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revision 1.1 by cnh, Thu Sep 27 17:45:03 2001 UTC revision 1.7 by cnh, Thu Oct 25 12:06:56 2001 UTC
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 %%%% \begin{document}  
 %%%%  
 %%%% \tableofcontents  
29    
30  \pagebreak  %tci%\begin{document}
31    
32    %tci%\tableofcontents
33    
 \part{MITgcm basics}  
34    
35  % Section: Overview  % Section: Overview
36    
# Line 78  MITgcm has a number of novel aspects: Line 54  MITgcm has a number of novel aspects:
54  \begin{itemize}  \begin{itemize}
55  \item it can be used to study both atmospheric and oceanic phenomena; one  \item it can be used to study both atmospheric and oceanic phenomena; one
56  hydrodynamical kernel is used to drive forward both atmospheric and oceanic  hydrodynamical kernel is used to drive forward both atmospheric and oceanic
57  models - see fig.1%  models - see fig \ref{fig:onemodel}
 \marginpar{  
 Fig.1 One model}\ref{fig:onemodel}  
58    
59  \begin{figure}  %% CNHbegin
60  \begin{center}  \input{part1/one_model_figure}
61  \resizebox{!}{4in}{  %% CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:onemodel}  
 \end{figure}  
62    
63  \item it has a non-hydrostatic capability and so can be used to study both  \item it has a non-hydrostatic capability and so can be used to study both
64  small-scale and large scale processes - see fig.2%  small-scale and large scale processes - see fig \ref{fig:all-scales}
 \marginpar{  
 Fig.2 All scales}\ref{fig:all-scales}  
   
   
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/scales.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:scales}  
 \end{figure}  
65    
66    %% CNHbegin
67    \input{part1/all_scales_figure}
68    %% CNHend
69    
70  \item finite volume techniques are employed yielding an intuitive  \item finite volume techniques are employed yielding an intuitive
71  discretization and support for the treatment of irregular geometries using  discretization and support for the treatment of irregular geometries using
72  orthogonal curvilinear grids and shaved cells - see fig.3%  orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes}
73  \marginpar{  
74  Fig.3 Finite volumes}\ref{fig:Finite volumes}  %% CNHbegin
75    \input{part1/fvol_figure}
76    %% CNHend
77    
78  \item tangent linear and adjoint counterparts are automatically maintained  \item tangent linear and adjoint counterparts are automatically maintained
79  along with the forward model, permitting sensitivity and optimization  along with the forward model, permitting sensitivity and optimization
# Line 134  listed in an Appendix. Line 88  listed in an Appendix.
88    
89  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
90  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.
 \pagebreak  
91    
92  % $Header$  % $Header$
93  % $Name$  % $Name$
# Line 143  give a feel for the wide range of proble Line 96  give a feel for the wide range of proble
96    
97  The MITgcm has been designed and used to model a wide range of phenomena,  The MITgcm has been designed and used to model a wide range of phenomena,
98  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
99  atmospheric winds - see fig.2\ref{fig:all-scales}. To give a flavor of the  atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the
100  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
101  of them here. A more detailed description of the underlying formulation,  of them here. A more detailed description of the underlying formulation,
102  numerical algorithm and implementation that lie behind these calculations is  numerical algorithm and implementation that lie behind these calculations is
103  given later. Indeed it is easy to reproduce the results shown here: simply  given later. Indeed many of the illustrative examples shown below can be
104  download the model (the minimum you need is a PC running linux, together  easily reproduced: simply download the model (the minimum you need is a PC
105  with a FORTRAN\ 77 compiler) and follow the examples.  running linux, together with a FORTRAN\ 77 compiler) and follow the examples
106    described in detail in the documentation.
107    
108  \subsection{Global atmosphere: `Held-Suarez' benchmark}  \subsection{Global atmosphere: `Held-Suarez' benchmark}
109    
110  Fig.E1a.\ref{fig:Held-Suarez} is an instaneous plot of the 500$mb$ height  A novel feature of MITgcm is its ability to simulate, using one basic algorithm,
111  field obtained using a 5-level version of the atmospheric pressure isomorph  both atmospheric and oceanographic flows at both small and large scales.
 run at 2.8$^{\circ }$ resolution. We see fully developed baroclinic eddies  
 along the northern hemisphere storm track. There are no mountains or  
 land-sea contrast in this calculation, but you can easily put them in. The  
 model is driven by relaxation to a radiative-convective equilibrium profile,  
 following the description set out in Held and Suarez; 1994 designed to test  
 atmospheric hydrodynamical cores - there are no mountains or land-sea  
 contrast. As decribed in Adcroft (2001), a `cubed sphere' is used to  
 descretize the globe permitting a uniform gridding and obviated the need to  
 fourier filter.  
   
 Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal  
 wind and meridional overturning streamfunction from the 5-level model.  
   
   
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps}  
  }  
 }  
 \end{center}  
 \label{fig:hscs}  
 \end{figure}  
   
   
 A regular spherical lat-lon grid can also be used.  
   
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps}  
  }  
 }  
 \end{center}  
 \label{fig:hslatlon}  
 \end{figure}  
112    
113  \subsection{Ocean gyres}  Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
114    temperature field obtained using the atmospheric isomorph of MITgcm run at
115    2.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
116    (blue) and warm air along an equatorial band (red). Fully developed
117    baroclinic eddies spawned in the northern hemisphere storm track are
118    evident. There are no mountains or land-sea contrast in this calculation,
119    but you can easily put them in. The model is driven by relaxation to a
120    radiative-convective equilibrium profile, following the description set out
121    in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
122    there are no mountains or land-sea contrast.
123    
124    %% CNHbegin
125    \input{part1/cubic_eddies_figure}
126    %% CNHend
127    
128    As described in Adcroft (2001), a `cubed sphere' is used to discretize the
129    globe permitting a uniform gridding and obviated the need to Fourier filter.
130    The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
131    grid, of which the cubed sphere is just one of many choices.
132    
133    Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
134    wind from a 20-level configuration of
135    the model. It compares favorable with more conventional spatial
136    discretization approaches. The two plots show the field calculated using the
137    cube-sphere grid and the flow calculated using a regular, spherical polar
138    latitude-longitude grid. Both grids are supported within the model.
139    
140    %% CNHbegin
141    \input{part1/hs_zave_u_figure}
142    %% CNHend
143    
144  \subsection{Global ocean circulation}  \subsection{Ocean gyres}
145    
146  Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$  Baroclinic instability is a ubiquitous process in the ocean, as well as the
147  global ocean model run with 15 vertical levels. The model is driven using  atmosphere. Ocean eddies play an important role in modifying the
148  monthly-mean winds with mixed boundary conditions on temperature and  hydrographic structure and current systems of the oceans. Coarse resolution
149  salinity at the surface. Fig.E2b shows the overturning (thermohaline)  models of the oceans cannot resolve the eddy field and yield rather broad,
150  circulation. Lopped cells are used to represent topography on a regular $%  diffusive patterns of ocean currents. But if the resolution of our models is
151  lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$.  increased until the baroclinic instability process is resolved, numerical
152    solutions of a different and much more realistic kind, can be obtained.
153    
154  \begin{figure}  Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity
155  \begin{center}  field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal
156  \resizebox{!}{4in}{  resolution on a $lat-lon$
157  % \rotatebox{90}{  grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
158    \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps}  (to avoid the converging of meridian in northern latitudes). 21 vertical
159  % }  levels are used in the vertical with a `lopped cell' representation of
160  }  topography. The development and propagation of anomalously warm and cold
161  \end{center}  eddies can be clearly seen in the Gulf Stream region. The transport of
162  \label{fig:horizcirc}  warm water northward by the mean flow of the Gulf Stream is also clearly
163  \end{figure}  visible.
