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1 afe 1.19 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.18 2004/03/23 15:29:39 afe Exp $
2 cnh 1.2 % $Name: $
3 cnh 1.1
4 adcroft 1.4 %tci%\documentclass[12pt]{book}
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17     %tci%%TCIDATA{Language=American English}
18    
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29    
30     %tci%\begin{document}
31    
32     %tci%\tableofcontents
33    
34    
35 cnh 1.1 % Section: Overview
36    
37 afe 1.19 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.18 2004/03/23 15:29:39 afe Exp $
38 cnh 1.1 % $Name: $
39    
40 cnh 1.16 This document provides the reader with the information necessary to
41 cnh 1.1 carry out numerical experiments using MITgcm. It gives a comprehensive
42     description of the continuous equations on which the model is based, the
43     numerical algorithms the model employs and a description of the associated
44     program code. Along with the hydrodynamical kernel, physical and
45     biogeochemical parameterizations of key atmospheric and oceanic processes
46     are available. A number of examples illustrating the use of the model in
47     both process and general circulation studies of the atmosphere and ocean are
48     also presented.
49    
50 cnh 1.16 \section{Introduction}
51 afe 1.18 \begin{rawhtml}
52 afe 1.19 <!-- CMIREDIR:innovations: -->
53 afe 1.18 \end{rawhtml}
54    
55 cnh 1.16
56 cnh 1.1 MITgcm has a number of novel aspects:
57    
58     \begin{itemize}
59     \item it can be used to study both atmospheric and oceanic phenomena; one
60     hydrodynamical kernel is used to drive forward both atmospheric and oceanic
61 cnh 1.7 models - see fig \ref{fig:onemodel}
62 cnh 1.1
63 cnh 1.3 %% CNHbegin
64     \input{part1/one_model_figure}
65     %% CNHend
66    
67 cnh 1.1 \item it has a non-hydrostatic capability and so can be used to study both
68 cnh 1.7 small-scale and large scale processes - see fig \ref{fig:all-scales}
69 cnh 1.1
70 cnh 1.3 %% CNHbegin
71     \input{part1/all_scales_figure}
72     %% CNHend
73    
74 cnh 1.1 \item finite volume techniques are employed yielding an intuitive
75     discretization and support for the treatment of irregular geometries using
76 cnh 1.7 orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes}
77 cnh 1.3
78     %% CNHbegin
79     \input{part1/fvol_figure}
80     %% CNHend
81 cnh 1.1
82     \item tangent linear and adjoint counterparts are automatically maintained
83     along with the forward model, permitting sensitivity and optimization
84     studies.
85    
86     \item the model is developed to perform efficiently on a wide variety of
87     computational platforms.
88     \end{itemize}
89    
90 cnh 1.16 Key publications reporting on and charting the development of the model are
91     \cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99}:
92 cnh 1.12
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 cnh 1.16 Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling
103 cnh 1.12 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 cnh 1.14 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 cnh 1.12
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 cnh 1.1
137     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.
139    
140 afe 1.19 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.18 2004/03/23 15:29:39 afe Exp $
141 cnh 1.1 % $Name: $
142    
143     \section{Illustrations of the model in action}
144    
145     The MITgcm has been designed and used to model a wide range of phenomena,
146     from convection on the scale of meters in the ocean to the global pattern of
147 cnh 1.7 atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the
148 cnh 1.1 kinds of problems the model has been used to study, we briefly describe some
149     of them here. A more detailed description of the underlying formulation,
150     numerical algorithm and implementation that lie behind these calculations is
151 cnh 1.2 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
153 cnh 1.10 running Linux, together with a FORTRAN\ 77 compiler) and follow the examples
154 cnh 1.2 described in detail in the documentation.
155 cnh 1.1
156     \subsection{Global atmosphere: `Held-Suarez' benchmark}
157 afe 1.18 \begin{rawhtml}
158 afe 1.19 <!-- CMIREDIR:atmospheric_example: -->
159 afe 1.18 \end{rawhtml}
160    
161    
162 cnh 1.1
163 cnh 1.7 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.
165 cnh 1.2
166 cnh 1.7 Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
167 cnh 1.2 temperature field obtained using the atmospheric isomorph of MITgcm run at
168     2.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
169     (blue) and warm air along an equatorial band (red). Fully developed
170     baroclinic eddies spawned in the northern hemisphere storm track are
171     evident. There are no mountains or land-sea contrast in this calculation,
172     but you can easily put them in. The model is driven by relaxation to a
173     radiative-convective equilibrium profile, following the description set out
174     in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
175     there are no mountains or land-sea contrast.
176    
177 cnh 1.3 %% CNHbegin
178     \input{part1/cubic_eddies_figure}
179     %% CNHend
180    
181 cnh 1.2 As described in Adcroft (2001), a `cubed sphere' is used to discretize the
182 cnh 1.10 globe permitting a uniform griding and obviated the need to Fourier filter.
183 cnh 1.2 The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
184     grid, of which the cubed sphere is just one of many choices.
185 cnh 1.1
186 cnh 1.7 Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
187     wind from a 20-level configuration of
188 cnh 1.2 the model. It compares favorable with more conventional spatial
189 cnh 1.7 discretization approaches. The two plots show the field calculated using the
190     cube-sphere grid and the flow calculated using a regular, spherical polar
191     latitude-longitude grid. Both grids are supported within the model.
192 cnh 1.1
193 cnh 1.3 %% CNHbegin
194     \input{part1/hs_zave_u_figure}
195     %% CNHend
196    
197 cnh 1.2 \subsection{Ocean gyres}
198 afe 1.18 \begin{rawhtml}
199 afe 1.19 <!-- CMIREDIR:oceanic_example: -->
200 afe 1.18 \end{rawhtml}
201     \begin{rawhtml}
202 afe 1.19 <!-- CMIREDIR:ocean_gyres: -->
203 afe 1.18 \end{rawhtml}
204 cnh 1.1
205 cnh 1.2 Baroclinic instability is a ubiquitous process in the ocean, as well as the
206     atmosphere. Ocean eddies play an important role in modifying the
207     hydrographic structure and current systems of the oceans. Coarse resolution
208     models of the oceans cannot resolve the eddy field and yield rather broad,
209     diffusive patterns of ocean currents. But if the resolution of our models is
210     increased until the baroclinic instability process is resolved, numerical
211     solutions of a different and much more realistic kind, can be obtained.
212    
213 cnh 1.7 Figure \ref{fig:ocean-gyres} shows the surface temperature and velocity
214     field obtained from MITgcm run at $\frac{1}{6}^{\circ }$ horizontal
215     resolution on a $lat-lon$
216 cnh 1.2 grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
217     (to avoid the converging of meridian in northern latitudes). 21 vertical
218     levels are used in the vertical with a `lopped cell' representation of
219     topography. The development and propagation of anomalously warm and cold
220 cnh 1.7 eddies can be clearly seen in the Gulf Stream region. The transport of
221 cnh 1.2 warm water northward by the mean flow of the Gulf Stream is also clearly
222     visible.
223 cnh 1.1
224 cnh 1.3 %% CNHbegin
225 cnh 1.11 \input{part1/atl6_figure}
226 cnh 1.3 %% CNHend
227    
228    
229 cnh 1.1 \subsection{Global ocean circulation}
230 afe 1.18 \begin{rawhtml}
231 afe 1.19 <!-- CMIREDIR:global_ocean_circulation: -->
232 afe 1.18 \end{rawhtml}
233 cnh 1.1
234 cnh 1.7 Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
235     the surface of a 4$^{\circ }$
236 cnh 1.2 global ocean model run with 15 vertical levels. Lopped cells are used to
237     represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
238     }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
239     mixed boundary conditions on temperature and salinity at the surface. The
240     transfer properties of ocean eddies, convection and mixing is parameterized
241     in this model.
242    
243 cnh 1.7 Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
244     circulation of the global ocean in Sverdrups.
