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revision 1.1 by cnh, Thu Sep 27 17:45:03 2001 UTC revision 1.3 by cnh, Wed Oct 10 16:48:45 2001 UTC
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2  %%%% % $Name$  % $Name$
3  %%%% %\usepackage{oldgerm}  %\usepackage{oldgerm}
4  %%%% % I commented the following because it introduced excessive white space  % I commented the following because it introduced excessive white space
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23  %%%% \documentclass[12pt]{book}  %%%% \documentclass[12pt]{book}
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37  %%%% %TCIDATA{Language=American English}  %%%% %TCIDATA{Language=American English}
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48  %%%% \input{tcilatex}  %%%% \input{tcilatex}
49  %%%%  %%%%
50  %%%% \begin{document}  %%%% \begin{document}
51  %%%%  %%%%
52  %%%% \tableofcontents  %%%% \tableofcontents
53    %%%%
54    %%%% \pagebreak
55    
56  \pagebreak  %%%% \part{MIT GCM basics}
   
 \part{MITgcm basics}  
57    
58  % Section: Overview  % Section: Overview
59    
# Line 78  MITgcm has a number of novel aspects: Line 77  MITgcm has a number of novel aspects:
77  \begin{itemize}  \begin{itemize}
78  \item it can be used to study both atmospheric and oceanic phenomena; one  \item it can be used to study both atmospheric and oceanic phenomena; one
79  hydrodynamical kernel is used to drive forward both atmospheric and oceanic  hydrodynamical kernel is used to drive forward both atmospheric and oceanic
80  models - see fig.1%  models - see fig%
81  \marginpar{  \marginpar{
82  Fig.1 One model}\ref{fig:onemodel}  Fig.1 One model}\ref{fig:onemodel}
83    
84  \begin{figure}  %% CNHbegin
85  \begin{center}  \input{part1/one_model_figure}
86  \resizebox{!}{4in}{  %% CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:onemodel}  
 \end{figure}  
87    
88  \item it has a non-hydrostatic capability and so can be used to study both  \item it has a non-hydrostatic capability and so can be used to study both
89  small-scale and large scale processes - see fig.2%  small-scale and large scale processes - see fig %
90  \marginpar{  \marginpar{
91  Fig.2 All scales}\ref{fig:all-scales}  Fig.2 All scales}\ref{fig:all-scales}
92    
93    %% CNHbegin
94  \begin{figure}  \input{part1/all_scales_figure}
95  \begin{center}  %% CNHend
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/scales.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:scales}  
 \end{figure}  
   
96    
97  \item finite volume techniques are employed yielding an intuitive  \item finite volume techniques are employed yielding an intuitive
98  discretization and support for the treatment of irregular geometries using  discretization and support for the treatment of irregular geometries using
99  orthogonal curvilinear grids and shaved cells - see fig.3%  orthogonal curvilinear grids and shaved cells - see fig %
100  \marginpar{  \marginpar{
101  Fig.3 Finite volumes}\ref{fig:Finite volumes}  Fig.3 Finite volumes}\ref{fig:finite-volumes}
102    
103    %% CNHbegin
104    \input{part1/fvol_figure}
105    %% CNHend
106    
107  \item tangent linear and adjoint counterparts are automatically maintained  \item tangent linear and adjoint counterparts are automatically maintained
108  along with the forward model, permitting sensitivity and optimization  along with the forward model, permitting sensitivity and optimization
# Line 147  atmospheric winds - see fig.2\ref{fig:al Line 130  atmospheric winds - see fig.2\ref{fig:al
130  kinds of problems the model has been used to study, we briefly describe some  kinds of problems the model has been used to study, we briefly describe some
131  of them here. A more detailed description of the underlying formulation,  of them here. A more detailed description of the underlying formulation,
132  numerical algorithm and implementation that lie behind these calculations is  numerical algorithm and implementation that lie behind these calculations is
133  given later. Indeed it is easy to reproduce the results shown here: simply  given later. Indeed many of the illustrative examples shown below can be
134  download the model (the minimum you need is a PC running linux, together  easily reproduced: simply download the model (the minimum you need is a PC
135  with a FORTRAN\ 77 compiler) and follow the examples.  running linux, together with a FORTRAN\ 77 compiler) and follow the examples
136    described in detail in the documentation.
