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1 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.20 2004/10/15 14:44:25 jmc Exp $
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4 %tci%\documentclass[12pt]{book}
5 %tci%\usepackage{amsmath}
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15 %tci%%TCIDATA{LastRevised=Thursday, October 04, 2001 14:41:22}
16 %tci%%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
17 %tci%%TCIDATA{Language=American English}
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28 %tci%\input{tcilatex}
29
30 %tci%\begin{document}
31
32 %tci%\tableofcontents
33
34
35 % Section: Overview
36
37 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.20 2004/10/15 14:44:25 jmc Exp $
38 % $Name: $
39
40 This document provides the reader with the information necessary to
41 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 \section{Introduction}
51 \begin{rawhtml}
52 <!-- CMIREDIR:innovations: -->
53 \end{rawhtml}
54
55
56 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 models - see fig \ref{fig:onemodel}
62
63 %% CNHbegin
64 \input{part1/one_model_figure}
65 %% CNHend
66
67 \item it has a non-hydrostatic capability and so can be used to study both
68 small-scale and large scale processes - see fig \ref{fig:all-scales}
69
70 %% CNHbegin
71 \input{part1/all_scales_figure}
72 %% CNHend
73
74 \item finite volume techniques are employed yielding an intuitive
75 discretization and support for the treatment of irregular geometries using
76 orthogonal curvilinear grids and shaved cells - see fig \ref{fig:finite-volumes}
77
78 %% CNHbegin
79 \input{part1/fvol_figure}
80 %% CNHend
81
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 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
93 \begin{verbatim}
94 Hill, C. and J. Marshall, (1995)
95 Application of a Parallel Navier-Stokes Model to Ocean Circulation in
96 Parallel Computational Fluid Dynamics
97 In Proceedings of Parallel Computational Fluid Dynamics: Implementations
98 and Results Using Parallel Computers, 545-552.
99 Elsevier Science B.V.: New York
100
101 Marshall, J., C. Hill, L. Perelman, and A. Adcroft, (1997)
102 Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling
103 J. Geophysical Res., 102(C3), 5733-5752.
104
105 Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, (1997)
106 A finite-volume, incompressible Navier Stokes model for studies of the ocean
107 on parallel computers,
108 J. Geophysical Res., 102(C3), 5753-5766.
109
110 Adcroft, A.J., Hill, C.N. and J. Marshall, (1997)
111 Representation of topography by shaved cells in a height coordinate ocean
112 model
113 Mon Wea Rev, vol 125, 2293-2315
114
115 Marshall, J., Jones, H. and C. Hill, (1998)
116 Efficient ocean modeling using non-hydrostatic algorithms
117 Journal of Marine Systems, 18, 115-134
118
119 Adcroft, A., Hill C. and J. Marshall: (1999)
120 A new treatment of the Coriolis terms in C-grid models at both high and low
121 resolutions,
122 Mon. Wea. Rev. Vol 127, pages 1928-1936
123
124 Hill, C, Adcroft,A., Jamous,D., and J. Marshall, (1999)
125 A Strategy for Terascale Climate Modeling.
126 In Proceedings of the Eighth ECMWF Workshop on the Use of Parallel Processors
127 in Meteorology, pages 406-425
128 World Scientific Publishing Co: UK
129
130 Marotzke, J, Giering,R., Zhang, K.Q., Stammer,D., Hill,C., and T.Lee, (1999)
131 Construction of the adjoint MIT ocean general circulation model and
132 application to Atlantic heat transport variability
133 J. Geophysical Res., 104(C12), 29,529-29,547.
134
135 \end{verbatim}
136
137 We begin by briefly showing some of the results of the model in action to
138 give a feel for the wide range of problems that can be addressed using it.
139
140 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.20 2004/10/15 14:44:25 jmc Exp $
141 % $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 atmospheric winds - see figure \ref{fig:all-scales}. To give a flavor of the
148 kinds of problems the model has been used to study, we briefly describe some
149 of them here. A more detailed description of the underlying formulation,
150 numerical algorithm and implementation that lie behind these calculations is
151 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 running Linux, together with a FORTRAN\ 77 compiler) and follow the examples
154 described in detail in the documentation.
155
156 \subsection{Global atmosphere: `Held-Suarez' benchmark}
157 \begin{rawhtml}
158 <!-- CMIREDIR:atmospheric_example: -->
159 \end{rawhtml}
160
161
162
163 A novel feature of MITgcm is its ability to simulate, using one basic algorithm,
164 both atmospheric and oceanographic flows at both small and large scales.
165
166 Figure \ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
167 temperature field obtained using the atmospheric isomorph of MITgcm run at
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 %% CNHbegin
178 \input{part1/cubic_eddies_figure}
179 %% CNHend
180
181 As described in Adcroft (2001), a `cubed sphere' is used to discretize the
182 globe permitting a uniform griding and obviated the need to Fourier filter.
183 The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
184 grid, of which the cubed sphere is just one of many choices.
185
186 Figure \ref{fig:hs_zave_u} shows the 5-year mean, zonally averaged zonal
187 wind from a 20-level configuration of
188 the model. It compares favorable with more conventional spatial
189 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
193 %% CNHbegin
194 \input{part1/hs_zave_u_figure}
195 %% CNHend
196
197 \subsection{Ocean gyres}
198 \begin{rawhtml}
199 <!-- CMIREDIR:oceanic_example: -->
200 \end{rawhtml}
201 \begin{rawhtml}
202 <!-- CMIREDIR:ocean_gyres: -->
203 \end{rawhtml}
204
205 Baroclinic instability is a ubiquitous process in the ocean, as well as the
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 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 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 eddies can be clearly seen in the Gulf Stream region. The transport of
221 warm water northward by the mean flow of the Gulf Stream is also clearly
222 visible.
223
224 %% CNHbegin
225 \input{part1/atl6_figure}
226 %% CNHend
227
228
229 \subsection{Global ocean circulation}
230 \begin{rawhtml}
231 <!-- CMIREDIR:global_ocean_circulation: -->
232 \end{rawhtml}
233
234 Figure \ref{fig:large-scale-circ} (top) shows the pattern of ocean currents at
235 the surface of a 4$^{\circ }$
236 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 Figure \ref{fig:large-scale-circ} (bottom) shows the meridional overturning
244 circulation of the global ocean in Sverdrups.