164    
165  \begin{figure}  %% CNHbegin
166  \begin{center}  \input{part1/ocean_gyres_figure}
167  \resizebox{!}{4in}{  %% CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:moc}  
 \end{figure}  
   
   
 \subsection{Flow over topography}  
   
 \subsection{Ocean convection}  
   
 Fig.E3 shows convection over a slope using the non-hydrostatic ocean  
 isomorph and lopped cells to respresent topography. .....The grid resolution  
 is  
168    
 \subsection{Boundary forced internal waves}  
169    
170  \subsection{Carbon outgassing sensitivity}  \subsection{Global ocean circulation}
171    
172  Fig.E4 shows....  Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
173    the surface of a 4$^{\circ }$
174    global ocean model run with 15 vertical levels. Lopped cells are used to
175    represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
176    }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
177    mixed boundary conditions on temperature and salinity at the surface. The
178    transfer properties of ocean eddies, convection and mixing is parameterized
179    in this model.
180    
181    Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
182    circulation of the global ocean in Sverdrups.
183    
184    %%CNHbegin
185    \input{part1/global_circ_figure}
186    %%CNHend
187    
188    \subsection{Convection and mixing over topography}
189    
190    Dense plumes generated by localized cooling on the continental shelf of the
191    ocean may be influenced by rotation when the deformation radius is smaller
192    than the width of the cooling region. Rather than gravity plumes, the
193    mechanism for moving dense fluid down the shelf is then through geostrophic
194    eddies. The simulation shown in the figure \ref{fig::convect-and-topo}
195    (blue is cold dense fluid, red is
196    warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
197    trigger convection by surface cooling. The cold, dense water falls down the
198    slope but is deflected along the slope by rotation. It is found that
199    entrainment in the vertical plane is reduced when rotational control is
200    strong, and replaced by lateral entrainment due to the baroclinic
201    instability of the along-slope current.
202    
203    %%CNHbegin
204    \input{part1/convect_and_topo}
205    %%CNHend
206    
207  \begin{figure}  \subsection{Boundary forced internal waves}
 \begin{center}  
 \resizebox{!}{4in}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps}  
 }  
 \end{center}  
 \label{fig:co2mrt}  
 \end{figure}  
208    
209    The unique ability of MITgcm to treat non-hydrostatic dynamics in the
210    presence of complex geometry makes it an ideal tool to study internal wave
211    dynamics and mixing in oceanic canyons and ridges driven by large amplitude
212    barotropic tidal currents imposed through open boundary conditions.
213    
214    Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope
215    topographic variations on
216    internal wave breaking - the cross-slope velocity is in color, the density
217    contoured. The internal waves are excited by application of open boundary
218    conditions on the left. They propagate to the sloping boundary (represented
219    using MITgcm's finite volume spatial discretization) where they break under
220    nonhydrostatic dynamics.
221    
222    %%CNHbegin
223    \input{part1/boundary_forced_waves}
224    %%CNHend
225    
226    \subsection{Parameter sensitivity using the adjoint of MITgcm}
227    
228    Forward and tangent linear counterparts of MITgcm are supported using an
229    `automatic adjoint compiler'. These can be used in parameter sensitivity and
230    data assimilation studies.
231    
232    As one example of application of the MITgcm adjoint, Figure \ref{fig:hf-sensitivity}
233    maps the gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
234    of the overturning streamfunction shown in figure \ref{fig:large-scale-circ}
235    at 60$^{\circ }$N and $
236    \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
237    a 100 year period. We see that $J$ is
238    sensitive to heat fluxes over the Labrador Sea, one of the important sources
239    of deep water for the thermohaline circulations. This calculation also
240    yields sensitivities to all other model parameters.
241    
242    %%CNHbegin
243    \input{part1/adj_hf_ocean_figure}
244    %%CNHend
245    
246    \subsection{Global state estimation of the ocean}
247    
248    An important application of MITgcm is in state estimation of the global
249    ocean circulation. An appropriately defined `cost function', which measures
250    the departure of the model from observations (both remotely sensed and
251    insitu) over an interval of time, is minimized by adjusting `control
252    parameters' such as air-sea fluxes, the wind field, the initial conditions
253    etc. Figure \ref{fig:assimilated-globes} shows an estimate of the time-mean
254    surface elevation of the ocean obtained by bringing the model in to
255    consistency with altimetric and in-situ observations over the period
256    1992-1997. {\bf CHANGE THIS TEXT - FIG FROM PATRICK/CARL/DETLEF}
257    
258    %% CNHbegin
259    \input{part1/globes_figure}
260    %% CNHend
261    
262    \subsection{Ocean biogeochemical cycles}
263    
264    MITgcm is being used to study global biogeochemical cycles in the ocean. For
265    example one can study the effects of interannual changes in meteorological
266    forcing and upper ocean circulation on the fluxes of carbon dioxide and
267    oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows
268    the annual air-sea flux of oxygen and its relation to density outcrops in
269    the southern oceans from a single year of a global, interannually varying
270    simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution
271    telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).
272    
273    %%CNHbegin
274    \input{part1/biogeo_figure}
275    %%CNHend
276    
277    \subsection{Simulations of laboratory experiments}
278    
279    Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
280    laboratory experiment enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
281    initially homogeneous tank of water ($1m$ in diameter) is driven from its
282    free surface by a rotating heated disk. The combined action of mechanical
283    and thermal forcing creates a lens of fluid which becomes baroclinically
284    unstable. The stratification and depth of penetration of the lens is
285    arrested by its instability in a process analogous to that which sets the
286    stratification of the ACC.
287    
288    %%CNHbegin
289    \input{part1/lab_figure}
290    %%CNHend
291    
292  % $Header$  % $Header$
293  % $Name$  % $Name$
# Line 262  Fig.E4 shows.... Line 296  Fig.E4 shows....
296    
297  To render atmosphere and ocean models from one dynamical core we exploit  To render atmosphere and ocean models from one dynamical core we exploit
298  `isomorphisms' between equation sets that govern the evolution of the  `isomorphisms' between equation sets that govern the evolution of the
299  respective fluids - see fig.4%  respective fluids - see figure \ref{fig:isomorphic-equations}.
300  \marginpar{  One system of hydrodynamical equations is written down
 Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down  
301  and encoded. The model variables have different interpretations depending on  and encoded. The model variables have different interpretations depending on
302  whether the atmosphere or ocean is being studied. Thus, for example, the  whether the atmosphere or ocean is being studied. Thus, for example, the
303  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
304  modeling the atmosphere and height, $z$, if we are modeling the ocean.  modeling the atmosphere (left hand side of figure \ref{fig:isomorphic-equations})
305    and height, $z$, if we are modeling the ocean (right hand side of figure
306    \ref{fig:isomorphic-equations}).