137    
138  \subsection{Global atmosphere: `Held-Suarez' benchmark}  \subsection{Global atmosphere: `Held-Suarez' benchmark}
139    
140  Fig.E1a.\ref{fig:Held-Suarez} is an instaneous plot of the 500$mb$ height  A novel feature of MITgcm is its ability to simulate both atmospheric and
141  field obtained using a 5-level version of the atmospheric pressure isomorph  oceanographic flows at both small and large scales.
 run at 2.8$^{\circ }$ resolution. We see fully developed baroclinic eddies  
 along the northern hemisphere storm track. There are no mountains or  
 land-sea contrast in this calculation, but you can easily put them in. The  
 model is driven by relaxation to a radiative-convective equilibrium profile,  
 following the description set out in Held and Suarez; 1994 designed to test  
 atmospheric hydrodynamical cores - there are no mountains or land-sea  
 contrast. As decribed in Adcroft (2001), a `cubed sphere' is used to  
 descretize the globe permitting a uniform gridding and obviated the need to  
 fourier filter.  
142    
143  Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal  Fig.E1a.\ref{fig:eddy_cs} shows an instantaneous plot of the 500$mb$
144  wind and meridional overturning streamfunction from the 5-level model.  temperature field obtained using the atmospheric isomorph of MITgcm run at
145    2.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
146    (blue) and warm air along an equatorial band (red). Fully developed
147  \begin{figure}  baroclinic eddies spawned in the northern hemisphere storm track are
148  \begin{center}  evident. There are no mountains or land-sea contrast in this calculation,
149  \resizebox{!}{4in}{  but you can easily put them in. The model is driven by relaxation to a
150   \rotatebox{90}{  radiative-convective equilibrium profile, following the description set out
151    \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps}  in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
152   }  there are no mountains or land-sea contrast.
153  }  
154  \end{center}  %% CNHbegin
155  \label{fig:hscs}  \input{part1/cubic_eddies_figure}
156  \end{figure}  %% CNHend
157    
158    As described in Adcroft (2001), a `cubed sphere' is used to discretize the
159    globe permitting a uniform gridding and obviated the need to fourier filter.
160    The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
161    grid, of which the cubed sphere is just one of many choices.
162    
163    Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
164    wind and meridional overturning streamfunction from a 20-level version of
165    the model. It compares favorable with more conventional spatial
166    discretization approaches.
167    
168  A regular spherical lat-lon grid can also be used.  A regular spherical lat-lon grid can also be used.
169    
170  \begin{figure}  %% CNHbegin
171  \begin{center}  \input{part1/hs_zave_u_figure}
172  \resizebox{!}{4in}{  %% CNHend
  \rotatebox{90}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps}  
  }  
 }  
 \end{center}  
 \label{fig:hslatlon}  
 \end{figure}  
173    
174  \subsection{Ocean gyres}  \subsection{Ocean gyres}
175    
176    Baroclinic instability is a ubiquitous process in the ocean, as well as the
177    atmosphere. Ocean eddies play an important role in modifying the
178    hydrographic structure and current systems of the oceans. Coarse resolution
179    models of the oceans cannot resolve the eddy field and yield rather broad,
180    diffusive patterns of ocean currents. But if the resolution of our models is
181    increased until the baroclinic instability process is resolved, numerical
182    solutions of a different and much more realistic kind, can be obtained.