245
246 %%CNHbegin
247 \input{part1/global_circ_figure}
248 %%CNHend
249
250 \subsection{Convection and mixing over topography}
251 \begin{rawhtml}
252 <!-- CMIREDIR:mixing_over_topography: -->
253 \end{rawhtml}
254
255
256 Dense plumes generated by localized cooling on the continental shelf of the
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 eddies. The simulation shown in the figure \ref{fig:convect-and-topo}
261 (blue is cold dense fluid, red is
262 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
269 %%CNHbegin
270 \input{part1/convect_and_topo}
271 %%CNHend
272
273 \subsection{Boundary forced internal waves}
274 \begin{rawhtml}
275 <!-- CMIREDIR:boundary_forced_internal_waves: -->
276 \end{rawhtml}
277
278 The unique ability of MITgcm to treat non-hydrostatic dynamics in the
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 Fig. \ref{fig:boundary-forced-wave} shows the influence of cross-slope
284 topographic variations on
285 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 conditions on the left. They propagate to the sloping boundary (represented
288 using MITgcm's finite volume spatial discretization) where they break under
289 nonhydrostatic dynamics.
290
291 %%CNHbegin
292 \input{part1/boundary_forced_waves}
293 %%CNHend
294
295 \subsection{Parameter sensitivity using the adjoint of MITgcm}
296 \begin{rawhtml}
297 <!-- CMIREDIR:parameter_sensitivity: -->
298 \end{rawhtml}
299
300 Forward and tangent linear counterparts of MITgcm are supported using an
301 `automatic adjoint compiler'. These can be used in parameter sensitivity and
302 data assimilation studies.
303
304 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 of the overturning stream-function shown in figure \ref{fig:large-scale-circ}
307 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 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 %%CNHbegin
315 \input{part1/adj_hf_ocean_figure}
316 %%CNHend
317
318 \subsection{Global state estimation of the ocean}
319 \begin{rawhtml}
320 <!-- CMIREDIR:global_state_estimation: -->
321 \end{rawhtml}
322
323
324 An important application of MITgcm is in state estimation of the global
325 ocean circulation. An appropriately defined `cost function', which measures
326 the departure of the model from observations (both remotely sensed and
327 in-situ) over an interval of time, is minimized by adjusting `control
328 parameters' such as air-sea fluxes, the wind field, the initial conditions
329 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 consistency with altimetric and in-situ observations over the period
333 1992-1997.
334
335 %% CNHbegin
336 \input{part1/assim_figure}
337 %% CNHend
338
339 \subsection{Ocean biogeochemical cycles}
340 \begin{rawhtml}
341 <!-- CMIREDIR:ocean_biogeo_cycles: -->
342 \end{rawhtml}
343
344 MITgcm is being used to study global biogeochemical cycles in the ocean. For
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 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
353 %%CNHbegin
354 \input{part1/biogeo_figure}
355 %%CNHend
356
357 \subsection{Simulations of laboratory experiments}
358 \begin{rawhtml}
359 <!-- CMIREDIR:classroom_exp: -->
360 \end{rawhtml}
361
362 Figure \ref{fig:lab-simulation} shows MITgcm being used to simulate a
363 laboratory experiment inquiring into the dynamics of the Antarctic Circumpolar Current (ACC). An
364 initially homogeneous tank of water ($1m$ in diameter) is driven from its
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 arrested by its instability in a process analogous to that which sets the
369 stratification of the ACC.
370
371 %%CNHbegin
372 \input{part1/lab_figure}
373 %%CNHend
374
375 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.20 2004/10/15 14:44:25 jmc Exp $
376 % $Name: $
377
378 \section{Continuous equations in `r' coordinates}
379 \begin{rawhtml}
380 <!-- CMIREDIR:z-p_isomorphism: -->
381 \end{rawhtml}
382
383 To render atmosphere and ocean models from one dynamical core we exploit
384 `isomorphisms' between equation sets that govern the evolution of the
385 respective fluids - see figure \ref{fig:isomorphic-equations}.
386 One system of hydrodynamical equations is written down
387 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 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 \ref{fig:isomorphic-equations}).
393
394 %%CNHbegin
395 \input{part1/zandpcoord_figure.tex}
396 %%CNHend
397
398 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 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
408 %%CNHbegin
409 \input{part1/vertcoord_figure.tex}
410 %%CNHend
411
412 \begin{equation}
413 \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 \text{ horizontal mtm} \label{eq:horizontal_mtm}
416 \end{equation}
417
418 \begin{equation}
419 \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
420 v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{
421 vertical mtm} \label{eq:vertical_mtm}
422 \end{equation}
423
424 \begin{equation}
425 \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
426 \partial r}=0\text{ continuity} \label{eq:continuity}
427 \end{equation}
428
429 \begin{equation}
430 b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state}
431 \end{equation}
432
433 \begin{equation}
434 \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
435 \label{eq:potential_temperature}
436 \end{equation}
437
438 \begin{equation}
439 \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
440 \label{eq:humidity_salt}
441 \end{equation}
442
443 Here:
444
445 \begin{equation*}
446 r\text{ is the vertical coordinate}
447 \end{equation*}
448
449 \begin{equation*}
450 \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{
451 is the total derivative}
452 \end{equation*}
453
454 \begin{equation*}
455 \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
456 \text{ is the `grad' operator}
457 \end{equation*}
458 with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}
459 \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
460 is a unit vector in the vertical
461
462 \begin{equation*}
463 t\text{ is time}
464 \end{equation*}
465
466 \begin{equation*}
467 \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the
468 velocity}
469 \end{equation*}
470
471 \begin{equation*}
472 \phi \text{ is the `pressure'/`geopotential'}
473 \end{equation*}
474
475 \begin{equation*}
476 \vec{\Omega}\text{ is the Earth's rotation}
477 \end{equation*}
478
479 \begin{equation*}
480 b\text{ is the `buoyancy'}
481 \end{equation*}
482
483 \begin{equation*}
484 \theta \text{ is potential temperature}
485 \end{equation*}
486
487 \begin{equation*}
488 S\text{ is specific humidity in the atmosphere; salinity in the ocean}
489 \end{equation*}
490
491 \begin{equation*}
492 \mathcal{F}_{\vec{\mathbf{v}}}\text{ are forcing and dissipation of }\vec{
493 \mathbf{v}}
494 \end{equation*}
495
496 \begin{equation*}
497 \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
498 \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 `physics' and forcing packages for atmosphere and ocean. These are described
506 in later chapters.