307    
308    %%CNHbegin
309    \input{part1/zandpcoord_figure.tex}
310    %%CNHend
311    
312  The state of the fluid at any time is characterized by the distribution of  The state of the fluid at any time is characterized by the distribution of
313  velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a  velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a
# Line 276  velocity $\vec{\mathbf{v}}$, active trac Line 315  velocity $\vec{\mathbf{v}}$, active trac
315  depend on $\theta $, $S$, and $p$. The equations that govern the evolution  depend on $\theta $, $S$, and $p$. The equations that govern the evolution
316  of these fields, obtained by applying the laws of classical mechanics and  of these fields, obtained by applying the laws of classical mechanics and
317  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of  thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
318  a generic vertical coordinate, $r$, see fig.5%  a generic vertical coordinate, $r$, so that the appropriate
319  \marginpar{  kinematic boundary conditions can be applied isomorphically
320  Fig.5 The vertical coordinate of model}:  see figure \ref{fig:zandp-vert-coord}.
321    
322  \begin{figure}  %%CNHbegin
323  \begin{center}  \input{part1/vertcoord_figure.tex}
324  \resizebox{!}{4in}{  %%CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/vertcoord.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:vertcoord}  
 \end{figure}  
325    
326  \begin{equation*}  \begin{equation*}
327  \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}}
328  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}%  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}
329  \text{ horizontal mtm}  \text{ horizontal mtm}
330  \end{equation*}  \end{equation*}
331    
332  \begin{equation*}  \begin{equation*}
333  \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{
334  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{  v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
335  vertical mtm}  vertical mtm}
336  \end{equation*}  \end{equation*}
337    
338  \begin{equation}  \begin{equation}
339  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{%  \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
340  \partial r}=0\text{ continuity}  \label{eq:continuous}  \partial r}=0\text{ continuity}  \label{eq:continuous}
341  \end{equation}  \end{equation}
342    
343  \begin{equation*}  \begin{equation*}
344  b=b(\theta ,S,r)\text{ equation of state}  b=b(\theta ,S,r)\text{ equation of state}
345  \end{equation*}  \end{equation*}
346    
347  \begin{equation*}  \begin{equation*}
348  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{  potential temperature}  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
349  \end{equation*}  \end{equation*}
350    
351  \begin{equation*}  \begin{equation*}
352  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{  humidity/salinity}  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
353  \end{equation*}  \end{equation*}
354    
355  Here:  Here:
356    
357  \begin{equation*}  \begin{equation*}
358  r\text{ is the vertical coordinate}  r\text{ is the vertical coordinate}
359  \end{equation*}  \end{equation*}
360    
361  \begin{equation*}  \begin{equation*}
362  \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{  \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{
363  is the total derivative}  is the total derivative}
364  \end{equation*}  \end{equation*}
365    
366  \begin{equation*}  \begin{equation*}
367  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
368  \text{ is the `grad' operator}  \text{ is the `grad' operator}
369  \end{equation*}  \end{equation*}
370  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}
371  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
372  is a unit vector in the vertical  is a unit vector in the vertical
373    
374  \begin{equation*}  \begin{equation*}
375  t\text{ is time}  t\text{ is time}
376  \end{equation*}  \end{equation*}
377    
378  \begin{equation*}  \begin{equation*}
379  \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the  \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the
380  velocity}  velocity}
381  \end{equation*}  \end{equation*}
382    
383  \begin{equation*}  \begin{equation*}
384  \phi \text{ is the `pressure'/`geopotential'}  \phi \text{ is the `pressure'/`geopotential'}
385  \end{equation*}  \end{equation*}
386    
387  \begin{equation*}  \begin{equation*}
388  \vec{\Omega}\text{ is the Earth's rotation}  \vec{\Omega}\text{ is the Earth's rotation}
389  \end{equation*}  \end{equation*}
390    
391  \begin{equation*}  \begin{equation*}
392  b\text{ is the `buoyancy'}  b\text{ is the `buoyancy'}
393  \end{equation*}  \end{equation*}
394    
395  \begin{equation*}  \begin{equation*}
396  \theta \text{ is potential temperature}  \theta \text{ is potential temperature}
397  \end{equation*}  \end{equation*}
398    
399  \begin{equation*}  \begin{equation*}
400  S\text{ is specific humidity in the atmosphere; salinity in the ocean}  S\text{ is specific humidity in the atmosphere; salinity in the ocean}
401  \end{equation*}  \end{equation*}
402    
403  \begin{equation*}  \begin{equation*}
404  \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{%  \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{
405  \mathbf{v}}  \mathbf{v}}
406  \end{equation*}  \end{equation*}
407    
408  \begin{equation*}  \begin{equation*}
409  \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }%  \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
 \theta  
410  \end{equation*}  \end{equation*}
411    
412  \begin{equation*}  \begin{equation*}
# Line 385  S\text{ is specific humidity in the atmo Line 414  S\text{ is specific humidity in the atmo
414  \end{equation*}  \end{equation*}
415    
416  The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by  The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by
417  extensive `physics' packages for atmosphere and ocean described in Chapter 6.  `physics' and forcing packages for atmosphere and ocean. These are described
418    in later chapters.
419    
420  \subsection{Kinematic Boundary conditions}  \subsection{Kinematic Boundary conditions}
421    
422  \subsubsection{vertical}  \subsubsection{vertical}
423    
424  at fixed and moving $r$ surfaces we set (see fig.5):  at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
425    
426  \begin{equation}  \begin{equation}
427  \dot{r}=0atr=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}  \dot{r}=0atr=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
# Line 406  at fixed and moving $r$ surfaces we set Line 436  at fixed and moving $r$ surfaces we set
436  Here  Here
437    
438  \begin{equation*}  \begin{equation*}
439  R_{moving}=R_{o}+\eta  R_{moving}=R_{o}+\eta
440  \end{equation*}  \end{equation*}
441  where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on  where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on
442  whether we are in the atmosphere or ocean) of the `moving surface' in the  whether we are in the atmosphere or ocean) of the `moving surface' in the
# Line 417  of motion. Line 447  of motion.
447    
448  \begin{equation}  \begin{equation}
449  \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0  \label{eq:noflow}  \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0  \label{eq:noflow}
450  \end{equation}%  \end{equation}
451  where $\vec{\mathbf{n}}$ is the normal to a solid boundary.  where $\vec{\mathbf{n}}$ is the normal to a solid boundary.
452    
453  \subsection{Atmosphere}  \subsection{Atmosphere}
454    
455  In the atmosphere, see fig.5, we interpret:  In the atmosphere, (see figure \ref{fig:zandp-vert-coord}), we interpret:
456    
457  \begin{equation}  \begin{equation}
458  r=p\text{ is the pressure}  \label{eq:atmos-r}  r=p\text{ is the pressure}  \label{eq:atmos-r}
# Line 454  where Line 484  where
484    
485  \begin{equation*}  \begin{equation*}
486  T\text{ is absolute temperature}  T\text{ is absolute temperature}
487  \end{equation*}%  \end{equation*}
488  \begin{equation*}  \begin{equation*}
489  p\text{ is the pressure}  p\text{ is the pressure}
490  \end{equation*}%  \end{equation*}
491  \begin{eqnarray*}  \begin{eqnarray*}
492  &&z\text{ is the height of the pressure surface} \\  &&z\text{ is the height of the pressure surface} \\
493  &&g\text{ is the acceleration due to gravity}  &&g\text{ is the acceleration due to gravity}
# Line 467  In the above the ideal gas law, $p=\rho Line 497  In the above the ideal gas law, $p=\rho
497  the Exner function $\Pi (p)$ given by (see Appendix Atmosphere)  the Exner function $\Pi (p)$ given by (see Appendix Atmosphere)
498  \begin{equation}  \begin{equation}
499  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{eq:exner}  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{eq:exner}
500  \end{equation}%  \end{equation}
501  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas
502  constant and $c_{p}$ the specific heat of air at constant pressure.  constant and $c_{p}$ the specific heat of air at constant pressure.
503    
504  At the top of the atmosphere (which is `fixed' in our $r$ coordinate):  At the top of the atmosphere (which is `fixed' in our $r$ coordinate):
505    
506  \begin{equation*}  \begin{equation*}
507  R_{fixed}=p_{top}=0  R_{fixed}=p_{top}=0
508  \end{equation*}  \end{equation*}
509  In a resting atmosphere the elevation of the mountains at the bottom is  In a resting atmosphere the elevation of the mountains at the bottom is
510  given by  given by
511  \begin{equation*}  \begin{equation*}
512  R_{moving}=R_{o}(x,y)=p_{o}(x,y)  R_{moving}=R_{o}(x,y)=p_{o}(x,y)
513  \end{equation*}  \end{equation*}
514  i.e. the (hydrostatic) pressure at the top of the mountains in a resting  i.e. the (hydrostatic) pressure at the top of the mountains in a resting
515  atmosphere.  atmosphere.