183    
184    Fig. ?.? shows the surface temperature and velocity field obtained from
185    MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
186    grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
187    (to avoid the converging of meridian in northern latitudes). 21 vertical
188    levels are used in the vertical with a `lopped cell' representation of
189    topography. The development and propagation of anomalously warm and cold
190    eddies can be clearly been seen in the Gulf Stream region. The transport of
191    warm water northward by the mean flow of the Gulf Stream is also clearly
192    visible.
193    
194    %% CNHbegin
195    \input{part1/ocean_gyres_figure}
196    %% CNHend
197    
198    
199  \subsection{Global ocean circulation}  \subsection{Global ocean circulation}
200    
201  Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$  Fig.E2a shows the pattern of ocean currents at the surface of a 4$^{\circ }$
202  global ocean model run with 15 vertical levels. The model is driven using  global ocean model run with 15 vertical levels. Lopped cells are used to
203  monthly-mean winds with mixed boundary conditions on temperature and  represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
204  salinity at the surface. Fig.E2b shows the overturning (thermohaline)  }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
205  circulation. Lopped cells are used to represent topography on a regular $%  mixed boundary conditions on temperature and salinity at the surface. The
206  lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$.  transfer properties of ocean eddies, convection and mixing is parameterized
207    in this model.
208    
209  \begin{figure}  Fig.E2b shows the meridional overturning circulation of the global ocean in
210  \begin{center}  Sverdrups.
211  \resizebox{!}{4in}{  
212  % \rotatebox{90}{  %%CNHbegin
213    \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps}  \input{part1/global_circ_figure}
214  % }  %%CNHend
215  }  
216  \end{center}  \subsection{Convection and mixing over topography}
217  \label{fig:horizcirc}  
218  \end{figure}  Dense plumes generated by localized cooling on the continental shelf of the
219    ocean may be influenced by rotation when the deformation radius is smaller
220  \begin{figure}  than the width of the cooling region. Rather than gravity plumes, the
221  \begin{center}  mechanism for moving dense fluid down the shelf is then through geostrophic
222  \resizebox{!}{4in}{  eddies. The simulation shown in the figure (blue is cold dense fluid, red is
223   \rotatebox{90}{  warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
224   \rotatebox{180}{  trigger convection by surface cooling. The cold, dense water falls down the
225    \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps}  slope but is deflected along the slope by rotation. It is found that
226   }  entrainment in the vertical plane is reduced when rotational control is
227   }  strong, and replaced by lateral entrainment due to the baroclinic
228  }  instability of the along-slope current.
229  \end{center}  
230  \label{fig:moc}  %%CNHbegin
231  \end{figure}  \input{part1/convect_and_topo}
232    %%CNHend
   
 \subsection{Flow over topography}  
   
 \subsection{Ocean convection}  
   
 Fig.E3 shows convection over a slope using the non-hydrostatic ocean  
 isomorph and lopped cells to respresent topography. .....The grid resolution  
 is  
233    
234  \subsection{Boundary forced internal waves}  \subsection{Boundary forced internal waves}
235    
236  \subsection{Carbon outgassing sensitivity}  The unique ability of MITgcm to treat non-hydrostatic dynamics in the
237    presence of complex geometry makes it an ideal tool to study internal wave
238  Fig.E4 shows....  dynamics and mixing in oceanic canyons and ridges driven by large amplitude
239    barotropic tidal currents imposed through open boundary conditions.
240  \begin{figure}  
241  \begin{center}  Fig. ?.? shows the influence of cross-slope topographic variations on
242  \resizebox{!}{4in}{  internal wave breaking - the cross-slope velocity is in color, the density
243    \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps}  contoured. The internal waves are excited by application of open boundary
244  }  conditions on the left.\ They propagate to the sloping boundary (represented
245  \end{center}  using MITgcm's finite volume spatial discretization) where they break under
246  \label{fig:co2mrt}  nonhydrostatic dynamics.