507
508 \subsection{Kinematic Boundary conditions}
509
510 \subsubsection{vertical}
511
512 at fixed and moving $r$ surfaces we set (see figure \ref{fig:zandp-vert-coord}):
513
514 \begin{equation}
515 \dot{r}=0 \text{\ at\ } r=R_{fixed}(x,y)\text{ (ocean bottom, top of the atmosphere)}
516 \label{eq:fixedbc}
517 \end{equation}
518
519 \begin{equation}
520 \dot{r}=\frac{Dr}{Dt} \text{\ at\ } r=R_{moving}\text{ \
521 (ocean surface,bottom of the atmosphere)} \label{eq:movingbc}
522 \end{equation}
523
524 Here
525
526 \begin{equation*}
527 R_{moving}=R_{o}+\eta
528 \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 \end{equation}
539 where $\vec{\mathbf{n}}$ is the normal to a solid boundary.
540
541 \subsection{Atmosphere}
542
543 In the atmosphere, (see figure \ref{fig:zandp-vert-coord}), we interpret:
544
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 \end{equation*}
576 \begin{equation*}
577 p\text{ is the pressure}
578 \end{equation*}
579 \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 \end{equation}
589 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 R_{fixed}=p_{top}=0
596 \end{equation*}
597 In a resting atmosphere the elevation of the mountains at the bottom is
598 given by
599 \begin{equation*}
600 R_{moving}=R_{o}(x,y)=p_{o}(x,y)
601 \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 Then the (hydrostatic form of) equations
615 (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yields a consistent
616 set of atmospheric equations which, for convenience, are written out
617 in $p$ coordinates in Appendix Atmosphere - see
618 eqs(\ref{eq:atmos-prime}).
619
620 \subsection{Ocean}
621
622 In the ocean we interpret:
623 \begin{eqnarray}
624 r &=&z\text{ is the height} \label{eq:ocean-z} \\
625 \dot{r} &=&\frac{Dz}{Dt}=w\text{ is the vertical velocity}
626 \label{eq:ocean-w} \\
627 \phi &=&\frac{p}{\rho _{c}}\text{ is the pressure} \label{eq:ocean-p} \\
628 b(\theta ,S,r) &=&\frac{g}{\rho _{c}}\left( \rho (\theta ,S,r)-\rho
629 _{c}\right) \text{ is the buoyancy} \label{eq:ocean-b}
630 \end{eqnarray}
631 where $\rho _{c}$ is a fixed reference density of water and $g$ is the
632 acceleration due to gravity.\noindent
633
634 In the above
635
636 At the bottom of the ocean: $R_{fixed}(x,y)=-H(x,y)$.
637
638 The surface of the ocean is given by: $R_{moving}=\eta $
639
640 The position of the resting free surface of the ocean is given by $
641 R_{o}=Z_{o}=0$.
642
643 Boundary conditions are:
644
645 \begin{eqnarray}
646 w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)} \label{eq:fixed-bc-ocean}
647 \\
648 w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)
649 \label{eq:moving-bc-ocean}}
650 \end{eqnarray}
651 where $\eta $ is the elevation of the free surface.
652
653 Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yield a consistent set
654 of oceanic equations
655 which, for convenience, are written out in $z$ coordinates in Appendix Ocean
656 - see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}).
657
658 \subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and
659 Non-hydrostatic forms}
660 \begin{rawhtml}
661 <!-- CMIREDIR:non_hydrostatic: -->
662 \end{rawhtml}
663
664
665 Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms:
666
667 \begin{equation}
668 \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)
669 \label{eq:phi-split}
670 \end{equation}
671 %and write eq(\ref{eq:incompressible}) in the form:
672 % ^- this eq is missing (jmc) ; replaced with:
673 and write eq( \ref{eq:horizontal_mtm}) in the form:
674
675 \begin{equation}
676 \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi
677 _{s}+\mathbf{\nabla }_{h}\phi _{hyd}+\epsilon _{nh}\mathbf{\nabla }_{h}\phi
678 _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}} \label{eq:mom-h}
679 \end{equation}
680
681 \begin{equation}
682 \frac{\partial \phi _{hyd}}{\partial r}=-b \label{eq:hydrostatic}
683 \end{equation}
684
685 \begin{equation}
686 \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{
687 \partial r}=G_{\dot{r}} \label{eq:mom-w}
688 \end{equation}
689 Here $\epsilon _{nh}$ is a non-hydrostatic parameter.
690
691 The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref
692 {eq:mom-h}) and (\ref{eq:mom-w}) represent advective, metric and Coriolis
693 terms in the momentum equations. In spherical coordinates they take the form
694 \footnote{
695 In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms
696 in (\ref{eq:gu-speherical}), (\ref{eq:gv-spherical}) and (\ref
697 {eq:gw-spherical}) are omitted; the singly-underlined terms are included in
698 the quasi-hydrostatic model (\textbf{QH}). The fully non-hydrostatic model (
699 \textbf{NH}) includes all terms.} - see Marshall et al 1997a for a full
700 discussion:
701
702 \begin{equation}
703 \left.
704 \begin{tabular}{l}
705 $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\
706 $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} $
707 \\
708 $-\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\right\} $
709 \\
710 $+\mathcal{F}_{u}$
711 \end{tabular}
712 \ \right\} \left\{
713 \begin{tabular}{l}
714 \textit{advection} \\
715 \textit{metric} \\
716 \textit{Coriolis} \\
717 \textit{\ Forcing/Dissipation}
718 \end{tabular}
719 \ \right. \qquad \label{eq:gu-speherical}
720 \end{equation}
721
722 \begin{equation}
723 \left.
724 \begin{tabular}{l}
725 $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
726 $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\}
727 $ \\
728 $-\left\{ -2\Omega u\sin \varphi \right\} $ \\
729 $+\mathcal{F}_{v}$
730 \end{tabular}
731 \ \right\} \left\{
732 \begin{tabular}{l}
733 \textit{advection} \\
734 \textit{metric} \\
735 \textit{Coriolis} \\
736 \textit{\ Forcing/Dissipation}
737 \end{tabular}
738 \ \right. \qquad \label{eq:gv-spherical}
739 \end{equation}
740 \qquad \qquad \qquad \qquad \qquad
741
742 \begin{equation}
743 \left.