245 cnh 1.2
246 cnh 1.3 %%CNHbegin
247     \input{part1/global_circ_figure}
248     %%CNHend
249    
250 cnh 1.2 \subsection{Convection and mixing over topography}
251 afe 1.18 \begin{rawhtml}
252 afe 1.19 <!-- CMIREDIR:mixing_over_topography: -->
253 afe 1.18 \end{rawhtml}
254    
255 cnh 1.2
256     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
258     than the width of the cooling region. Rather than gravity plumes, the
259     mechanism for moving dense fluid down the shelf is then through geostrophic
260 adcroft 1.9 eddies. The simulation shown in the figure \ref{fig:convect-and-topo}
261 cnh 1.7 (blue is cold dense fluid, red is
262 cnh 1.2 warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
263     trigger convection by surface cooling. The cold, dense water falls down the
264     slope but is deflected along the slope by rotation. It is found that
265     entrainment in the vertical plane is reduced when rotational control is
266     strong, and replaced by lateral entrainment due to the baroclinic
267     instability of the along-slope current.
268 cnh 1.1
269 cnh 1.3 %%CNHbegin
270     \input{part1/convect_and_topo}
271     %%CNHend
272    
273 cnh 1.1 \subsection{Boundary forced internal waves}
274 afe 1.18 \begin{rawhtml}
275 afe 1.19 <!-- CMIREDIR:boundary_forced_internal_waves: -->
276 afe 1.18 \end{rawhtml}
277 cnh 1.1
278 cnh 1.2 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
280     dynamics and mixing in oceanic canyons and ridges driven by large amplitude
281     barotropic tidal currents imposed through open boundary conditions.
282    
283 cnh 1.7 Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope
284     topographic variations on
285 cnh 1.2 internal wave breaking - the cross-slope velocity is in color, the density
286     contoured. The internal waves are excited by application of open boundary
287 cnh 1.7 conditions on the left. They propagate to the sloping boundary (represented
288 cnh 1.2 using MITgcm's finite volume spatial discretization) where they break under
289     nonhydrostatic dynamics.
290    
291 cnh 1.3 %%CNHbegin
292     \input{part1/boundary_forced_waves}
293     %%CNHend
294    
295 cnh 1.2 \subsection{Parameter sensitivity using the adjoint of MITgcm}
296 afe 1.18 \begin{rawhtml}
297 afe 1.19 <!-- CMIREDIR:parameter_sensitivity: -->
298 afe 1.18 \end{rawhtml}
299 cnh 1.2
300     Forward and tangent linear counterparts of MITgcm are supported using an
301     `automatic adjoint compiler'. These can be used in parameter sensitivity and
302     data assimilation studies.
303    
304 cnh 1.7 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
306 cnh 1.10 of the overturning stream-function shown in figure \ref{fig:large-scale-circ}
307 cnh 1.7 at 60$^{\circ }$N and $
308     \mathcal{H}(\lambda,\varphi)$ is the mean, local air-sea heat flux over
309     a 100 year period. We see that $J$ is
310 cnh 1.2 sensitive to heat fluxes over the Labrador Sea, one of the important sources
311     of deep water for the thermohaline circulations. This calculation also
312     yields sensitivities to all other model parameters.
313    
314 cnh 1.3 %%CNHbegin
315     \input{part1/adj_hf_ocean_figure}
316     %%CNHend
317    
318 cnh 1.2 \subsection{Global state estimation of the ocean}
319 afe 1.18 \begin{rawhtml}
320 afe 1.19 <!-- CMIREDIR:global_state_estimation: -->
321 afe 1.18 \end{rawhtml}
322    
323 cnh 1.2
324     An important application of MITgcm is in state estimation of the global
325     ocean circulation. An appropriately defined `cost function', which measures
326     the departure of the model from observations (both remotely sensed and
327 cnh 1.10 in-situ) over an interval of time, is minimized by adjusting `control
328 cnh 1.2 parameters' such as air-sea fluxes, the wind field, the initial conditions
329 cnh 1.15 etc. Figure \ref{fig:assimilated-globes} shows the large scale planetary
330     circulation and a Hopf-Muller plot of Equatorial sea-surface height.
331     Both are obtained from assimilation bringing the model in to
332 cnh 1.7 consistency with altimetric and in-situ observations over the period
333 cnh 1.15 1992-1997.
334 cnh 1.2
335 cnh 1.3 %% CNHbegin
336 cnh 1.13 \input{part1/assim_figure}
337 cnh 1.3 %% CNHend
338    
339 cnh 1.2 \subsection{Ocean biogeochemical cycles}
340 afe 1.18 \begin{rawhtml}
341 afe 1.19 <!-- CMIREDIR:ocean_biogeo_cycles: -->
342 afe 1.18 \end{rawhtml}
343 cnh 1.2
344     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
346     forcing and upper ocean circulation on the fluxes of carbon dioxide and
347 cnh 1.7 oxygen between the ocean and atmosphere. Figure \ref{fig:biogeo} shows
348     the annual air-sea flux of oxygen and its relation to density outcrops in
349     the southern oceans from a single year of a global, interannually varying
350     simulation. The simulation is run at $1^{\circ}\times1^{\circ}$ resolution
351     telescoping to $\frac{1}{3}^{\circ}\times\frac{1}{3}^{\circ}$ in the tropics (not shown).
352 cnh 1.2
353 cnh 1.3 %%CNHbegin
354     \input{part1/biogeo_figure}
355     %%CNHend
356 cnh 1.2
357     \subsection{Simulations of laboratory experiments}
358 afe 1.18 \begin{rawhtml}
359 afe 1.19 <!-- CMIREDIR:classroom_exp: -->
360 afe 1.18 \end{rawhtml}
361 cnh 1.2
362 cnh 1.7 Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
363 edhill 1.17 laboratory experiment inquiring into the dynamics of the Antarctic Circumpolar Current (ACC). An
364 cnh 1.2 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
366     and thermal forcing creates a lens of fluid which becomes baroclinically
367     unstable. The stratification and depth of penetration of the lens is
368 cnh 1.7 arrested by its instability in a process analogous to that which sets the
369 cnh 1.2 stratification of the ACC.
370 cnh 1.1
371 cnh 1.3 %%CNHbegin
372     \input{part1/lab_figure}
373     %%CNHend
374    
375 afe 1.19 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.18 2004/03/23 15:29:39 afe Exp $
376 cnh 1.1 % $Name: $
377    
378     \section{Continuous equations in `r' coordinates}
379 afe 1.18 \begin{rawhtml}
380 afe 1.19 <!-- CMIREDIR:z-p_isomorphism: -->
381 afe 1.18 \end{rawhtml}
382 cnh 1.1
383     To render atmosphere and ocean models from one dynamical core we exploit
384     `isomorphisms' between equation sets that govern the evolution of the
385 cnh 1.7 respective fluids - see figure \ref{fig:isomorphic-equations}.
386     One system of hydrodynamical equations is written down
387 cnh 1.1 and encoded. The model variables have different interpretations depending on
388     whether the atmosphere or ocean is being studied. Thus, for example, the
389     vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
390 edhill 1.17 modeling the atmosphere (right hand side of figure \ref{fig:isomorphic-equations})
391     and height, $z$, if we are modeling the ocean (left hand side of figure
392 cnh 1.7 \ref{fig:isomorphic-equations}).
393 cnh 1.1
394 cnh 1.3 %%CNHbegin
395     \input{part1/zandpcoord_figure.tex}
396     %%CNHend
397    
398 cnh 1.1 The state of the fluid at any time is characterized by the distribution of
399     velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a
400     `geopotential' $\phi $ and density $\rho =\rho (\theta ,S,p)$ which may
401     depend on $\theta $, $S$, and $p$. The equations that govern the evolution
402     of these fields, obtained by applying the laws of classical mechanics and
403     thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of
404 cnh 1.7 a generic vertical coordinate, $r$, so that the appropriate
405     kinematic boundary conditions can be applied isomorphically
406     see figure \ref{fig:zandp-vert-coord}.