247  \end{figure}  
248    %%CNHbegin
249    \input{part1/boundary_forced_waves}
250    %%CNHend
251    
252    \subsection{Parameter sensitivity using the adjoint of MITgcm}
253    
254    Forward and tangent linear counterparts of MITgcm are supported using an
255    `automatic adjoint compiler'. These can be used in parameter sensitivity and
256    data assimilation studies.
257    
258    As one example of application of the MITgcm adjoint, Fig.E4 maps the
259    gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
260    of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $%
261    \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
262    sensitive to heat fluxes over the Labrador Sea, one of the important sources
263    of deep water for the thermohaline circulations. This calculation also
264    yields sensitivities to all other model parameters.
265    
266    %%CNHbegin
267    \input{part1/adj_hf_ocean_figure}
268    %%CNHend
269    
270    \subsection{Global state estimation of the ocean}
271    
272    An important application of MITgcm is in state estimation of the global
273    ocean circulation. An appropriately defined `cost function', which measures
274    the departure of the model from observations (both remotely sensed and
275    insitu) over an interval of time, is minimized by adjusting `control
276    parameters' such as air-sea fluxes, the wind field, the initial conditions
277    etc. Figure ?.? shows an estimate of the time-mean surface elevation of the
278    ocean obtained by bringing the model in to consistency with altimetric and
279    in-situ observations over the period 1992-1997.
280    
281    %% CNHbegin
282    \input{part1/globes_figure}
283    %% CNHend
284    
285    \subsection{Ocean biogeochemical cycles}
286    
287    MITgcm is being used to study global biogeochemical cycles in the ocean. For
288    example one can study the effects of interannual changes in meteorological
289    forcing and upper ocean circulation on the fluxes of carbon dioxide and
290    oxygen between the ocean and atmosphere. The figure shows the annual air-sea
291    flux of oxygen and its relation to density outcrops in the southern oceans
292    from a single year of a global, interannually varying simulation.
293    
294    %%CNHbegin
295    \input{part1/biogeo_figure}
296    %%CNHend
297    
298    \subsection{Simulations of laboratory experiments}
299    
300    Figure ?.? shows MITgcm being used to simulate a laboratory experiment
301    enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
302    initially homogeneous tank of water ($1m$ in diameter) is driven from its
303    free surface by a rotating heated disk. The combined action of mechanical
304    and thermal forcing creates a lens of fluid which becomes baroclinically
305    unstable. The stratification and depth of penetration of the lens is
306    arrested by its instability in a process analogous to that whic sets the
307    stratification of the ACC.
308    
309    %%CNHbegin
310    \input{part1/lab_figure}
311    %%CNHend
312    
313  % $Header$  % $Header$
314  % $Name$  % $Name$
# Line 270  whether the atmosphere or ocean is being Line 325  whether the atmosphere or ocean is being
325  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are  vertical coordinate `$r$' is interpreted as pressure, $p$, if we are
326  modeling the atmosphere and height, $z$, if we are modeling the ocean.  modeling the atmosphere and height, $z$, if we are modeling the ocean.