744 \begin{tabular}{l}
745 $G_{\dot{r}}=-\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}}$ \\
746 $+\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} $ \\
747 ${+}\underline{{2\Omega u\cos \varphi}}$ \\
748 $\underline{\underline{\mathcal{F}_{\dot{r}}}}$
749 \end{tabular}
750 \ \right\} \left\{
751 \begin{tabular}{l}
752 \textit{advection} \\
753 \textit{metric} \\
754 \textit{Coriolis} \\
755 \textit{\ Forcing/Dissipation}
756 \end{tabular}
757 \ \right. \label{eq:gw-spherical}
758 \end{equation}
759 \qquad \qquad \qquad \qquad \qquad
760
761 In the above `${r}$' is the distance from the center of the earth and `$\varphi$
762 ' is latitude.
763
764 Grad and div operators in spherical coordinates are defined in appendix
765 OPERATORS.
766
767 %%CNHbegin
768 \input{part1/sphere_coord_figure.tex}
769 %%CNHend
770
771 \subsubsection{Shallow atmosphere approximation}
772
773 Most models are based on the `hydrostatic primitive equations' (HPE's) in
774 which the vertical momentum equation is reduced to a statement of
775 hydrostatic balance and the `traditional approximation' is made in which the
776 Coriolis force is treated approximately and the shallow atmosphere
777 approximation is made.\ The MITgcm need not make the `traditional
778 approximation'. To be able to support consistent non-hydrostatic forms the
779 shallow atmosphere approximation can be relaxed - when dividing through by $
780 r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
781 the radius of the earth.
782
783 \subsubsection{Hydrostatic and quasi-hydrostatic forms}
784 \label{sec:hydrostatic_and_quasi-hydrostatic_forms}
785
786 These are discussed at length in Marshall et al (1997a).
787
788 In the `hydrostatic primitive equations' (\textbf{HPE)} all the underlined
789 terms in Eqs. (\ref{eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical})
790 are neglected and `${r}$' is replaced by `$a$', the mean radius of the
791 earth. Once the pressure is found at one level - e.g. by inverting a 2-d
792 Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be
793 computed at all other levels by integration of the hydrostatic relation, eq(
794 \ref{eq:hydrostatic}).
795
796 In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
797 gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
798 \varphi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
799 contribution to the pressure field: only the terms underlined twice in Eqs. (
800 \ref{eq:gu-speherical}$\rightarrow $\ \ref{eq:gw-spherical}) are set to zero
801 and, simultaneously, the shallow atmosphere approximation is relaxed. In
802 \textbf{QH}\ \textit{all} the metric terms are retained and the full
803 variation of the radial position of a particle monitored. The \textbf{QH}\
804 vertical momentum equation (\ref{eq:mom-w}) becomes:
805
806 \begin{equation*}
807 \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi
808 \end{equation*}
809 making a small correction to the hydrostatic pressure.
810
811 \textbf{QH} has good energetic credentials - they are the same as for
812 \textbf{HPE}. Importantly, however, it has the same angular momentum
813 principle as the full non-hydrostatic model (\textbf{NH)} - see Marshall
814 et.al., 1997a. As in \textbf{HPE }only a 2-d elliptic problem need be solved.
815
816 \subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms}
817
818 The MIT model presently supports a full non-hydrostatic ocean isomorph, but
819 only a quasi-non-hydrostatic atmospheric isomorph.
820
821 \paragraph{Non-hydrostatic Ocean}
822
823 In the non-hydrostatic ocean model all terms in equations Eqs.(\ref
824 {eq:gu-speherical} $\rightarrow $\ \ref{eq:gw-spherical}) are retained. A
825 three dimensional elliptic equation must be solved subject to Neumann
826 boundary conditions (see below). It is important to note that use of the
827 full \textbf{NH} does not admit any new `fast' waves in to the system - the
828 incompressible condition eq(\ref{eq:continuity}) has already filtered out
829 acoustic modes. It does, however, ensure that the gravity waves are treated
830 accurately with an exact dispersion relation. The \textbf{NH} set has a
831 complete angular momentum principle and consistent energetics - see White
832 and Bromley, 1995; Marshall et.al.\ 1997a.
833
834 \paragraph{Quasi-nonhydrostatic Atmosphere}
835
836 In the non-hydrostatic version of our atmospheric model we approximate $\dot{
837 r}$ in the vertical momentum eqs(\ref{eq:mom-w}) and (\ref{eq:gv-spherical})
838 (but only here) by:
839
840 \begin{equation}
841 \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt} \label{eq:quasi-nh-w}
842 \end{equation}
843 where $p_{hy}$ is the hydrostatic pressure.
844
845 \subsubsection{Summary of equation sets supported by model}
846
847 \paragraph{Atmosphere}
848
849 Hydrostatic, and quasi-hydrostatic and quasi non-hydrostatic forms of the
850 compressible non-Boussinesq equations in $p-$coordinates are supported.
851
852 \subparagraph{Hydrostatic and quasi-hydrostatic}
853
854 The hydrostatic set is written out in $p-$coordinates in appendix Atmosphere
855 - see eq(\ref{eq:atmos-prime}).
856
857 \subparagraph{Quasi-nonhydrostatic}
858
859 A quasi-nonhydrostatic form is also supported.
860
861 \paragraph{Ocean}
862
863 \subparagraph{Hydrostatic and quasi-hydrostatic}
864
865 Hydrostatic, and quasi-hydrostatic forms of the incompressible Boussinesq
866 equations in $z-$coordinates are supported.
867
868 \subparagraph{Non-hydrostatic}
869
870 Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
871 coordinates are supported - see eqs(\ref{eq:ocean-mom}) to (\ref
872 {eq:ocean-salt}).
873
874 \subsection{Solution strategy}
875
876 The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
877 NH} models is summarized in Figure \ref{fig:solution-strategy}.
878 Under all dynamics, a 2-d elliptic equation is
879 first solved to find the surface pressure and the hydrostatic pressure at
880 any level computed from the weight of fluid above. Under \textbf{HPE} and
881 \textbf{QH} dynamics, the horizontal momentum equations are then stepped
882 forward and $\dot{r}$ found from continuity. Under \textbf{NH} dynamics a
883 3-d elliptic equation must be solved for the non-hydrostatic pressure before
884 stepping forward the horizontal momentum equations; $\dot{r}$ is found by
885 stepping forward the vertical momentum equation.