407 cnh 1.1
408 cnh 1.3 %%CNHbegin
409     \input{part1/vertcoord_figure.tex}
410     %%CNHend
411    
412 cnh 1.1 \begin{equation*}
413 adcroft 1.4 \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}}}
415 cnh 1.8 \text{ horizontal mtm} \label{eq:horizontal_mtm}
416 cnh 1.1 \end{equation*}
417    
418 cnh 1.8 \begin{equation}
419 adcroft 1.4 \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
420 cnh 1.1 v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
421 cnh 1.8 vertical mtm} \label{eq:vertical_mtm}
422     \end{equation}
423 cnh 1.1
424     \begin{equation}
425 adcroft 1.4 \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
426 cnh 1.8 \partial r}=0\text{ continuity} \label{eq:continuity}
427 cnh 1.1 \end{equation}
428    
429 cnh 1.8 \begin{equation}
430     b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
431     \end{equation}
432 cnh 1.1
433 cnh 1.8 \begin{equation}
434 cnh 1.2 \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
435 cnh 1.8 \label{eq:potential_temperature}
436     \end{equation}
437 cnh 1.1
438 cnh 1.8 \begin{equation}
439 cnh 1.2 \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
440 adcroft 1.9 \label{eq:humidity_salt}
441 cnh 1.8 \end{equation}
442 cnh 1.1
443     Here:
444    
445     \begin{equation*}
446 cnh 1.2 r\text{ is the vertical coordinate}
447 cnh 1.1 \end{equation*}
448    
449     \begin{equation*}
450     \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{
451 cnh 1.2 is the total derivative}
452 cnh 1.1 \end{equation*}
453    
454     \begin{equation*}
455 adcroft 1.4 \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
456 cnh 1.2 \text{ is the `grad' operator}
457 cnh 1.1 \end{equation*}
458 adcroft 1.4 with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}
459 cnh 1.1 \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
460     is a unit vector in the vertical
461    
462     \begin{equation*}
463 cnh 1.2 t\text{ is time}
464 cnh 1.1 \end{equation*}
465    
466     \begin{equation*}
467     \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the
468 cnh 1.2 velocity}
469 cnh 1.1 \end{equation*}
470    
471     \begin{equation*}
472 cnh 1.2 \phi \text{ is the `pressure'/`geopotential'}
473 cnh 1.1 \end{equation*}
474    
475     \begin{equation*}
476 cnh 1.2 \vec{\Omega}\text{ is the Earth's rotation}
477 cnh 1.1 \end{equation*}
478    
479     \begin{equation*}
480 cnh 1.2 b\text{ is the `buoyancy'}
481 cnh 1.1 \end{equation*}
482    
483     \begin{equation*}
484 cnh 1.2 \theta \text{ is potential temperature}
485 cnh 1.1 \end{equation*}
486    
487     \begin{equation*}
488 cnh 1.2 S\text{ is specific humidity in the atmosphere; salinity in the ocean}
489 cnh 1.1 \end{equation*}
490    
491     \begin{equation*}
492 adcroft 1.4 \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{
493 cnh 1.1 \mathbf{v}}
494     \end{equation*}
495    
496     \begin{equation*}
497 cnh 1.2 \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
498 cnh 1.1 \end{equation*}
499    
500     \begin{equation*}
501     \mathcal{Q}_{S}\mathcal{\ }\text{are forcing and dissipation of }S
502     \end{equation*}
503    
504     The $\mathcal{F}^{\prime }s$ and $\mathcal{Q}^{\prime }s$ are provided by
505 cnh 1.7 `physics' and forcing packages for atmosphere and ocean. These are described
506     in later chapters.
507 cnh 1.1
508     \subsection{Kinematic Boundary conditions}
509    
510     \subsubsection{vertical}
511    
512 cnh 1.7 at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
513 cnh 1.1
514     \begin{equation}
515 edhill 1.17 \dot{r}=0 \text{\ at\ } r=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
516 cnh 1.1 \label{eq:fixedbc}
517     \end{equation}
518    
519     \begin{equation}
520 edhill 1.17 \dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
521 cnh 1.10 (ocean surface,bottom of the atmosphere)} \label{eq:movingbc}
522 cnh 1.1 \end{equation}
523    
524     Here
525    
526     \begin{equation*}
527 cnh 1.2 R_{moving}=R_{o}+\eta
528 cnh 1.1 \end{equation*}
529     where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on
530     whether we are in the atmosphere or ocean) of the `moving surface' in the
531     resting fluid and $\eta $ is the departure from $R_{o}(x,y)$ in the presence
532     of motion.
533    
534     \subsubsection{horizontal}
535    
536     \begin{equation}
537     \vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0 \label{eq:noflow}
538 adcroft 1.4 \end{equation}
539 cnh 1.1 where $\vec{\mathbf{n}}$ is the normal to a solid boundary.
540    
541     \subsection{Atmosphere}
542    
543 cnh 1.7 In the atmosphere, (see figure \ref{fig:zandp-vert-coord}), we interpret:
544 cnh 1.1
545     \begin{equation}
546     r=p\text{ is the pressure} \label{eq:atmos-r}
547     \end{equation}
548    
549     \begin{equation}
550     \dot{r}=\frac{Dp}{Dt}=\omega \text{ is the vertical velocity in }p\text{
551     coordinates} \label{eq:atmos-omega}
552     \end{equation}
553    
554     \begin{equation}
555     \phi =g\,z\text{ is the geopotential height} \label{eq:atmos-phi}
556     \end{equation}
557    
558     \begin{equation}
559     b=\frac{\partial \Pi }{\partial p}\theta \text{ is the buoyancy}
560     \label{eq:atmos-b}
561     \end{equation}
562    
563     \begin{equation}
564     \theta =T(\frac{p_{c}}{p})^{\kappa }\text{ is potential temperature}
565     \label{eq:atmos-theta}
566     \end{equation}
567    
568     \begin{equation}
569     S=q,\text{ is the specific humidity} \label{eq:atmos-s}
570     \end{equation}
571     where
572    
573     \begin{equation*}
574     T\text{ is absolute temperature}
575 adcroft 1.4 \end{equation*}
576 cnh 1.1 \begin{equation*}
577     p\text{ is the pressure}
578 adcroft 1.4 \end{equation*}
579 cnh 1.1 \begin{eqnarray*}
580     &&z\text{ is the height of the pressure surface} \\
581     &&g\text{ is the acceleration due to gravity}
582     \end{eqnarray*}
583    
584     In the above the ideal gas law, $p=\rho RT$, has been expressed in terms of
585     the Exner function $\Pi (p)$ given by (see Appendix Atmosphere)
586     \begin{equation}
587     \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa } \label{eq:exner}
588 adcroft 1.4 \end{equation}
589 cnh 1.1 where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas
590     constant and $c_{p}$ the specific heat of air at constant pressure.
591    
592     At the top of the atmosphere (which is `fixed' in our $r$ coordinate):
593    
594     \begin{equation*}
595 cnh 1.2 R_{fixed}=p_{top}=0
596 cnh 1.1 \end{equation*}
597     In a resting atmosphere the elevation of the mountains at the bottom is
598     given by
599     \begin{equation*}
600 cnh 1.2 R_{moving}=R_{o}(x,y)=p_{o}(x,y)
601 cnh 1.1 \end{equation*}
602     i.e. the (hydrostatic) pressure at the top of the mountains in a resting
603     atmosphere.
604    
605     The boundary conditions at top and bottom are given by:
606    
607     \begin{eqnarray}
608     &&\omega =0~\text{at }r=R_{fixed} \text{ (top of the atmosphere)}
609     \label{eq:fixed-bc-atmos} \\
610     \omega &=&\frac{Dp_{s}}{Dt}\text{; at }r=R_{moving}\text{ (bottom of the
611     atmosphere)} \label{eq:moving-bc-atmos}
612     \end{eqnarray}
613    
614 adcroft 1.9 Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt})
615 cnh 1.8 yields a consistent set of atmospheric equations which, for convenience, are written out in $p$
616 cnh 1.1 coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}).