327    
328    %%CNHbegin
329    \input{part1/zandpcoord_figure.tex}
330    %%CNHend
331    
332  The state of the fluid at any time is characterized by the distribution of  The state of the fluid at any time is characterized by the distribution of
333  velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a  velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a
334  `geopotential' $\phi $ and density $\rho =\rho (\theta ,S,p)$ which may  `geopotential' $\phi $ and density $\rho =\rho (\theta ,S,p)$ which may
# Line 280  a generic vertical coordinate, $r$, see Line 339  a generic vertical coordinate, $r$, see
339  \marginpar{  \marginpar{
340  Fig.5 The vertical coordinate of model}:  Fig.5 The vertical coordinate of model}:
341    
342  \begin{figure}  %%CNHbegin
343  \begin{center}  \input{part1/vertcoord_figure.tex}
344  \resizebox{!}{4in}{  %%CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/vertcoord.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:vertcoord}  
 \end{figure}  
345    
346  \begin{equation*}  \begin{equation*}
347  \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}%
# Line 311  vertical mtm} Line 361  vertical mtm}
361  \end{equation}  \end{equation}
362    
363  \begin{equation*}  \begin{equation*}
364  b=b(\theta ,S,r)\text{ equation of state}  b=b(\theta ,S,r)\text{ equation of state}
365  \end{equation*}  \end{equation*}
366    
367  \begin{equation*}  \begin{equation*}
368  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{  potential temperature}  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
369  \end{equation*}  \end{equation*}
370    
371  \begin{equation*}  \begin{equation*}
372  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{  humidity/salinity}  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
373  \end{equation*}  \end{equation*}
374    
375  Here:  Here:
376    
377  \begin{equation*}  \begin{equation*}
378  r\text{ is the vertical coordinate}  r\text{ is the vertical coordinate}
379  \end{equation*}  \end{equation*}
380    
381  \begin{equation*}  \begin{equation*}
382  \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{  \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{
383  is the total derivative}  is the total derivative}
384  \end{equation*}  \end{equation*}
385    
386  \begin{equation*}  \begin{equation*}
387  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%
388  \text{ is the `grad' operator}  \text{ is the `grad' operator}
389  \end{equation*}  \end{equation*}
390  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%
391  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
392  is a unit vector in the vertical  is a unit vector in the vertical
393    
394  \begin{equation*}  \begin{equation*}
395  t\text{ is time}  t\text{ is time}
396  \end{equation*}  \end{equation*}
397    
398  \begin{equation*}  \begin{equation*}
399  \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the  \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the
400  velocity}  velocity}
401  \end{equation*}  \end{equation*}
402    
403  \begin{equation*}  \begin{equation*}
404  \phi \text{ is the `pressure'/`geopotential'}  \phi \text{ is the `pressure'/`geopotential'}
405  \end{equation*}  \end{equation*}
406    
407  \begin{equation*}  \begin{equation*}
408  \vec{\Omega}\text{ is the Earth's rotation}  \vec{\Omega}\text{ is the Earth's rotation}
409  \end{equation*}  \end{equation*}
410    
411  \begin{equation*}  \begin{equation*}
412  b\text{ is the `buoyancy'}  b\text{ is the `buoyancy'}
413  \end{equation*}  \end{equation*}
414    
415  \begin{equation*}  \begin{equation*}
416  \theta \text{ is potential temperature}  \theta \text{ is potential temperature}
417  \end{equation*}  \end{equation*}
418    
419  \begin{equation*}  \begin{equation*}
420  S\text{ is specific humidity in the atmosphere; salinity in the ocean}  S\text{ is specific humidity in the atmosphere; salinity in the ocean}
421  \end{equation*}  \end{equation*}
422    
423  \begin{equation*}  \begin{equation*}
# Line 376  S\text{ is specific humidity in the atmo Line 426  S\text{ is specific humidity in the atmo
426  \end{equation*}  \end{equation*}
427    
428  \begin{equation*}  \begin{equation*}
429  \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }%  \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
 \theta  
430  \end{equation*}  \end{equation*}
431    
432  \begin{equation*}  \begin{equation*}
# Line 406  at fixed and moving $r$ surfaces we set Line 455  at fixed and moving $r$ surfaces we set
455  Here  Here
456    
457  \begin{equation*}  \begin{equation*}
458  R_{moving}=R_{o}+\eta  R_{moving}=R_{o}+\eta
459  \end{equation*}  \end{equation*}
460  where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on  where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on
461  whether we are in the atmosphere or ocean) of the `moving surface' in the  whether we are in the atmosphere or ocean) of the `moving surface' in the
# Line 474  constant and $c_{p}$ the specific heat o Line 523  constant and $c_{p}$ the specific heat o
523  At the top of the atmosphere (which is `fixed' in our $r$ coordinate):  At the top of the atmosphere (which is `fixed' in our $r$ coordinate):
524    
525  \begin{equation*}  \begin{equation*}
526  R_{fixed}=p_{top}=0  R_{fixed}=p_{top}=0
527  \end{equation*}  \end{equation*}
528  In a resting atmosphere the elevation of the mountains at the bottom is  In a resting atmosphere the elevation of the mountains at the bottom is
529  given by  given by
530  \begin{equation*}  \begin{equation*}
531  R_{moving}=R_{o}(x,y)=p_{o}(x,y)  R_{moving}=R_{o}(x,y)=p_{o}(x,y)
532  \end{equation*}  \end{equation*}
533  i.e. the (hydrostatic) pressure at the top of the mountains in a resting  i.e. the (hydrostatic) pressure at the top of the mountains in a resting
534  atmosphere.  atmosphere.