# Line 517  At the bottom of the ocean: $R_{fixed}(x Line 547  At the bottom of the ocean: $R_{fixed}(x
547    
548  The surface of the ocean is given by: $R_{moving}=\eta $  The surface of the ocean is given by: $R_{moving}=\eta $
549    
550  The position of the resting free surface of the ocean is given by $%  The position of the resting free surface of the ocean is given by $
551  R_{o}=Z_{o}=0$.  R_{o}=Z_{o}=0$.
552    
553  Boundary conditions are:  Boundary conditions are:
# Line 525  Boundary conditions are: Line 555  Boundary conditions are:
555  \begin{eqnarray}  \begin{eqnarray}
556  w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)}  \label{eq:fixed-bc-ocean}  w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)}  \label{eq:fixed-bc-ocean}
557  \\  \\
558  w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) %  w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)
559  \label{eq:moving-bc-ocean}}  \label{eq:moving-bc-ocean}}
560  \end{eqnarray}  \end{eqnarray}
561  where $\eta $ is the elevation of the free surface.  where $\eta $ is the elevation of the free surface.
# Line 542  Let us separate $\phi $ in to surface, h Line 572  Let us separate $\phi $ in to surface, h
572  \begin{equation}  \begin{equation}
573  \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)
574  \label{eq:phi-split}  \label{eq:phi-split}
575  \end{equation}%  \end{equation}
576  and write eq(\ref{incompressible}a,b) in the form:  and write eq(\ref{incompressible}a,b) in the form:
577    
578  \begin{equation}  \begin{equation}
# Line 556  _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l Line 586  _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
586  \end{equation}  \end{equation}
587    
588  \begin{equation}  \begin{equation}
589  \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{%  \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{
590  \partial r}=G_{\dot{r}}  \label{eq:mom-w}  \partial r}=G_{\dot{r}}  \label{eq:mom-w}
591  \end{equation}  \end{equation}
592  Here $\epsilon _{nh}$ is a non-hydrostatic parameter.  Here $\epsilon _{nh}$ is a non-hydrostatic parameter.
593    
594  The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref%  The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref
595  {eq:mom-h}) and (\ref{eq:mom-w}) represent advective, metric and Coriolis  {eq:mom-h}) and (\ref{eq:mom-w}) represent advective, metric and Coriolis
596  terms in the momentum equations. In spherical coordinates they take the form%  terms in the momentum equations. In spherical coordinates they take the form
597  \footnote{%  \footnote{
598  In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms  In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms
599  in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref%  in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref
600  {eq:gw-spherical}) are omitted; the singly-underlined terms are included in  {eq:gw-spherical}) are omitted; the singly-underlined terms are included in
601  the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (%  the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (
602  \textbf{NH}) includes all terms.} - see Marshall et al 1997a for a full  \textbf{NH}) includes all terms.} - see Marshall et al 1997a for a full
603  discussion:  discussion:
604    
# Line 576  discussion: Line 606  discussion:
606  \left.  \left.
607  \begin{tabular}{l}  \begin{tabular}{l}
608  $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\  $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\
609  $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan lat}{{r}}\right\} $  $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} $
610  \\  \\
611  $-\left\{ -2\Omega v\sin lat+\underline{2\Omega \dot{r}\cos lat}\right\} $  $-\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\right\} $
612  \\  \\
613  $+\mathcal{F}_{u}$%  $+\mathcal{F}_{u}$
614  \end{tabular}%  \end{tabular}
615  \ \right\} \left\{  \ \right\} \left\{
616  \begin{tabular}{l}  \begin{tabular}{l}
617  \textit{advection} \\  \textit{advection} \\
618  \textit{metric} \\  \textit{metric} \\
619  \textit{Coriolis} \\  \textit{Coriolis} \\
620  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
621  \end{tabular}%  \end{tabular}
622  \ \right. \qquad   \label{eq:gu-speherical}  \ \right. \qquad  \label{eq:gu-speherical}
623  \end{equation}  \end{equation}
624    
625  \begin{equation}  \begin{equation}
626  \left.  \left.
627  \begin{tabular}{l}  \begin{tabular}{l}
628  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
629  $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}}\right\}  $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\}
630  $ \\  $ \\
631  $-\left\{ -2\Omega u\sin lat\right\} $ \\  $-\left\{ -2\Omega u\sin \varphi \right\} $ \\
632  $+\mathcal{F}_{v}$%  $+\mathcal{F}_{v}$
633  \end{tabular}%  \end{tabular}
634  \ \right\} \left\{  \ \right\} \left\{
635  \begin{tabular}{l}  \begin{tabular}{l}
636  \textit{advection} \\  \textit{advection} \\
637  \textit{metric} \\  \textit{metric} \\
638  \textit{Coriolis} \\  \textit{Coriolis} \\
639  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
640  \end{tabular}%  \end{tabular}
641  \ \right. \qquad   \label{eq:gv-spherical}  \ \right. \qquad  \label{eq:gv-spherical}
642  \end{equation}%  \end{equation}
643  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
644    
645  \begin{equation}  \begin{equation}
646  \left.  \left.
647  \begin{tabular}{l}  \begin{tabular}{l}
648  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\  $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
649  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\  $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
650  ${+}\underline{{2\Omega u\cos lat}}$ \\  ${+}\underline{{2\Omega u\cos \varphi}}$ \\
651  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$%  $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
652  \end{tabular}%  \end{tabular}
653  \ \right\} \left\{  \ \right\} \left\{
654  \begin{tabular}{l}  \begin{tabular}{l}
655  \textit{advection} \\  \textit{advection} \\
656  \textit{metric} \\  \textit{metric} \\
657  \textit{Coriolis} \\  \textit{Coriolis} \\
658  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}
659  \end{tabular}%  \end{tabular}
660  \ \right.   \label{eq:gw-spherical}  \ \right.  \label{eq:gw-spherical}
661  \end{equation}%  \end{equation}
662  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
663    
664  In the above `${r}$' is the distance from the center of the earth and `$lat$%  In the above `${r}$' is the distance from the center of the earth and `$\varphi$
665  ' is latitude.  ' is latitude.
666    
667  Grad and div operators in spherical coordinates are defined in appendix  Grad and div operators in spherical coordinates are defined in appendix
668  OPERATORS.%  OPERATORS.
669  \marginpar{  \marginpar{
670  Fig.6 Spherical polar coordinate system.}  Fig.6 Spherical polar coordinate system.}
671    
672  \begin{figure}  %%CNHbegin
673  \begin{center}  \input{part1/sphere_coord_figure.tex}
674  \resizebox{!}{4in}{  %%CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:spcoord}  
 \end{figure}  
   
675    
676  \subsubsection{Shallow atmosphere approximation}  \subsubsection{Shallow atmosphere approximation}
677    
# Line 661  hydrostatic balance and the `traditional Line 681  hydrostatic balance and the `traditional
681  Coriolis force is treated approximately and the shallow atmosphere  Coriolis force is treated approximately and the shallow atmosphere
682  approximation is made.\ The MITgcm need not make the `traditional  approximation is made.\ The MITgcm need not make the `traditional
683  approximation'. To be able to support consistent non-hydrostatic forms the  approximation'. To be able to support consistent non-hydrostatic forms the
684  shallow atmosphere approximation can be relaxed - when dividing through by $r  shallow atmosphere approximation can be relaxed - when dividing through by $
685  $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,  r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
686  the radius of the earth.  the radius of the earth.