886
887 %%CNHbegin
888 \input{part1/solution_strategy_figure.tex}
889 %%CNHend
890
891 There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
892 course, some complication that goes with the inclusion of $\cos \varphi \ $
893 Coriolis terms and the relaxation of the shallow atmosphere approximation.
894 But this leads to negligible increase in computation. In \textbf{NH}, in
895 contrast, one additional elliptic equation - a three-dimensional one - must
896 be inverted for $p_{nh}$. However the `overhead' of the \textbf{NH} model is
897 essentially negligible in the hydrostatic limit (see detailed discussion in
898 Marshall et al, 1997) resulting in a non-hydrostatic algorithm that, in the
899 hydrostatic limit, is as computationally economic as the \textbf{HPEs}.
900
901 \subsection{Finding the pressure field}
902 \label{sec:finding_the_pressure_field}
903
904 Unlike the prognostic variables $u$, $v$, $w$, $\theta $ and $S$, the
905 pressure field must be obtained diagnostically. We proceed, as before, by
906 dividing the total (pressure/geo) potential in to three parts, a surface
907 part, $\phi _{s}(x,y)$, a hydrostatic part $\phi _{hyd}(x,y,r)$ and a
908 non-hydrostatic part $\phi _{nh}(x,y,r)$, as in (\ref{eq:phi-split}), and
909 writing the momentum equation as in (\ref{eq:mom-h}).
910
911 \subsubsection{Hydrostatic pressure}
912
913 Hydrostatic pressure is obtained by integrating (\ref{eq:hydrostatic})
914 vertically from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield:
915
916 \begin{equation*}
917 \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}
918 \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
919 \end{equation*}
920 and so
921
922 \begin{equation}
923 \phi _{hyd}(x,y,r)=\int_{r}^{R_{o}}bdr \label{eq:hydro-phi}
924 \end{equation}
925
926 The model can be easily modified to accommodate a loading term (e.g
927 atmospheric pressure pushing down on the ocean's surface) by setting:
928
929 \begin{equation}
930 \phi _{hyd}(r=R_{o})=loading \label{eq:loading}
931 \end{equation}
932
933 \subsubsection{Surface pressure}
934
935 The surface pressure equation can be obtained by integrating continuity,
936 (\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$
937
938 \begin{equation*}
939 \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
940 }_{h}+\partial _{r}\dot{r}\right) dr=0
941 \end{equation*}
942
943 Thus:
944
945 \begin{equation*}
946 \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
947 +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}
948 _{h}dr=0
949 \end{equation*}
950 where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $
951 r $. The above can be rearranged to yield, using Leibnitz's theorem:
952
953 \begin{equation}
954 \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot
955 \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=\text{source}
956 \label{eq:free-surface}
957 \end{equation}
958 where we have incorporated a source term.
959
960 Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential
961 (atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can
962 be written
963 \begin{equation}
964 \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
965 \label{eq:phi-surf}
966 \end{equation}
967 where $b_{s}$ is the buoyancy at the surface.
968
969 In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref
970 {eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d
971 elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free
972 surface' and `rigid lid' approaches are available.
973
974 \subsubsection{Non-hydrostatic pressure}
975
976 Taking the horizontal divergence of (\ref{eq:mom-h}) and adding
977 $\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation
978 (\ref{eq:continuity}), we deduce that:
979
980 \begin{equation}
981 \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{
982 \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .
983 \vec{\mathbf{F}} \label{eq:3d-invert}
984 \end{equation}
985
986 For a given rhs this 3-d elliptic equation must be inverted for $\phi _{nh}$
987 subject to appropriate choice of boundary conditions. This method is usually
988 called \textit{The Pressure Method} [Harlow and Welch, 1965; Williams, 1969;
989 Potter, 1976]. In the hydrostatic primitive equations case (\textbf{HPE}),
990 the 3-d problem does not need to be solved.
991
992 \paragraph{Boundary Conditions}
993
994 We apply the condition of no normal flow through all solid boundaries - the
995 coasts (in the ocean) and the bottom:
996
997 \begin{equation}
998 \vec{\mathbf{v}}.\widehat{n}=0 \label{nonormalflow}
999 \end{equation}
1000 where $\widehat{n}$ is a vector of unit length normal to the boundary. The
1001 kinematic condition (\ref{nonormalflow}) is also applied to the vertical
1002 velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $
1003 \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the
1004 tangential component of velocity, $v_{T}$, at all solid boundaries,
1005 depending on the form chosen for the dissipative terms in the momentum
1006 equations - see below.
1007
1008 Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that:
1009
1010 \begin{equation}
1011 \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
1012 \label{eq:inhom-neumann-nh}
1013 \end{equation}
1014 where
1015
1016 \begin{equation*}
1017 \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi
1018 _{s}+\mathbf{\nabla }\phi _{hyd}\right)
1019 \end{equation*}
1020 presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
1021 (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
1022 exploit classical 3D potential theory and, by introducing an appropriately
1023 chosen $\delta $-function sheet of `source-charge', replace the
1024 inhomogeneous boundary condition on pressure by a homogeneous one. The
1025 source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $
1026 \vec{\mathbf{F}}.$ By simultaneously setting $
1027 \begin{array}{l}
1028 \widehat{n}.\vec{\mathbf{F}}
1029 \end{array}
1030 =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
1031 self-consistent but simpler homogenized Elliptic problem is obtained:
1032
1033 \begin{equation*}
1034 \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
1035 \end{equation*}
1036 where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
1037 that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref
1038 {eq:inhom-neumann-nh}) the modified boundary condition becomes:
1039
1040 \begin{equation}
1041 \widehat{n}.\nabla \phi _{nh}=0 \label{eq:hom-neumann-nh}
1042 \end{equation}
1043
1044 If the flow is `close' to hydrostatic balance then the 3-d inversion
1045 converges rapidly because $\phi _{nh}\ $is then only a small correction to
1046 the hydrostatic pressure field (see the discussion in Marshall et al, a,b).
1047
1048 The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh})
1049 does not vanish at $r=R_{moving}$, and so refines the pressure there.
1050
1051 \subsection{Forcing/dissipation}
1052
1053 \subsubsection{Forcing}
1054
1055 The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by
1056 `physics packages' and forcing packages. These are described later on.