# Line 589  $+\mathcal{F}_{u}$% Line 638  $+\mathcal{F}_{u}$%
638  \textit{Coriolis} \\  \textit{Coriolis} \\
639  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}%
640  \end{tabular}%  \end{tabular}%
641  \ \right. \qquad   \label{eq:gu-speherical}  \ \right. \qquad  \label{eq:gu-speherical}
642  \end{equation}  \end{equation}
643    
644  \begin{equation}  \begin{equation}
# Line 608  $+\mathcal{F}_{v}$% Line 657  $+\mathcal{F}_{v}$%
657  \textit{Coriolis} \\  \textit{Coriolis} \\
658  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}%
659  \end{tabular}%  \end{tabular}%
660  \ \right. \qquad   \label{eq:gv-spherical}  \ \right. \qquad  \label{eq:gv-spherical}
661  \end{equation}%  \end{equation}%
662  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
663    
664  \begin{equation}  \begin{equation}
665  \left.  \left.
# Line 627  $\underline{\underline{\mathcal{F}_{\dot Line 676  $\underline{\underline{\mathcal{F}_{\dot
676  \textit{Coriolis} \\  \textit{Coriolis} \\
677  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}%
678  \end{tabular}%  \end{tabular}%
679  \ \right.   \label{eq:gw-spherical}  \ \right.  \label{eq:gw-spherical}
680  \end{equation}%  \end{equation}%
681  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
682    
683  In the above `${r}$' is the distance from the center of the earth and `$lat$%  In the above `${r}$' is the distance from the center of the earth and `$lat$%
684  ' is latitude.  ' is latitude.
# Line 639  OPERATORS.% Line 688  OPERATORS.%
688  \marginpar{  \marginpar{
689  Fig.6 Spherical polar coordinate system.}  Fig.6 Spherical polar coordinate system.}
690    
691  \begin{figure}  %%CNHbegin
692  \begin{center}  \input{part1/sphere_coord_figure.tex}
693  \resizebox{!}{4in}{  %%CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:spcoord}  
 \end{figure}  
   
694    
695  \subsubsection{Shallow atmosphere approximation}  \subsubsection{Shallow atmosphere approximation}
696    
# Line 661  hydrostatic balance and the `traditional Line 700  hydrostatic balance and the `traditional
700  Coriolis force is treated approximately and the shallow atmosphere  Coriolis force is treated approximately and the shallow atmosphere
701  approximation is made.\ The MITgcm need not make the `traditional  approximation is made.\ The MITgcm need not make the `traditional
702  approximation'. To be able to support consistent non-hydrostatic forms the  approximation'. To be able to support consistent non-hydrostatic forms the
703  shallow atmosphere approximation can be relaxed - when dividing through by $r  shallow atmosphere approximation can be relaxed - when dividing through by $%
704  $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,  r $ in, for example, (\ref{eq:gu-speherical}), we do not replace $r$ by $a$,
705  the radius of the earth.  the radius of the earth.