617    
618     \subsection{Ocean}
619    
620     In the ocean we interpret:
621     \begin{eqnarray}
622     r &=&z\text{ is the height} \label{eq:ocean-z} \\
623     \dot{r} &=&\frac{Dz}{Dt}=w\text{ is the vertical velocity}
624     \label{eq:ocean-w} \\
625     \phi &=&\frac{p}{\rho _{c}}\text{ is the pressure} \label{eq:ocean-p} \\
626     b(\theta ,S,r) &=&\frac{g}{\rho _{c}}\left( \rho (\theta ,S,r)-\rho
627     _{c}\right) \text{ is the buoyancy} \label{eq:ocean-b}
628     \end{eqnarray}
629     where $\rho _{c}$ is a fixed reference density of water and $g$ is the
630     acceleration due to gravity.\noindent
631    
632     In the above
633    
634     At the bottom of the ocean: $R_{fixed}(x,y)=-H(x,y)$.
635    
636     The surface of the ocean is given by: $R_{moving}=\eta $
637    
638 adcroft 1.4 The position of the resting free surface of the ocean is given by $
639 cnh 1.1 R_{o}=Z_{o}=0$.
640    
641     Boundary conditions are:
642    
643     \begin{eqnarray}
644     w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)} \label{eq:fixed-bc-ocean}
645     \\
646 adcroft 1.4 w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)
647 cnh 1.1 \label{eq:moving-bc-ocean}}
648     \end{eqnarray}
649     where $\eta $ is the elevation of the free surface.
650    
651 adcroft 1.9 Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yield a consistent set
652 cnh 1.8 of oceanic equations
653 cnh 1.1 which, for convenience, are written out in $z$ coordinates in Appendix Ocean
654     - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
655    
656     \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
657     Non-hydrostatic forms}
658 afe 1.18 \begin{rawhtml}
659 afe 1.19 <!-- CMIREDIR:non_hydrostatic: -->
660 afe 1.18 \end{rawhtml}
661    
662 cnh 1.1
663     Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
664    
665     \begin{equation}
666     \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
667     \label{eq:phi-split}
668 adcroft 1.4 \end{equation}
669 cnh 1.8 and write eq(\ref{eq:incompressible}) in the form:
670 cnh 1.1
671     \begin{equation}
672     \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
673     _{s}+\mathbf{\nabla }_{h}\phi _{hyd}+\epsilon _{nh}\mathbf{\nabla }_{h}\phi
674     _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}} \label{eq:mom-h}
675     \end{equation}
676    
677     \begin{equation}
678     \frac{\partial \phi _{hyd}}{\partial r}=-b \label{eq:hydrostatic}
679     \end{equation}
680    
681     \begin{equation}
682 adcroft 1.4 \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{
683 cnh 1.1 \partial r}=G_{\dot{r}} \label{eq:mom-w}
684     \end{equation}
685     Here $\epsilon _{nh}$ is a non-hydrostatic parameter.
686    
687 adcroft 1.4 The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref
688 cnh 1.1 {eq:mom-h}) and (\ref{eq:mom-w}) represent advective, metric and Coriolis
689 adcroft 1.4 terms in the momentum equations. In spherical coordinates they take the form
690     \footnote{
691 cnh 1.1 In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms
692 adcroft 1.4 in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref
693 cnh 1.1 {eq:gw-spherical}) are omitted; the singly-underlined terms are included in
694 adcroft 1.4 the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (
695 cnh 1.1 \textbf{NH}) includes all terms.} - see Marshall et al 1997a for a full
696     discussion:
697    
698     \begin{equation}
699     \left.
700     \begin{tabular}{l}
701     $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\
702 cnh 1.6 $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} $
703 cnh 1.1 \\
704 cnh 1.6 $-\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\right\} $
705 cnh 1.1 \\
706 adcroft 1.4 $+\mathcal{F}_{u}$
707     \end{tabular}
708 cnh 1.1 \ \right\} \left\{
709     \begin{tabular}{l}
710     \textit{advection} \\
711     \textit{metric} \\
712     \textit{Coriolis} \\
713 adcroft 1.4 \textit{\ Forcing/Dissipation}
714     \end{tabular}
715 cnh 1.2 \ \right. \qquad \label{eq:gu-speherical}
716 cnh 1.1 \end{equation}
717    
718     \begin{equation}
719     \left.
720     \begin{tabular}{l}
721     $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
722 cnh 1.6 $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\}
723 cnh 1.1 $ \\
724 cnh 1.6 $-\left\{ -2\Omega u\sin \varphi \right\} $ \\
725 adcroft 1.4 $+\mathcal{F}_{v}$
726     \end{tabular}
727 cnh 1.1 \ \right\} \left\{
728     \begin{tabular}{l}
729     \textit{advection} \\
730     \textit{metric} \\
731     \textit{Coriolis} \\
732 adcroft 1.4 \textit{\ Forcing/Dissipation}
733     \end{tabular}
734 cnh 1.2 \ \right. \qquad \label{eq:gv-spherical}
735 adcroft 1.4 \end{equation}
736 cnh 1.2 \qquad \qquad \qquad \qquad \qquad
737 cnh 1.1
738     \begin{equation}
739     \left.
740     \begin{tabular}{l}
741     $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
742     $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
743 cnh 1.6 ${+}\underline{{2\Omega u\cos \varphi}}$ \\
744 adcroft 1.4 $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
745     \end{tabular}
746 cnh 1.1 \ \right\} \left\{
747     \begin{tabular}{l}
748     \textit{advection} \\
749     \textit{metric} \\
750     \textit{Coriolis} \\
751 adcroft 1.4 \textit{\ Forcing/Dissipation}
752     \end{tabular}
753 cnh 1.2 \ \right. \label{eq:gw-spherical}
754 adcroft 1.4 \end{equation}
755 cnh 1.2 \qquad \qquad \qquad \qquad \qquad
756 cnh 1.1
757 cnh 1.6 In the above `${r}$' is the distance from the center of the earth and `$\varphi$
758 cnh 1.1 ' is latitude.
759    
760     Grad and div operators in spherical coordinates are defined in appendix
761 adcroft 1.4 OPERATORS.
762 cnh 1.1
763 cnh 1.3 %%CNHbegin
764     \input{part1/sphere_coord_figure.tex}
765     %%CNHend
766    
767 cnh 1.1 \subsubsection{Shallow atmosphere approximation}
768    
769     Most models are based on the `hydrostatic primitive equations' (HPE's) in
770     which the vertical momentum equation is reduced to a statement of
771     hydrostatic balance and the `traditional approximation' is made in which the
772     Coriolis force is treated approximately and the shallow atmosphere
773     approximation is made.\ The MITgcm need not make the `traditional
774     approximation'. To be able to support consistent non-hydrostatic forms the
775 adcroft 1.4 shallow atmosphere approximation can be relaxed - when dividing through by $
776 cnh 1.2 r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
777 cnh 1.1 the radius of the earth.
778    
779     \subsubsection{Hydrostatic and quasi-hydrostatic forms}
780 cnh 1.7 \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
781 cnh 1.1
782     These are discussed at length in Marshall et al (1997a).
783    
784     In the `hydrostatic primitive equations' (\textbf{HPE)} all the underlined
785     terms in Eqs. (\ref{eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical})
786     are neglected and `${r}$' is replaced by `$a$', the mean radius of the
787     earth. Once the pressure is found at one level - e.g. by inverting a 2-d
788     Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be
789 adcroft 1.4 computed at all other levels by integration of the hydrostatic relation, eq(
790 cnh 1.1 \ref{eq:hydrostatic}).