687    
688  \subsubsection{Hydrostatic and quasi-hydrostatic forms}  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
689    \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
690    
691  These are discussed at length in Marshall et al (1997a).  These are discussed at length in Marshall et al (1997a).
692    
# Line 674  terms in Eqs. (\ref{eq:gu-speherical} $\ Line 695  terms in Eqs. (\ref{eq:gu-speherical} $\
695  are neglected and `${r}$' is replaced by `$a$', the mean radius of the  are neglected and `${r}$' is replaced by `$a$', the mean radius of the
696  earth. Once the pressure is found at one level - e.g. by inverting a 2-d  earth. Once the pressure is found at one level - e.g. by inverting a 2-d
697  Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be  Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be
698  computed at all other levels by integration of the hydrostatic relation, eq(%  computed at all other levels by integration of the hydrostatic relation, eq(
699  \ref{eq:hydrostatic}).  \ref{eq:hydrostatic}).
700    
701  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
702  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
703  \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic  \varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
704  contribution to the pressure field: only the terms underlined twice in Eqs. (%  contribution to the pressure field: only the terms underlined twice in Eqs. (
705  \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero  \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero
706  and, simultaneously, the shallow atmosphere approximation is relaxed. In  and, simultaneously, the shallow atmosphere approximation is relaxed. In
707  \textbf{QH}\ \textit{all} the metric terms are retained and the full  \textbf{QH}\ \textit{all} the metric terms are retained and the full
# Line 688  variation of the radial position of a pa Line 709  variation of the radial position of a pa
709  vertical momentum equation (\ref{eq:mom-w}) becomes:  vertical momentum equation (\ref{eq:mom-w}) becomes:
710    
711  \begin{equation*}  \begin{equation*}
712  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi
713  \end{equation*}  \end{equation*}
714  making a small correction to the hydrostatic pressure.  making a small correction to the hydrostatic pressure.
715    
# Line 704  only a quasi-non-hydrostatic atmospheric Line 725  only a quasi-non-hydrostatic atmospheric
725    
726  \paragraph{Non-hydrostatic Ocean}  \paragraph{Non-hydrostatic Ocean}
727    
728  In the non-hydrostatic ocean model all terms in equations Eqs.(\ref%  In the non-hydrostatic ocean model all terms in equations Eqs.(\ref
729  {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A  {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A
730  three dimensional elliptic equation must be solved subject to Neumann  three dimensional elliptic equation must be solved subject to Neumann
731  boundary conditions (see below). It is important to note that use of the  boundary conditions (see below). It is important to note that use of the
# Line 717  and Bromley, 1995; Marshall et.al.\ 1997 Line 738  and Bromley, 1995; Marshall et.al.\ 1997
738    
739  \paragraph{Quasi-nonhydrostatic Atmosphere}  \paragraph{Quasi-nonhydrostatic Atmosphere}
740    
741  In the non-hydrostatic version of our atmospheric model we approximate $\dot{%  In the non-hydrostatic version of our atmospheric model we approximate $\dot{
742  r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical})  r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical})
743  (but only here) by:  (but only here) by:
744    
745  \begin{equation}  \begin{equation}
746  \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt}  \label{eq:quasi-nh-w}  \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt}  \label{eq:quasi-nh-w}
747  \end{equation}%  \end{equation}
748  where $p_{hy}$ is the hydrostatic pressure.  where $p_{hy}$ is the hydrostatic pressure.
749    
750  \subsubsection{Summary of equation sets supported by model}  \subsubsection{Summary of equation sets supported by model}
# Line 751  equations in $z-$coordinates are support Line 772  equations in $z-$coordinates are support
772    
773  \subparagraph{Non-hydrostatic}  \subparagraph{Non-hydrostatic}
774    
775  Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$%  Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
776  coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref%  coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref
777  {eq:ocean-salt}).  {eq:ocean-salt}).
778    
779  \subsection{Solution strategy}  \subsection{Solution strategy}
780    
781  The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{%  The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
782  NH} models is summarized in Fig.7.%  NH} models is summarized in Fig.7.
783  \marginpar{  \marginpar{
784  Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is  Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is
785  first solved to find the surface pressure and the hydrostatic pressure at  first solved to find the surface pressure and the hydrostatic pressure at
# Line 769  forward and $\dot{r}$ found from continu Line 790  forward and $\dot{r}$ found from continu
790  stepping forward the horizontal momentum equations; $\dot{r}$ is found by  stepping forward the horizontal momentum equations; $\dot{r}$ is found by
791  stepping forward the vertical momentum equation.  stepping forward the vertical momentum equation.
792    
793  \begin{figure}  %%CNHbegin
794  \begin{center}  \input{part1/solution_strategy_figure.tex}
795  \resizebox{!}{4in}{  %%CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:solnstart}  
 \end{figure}  
   
796    
797  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
798  course, some complication that goes with the inclusion of $\cos \phi \ $%  course, some complication that goes with the inclusion of $\cos \varphi \ $
799  Coriolis terms and the relaxation of the shallow atmosphere approximation.  Coriolis terms and the relaxation of the shallow atmosphere approximation.
800  But this leads to negligible increase in computation. In \textbf{NH}, in  But this leads to negligible increase in computation. In \textbf{NH}, in
801  contrast, one additional elliptic equation - a three-dimensional one - must  contrast, one additional elliptic equation - a three-dimensional one - must
# Line 794  Marshall et al, 1997) resulting in a non Line 805  Marshall et al, 1997) resulting in a non
805  hydrostatic limit, is as computationally economic as the \textbf{HPEs}.  hydrostatic limit, is as computationally economic as the \textbf{HPEs}.
806    
807  \subsection{Finding the pressure field}  \subsection{Finding the pressure field}
808    \label{sec:finding_the_pressure_field}
809    
810  Unlike the prognostic variables $u$, $v$, $w$, $\theta $ and $S$, the  Unlike the prognostic variables $u$, $v$, $w$, $\theta $ and $S$, the
811  pressure field must be obtained diagnostically. We proceed, as before, by  pressure field must be obtained diagnostically. We proceed, as before, by
# Line 808  Hydrostatic pressure is obtained by inte Line 820  Hydrostatic pressure is obtained by inte
820  vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:  vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:
821    
822  \begin{equation*}  \begin{equation*}
823  \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}%  \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}
824  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
825  \end{equation*}  \end{equation*}
826  and so  and so
827    
# Line 826  atmospheric pressure pushing down on the Line 838  atmospheric pressure pushing down on the
838    
839  \subsubsection{Surface pressure}  \subsubsection{Surface pressure}
840    
841  The surface pressure equation can be obtained by integrating continuity, (%  The surface pressure equation can be obtained by integrating continuity, (
842  \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$  \ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
843    
844  \begin{equation*}  \begin{equation*}
845  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}%  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
846  }_{h}+\partial _{r}\dot{r}\right) dr=0  }_{h}+\partial _{r}\dot{r}\right) dr=0
847  \end{equation*}  \end{equation*}
848    
849  Thus:  Thus:
850    
851  \begin{equation*}  \begin{equation*}
852  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
853  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}
854  _{h}dr=0  _{h}dr=0
855  \end{equation*}  \end{equation*}
856  where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $%  where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $
857  r $. The above can be rearranged to yield, using Leibnitz's theorem:  r $. The above can be rearranged to yield, using Leibnitz's theorem:
858    
859  \begin{equation}  \begin{equation}
860  \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot  \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot
861  \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}  \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}
862  \label{eq:free-surface}  \label{eq:free-surface}
863  \end{equation}%  \end{equation}
864  where we have incorporated a source term.  where we have incorporated a source term.
865    
866  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential  Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
867  (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can  (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
868  be written  be written
869  \begin{equation}  \begin{equation}
870  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
871  \label{eq:phi-surf}  \label{eq:phi-surf}
872  \end{equation}%  \end{equation}
873  where $b_{s}$ is the buoyancy at the surface.  where $b_{s}$ is the buoyancy at the surface.
874    
875  In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref%  In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref
876  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d  {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
877  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free  elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
878  surface' and `rigid lid' approaches are available.  surface' and `rigid lid' approaches are available.