1057
1058 \subsubsection{Dissipation}
1059
1060 \paragraph{Momentum}
1061
1062 Many forms of momentum dissipation are available in the model. Laplacian and
1063 biharmonic frictions are commonly used:
1064
1065 \begin{equation}
1066 D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}
1067 +A_{4}\nabla _{h}^{4}v \label{eq:dissipation}
1068 \end{equation}
1069 where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity
1070 coefficients and $A_{4}\ $is the horizontal coefficient for biharmonic
1071 friction. These coefficients are the same for all velocity components.
1072
1073 \paragraph{Tracers}
1074
1075 The mixing terms for the temperature and salinity equations have a similar
1076 form to that of momentum except that the diffusion tensor can be
1077 non-diagonal and have varying coefficients. $\qquad $
1078 \begin{equation}
1079 D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla
1080 _{h}^{4}(T,S) \label{eq:diffusion}
1081 \end{equation}
1082 where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $
1083 horizontal coefficient for biharmonic diffusion. In the simplest case where
1084 the subgrid-scale fluxes of heat and salt are parameterized with constant
1085 horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$,
1086 reduces to a diagonal matrix with constant coefficients:
1087
1088 \begin{equation}
1089 \qquad \qquad \qquad \qquad K=\left(
1090 \begin{array}{ccc}
1091 K_{h} & 0 & 0 \\
1092 0 & K_{h} & 0 \\
1093 0 & 0 & K_{v}
1094 \end{array}
1095 \right) \qquad \qquad \qquad \label{eq:diagonal-diffusion-tensor}
1096 \end{equation}
1097 where $K_{h}\ $and $K_{v}\ $are the horizontal and vertical diffusion
1098 coefficients. These coefficients are the same for all tracers (temperature,
1099 salinity ... ).
1100
1101 \subsection{Vector invariant form}
1102
1103 For some purposes it is advantageous to write momentum advection in
1104 eq(\ref {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the
1105 (so-called) `vector invariant' form:
1106
1107 \begin{equation}
1108 \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
1109 +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla
1110 \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
1111 \label{eq:vi-identity}
1112 \end{equation}
1113 This permits alternative numerical treatments of the non-linear terms based
1114 on their representation as a vorticity flux. Because gradients of coordinate
1115 vectors no longer appear on the rhs of (\ref{eq:vi-identity}), explicit
1116 representation of the metric terms in (\ref{eq:gu-speherical}), (\ref
1117 {eq:gv-spherical}) and (\ref{eq:gw-spherical}), can be avoided: information
1118 about the geometry is contained in the areas and lengths of the volumes used
1119 to discretize the model.
1120
1121 \subsection{Adjoint}
1122
1123 Tangent linear and adjoint counterparts of the forward model are described
1124 in Chapter 5.
1125
1126 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.20 2004/10/15 14:44:25 jmc Exp $
1127 % $Name: $
1128
1129 \section{Appendix ATMOSPHERE}
1130
1131 \subsection{Hydrostatic Primitive Equations for the Atmosphere in pressure
1132 coordinates}
1133
1134 \label{sect-hpe-p}
1135
1136 The hydrostatic primitive equations (HPEs) in p-coordinates are:
1137 \begin{eqnarray}
1138 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1139 _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
1140 \label{eq:atmos-mom} \\
1141 \frac{\partial \phi }{\partial p}+\alpha &=&0 \label{eq-p-hydro-start} \\
1142 \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1143 \partial p} &=&0 \label{eq:atmos-cont} \\
1144 p\alpha &=&RT \label{eq:atmos-eos} \\
1145 c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q} \label{eq:atmos-heat}
1146 \end{eqnarray}
1147 where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
1148 surfaces) component of velocity,$\frac{D}{Dt}=\vec{\mathbf{v}}_{h}\cdot
1149 \mathbf{\nabla }_{p}+\omega \frac{\partial }{\partial p}$ is the total
1150 derivative, $f=2\Omega \sin \varphi$ is the Coriolis parameter, $\phi =gz$ is
1151 the geopotential, $\alpha =1/\rho $ is the specific volume, $\omega =\frac{Dp
1152 }{Dt}$ is the vertical velocity in the $p-$coordinate. Equation(\ref
1153 {eq:atmos-heat}) is the first law of thermodynamics where internal energy $
1154 e=c_{v}T$, $T$ is temperature, $Q$ is the rate of heating per unit mass and $
1155 p\frac{D\alpha }{Dt}$ is the work done by the fluid in compressing.
1156
1157 It is convenient to cast the heat equation in terms of potential temperature
1158 $\theta $ so that it looks more like a generic conservation law.
1159 Differentiating (\ref{eq:atmos-eos}) we get:
1160 \begin{equation*}
1161 p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
1162 \end{equation*}
1163 which, when added to the heat equation (\ref{eq:atmos-heat}) and using $
1164 c_{p}=c_{v}+R$, gives:
1165 \begin{equation}
1166 c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
1167 \label{eq-p-heat-interim}
1168 \end{equation}
1169 Potential temperature is defined:
1170 \begin{equation}
1171 \theta =T(\frac{p_{c}}{p})^{\kappa } \label{eq:potential-temp}
1172 \end{equation}
1173 where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$. For convenience
1174 we will make use of the Exner function $\Pi (p)$ which defined by:
1175 \begin{equation}
1176 \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa } \label{Exner}
1177 \end{equation}
1178 The following relations will be useful and are easily expressed in terms of
1179 the Exner function:
1180 \begin{equation*}
1181 c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi
1182 }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
1183 \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}
1184 \frac{Dp}{Dt}
1185 \end{equation*}
1186 where $b=\frac{\partial \ \Pi }{\partial p}\theta $ is the buoyancy.
1187
1188 The heat equation is obtained by noting that
1189 \begin{equation*}
1190 c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta
1191 \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
1192 \end{equation*}
1193 and on substituting into (\ref{eq-p-heat-interim}) gives:
1194 \begin{equation}
1195 \Pi \frac{D\theta }{Dt}=\mathcal{Q}
1196 \label{eq:potential-temperature-equation}
1197 \end{equation}
1198 which is in conservative form.
1199
1200 For convenience in the model we prefer to step forward (\ref
1201 {eq:potential-temperature-equation}) rather than (\ref{eq:atmos-heat}).