791    
792     In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
793     gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
794 cnh 1.6 \varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
795 adcroft 1.4 contribution to the pressure field: only the terms underlined twice in Eqs. (
796 cnh 1.1 \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero
797     and, simultaneously, the shallow atmosphere approximation is relaxed. In
798     \textbf{QH}\ \textit{all} the metric terms are retained and the full
799     variation of the radial position of a particle monitored. The \textbf{QH}\
800     vertical momentum equation (\ref{eq:mom-w}) becomes:
801    
802     \begin{equation*}
803 cnh 1.6 \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi
804 cnh 1.1 \end{equation*}
805     making a small correction to the hydrostatic pressure.
806    
807     \textbf{QH} has good energetic credentials - they are the same as for
808     \textbf{HPE}. Importantly, however, it has the same angular momentum
809     principle as the full non-hydrostatic model (\textbf{NH)} - see Marshall
810     et.al., 1997a. As in \textbf{HPE }only a 2-d elliptic problem need be solved.
811    
812     \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}
813    
814     The MIT model presently supports a full non-hydrostatic ocean isomorph, but
815     only a quasi-non-hydrostatic atmospheric isomorph.
816    
817     \paragraph{Non-hydrostatic Ocean}
818    
819 adcroft 1.4 In the non-hydrostatic ocean model all terms in equations Eqs.(\ref
820 cnh 1.1 {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A
821     three dimensional elliptic equation must be solved subject to Neumann
822     boundary conditions (see below). It is important to note that use of the
823     full \textbf{NH} does not admit any new `fast' waves in to the system - the
824 cnh 1.8 incompressible condition eq(\ref{eq:continuity}) has already filtered out
825 cnh 1.1 acoustic modes. It does, however, ensure that the gravity waves are treated
826     accurately with an exact dispersion relation. The \textbf{NH} set has a
827     complete angular momentum principle and consistent energetics - see White
828     and Bromley, 1995; Marshall et.al.\ 1997a.
829    
830     \paragraph{Quasi-nonhydrostatic Atmosphere}
831    
832 adcroft 1.4 In the non-hydrostatic version of our atmospheric model we approximate $\dot{
833 cnh 1.1 r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical})
834     (but only here) by:
835    
836     \begin{equation}
837     \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt} \label{eq:quasi-nh-w}
838 adcroft 1.4 \end{equation}
839 cnh 1.1 where $p_{hy}$ is the hydrostatic pressure.
840    
841     \subsubsection{Summary of equation sets supported by model}
842    
843     \paragraph{Atmosphere}
844    
845     Hydrostatic, and quasi-hydrostatic and quasi non-hydrostatic forms of the
846     compressible non-Boussinesq equations in $p-$coordinates are supported.
847    
848     \subparagraph{Hydrostatic and quasi-hydrostatic}
849    
850     The hydrostatic set is written out in $p-$coordinates in appendix Atmosphere
851     - see eq(\ref{eq:atmos-prime}).
852    
853     \subparagraph{Quasi-nonhydrostatic}
854    
855     A quasi-nonhydrostatic form is also supported.
856    
857     \paragraph{Ocean}
858    
859     \subparagraph{Hydrostatic and quasi-hydrostatic}
860    
861     Hydrostatic, and quasi-hydrostatic forms of the incompressible Boussinesq
862     equations in $z-$coordinates are supported.
863    
864     \subparagraph{Non-hydrostatic}
865    
866 adcroft 1.4 Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
867     coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref
868 cnh 1.1 {eq:ocean-salt}).
869    
870     \subsection{Solution strategy}
871    
872 adcroft 1.4 The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
873 cnh 1.8 NH} models is summarized in Figure \ref{fig:solution-strategy}.
874     Under all dynamics, a 2-d elliptic equation is
875 cnh 1.1 first solved to find the surface pressure and the hydrostatic pressure at
876     any level computed from the weight of fluid above. Under \textbf{HPE} and
877     \textbf{QH} dynamics, the horizontal momentum equations are then stepped
878     forward and $\dot{r}$ found from continuity. Under \textbf{NH} dynamics a
879     3-d elliptic equation must be solved for the non-hydrostatic pressure before
880     stepping forward the horizontal momentum equations; $\dot{r}$ is found by
881     stepping forward the vertical momentum equation.
882    
883 cnh 1.3 %%CNHbegin
884     \input{part1/solution_strategy_figure.tex}
885     %%CNHend
886    
887 cnh 1.1 There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
888 cnh 1.6 course, some complication that goes with the inclusion of $\cos \varphi \ $
889 cnh 1.1 Coriolis terms and the relaxation of the shallow atmosphere approximation.
890     But this leads to negligible increase in computation. In \textbf{NH}, in
891     contrast, one additional elliptic equation - a three-dimensional one - must
892     be inverted for $p_{nh}$. However the `overhead' of the \textbf{NH} model is
893     essentially negligible in the hydrostatic limit (see detailed discussion in
894     Marshall et al, 1997) resulting in a non-hydrostatic algorithm that, in the
895     hydrostatic limit, is as computationally economic as the \textbf{HPEs}.
896    
897     \subsection{Finding the pressure field}
898 cnh 1.7 \label{sec:finding_the_pressure_field}
899 cnh 1.1
900     Unlike the prognostic variables $u$, $v$, $w$, $\theta $ and $S$, the
901     pressure field must be obtained diagnostically. We proceed, as before, by
902     dividing the total (pressure/geo) potential in to three parts, a surface
903     part, $\phi _{s}(x,y)$, a hydrostatic part $\phi _{hyd}(x,y,r)$ and a
904     non-hydrostatic part $\phi _{nh}(x,y,r)$, as in (\ref{eq:phi-split}), and
905     writing the momentum equation as in (\ref{eq:mom-h}).
906    
907     \subsubsection{Hydrostatic pressure}
908    
909     Hydrostatic pressure is obtained by integrating (\ref{eq:hydrostatic})
910     vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:
911    
912     \begin{equation*}
913 adcroft 1.4 \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}
914 cnh 1.2 \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
915 cnh 1.1 \end{equation*}
916     and so
917    
918     \begin{equation}
919     \phi _{hyd}(x,y,r)=\int_{r}^{R_{o}}bdr \label{eq:hydro-phi}
920     \end{equation}
921    
922     The model can be easily modified to accommodate a loading term (e.g
923     atmospheric pressure pushing down on the ocean's surface) by setting:
924    
925     \begin{equation}
926     \phi _{hyd}(r=R_{o})=loading \label{eq:loading}
927     \end{equation}
928    
929     \subsubsection{Surface pressure}
930    
931 cnh 1.8 The surface pressure equation can be obtained by integrating continuity,
932     (\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
933 cnh 1.1
934     \begin{equation*}
935 adcroft 1.4 \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
936 cnh 1.2 }_{h}+\partial _{r}\dot{r}\right) dr=0
937 cnh 1.1 \end{equation*}
938    
939     Thus:
940    
941     \begin{equation*}
942     \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
943 adcroft 1.4 +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}
944 cnh 1.2 _{h}dr=0
945 cnh 1.1 \end{equation*}
946 adcroft 1.4 where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $
947 cnh 1.1 r $. The above can be rearranged to yield, using Leibnitz's theorem:
948    
949     \begin{equation}
950     \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot
951     \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}
952     \label{eq:free-surface}
953 adcroft 1.4 \end{equation}
954 cnh 1.1 where we have incorporated a source term.
955    
956     Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
957 cnh 1.8 (atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
958 cnh 1.1 be written
959     \begin{equation}
960 cnh 1.2 \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
961 cnh 1.1 \label{eq:phi-surf}
962 adcroft 1.4 \end{equation}
963 cnh 1.1 where $b_{s}$ is the buoyancy at the surface.
964    
965 cnh 1.8 In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref
966 cnh 1.1 {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
967     elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
968     surface' and `rigid lid' approaches are available.
969    
970     \subsubsection{Non-hydrostatic pressure}
971    
972 cnh 1.8 Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
973     $\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
974     (\ref{eq:continuity}), we deduce that:
975 cnh 1.1
976     \begin{equation}
977 adcroft 1.4 \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
978     \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .
979 cnh 1.1 \vec{\mathbf{F}} \label{eq:3d-invert}
980     \end{equation}
981    
982     For a given rhs this 3-d elliptic equation must be inverted for $\phi _{nh}$
983     subject to appropriate choice of boundary conditions. This method is usually
984     called \textit{The Pressure Method} [Harlow and Welch, 1965; Williams, 1969;
985     Potter, 1976]. In the hydrostatic primitive equations case (\textbf{HPE}),
986     the 3-d problem does not need to be solved.
987    
988     \paragraph{Boundary Conditions}
989    
990     We apply the condition of no normal flow through all solid boundaries - the
991     coasts (in the ocean) and the bottom:
992    
993     \begin{equation}
994     \vec{\mathbf{v}}.\widehat{n}=0 \label{nonormalflow}
995     \end{equation}
996     where $\widehat{n}$ is a vector of unit length normal to the boundary. The
997     kinematic condition (\ref{nonormalflow}) is also applied to the vertical
998 adcroft 1.4 velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $
999 cnh 1.1 \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the
1000     tangential component of velocity, $v_{T}$, at all solid boundaries,
1001     depending on the form chosen for the dissipative terms in the momentum
1002     equations - see below.
1003    
1004 cnh 1.8 Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
1005 cnh 1.1
1006     \begin{equation}
1007     \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
1008     \label{eq:inhom-neumann-nh}
1009     \end{equation}
1010     where
1011    
1012     \begin{equation*}
1013     \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi
1014     _{s}+\mathbf{\nabla }\phi _{hyd}\right)
1015 adcroft 1.4 \end{equation*}
1016 cnh 1.1 presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
1017     (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
1018     exploit classical 3D potential theory and, by introducing an appropriately
1019 cnh 1.2 chosen $\delta $-function sheet of `source-charge', replace the
1020     inhomogeneous boundary condition on pressure by a homogeneous one. The
1021 adcroft 1.4 source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $
1022     \vec{\mathbf{F}}.$ By simultaneously setting $
1023 cnh 1.1 \begin{array}{l}
1024 adcroft 1.4 \widehat{n}.\vec{\mathbf{F}}
1025     \end{array}
1026 cnh 1.1 =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
1027 cnh 1.2 self-consistent but simpler homogenized Elliptic problem is obtained:
1028 cnh 1.1
1029     \begin{equation*}
1030 cnh 1.2 \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
1031 adcroft 1.4 \end{equation*}
1032 cnh 1.1 where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
1033 adcroft 1.4 that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref
1034 cnh 1.1 {eq:inhom-neumann-nh}) the modified boundary condition becomes:
1035    
1036     \begin{equation}
1037     \widehat{n}.\nabla \phi _{nh}=0 \label{eq:hom-neumann-nh}
1038     \end{equation}
1039    
1040     If the flow is `close' to hydrostatic balance then the 3-d inversion
1041     converges rapidly because $\phi _{nh}\ $is then only a small correction to
1042     the hydrostatic pressure field (see the discussion in Marshall et al, a,b).
1043    
1044 cnh 1.8 The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh})
1045 cnh 1.1 does not vanish at $r=R_{moving}$, and so refines the pressure there.
1046    
1047     \subsection{Forcing/dissipation}
1048    
1049     \subsubsection{Forcing}
1050    
1051     The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by
1052 cnh 1.8 `physics packages' and forcing packages. These are described later on.
1053 cnh 1.1
1054     \subsubsection{Dissipation}
1055    
1056     \paragraph{Momentum}
1057    
1058     Many forms of momentum dissipation are available in the model. Laplacian and
1059     biharmonic frictions are commonly used:
1060    
1061     \begin{equation}
1062 adcroft 1.4 D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}
1063 cnh 1.1 +A_{4}\nabla _{h}^{4}v \label{eq:dissipation}
1064     \end{equation}
1065     where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity
1066     coefficients and $A_{4}\ $is the horizontal coefficient for biharmonic
1067     friction. These coefficients are the same for all velocity components.
1068    
1069     \paragraph{Tracers}
1070    
1071     The mixing terms for the temperature and salinity equations have a similar
1072     form to that of momentum except that the diffusion tensor can be
1073 adcroft 1.4 non-diagonal and have varying coefficients. $\qquad $
1074 cnh 1.1 \begin{equation}
1075     D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
1076     _{h}^{4}(T,S) \label{eq:diffusion}
1077     \end{equation}
1078 adcroft 1.4 where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $
1079 cnh 1.1 horizontal coefficient for biharmonic diffusion. In the simplest case where
1080     the subgrid-scale fluxes of heat and salt are parameterized with constant
1081     horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,
1082     reduces to a diagonal matrix with constant coefficients:
1083    
1084     \begin{equation}
1085     \qquad \qquad \qquad \qquad K=\left(
1086     \begin{array}{ccc}
1087     K_{h} & 0 & 0 \\
1088     0 & K_{h} & 0 \\
1089 adcroft 1.4 0 & 0 & K_{v}
1090 cnh 1.1 \end{array}
1091     \right) \qquad \qquad \qquad \label{eq:diagonal-diffusion-tensor}
1092     \end{equation}
1093     where $K_{h}\ $and $K_{v}\ $are the horizontal and vertical diffusion
1094     coefficients. These coefficients are the same for all tracers (temperature,
1095     salinity ... ).
1096    
1097     \subsection{Vector invariant form}
1098    
1099 adcroft 1.4 For some purposes it is advantageous to write momentum advection in eq(\ref
1100 cnh 1.8 {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the (so-called) `vector invariant' form:
1101 cnh 1.1
1102     \begin{equation}
1103 adcroft 1.4 \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
1104     +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla
1105 cnh 1.2 \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
1106 cnh 1.1 \label{eq:vi-identity}
1107 adcroft 1.4 \end{equation}
1108 cnh 1.1 This permits alternative numerical treatments of the non-linear terms based
1109     on their representation as a vorticity flux. Because gradients of coordinate
1110     vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit
1111 adcroft 1.4 representation of the metric terms in (\ref{eq:gu-speherical}), (\ref
1112 cnh 1.1 {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information
1113     about the geometry is contained in the areas and lengths of the volumes used
1114     to discretize the model.
1115    
1116     \subsection{Adjoint}
1117    
1118 cnh 1.8 Tangent linear and adjoint counterparts of the forward model are described
1119 cnh 1.2 in Chapter 5.
1120 cnh 1.1
1121 afe 1.19 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.18 2004/03/23 15:29:39 afe Exp $
1122 cnh 1.1 % $Name: $
1123    
1124     \section{Appendix ATMOSPHERE}
1125    
1126     \subsection{Hydrostatic Primitive Equations for the Atmosphere in pressure
1127     coordinates}
1128    
1129     \label{sect-hpe-p}
1130    
1131     The hydrostatic primitive equations (HPEs) in p-coordinates are:
1132     \begin{eqnarray}
1133 adcroft 1.4 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1134 cnh 1.2 _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
1135 cnh 1.1 \label{eq:atmos-mom} \\
1136 cnh 1.2 \frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\
1137 adcroft 1.4 \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1138 cnh 1.1 \partial p} &=&0 \label{eq:atmos-cont} \\
1139 cnh 1.2 p\alpha &=&RT \label{eq:atmos-eos} \\
1140 cnh 1.1 c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} \label{eq:atmos-heat}
1141 adcroft 1.4 \end{eqnarray}
1142 cnh 1.1 where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
1143     surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
1144     \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
1145 cnh 1.6 derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter, $\phi =gz$ is
1146 adcroft 1.4 the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
1147     }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
1148     {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
1149     e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $
1150 cnh 1.1 p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.
1151    
1152     It is convenient to cast the heat equation in terms of potential temperature
1153     $\theta $ so that it looks more like a generic conservation law.