879    
880  \subsubsection{Non-hydrostatic pressure}  \subsubsection{Non-hydrostatic pressure}
881    
882  Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{%  Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
883  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
884  (\ref{incompressible}), we deduce that:  (\ref{incompressible}), we deduce that:
885    
886  \begin{equation}  \begin{equation}
887  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{%  \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
888  \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .%  \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .
889  \vec{\mathbf{F}}  \label{eq:3d-invert}  \vec{\mathbf{F}}  \label{eq:3d-invert}
890  \end{equation}  \end{equation}
891    
# Line 893  coasts (in the ocean) and the bottom: Line 905  coasts (in the ocean) and the bottom:
905  \end{equation}  \end{equation}
906  where $\widehat{n}$ is a vector of unit length normal to the boundary. The  where $\widehat{n}$ is a vector of unit length normal to the boundary. The
907  kinematic condition (\ref{nonormalflow}) is also applied to the vertical  kinematic condition (\ref{nonormalflow}) is also applied to the vertical
908  velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $%  velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $
909  \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the  \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the
910  tangential component of velocity, $v_{T}$, at all solid boundaries,  tangential component of velocity, $v_{T}$, at all solid boundaries,
911  depending on the form chosen for the dissipative terms in the momentum  depending on the form chosen for the dissipative terms in the momentum
# Line 910  where Line 922  where
922  \begin{equation*}  \begin{equation*}
923  \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi  \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi
924  _{s}+\mathbf{\nabla }\phi _{hyd}\right)  _{s}+\mathbf{\nabla }\phi _{hyd}\right)
925  \end{equation*}%  \end{equation*}
926  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
927  (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can  (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
928  exploit classical 3D potential theory and, by introducing an appropriately  exploit classical 3D potential theory and, by introducing an appropriately
929  chosen $\delta $-function sheet of `source-charge', replace the inhomogenous  chosen $\delta $-function sheet of `source-charge', replace the
930  boundary condition on pressure by a homogeneous one. The source term $rhs$  inhomogeneous boundary condition on pressure by a homogeneous one. The
931  in (\ref{eq:3d-invert}) is the divergence of the vector $\vec{\mathbf{F}}.$  source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $
932  By simultaneously setting $%  \vec{\mathbf{F}}.$ By simultaneously setting $
933  \begin{array}{l}  \begin{array}{l}
934  \widehat{n}.\vec{\mathbf{F}}%  \widehat{n}.\vec{\mathbf{F}}
935  \end{array}%  \end{array}
936  =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following  =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
937  self-consistent but simpler homogenised Elliptic problem is obtained:  self-consistent but simpler homogenized Elliptic problem is obtained:
938    
939  \begin{equation*}  \begin{equation*}
940  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
941  \end{equation*}%  \end{equation*}
942  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
943  that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref%  that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref
944  {eq:inhom-neumann-nh}) the modified boundary condition becomes:  {eq:inhom-neumann-nh}) the modified boundary condition becomes:
945    
946  \begin{equation}  \begin{equation}
# Line 957  Many forms of momentum dissipation are a Line 969  Many forms of momentum dissipation are a
969  biharmonic frictions are commonly used:  biharmonic frictions are commonly used:
970    
971  \begin{equation}  \begin{equation}
972  D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}%  D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}
973  +A_{4}\nabla _{h}^{4}v  \label{eq:dissipation}  +A_{4}\nabla _{h}^{4}v  \label{eq:dissipation}
974  \end{equation}  \end{equation}
975  where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity  where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity
# Line 968  friction. These coefficients are the sam Line 980  friction. These coefficients are the sam
980    
981  The mixing terms for the temperature and salinity equations have a similar  The mixing terms for the temperature and salinity equations have a similar
982  form to that of momentum except that the diffusion tensor can be  form to that of momentum except that the diffusion tensor can be
983  non-diagonal and have varying coefficients. $\qquad $%  non-diagonal and have varying coefficients. $\qquad $
984  \begin{equation}  \begin{equation}
985  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
986  _{h}^{4}(T,S)  \label{eq:diffusion}  _{h}^{4}(T,S)  \label{eq:diffusion}
987  \end{equation}  \end{equation}
988  where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%  where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $
989  horizontal coefficient for biharmonic diffusion. In the simplest case where  horizontal coefficient for biharmonic diffusion. In the simplest case where
990  the subgrid-scale fluxes of heat and salt are parameterized with constant  the subgrid-scale fluxes of heat and salt are parameterized with constant
991  horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,  horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,
# Line 984  reduces to a diagonal matrix with consta Line 996  reduces to a diagonal matrix with consta
996  \begin{array}{ccc}  \begin{array}{ccc}
997  K_{h} & 0 & 0 \\  K_{h} & 0 & 0 \\
998  0 & K_{h} & 0 \\  0 & K_{h} & 0 \\
999  0 & 0 & K_{v}%  0 & 0 & K_{v}
1000  \end{array}  \end{array}
1001  \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}  \right) \qquad \qquad \qquad  \label{eq:diagonal-diffusion-tensor}
1002  \end{equation}  \end{equation}
# Line 994  salinity ... ). Line 1006  salinity ... ).
1006    
1007  \subsection{Vector invariant form}  \subsection{Vector invariant form}
1008    
1009  For some purposes it is advantageous to write momentum advection in eq(\ref%  For some purposes it is advantageous to write momentum advection in eq(\ref
1010  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
1011    
1012  \begin{equation}  \begin{equation}
1013  \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}
1014  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla
1015  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
1016  \label{eq:vi-identity}  \label{eq:vi-identity}
1017  \end{equation}%  \end{equation}
1018  This permits alternative numerical treatments of the non-linear terms based  This permits alternative numerical treatments of the non-linear terms based
1019  on their representation as a vorticity flux. Because gradients of coordinate  on their representation as a vorticity flux. Because gradients of coordinate
1020  vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit  vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit
1021  representation of the metric terms in (\ref{eq:gu-speherical}), (\ref%  representation of the metric terms in (\ref{eq:gu-speherical}), (\ref
1022  {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information  {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information
1023  about the geometry is contained in the areas and lengths of the volumes used  about the geometry is contained in the areas and lengths of the volumes used
1024  to discretize the model.  to discretize the model.
1025    
1026  \subsection{Adjoint}  \subsection{Adjoint}
1027    
1028  Tangent linear and adoint counterparts of the forward model and described in  Tangent linear and adjoint counterparts of the forward model and described
1029  Chapter 5.  in Chapter 5.
1030    
1031  % $Header$  % $Header$
1032  % $Name$  % $Name$
# Line 1028  coordinates} Line 1040  coordinates}
1040    
1041  The hydrostatic primitive equations (HPEs) in p-coordinates are:  The hydrostatic primitive equations (HPEs) in p-coordinates are:
1042  \begin{eqnarray}  \begin{eqnarray}
1043  \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}}
1044  _{h}+\mathbf{\nabla }_{p}\phi  &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
1045  \label{eq:atmos-mom} \\  \label{eq:atmos-mom} \\
1046  \frac{\partial \phi }{\partial p}+\alpha  &=&0  \label{eq-p-hydro-start} \\  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
1047  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1048  \partial p} &=&0  \label{eq:atmos-cont} \\  \partial p} &=&0  \label{eq:atmos-cont} \\
1049  p\alpha  &=&RT  \label{eq:atmos-eos} \\  p\alpha &=&RT  \label{eq:atmos-eos} \\
1050  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}
1051  \end{eqnarray}%  \end{eqnarray}
1052  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
1053  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot  surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
1054  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total  \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
1055  derivative, $f=2\Omega \sin lat$ is the Coriolis parameter, $\phi =gz$ is  derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter, $\phi =gz$ is
1056  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp%  the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
1057  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref%  }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
1058  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $%  {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
1059  e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $%  e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $
1060  p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.  p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.