706    
707  \subsubsection{Hydrostatic and quasi-hydrostatic forms}  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
# Line 688  variation of the radial position of a pa Line 727  variation of the radial position of a pa
727  vertical momentum equation (\ref{eq:mom-w}) becomes:  vertical momentum equation (\ref{eq:mom-w}) becomes:
728    
729  \begin{equation*}  \begin{equation*}
730  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat
731  \end{equation*}  \end{equation*}
732  making a small correction to the hydrostatic pressure.  making a small correction to the hydrostatic pressure.
733    
# Line 769  forward and $\dot{r}$ found from continu Line 808  forward and $\dot{r}$ found from continu
808  stepping forward the horizontal momentum equations; $\dot{r}$ is found by  stepping forward the horizontal momentum equations; $\dot{r}$ is found by
809  stepping forward the vertical momentum equation.  stepping forward the vertical momentum equation.
810    
811  \begin{figure}  %%CNHbegin
812  \begin{center}  \input{part1/solution_strategy_figure.tex}
813  \resizebox{!}{4in}{  %%CNHend
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:solnstart}  
 \end{figure}  
   
814    
815  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
816  course, some complication that goes with the inclusion of $\cos \phi \ $%  course, some complication that goes with the inclusion of $\cos \phi \ $%
# Line 809  vertically from $r=R_{o}$ where $\phi _{ Line 838  vertically from $r=R_{o}$ where $\phi _{
838    
839  \begin{equation*}  \begin{equation*}
840  \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}%  \int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd}%
841  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
842  \end{equation*}  \end{equation*}
843  and so  and so
844    
# Line 831  The surface pressure equation can be obt Line 860  The surface pressure equation can be obt
860    
861  \begin{equation*}  \begin{equation*}
862  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}%  \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}%
863  }_{h}+\partial _{r}\dot{r}\right) dr=0  }_{h}+\partial _{r}\dot{r}\right) dr=0
864  \end{equation*}  \end{equation*}
865    
866  Thus:  Thus:
# Line 839  Thus: Line 868  Thus:
868  \begin{equation*}  \begin{equation*}
869  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
870  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%
871  _{h}dr=0  _{h}dr=0
872  \end{equation*}  \end{equation*}
873  where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $%  where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $%
874  r $. The above can be rearranged to yield, using Leibnitz's theorem:  r $. The above can be rearranged to yield, using Leibnitz's theorem:
# Line 855  Whether $\phi $ is pressure (ocean model Line 884  Whether $\phi $ is pressure (ocean model
884  (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can  (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
885  be written  be written
886  \begin{equation}  \begin{equation}
887  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)  \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)
888  \label{eq:phi-surf}  \label{eq:phi-surf}
889  \end{equation}%  \end{equation}%
890  where $b_{s}$ is the buoyancy at the surface.  where $b_{s}$ is the buoyancy at the surface.
# Line 914  _{s}+\mathbf{\nabla }\phi _{hyd}\right) Line 943  _{s}+\mathbf{\nabla }\phi _{hyd}\right)
943  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
944  (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can  (\ref{eq:3d-invert}). As shown, for example, by Williams (1969), one can
945  exploit classical 3D potential theory and, by introducing an appropriately  exploit classical 3D potential theory and, by introducing an appropriately
946  chosen $\delta $-function sheet of `source-charge', replace the inhomogenous  chosen $\delta $-function sheet of `source-charge', replace the
947  boundary condition on pressure by a homogeneous one. The source term $rhs$  inhomogeneous boundary condition on pressure by a homogeneous one. The
948  in (\ref{eq:3d-invert}) is the divergence of the vector $\vec{\mathbf{F}}.$  source term $rhs$ in (\ref{eq:3d-invert}) is the divergence of the vector $%
949  By simultaneously setting $%  \vec{\mathbf{F}}.$ By simultaneously setting $%
950  \begin{array}{l}  \begin{array}{l}
951  \widehat{n}.\vec{\mathbf{F}}%  \widehat{n}.\vec{\mathbf{F}}%
952  \end{array}%  \end{array}%
953  =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following  =0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following
954  self-consistent but simpler homogenised Elliptic problem is obtained:  self-consistent but simpler homogenized Elliptic problem is obtained:
955    
956  \begin{equation*}  \begin{equation*}
957  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
958  \end{equation*}%  \end{equation*}%
959  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
960  that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref%  that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref%
# Line 1000  For some purposes it is advantageous to Line 1029  For some purposes it is advantageous to
1029  \begin{equation}  \begin{equation}
1030  \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}%  \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}%
1031  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %
1032  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
1033  \label{eq:vi-identity}  \label{eq:vi-identity}
1034  \end{equation}%  \end{equation}%
1035  This permits alternative numerical treatments of the non-linear terms based  This permits alternative numerical treatments of the non-linear terms based