1154     Differentiating (\ref{eq:atmos-eos}) we get:
1155     \begin{equation*}
1156     p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
1157 adcroft 1.4 \end{equation*}
1158     which, when added to the heat equation (\ref{eq:atmos-heat}) and using $
1159 cnh 1.1 c_{p}=c_{v}+R$, gives:
1160     \begin{equation}
1161     c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
1162     \label{eq-p-heat-interim}
1163 adcroft 1.4 \end{equation}
1164 cnh 1.1 Potential temperature is defined:
1165     \begin{equation}
1166     \theta =T(\frac{p_{c}}{p})^{\kappa } \label{eq:potential-temp}
1167 adcroft 1.4 \end{equation}
1168 cnh 1.1 where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience
1169     we will make use of the Exner function $\Pi (p)$ which defined by:
1170     \begin{equation}
1171     \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa } \label{Exner}
1172 adcroft 1.4 \end{equation}
1173 cnh 1.1 The following relations will be useful and are easily expressed in terms of
1174     the Exner function:
1175     \begin{equation*}
1176     c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi
1177 adcroft 1.4 }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
1178     \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}
1179 cnh 1.1 \frac{Dp}{Dt}
1180 adcroft 1.4 \end{equation*}
1181 cnh 1.1 where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.
1182    
1183     The heat equation is obtained by noting that
1184     \begin{equation*}
1185     c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta
1186 cnh 1.2 \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
1187 cnh 1.1 \end{equation*}
1188     and on substituting into (\ref{eq-p-heat-interim}) gives:
1189     \begin{equation}
1190     \Pi \frac{D\theta }{Dt}=\mathcal{Q}
1191     \label{eq:potential-temperature-equation}
1192     \end{equation}
1193     which is in conservative form.
1194    
1195 adcroft 1.4 For convenience in the model we prefer to step forward (\ref
1196 cnh 1.1 {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).
1197    
1198     \subsubsection{Boundary conditions}
1199    
1200     The upper and lower boundary conditions are :
1201     \begin{eqnarray}
1202     \mbox{at the top:}\;\;p=0 &&\text{, }\omega =\frac{Dp}{Dt}=0 \\
1203     \mbox{at the surface:}\;\;p=p_{s} &&\text{, }\phi =\phi _{topo}=g~Z_{topo}
1204     \label{eq:boundary-condition-atmosphere}
1205     \end{eqnarray}
1206     In $p$-coordinates, the upper boundary acts like a solid boundary ($\omega
1207     =0 $); in $z$-coordinates and the lower boundary is analogous to a free
1208     surface ($\phi $ is imposed and $\omega \neq 0$).
1209    
1210     \subsubsection{Splitting the geo-potential}
1211    
1212     For the purposes of initialization and reducing round-off errors, the model
1213     deals with perturbations from reference (or ``standard'') profiles. For
1214     example, the hydrostatic geopotential associated with the resting atmosphere
1215     is not dynamically relevant and can therefore be subtracted from the
1216     equations. The equations written in terms of perturbations are obtained by
1217     substituting the following definitions into the previous model equations:
1218     \begin{eqnarray}
1219     \theta &=&\theta _{o}+\theta ^{\prime } \label{eq:atmos-ref-prof-theta} \\
1220     \alpha &=&\alpha _{o}+\alpha ^{\prime } \label{eq:atmos-ref-prof-alpha} \\
1221     \phi &=&\phi _{o}+\phi ^{\prime } \label{eq:atmos-ref-prof-phi}
1222     \end{eqnarray}
1223     The reference state (indicated by subscript ``0'') corresponds to
1224     horizontally homogeneous atmosphere at rest ($\theta _{o},\alpha _{o},\phi
1225     _{o}$) with surface pressure $p_{o}(x,y)$ that satisfies $\phi
1226     _{o}(p_{o})=g~Z_{topo}$, defined:
1227     \begin{eqnarray*}
1228     \theta _{o}(p) &=&f^{n}(p) \\
1229     \alpha _{o}(p) &=&\Pi _{p}\theta _{o} \\
1230     \phi _{o}(p) &=&\phi _{topo}-\int_{p_{0}}^{p}\alpha _{o}dp
1231     \end{eqnarray*}
1232     %\begin{eqnarray*}
1233     %\phi'_\alpha & = & \int^p_{p_o} (\alpha_o -\alpha) dp \\
1234     %\phi'_s(x,y,t) & = & \int_{p_o}^{p_s} \alpha dp
1235     %\end{eqnarray*}
1236    
1237     The final form of the HPE's in p coordinates is then:
1238     \begin{eqnarray}
1239 adcroft 1.4 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1240 cnh 1.8 _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-prime} \\
1241 cnh 1.1 \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
1242 adcroft 1.4 \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1243 cnh 1.1 \partial p} &=&0 \\
1244     \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
1245 cnh 1.8 \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }
1246 cnh 1.1 \end{eqnarray}
1247    
1248 afe 1.19 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.18 2004/03/23 15:29:39 afe Exp $
1249 cnh 1.1 % $Name: $
1250    
1251     \section{Appendix OCEAN}
1252    
1253     \subsection{Equations of motion for the ocean}
1254    
1255     We review here the method by which the standard (Boussinesq, incompressible)
1256     HPE's for the ocean written in z-coordinates are obtained. The
1257     non-Boussinesq equations for oceanic motion are:
1258     \begin{eqnarray}
1259 adcroft 1.4 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1260 cnh 1.1 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\
1261     \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
1262     &=&\epsilon _{nh}\mathcal{F}_{w} \\
1263 adcroft 1.4 \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
1264 cnh 1.8 _{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
1265     \rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
1266     \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
1267     \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zns-salt}
1268     \label{eq:non-boussinesq}
1269 adcroft 1.4 \end{eqnarray}
1270 cnh 1.1 These equations permit acoustics modes, inertia-gravity waves,
1271 cnh 1.10 non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
1272 cnh 1.1 mode. As written, they cannot be integrated forward consistently - if we
1273     step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
1274 adcroft 1.4 consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
1275 cnh 1.1 {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
1276     therefore necessary to manipulate the system as follows. Differentiating the
1277     EOS (equation of state) gives:
1278    
1279     \begin{equation}
1280     \frac{D\rho }{Dt}=\left. \frac{\partial \rho }{\partial \theta }\right|
1281     _{S,p}\frac{D\theta }{Dt}+\left. \frac{\partial \rho }{\partial S}\right|
1282     _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\partial \rho }{\partial p}\right|
1283     _{\theta ,S}\frac{Dp}{Dt} \label{EOSexpansion}
1284     \end{equation}
1285    
1286     Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the
1287 cnh 1.8 reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref{eq-zns-cont} gives:
1288 cnh 1.1 \begin{equation}
1289 adcroft 1.4 \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
1290 cnh 1.1 v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure}
1291     \end{equation}
1292     where we have used an approximation sign to indicate that we have assumed
1293     adiabatic motion, dropping the $\frac{D\theta }{Dt}$ and $\frac{DS}{Dt}$.
1294     Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that
1295     can be explicitly integrated forward:
1296     \begin{eqnarray}
1297 adcroft 1.4 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1298 cnh 1.1 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1299     \label{eq-cns-hmom} \\
1300     \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
1301     &=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-cns-hydro} \\
1302 adcroft 1.4 \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
1303 cnh 1.1 v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-cns-cont} \\
1304     \rho &=&\rho (\theta ,S,p) \label{eq-cns-eos} \\
1305     \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-cns-heat} \\
1306     \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-cns-salt}
1307     \end{eqnarray}
1308    
1309     \subsubsection{Compressible z-coordinate equations}
1310    
1311     Here we linearize the acoustic modes by replacing $\rho $ with $\rho _{o}(z)$
1312     wherever it appears in a product (ie. non-linear term) - this is the
1313     `Boussinesq assumption'. The only term that then retains the full variation
1314     in $\rho $ is the gravitational acceleration:
1315     \begin{eqnarray}
1316 adcroft 1.4 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1317 cnh 1.1 _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1318     \label{eq-zcb-hmom} \\
1319 adcroft 1.4 \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
1320 cnh 1.1 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1321     \label{eq-zcb-hydro} \\
1322 adcroft 1.4 \frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{
1323 cnh 1.1 \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zcb-cont} \\
1324     \rho &=&\rho (\theta ,S,p) \label{eq-zcb-eos} \\
1325     \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zcb-heat} \\
1326     \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zcb-salt}
1327     \end{eqnarray}
1328     These equations still retain acoustic modes. But, because the
1329 adcroft 1.4 ``compressible'' terms are linearized, the pressure equation \ref
1330 cnh 1.1 {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
1331     term appears as a Helmholtz term in the non-hydrostatic pressure equation).