1061    
1062  It is convenient to cast the heat equation in terms of potential temperature  It is convenient to cast the heat equation in terms of potential temperature
# Line 1052  $\theta $ so that it looks more like a g Line 1064  $\theta $ so that it looks more like a g
1064  Differentiating (\ref{eq:atmos-eos}) we get:  Differentiating (\ref{eq:atmos-eos}) we get:
1065  \begin{equation*}  \begin{equation*}
1066  p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}  p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
1067  \end{equation*}%  \end{equation*}
1068  which, when added to the heat equation (\ref{eq:atmos-heat}) and using $%  which, when added to the heat equation (\ref{eq:atmos-heat}) and using $
1069  c_{p}=c_{v}+R$, gives:  c_{p}=c_{v}+R$, gives:
1070  \begin{equation}  \begin{equation}
1071  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}  c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
1072  \label{eq-p-heat-interim}  \label{eq-p-heat-interim}
1073  \end{equation}%  \end{equation}
1074  Potential temperature is defined:  Potential temperature is defined:
1075  \begin{equation}  \begin{equation}
1076  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}  \theta =T(\frac{p_{c}}{p})^{\kappa }  \label{eq:potential-temp}
1077  \end{equation}%  \end{equation}
1078  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience  where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience
1079  we will make use of the Exner function $\Pi (p)$ which defined by:  we will make use of the Exner function $\Pi (p)$ which defined by:
1080  \begin{equation}  \begin{equation}
1081  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{Exner}  \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }  \label{Exner}
1082  \end{equation}%  \end{equation}
1083  The following relations will be useful and are easily expressed in terms of  The following relations will be useful and are easily expressed in terms of
1084  the Exner function:  the Exner function:
1085  \begin{equation*}  \begin{equation*}
1086  c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi  c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi
1087  }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{%  }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
1088  \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}%  \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}
1089  \frac{Dp}{Dt}  \frac{Dp}{Dt}
1090  \end{equation*}%  \end{equation*}
1091  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.  where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.
1092    
1093  The heat equation is obtained by noting that  The heat equation is obtained by noting that
1094  \begin{equation*}  \begin{equation*}
1095  c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta  c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta
1096  \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}  \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
1097  \end{equation*}  \end{equation*}
1098  and on substituting into (\ref{eq-p-heat-interim}) gives:  and on substituting into (\ref{eq-p-heat-interim}) gives:
1099  \begin{equation}  \begin{equation}
# Line 1090  and on substituting into (\ref{eq-p-heat Line 1102  and on substituting into (\ref{eq-p-heat
1102  \end{equation}  \end{equation}
1103  which is in conservative form.  which is in conservative form.
1104    
1105  For convenience in the model we prefer to step forward (\ref%  For convenience in the model we prefer to step forward (\ref
1106  {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).  {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).
1107    
1108  \subsubsection{Boundary conditions}  \subsubsection{Boundary conditions}
# Line 1134  _{o}(p_{o})=g~Z_{topo}$, defined: Line 1146  _{o}(p_{o})=g~Z_{topo}$, defined:
1146    
1147  The final form of the HPE's in p coordinates is then:  The final form of the HPE's in p coordinates is then:
1148  \begin{eqnarray}  \begin{eqnarray}
1149  \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}}
1150  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\  _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\
1151  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\  \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
1152  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1153  \partial p} &=&0 \\  \partial p} &=&0 \\
1154  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\  \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
1155  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}  \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }  \label{eq:atmos-prime}
# Line 1154  We review here the method by which the s Line 1166  We review here the method by which the s
1166  HPE's for the ocean written in z-coordinates are obtained. The  HPE's for the ocean written in z-coordinates are obtained. The
1167  non-Boussinesq equations for oceanic motion are:  non-Boussinesq equations for oceanic motion are:
1168  \begin{eqnarray}  \begin{eqnarray}
1169  \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}}
1170  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\
1171  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
1172  &=&\epsilon _{nh}\mathcal{F}_{w} \\  &=&\epsilon _{nh}\mathcal{F}_{w} \\
1173  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
1174  _{h}+\frac{\partial w}{\partial z} &=&0 \\  _{h}+\frac{\partial w}{\partial z} &=&0 \\
1175  \rho  &=&\rho (\theta ,S,p) \\  \rho &=&\rho (\theta ,S,p) \\
1176  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
1177  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
1178  \end{eqnarray}%  \end{eqnarray}
1179  These equations permit acoustics modes, inertia-gravity waves,  These equations permit acoustics modes, inertia-gravity waves,
1180  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline  non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline
1181  mode. As written, they cannot be integrated forward consistently - if we  mode. As written, they cannot be integrated forward consistently - if we
1182  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
1183  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref%  consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
1184  {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is  {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
1185  therefore necessary to manipulate the system as follows. Differentiating the  therefore necessary to manipulate the system as follows. Differentiating the
1186  EOS (equation of state) gives:  EOS (equation of state) gives:
# Line 1181  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp Line 1193  _{\theta ,S}\frac{Dp}{Dt}  \label{EOSexp
1193  \end{equation}  \end{equation}
1194    
1195  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 the
1196  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref%  reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref
1197  {eq-zns-cont} gives:  {eq-zns-cont} gives:
1198  \begin{equation}  \begin{equation}
1199  \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{
1200  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}  v}}+\partial _{z}w\approx 0  \label{eq-zns-pressure}
1201  \end{equation}  \end{equation}
1202  where we have used an approximation sign to indicate that we have assumed  where we have used an approximation sign to indicate that we have assumed
# Line 1192  adiabatic motion, dropping the $\frac{D\ Line 1204  adiabatic motion, dropping the $\frac{D\
1204  Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that  Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that
1205  can be explicitly integrated forward:  can be explicitly integrated forward:
1206  \begin{eqnarray}  \begin{eqnarray}
1207  \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}}
1208  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1209  \label{eq-cns-hmom} \\  \label{eq-cns-hmom} \\
1210  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}  \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
1211  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\  &=&\epsilon _{nh}\mathcal{F}_{w}  \label{eq-cns-hydro} \\
1212  \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{
1213  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\  v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-cns-cont} \\
1214  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\  \rho &=&\rho (\theta ,S,p)  \label{eq-cns-eos} \\
1215  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-cns-heat} \\
# Line 1211  wherever it appears in a product (ie. no Line 1223  wherever it appears in a product (ie. no
1223  `Boussinesq assumption'. The only term that then retains the full variation  `Boussinesq assumption'. The only term that then retains the full variation
1224  in $\rho $ is the gravitational acceleration:  in $\rho $ is the gravitational acceleration:
1225  \begin{eqnarray}  \begin{eqnarray}
1226  \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}}
1227  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1228  \label{eq-zcb-hmom} \\  \label{eq-zcb-hmom} \\
1229  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
1230  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1231  \label{eq-zcb-hydro} \\  \label{eq-zcb-hydro} \\
1232  \frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{%  \frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{
1233  \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-zcb-cont} \\  \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0  \label{eq-zcb-cont} \\
1234  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\  \rho &=&\rho (\theta ,S,p)  \label{eq-zcb-eos} \\
1235  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zcb-heat} \\
1236  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq-zcb-salt}
1237  \end{eqnarray}  \end{eqnarray}
1238  These equations still retain acoustic modes. But, because the  These equations still retain acoustic modes. But, because the
1239  ``compressible'' terms are linearized, the pressure equation \ref%  ``compressible'' terms are linearized, the pressure equation \ref
1240  {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent  {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
1241  term appears as a Helmholtz term in the non-hydrostatic pressure equation).  term appears as a Helmholtz term in the non-hydrostatic pressure equation).