1202
1203 \subsubsection{Boundary conditions}
1204
1205 The upper and lower boundary conditions are :
1206 \begin{eqnarray}
1207 \mbox{at the top:}\;\;p=0 &&\text{, }\omega =\frac{Dp}{Dt}=0 \\
1208 \mbox{at the surface:}\;\;p=p_{s} &&\text{, }\phi =\phi _{topo}=g~Z_{topo}
1209 \label{eq:boundary-condition-atmosphere}
1210 \end{eqnarray}
1211 In $p$-coordinates, the upper boundary acts like a solid boundary ($\omega
1212 =0 $); in $z$-coordinates and the lower boundary is analogous to a free
1213 surface ($\phi $ is imposed and $\omega \neq 0$).
1214
1215 \subsubsection{Splitting the geo-potential}
1216
1217 For the purposes of initialization and reducing round-off errors, the model
1218 deals with perturbations from reference (or ``standard'') profiles. For
1219 example, the hydrostatic geopotential associated with the resting atmosphere
1220 is not dynamically relevant and can therefore be subtracted from the
1221 equations. The equations written in terms of perturbations are obtained by
1222 substituting the following definitions into the previous model equations:
1223 \begin{eqnarray}
1224 \theta &=&\theta _{o}+\theta ^{\prime } \label{eq:atmos-ref-prof-theta} \\
1225 \alpha &=&\alpha _{o}+\alpha ^{\prime } \label{eq:atmos-ref-prof-alpha} \\
1226 \phi &=&\phi _{o}+\phi ^{\prime } \label{eq:atmos-ref-prof-phi}
1227 \end{eqnarray}
1228 The reference state (indicated by subscript ``0'') corresponds to
1229 horizontally homogeneous atmosphere at rest ($\theta _{o},\alpha _{o},\phi
1230 _{o}$) with surface pressure $p_{o}(x,y)$ that satisfies $\phi
1231 _{o}(p_{o})=g~Z_{topo}$, defined:
1232 \begin{eqnarray*}
1233 \theta _{o}(p) &=&f^{n}(p) \\
1234 \alpha _{o}(p) &=&\Pi _{p}\theta _{o} \\
1235 \phi _{o}(p) &=&\phi _{topo}-\int_{p_{0}}^{p}\alpha _{o}dp
1236 \end{eqnarray*}
1237 %\begin{eqnarray*}
1238 %\phi'_\alpha & = & \int^p_{p_o} (\alpha_o -\alpha) dp \\
1239 %\phi'_s(x,y,t) & = & \int_{p_o}^{p_s} \alpha dp
1240 %\end{eqnarray*}
1241
1242 The final form of the HPE's in p coordinates is then:
1243 \begin{eqnarray}
1244 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1245 _{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}}
1246 \label{eq:atmos-prime} \\
1247 \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\
1248 \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
1249 \partial p} &=&0 \\
1250 \frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\
1251 \frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi }
1252 \end{eqnarray}
1253
1254 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.20 2004/10/15 14:44:25 jmc Exp $
1255 % $Name: $
1256
1257 \section{Appendix OCEAN}
1258
1259 \subsection{Equations of motion for the ocean}
1260
1261 We review here the method by which the standard (Boussinesq, incompressible)
1262 HPE's for the ocean written in z-coordinates are obtained. The
1263 non-Boussinesq equations for oceanic motion are:
1264 \begin{eqnarray}
1265 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1266 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} \\
1267 \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
1268 &=&\epsilon _{nh}\mathcal{F}_{w} \\
1269 \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}
1270 _{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\
1271 \rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\
1272 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\
1273 \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zns-salt}
1274 \label{eq:non-boussinesq}
1275 \end{eqnarray}
1276 These equations permit acoustics modes, inertia-gravity waves,
1277 non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline
1278 mode. As written, they cannot be integrated forward consistently - if we
1279 step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be
1280 consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref
1281 {eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is
1282 therefore necessary to manipulate the system as follows. Differentiating the
1283 EOS (equation of state) gives:
1284
1285 \begin{equation}
1286 \frac{D\rho }{Dt}=\left. \frac{\partial \rho }{\partial \theta }\right|
1287 _{S,p}\frac{D\theta }{Dt}+\left. \frac{\partial \rho }{\partial S}\right|
1288 _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\partial \rho }{\partial p}\right|
1289 _{\theta ,S}\frac{Dp}{Dt} \label{EOSexpansion}
1290 \end{equation}
1291
1292 Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is
1293 the reciprocal of the sound speed ($c_{s}$) squared. Substituting into
1294 \ref{eq-zns-cont} gives:
1295 \begin{equation}
1296 \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
1297 v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure}
1298 \end{equation}
1299 where we have used an approximation sign to indicate that we have assumed
1300 adiabatic motion, dropping the $\frac{D\theta }{Dt}$ and $\frac{DS}{Dt}$.
1301 Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that
1302 can be explicitly integrated forward:
1303 \begin{eqnarray}
1304 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1305 _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1306 \label{eq-cns-hmom} \\
1307 \epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z}
1308 &=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-cns-hydro} \\
1309 \frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{
1310 v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-cns-cont} \\
1311 \rho &=&\rho (\theta ,S,p) \label{eq-cns-eos} \\
1312 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-cns-heat} \\
1313 \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-cns-salt}
1314 \end{eqnarray}
1315
1316 \subsubsection{Compressible z-coordinate equations}
1317
1318 Here we linearize the acoustic modes by replacing $\rho $ with $\rho _{o}(z)$
1319 wherever it appears in a product (ie. non-linear term) - this is the
1320 `Boussinesq assumption'. The only term that then retains the full variation
1321 in $\rho $ is the gravitational acceleration:
1322 \begin{eqnarray}
1323 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1324 _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1325 \label{eq-zcb-hmom} \\
1326 \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
1327 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1328 \label{eq-zcb-hydro} \\
1329 \frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{
1330 \mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zcb-cont} \\
1331 \rho &=&\rho (\theta ,S,p) \label{eq-zcb-eos} \\
1332 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zcb-heat} \\
1333 \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zcb-salt}
1334 \end{eqnarray}
1335 These equations still retain acoustic modes. But, because the
1336 ``compressible'' terms are linearized, the pressure equation \ref
1337 {eq-zcb-cont} can be integrated implicitly with ease (the time-dependent
1338 term appears as a Helmholtz term in the non-hydrostatic pressure equation).