1332     These are the \emph{truly} compressible Boussinesq equations. Note that the
1333     EOS must have the same pressure dependency as the linearized pressure term,
1334 adcroft 1.4 ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{
1335 cnh 1.1 c_{s}^{2}}$, for consistency.
1336    
1337     \subsubsection{`Anelastic' z-coordinate equations}
1338    
1339     The anelastic approximation filters the acoustic mode by removing the
1340 adcroft 1.4 time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}
1341     ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}
1342 cnh 1.1 \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
1343     continuity and EOS. A better solution is to change the dependency on
1344     pressure in the EOS by splitting the pressure into a reference function of
1345     height and a perturbation:
1346     \begin{equation*}
1347 cnh 1.2 \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
1348 cnh 1.1 \end{equation*}
1349     Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
1350     differentiating the EOS, the continuity equation then becomes:
1351     \begin{equation*}
1352 adcroft 1.4 \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{
1353     Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+
1354 cnh 1.2 \frac{\partial w}{\partial z}=0
1355 cnh 1.1 \end{equation*}
1356     If the time- and space-scales of the motions of interest are longer than
1357 adcroft 1.4 those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},
1358 cnh 1.1 \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and
1359 adcroft 1.4 $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{
1360 cnh 1.1 Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
1361     ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
1362     _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
1363     and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the
1364     anelastic continuity equation:
1365     \begin{equation}
1366 adcroft 1.4 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-
1367 cnh 1.1 \frac{g}{c_{s}^{2}}w=0 \label{eq-za-cont1}
1368     \end{equation}
1369     A slightly different route leads to the quasi-Boussinesq continuity equation
1370 adcroft 1.4 where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
1371     \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
1372 cnh 1.1 _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:
1373     \begin{equation}
1374 adcroft 1.4 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
1375 cnh 1.1 \partial \left( \rho _{o}w\right) }{\partial z}=0 \label{eq-za-cont2}
1376     \end{equation}
1377     Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same
1378     equation if:
1379     \begin{equation}
1380     \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}
1381     \end{equation}
1382     Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
1383 adcroft 1.4 and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{
1384 cnh 1.1 g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
1385     full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
1386     then:
1387     \begin{eqnarray}
1388 adcroft 1.4 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1389 cnh 1.1 _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1390     \label{eq-zab-hmom} \\
1391 adcroft 1.4 \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
1392 cnh 1.1 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1393     \label{eq-zab-hydro} \\
1394 adcroft 1.4 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
1395 cnh 1.1 \partial \left( \rho _{o}w\right) }{\partial z} &=&0 \label{eq-zab-cont} \\
1396     \rho &=&\rho (\theta ,S,p_{o}(z)) \label{eq-zab-eos} \\
1397     \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zab-heat} \\
1398     \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zab-salt}
1399     \end{eqnarray}
1400    
1401     \subsubsection{Incompressible z-coordinate equations}
1402    
1403     Here, the objective is to drop the depth dependence of $\rho _{o}$ and so,
1404     technically, to also remove the dependence of $\rho $ on $p_{o}$. This would
1405     yield the ``truly'' incompressible Boussinesq equations:
1406     \begin{eqnarray}
1407 adcroft 1.4 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1408 cnh 1.1 _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1409     \label{eq-ztb-hmom} \\
1410 adcroft 1.4 \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}
1411 cnh 1.1 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1412     \label{eq-ztb-hydro} \\
1413     \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
1414     &=&0 \label{eq-ztb-cont} \\
1415     \rho &=&\rho (\theta ,S) \label{eq-ztb-eos} \\
1416     \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-ztb-heat} \\
1417     \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-ztb-salt}
1418     \end{eqnarray}
1419     where $\rho _{c}$ is a constant reference density of water.
1420    
1421     \subsubsection{Compressible non-divergent equations}
1422    
1423     The above ``incompressible'' equations are incompressible in both the flow
1424     and the density. In many oceanic applications, however, it is important to
1425     retain compressibility effects in the density. To do this we must split the
1426     density thus:
1427     \begin{equation*}
1428     \rho =\rho _{o}+\rho ^{\prime }
1429 adcroft 1.4 \end{equation*}
1430 cnh 1.1 We then assert that variations with depth of $\rho _{o}$ are unimportant
1431     while the compressible effects in $\rho ^{\prime }$ are:
1432     \begin{equation*}
1433     \rho _{o}=\rho _{c}
1434 adcroft 1.4 \end{equation*}
1435 cnh 1.1 \begin{equation*}
1436     \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}
1437 adcroft 1.4 \end{equation*}
1438 cnh 1.1 This then yields what we can call the semi-compressible Boussinesq
1439     equations:
1440     \begin{eqnarray}
1441 adcroft 1.4 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1442     _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{
1443 cnh 1.1 \mathcal{F}}} \label{eq:ocean-mom} \\
1444     \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
1445     _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1446     \label{eq:ocean-wmom} \\
1447     \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
1448     &=&0 \label{eq:ocean-cont} \\
1449     \rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c} \label{eq:ocean-eos}
1450     \\
1451     \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq:ocean-theta} \\
1452     \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:ocean-salt}
1453 adcroft 1.4 \end{eqnarray}
1454 cnh 1.1 Note that the hydrostatic pressure of the resting fluid, including that
1455     associated with $\rho _{c}$, is subtracted out since it has no effect on the
1456     dynamics.
1457    
1458     Though necessary, the assumptions that go into these equations are messy
1459     since we essentially assume a different EOS for the reference density and
1460     the perturbation density. Nevertheless, it is the hydrostatic ($\epsilon
1461     _{nh}=0$ form of these equations that are used throughout the ocean modeling
1462     community and referred to as the primitive equations (HPE).
1463    
1464 afe 1.19 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.18 2004/03/23 15:29:39 afe Exp $
1465 cnh 1.1 % $Name: $
1466    
1467     \section{Appendix:OPERATORS}
1468    
1469     \subsection{Coordinate systems}
1470    
1471     \subsubsection{Spherical coordinates}
1472    
1473     In spherical coordinates, the velocity components in the zonal, meridional
1474     and vertical direction respectively, are given by (see Fig.2) :
1475    
1476     \begin{equation*}
1477 cnh 1.6 u=r\cos \varphi \frac{D\lambda }{Dt}
1478 cnh 1.1 \end{equation*}
1479    
1480     \begin{equation*}
1481 cnh 1.6 v=r\frac{D\varphi }{Dt}\qquad
1482 cnh 1.1 \end{equation*}
1483     $\qquad \qquad \qquad \qquad $
1484    
1485     \begin{equation*}
1486 cnh 1.2 \dot{r}=\frac{Dr}{Dt}
1487 cnh 1.1 \end{equation*}
1488    
1489 cnh 1.6 Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
1490 cnh 1.1 distance of the particle from the center of the earth, $\Omega $ is the
1491     angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
1492    
1493     The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in
1494     spherical coordinates:
1495    
1496     \begin{equation*}
1497 cnh 1.6 \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda }
1498     ,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r}
1499 cnh 1.2 \right)
1500 cnh 1.1 \end{equation*}
1501    
1502     \begin{equation*}
1503 cnh 1.6 \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
1504     \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
1505 cnh 1.2 +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
1506 cnh 1.1 \end{equation*}
1507    
1508 adcroft 1.4 %tci%\end{document}

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