1242  These are the \emph{truly} compressible Boussinesq equations. Note that the  These are the \emph{truly} compressible Boussinesq equations. Note that the
1243  EOS must have the same pressure dependency as the linearized pressure term,  EOS must have the same pressure dependency as the linearized pressure term,
1244  ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{%  ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{
1245  c_{s}^{2}}$, for consistency.  c_{s}^{2}}$, for consistency.
1246    
1247  \subsubsection{`Anelastic' z-coordinate equations}  \subsubsection{`Anelastic' z-coordinate equations}
1248    
1249  The anelastic approximation filters the acoustic mode by removing the  The anelastic approximation filters the acoustic mode by removing the
1250  time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}%  time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}
1251  ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}%  ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}
1252  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between  \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
1253  continuity and EOS. A better solution is to change the dependency on  continuity and EOS. A better solution is to change the dependency on
1254  pressure in the EOS by splitting the pressure into a reference function of  pressure in the EOS by splitting the pressure into a reference function of
1255  height and a perturbation:  height and a perturbation:
1256  \begin{equation*}  \begin{equation*}
1257  \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })  \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
1258  \end{equation*}  \end{equation*}
1259  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
1260  differentiating the EOS, the continuity equation then becomes:  differentiating the EOS, the continuity equation then becomes:
1261  \begin{equation*}  \begin{equation*}
1262  \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%  \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{
1263  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+
1264  \frac{\partial w}{\partial z}=0  \frac{\partial w}{\partial z}=0
1265  \end{equation*}  \end{equation*}
1266  If the time- and space-scales of the motions of interest are longer than  If the time- and space-scales of the motions of interest are longer than
1267  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},
1268  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and  \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and
1269  $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{%  $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{
1270  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta  Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
1271  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon  ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
1272  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation  _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
1273  and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the  and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the
1274  anelastic continuity equation:  anelastic continuity equation:
1275  \begin{equation}  \begin{equation}
1276  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-%  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-
1277  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}  \frac{g}{c_{s}^{2}}w=0  \label{eq-za-cont1}
1278  \end{equation}  \end{equation}
1279  A slightly different route leads to the quasi-Boussinesq continuity equation  A slightly different route leads to the quasi-Boussinesq continuity equation
1280  where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+%  where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
1281  \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }%  \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
1282  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:  _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:
1283  \begin{equation}  \begin{equation}
1284  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
1285  \partial \left( \rho _{o}w\right) }{\partial z}=0  \label{eq-za-cont2}  \partial \left( \rho _{o}w\right) }{\partial z}=0  \label{eq-za-cont2}
1286  \end{equation}  \end{equation}
1287  Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same  Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same
# Line 1278  equation if: Line 1290  equation if:
1290  \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}  \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}
1291  \end{equation}  \end{equation}
1292  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$  Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
1293  and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{%  and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{
1294  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The  g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
1295  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are  full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
1296  then:  then:
1297  \begin{eqnarray}  \begin{eqnarray}
1298  \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}}
1299  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1300  \label{eq-zab-hmom} \\  \label{eq-zab-hmom} \\
1301  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}%  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
1302  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1303  \label{eq-zab-hydro} \\  \label{eq-zab-hydro} \\
1304  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{%  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
1305  \partial \left( \rho _{o}w\right) }{\partial z} &=&0  \label{eq-zab-cont} \\  \partial \left( \rho _{o}w\right) }{\partial z} &=&0  \label{eq-zab-cont} \\
1306  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\  \rho &=&\rho (\theta ,S,p_{o}(z))  \label{eq-zab-eos} \\
1307  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq-zab-heat} \\
# Line 1302  Here, the objective is to drop the depth Line 1314  Here, the objective is to drop the depth
1314  technically, to also remove the dependence of $\rho $ on $p_{o}$. This would  technically, to also remove the dependence of $\rho $ on $p_{o}$. This would
1315  yield the ``truly'' incompressible Boussinesq equations:  yield the ``truly'' incompressible Boussinesq equations:
1316  \begin{eqnarray}  \begin{eqnarray}
1317  \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}}
1318  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1319  \label{eq-ztb-hmom} \\  \label{eq-ztb-hmom} \\
1320  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}%  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}
1321  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1322  \label{eq-ztb-hydro} \\  \label{eq-ztb-hydro} \\
1323  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}  \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
# Line 1324  retain compressibility effects in the de Line 1336  retain compressibility effects in the de
1336  density thus:  density thus:
1337  \begin{equation*}  \begin{equation*}
1338  \rho =\rho _{o}+\rho ^{\prime }  \rho =\rho _{o}+\rho ^{\prime }
1339  \end{equation*}%  \end{equation*}
1340  We then assert that variations with depth of $\rho _{o}$ are unimportant  We then assert that variations with depth of $\rho _{o}$ are unimportant
1341  while the compressible effects in $\rho ^{\prime }$ are:  while the compressible effects in $\rho ^{\prime }$ are:
1342  \begin{equation*}  \begin{equation*}
1343  \rho _{o}=\rho _{c}  \rho _{o}=\rho _{c}
1344  \end{equation*}%  \end{equation*}
1345  \begin{equation*}  \begin{equation*}
1346  \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}  \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}
1347  \end{equation*}%  \end{equation*}
1348  This then yields what we can call the semi-compressible Boussinesq  This then yields what we can call the semi-compressible Boussinesq
1349  equations:  equations:
1350  \begin{eqnarray}  \begin{eqnarray}
1351  \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}}
1352  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{%  _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{
1353  \mathcal{F}}}  \label{eq:ocean-mom} \\  \mathcal{F}}}  \label{eq:ocean-mom} \\
1354  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho  \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
1355  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}  _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
# Line 1348  _{c}}\frac{\partial p^{\prime }}{\partia Line 1360  _{c}}\frac{\partial p^{\prime }}{\partia
1360  \\  \\
1361  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta }  \label{eq:ocean-theta} \\
1362  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:ocean-salt}
1363  \end{eqnarray}%  \end{eqnarray}
1364  Note that the hydrostatic pressure of the resting fluid, including that  Note that the hydrostatic pressure of the resting fluid, including that
1365  associated with $\rho _{c}$, is subtracted out since it has no effect on the  associated with $\rho _{c}$, is subtracted out since it has no effect on the
1366  dynamics.  dynamics.
# Line 1372  In spherical coordinates, the velocity c Line 1384  In spherical coordinates, the velocity c
1384  and vertical direction respectively, are given by (see Fig.2) :  and vertical direction respectively, are given by (see Fig.2) :
1385    
1386  \begin{equation*}  \begin{equation*}
1387  u=r\cos \phi \frac{D\lambda }{Dt}  u=r\cos \varphi \frac{D\lambda }{Dt}
1388  \end{equation*}  \end{equation*}
1389    
1390  \begin{equation*}  \begin{equation*}
1391  v=r\frac{D\phi }{Dt}\qquad  v=r\frac{D\varphi }{Dt}\qquad
1392  \end{equation*}  \end{equation*}
1393  $\qquad \qquad \qquad \qquad $  $\qquad \qquad \qquad \qquad $
1394    
1395  \begin{equation*}  \begin{equation*}
1396  \dot{r}=\frac{Dr}{Dt}  \dot{r}=\frac{Dr}{Dt}
1397  \end{equation*}  \end{equation*}
1398    
1399  Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial  Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
1400  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
1401  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.
1402    
# Line 1392  The `grad' ($\nabla $) and `div' ($\nabl Line 1404  The `grad' ($\nabla $) and `div' ($\nabl
1404  spherical coordinates:  spherical coordinates:
1405    
1406  \begin{equation*}  \begin{equation*}
1407  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%  \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda }
1408  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%  ,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r}
1409  \right)  \right)
1410  \end{equation*}  \end{equation*}
1411    
1412  \begin{equation*}  \begin{equation*}
1413  \nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial  \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
1414  \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}  \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
1415  +\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}
1416  \end{equation*}  \end{equation*}
1417    
1418  %%%% \end{document}  %tci%\end{document}
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