ... ).
1100    
1101  \subsection{Vector invariant form}  \subsection{Vector invariant form}
1102    
1103  For some purposes it is advantageous to write momentum advection in eq(\ref  For some purposes it is advantageous to write momentum advection in
1104  {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
1105    (so-called) `vector invariant' form:
1106    
1107  \begin{equation}  \begin{equation}
1108  \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 1213  In $p$-coordinates, the upper boundary a
1213  surface ($\phi $ is imposed and $\omega \neq 0$).  surface ($\phi $ is imposed and $\omega \neq 0$).
1214    
1215  \subsubsection{Splitting the geo-potential}  \subsubsection{Splitting the geo-potential}
1216    \label{sec:hpe-p-geo-potential-split}
1217    
1218  For the purposes of initialization and reducing round-off errors, the model  For the purposes of initialization and reducing round-off errors, the model
1219  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}$, defined: Line 1243  _{o}(p_{o})=g~Z_{topo}$, defined:
1243  The final form of the HPE's in p coordinates is then:  The final form of the HPE's in p coordinates is then:
1244  \begin{eqnarray}  \begin{eqnarray}
1245  \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}}
1246  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-prime} \\  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}}
1247    \label{eq:atmos-prime} \\
1248  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
1249  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1250  \partial p} &=&0 \\  \partial p} &=&0 \\
# Line 1178  _{h}+\frac{\partial w}{\partial z} &=&0 Line 1275  _{h}+\frac{\partial w}{\partial z} &=&0
1275  \label{eq:non-boussinesq}  \label{eq:non-boussinesq}
1276  \end{eqnarray}  \end{eqnarray}
1277  These equations permit acoustics modes, inertia-gravity waves,  These equations permit acoustics modes, inertia-gravity waves,
1278  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
1279  mode. As written, they cannot be integrated forward consistently - if we  mode. As written, they cannot be integrated forward consistently - if we
1280  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
1281  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 1193  _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\ Line 1290  _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\
1290  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexpansion}
1291  \end{equation}  \end{equation}
1292    
1293  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
1294  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
1295    \ref{eq-zns-cont} gives:
1296  \begin{equation}  \begin{equation}
1297  \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{
1298  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
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