1339 These are the \emph{truly} compressible Boussinesq equations. Note that the
1340 EOS must have the same pressure dependency as the linearized pressure term,
1341 ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{
1342 c_{s}^{2}}$, for consistency.
1343
1344 \subsubsection{`Anelastic' z-coordinate equations}
1345
1346 The anelastic approximation filters the acoustic mode by removing the
1347 time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}
1348 ). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}
1349 \frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between
1350 continuity and EOS. A better solution is to change the dependency on
1351 pressure in the EOS by splitting the pressure into a reference function of
1352 height and a perturbation:
1353 \begin{equation*}
1354 \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
1355 \end{equation*}
1356 Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
1357 differentiating the EOS, the continuity equation then becomes:
1358 \begin{equation*}
1359 \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{
1360 Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+
1361 \frac{\partial w}{\partial z}=0
1362 \end{equation*}
1363 If the time- and space-scales of the motions of interest are longer than
1364 those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},
1365 \mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and
1366 $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{
1367 Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta
1368 ,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon
1369 _{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation
1370 and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the
1371 anelastic continuity equation:
1372 \begin{equation}
1373 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-
1374 \frac{g}{c_{s}^{2}}w=0 \label{eq-za-cont1}
1375 \end{equation}
1376 A slightly different route leads to the quasi-Boussinesq continuity equation
1377 where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+
1378 \mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }
1379 _{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding:
1380 \begin{equation}
1381 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
1382 \partial \left( \rho _{o}w\right) }{\partial z}=0 \label{eq-za-cont2}
1383 \end{equation}
1384 Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same
1385 equation if:
1386 \begin{equation}
1387 \frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}}
1388 \end{equation}
1389 Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$
1390 and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{
1391 g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The
1392 full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are
1393 then:
1394 \begin{eqnarray}
1395 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1396 _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1397 \label{eq-zab-hmom} \\
1398 \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}
1399 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1400 \label{eq-zab-hydro} \\
1401 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{
1402 \partial \left( \rho _{o}w\right) }{\partial z} &=&0 \label{eq-zab-cont} \\
1403 \rho &=&\rho (\theta ,S,p_{o}(z)) \label{eq-zab-eos} \\
1404 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zab-heat} \\
1405 \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zab-salt}
1406 \end{eqnarray}
1407
1408 \subsubsection{Incompressible z-coordinate equations}
1409
1410 Here, the objective is to drop the depth dependence of $\rho _{o}$ and so,
1411 technically, to also remove the dependence of $\rho $ on $p_{o}$. This would
1412 yield the ``truly'' incompressible Boussinesq equations:
1413 \begin{eqnarray}
1414 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1415 _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}}
1416 \label{eq-ztb-hmom} \\
1417 \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}
1418 \frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1419 \label{eq-ztb-hydro} \\
1420 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
1421 &=&0 \label{eq-ztb-cont} \\
1422 \rho &=&\rho (\theta ,S) \label{eq-ztb-eos} \\
1423 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-ztb-heat} \\
1424 \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-ztb-salt}
1425 \end{eqnarray}
1426 where $\rho _{c}$ is a constant reference density of water.
1427
1428 \subsubsection{Compressible non-divergent equations}
1429
1430 The above ``incompressible'' equations are incompressible in both the flow
1431 and the density. In many oceanic applications, however, it is important to
1432 retain compressibility effects in the density. To do this we must split the
1433 density thus:
1434 \begin{equation*}
1435 \rho =\rho _{o}+\rho ^{\prime }
1436 \end{equation*}
1437 We then assert that variations with depth of $\rho _{o}$ are unimportant
1438 while the compressible effects in $\rho ^{\prime }$ are:
1439 \begin{equation*}
1440 \rho _{o}=\rho _{c}
1441 \end{equation*}
1442 \begin{equation*}
1443 \rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}
1444 \end{equation*}
1445 This then yields what we can call the semi-compressible Boussinesq
1446 equations:
1447 \begin{eqnarray}
1448 \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}
1449 _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{
1450 \mathcal{F}}} \label{eq:ocean-mom} \\
1451 \epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho
1452 _{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w}
1453 \label{eq:ocean-wmom} \\
1454 \mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}
1455 &=&0 \label{eq:ocean-cont} \\
1456 \rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c} \label{eq:ocean-eos}
1457 \\
1458 \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq:ocean-theta} \\
1459 \frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:ocean-salt}
1460 \end{eqnarray}
1461 Note that the hydrostatic pressure of the resting fluid, including that
1462 associated with $\rho _{c}$, is subtracted out since it has no effect on the
1463 dynamics.
1464
1465 Though necessary, the assumptions that go into these equations are messy
1466 since we essentially assume a different EOS for the reference density and
1467 the perturbation density. Nevertheless, it is the hydrostatic ($\epsilon
1468 _{nh}=0$ form of these equations that are used throughout the ocean modeling
1469 community and referred to as the primitive equations (HPE).
1470
1471 % $Header: /u/gcmpack/manual/part1/manual.tex,v 1.20 2004/10/15 14:44:25 jmc Exp $
1472 % $Name: $
1473
1474 \section{Appendix:OPERATORS}
1475
1476 \subsection{Coordinate systems}
1477
1478 \subsubsection{Spherical coordinates}
1479
1480 In spherical coordinates, the velocity components in the zonal, meridional
1481 and vertical direction respectively, are given by (see Fig.2) :
1482
1483 \begin{equation*}
1484 u=r\cos \varphi \frac{D\lambda }{Dt}
1485 \end{equation*}
1486
1487 \begin{equation*}
1488 v=r\frac{D\varphi }{Dt}\qquad
1489 \end{equation*}
1490 $\qquad \qquad \qquad \qquad $
1491
1492 \begin{equation*}
1493 \dot{r}=\frac{Dr}{Dt}
1494 \end{equation*}
1495
1496 Here $\varphi $ is the latitude, $\lambda $ the longitude, $r$ the radial
1497 distance of the particle from the center of the earth, $\Omega $ is the
1498 angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
1499
1500 The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in
1501 spherical coordinates:
1502
1503 \begin{equation*}
1504 \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda }
1505 ,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r}
1506 \right)
1507 \end{equation*}
1508
1509 \begin{equation*}
1510 \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
1511 \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
1512 +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
1513 \end{equation*}
1514
1515 %tci%\end{document}

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