# Line 1013  to discretize the model. Line 1042  to discretize the model.
1042    
1043  \subsection{Adjoint}  \subsection{Adjoint}
1044    
1045  Tangent linear and adoint counterparts of the forward model and described in  Tangent linear and adjoint counterparts of the forward model and described
1046  Chapter 5.  in Chapter 5.
1047    
1048  % $Header$  % $Header$
1049  % $Name$  % $Name$
# Line 1029  coordinates} Line 1058  coordinates}
1058  The hydrostatic primitive equations (HPEs) in p-coordinates are:  The hydrostatic primitive equations (HPEs) in p-coordinates are:
1059  \begin{eqnarray}  \begin{eqnarray}
1060  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%  \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}%
1061  _{h}+\mathbf{\nabla }_{p}\phi  &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
1062  \label{eq:atmos-mom} \\  \label{eq:atmos-mom} \\
1063  \frac{\partial \phi }{\partial p}+\alpha  &=&0  \label{eq-p-hydro-start} \\  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
1064  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%
1065  \partial p} &=&0  \label{eq:atmos-cont} \\  \partial p} &=&0  \label{eq:atmos-cont} \\
1066  p\alpha  &=&RT  \label{eq:atmos-eos} \\  p\alpha &=&RT  \label{eq:atmos-eos} \\
1067  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}  c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} &=&\mathcal{Q}  \label{eq:atmos-heat}
1068  \end{eqnarray}%  \end{eqnarray}%
1069  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure  where $\vec{\mathbf{v}}_{h}=(u,v,0)$ is the `horizontal' (on pressure
# Line 1081  where $b=\frac{\partial \ \Pi }{\partial Line 1110  where $b=\frac{\partial \ \Pi }{\partial
1110  The heat equation is obtained by noting that  The heat equation is obtained by noting that
1111  \begin{equation*}  \begin{equation*}
1112  c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta  c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta
1113  \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}  \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
1114  \end{equation*}  \end{equation*}
1115  and on substituting into (\ref{eq-p-heat-interim}) gives:  and on substituting into (\ref{eq-p-heat-interim}) gives:
1116  \begin{equation}  \begin{equation}
# Line 1160  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z} Line 1189  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}
1189  &=&\epsilon _{nh}\mathcal{F}_{w} \\  &=&\epsilon _{nh}\mathcal{F}_{w} \\
1190  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%  \frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}%
1191  _{h}+\frac{\partial w}{\partial z} &=&0 \\  _{h}+\frac{\partial w}{\partial z} &=&0 \\
1192  \rho  &=&\rho (\theta ,S,p) \\  \rho &=&\rho (\theta ,S,p) \\
1193  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
1194  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
1195  \end{eqnarray}%  \end{eqnarray}%
# Line 1242  continuity and EOS. A better solution is Line 1271  continuity and EOS. A better solution is
1271  pressure in the EOS by splitting the pressure into a reference function of  pressure in the EOS by splitting the pressure into a reference function of
1272  height and a perturbation:  height and a perturbation:
1273  \begin{equation*}  \begin{equation*}
1274  \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })  \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
1275  \end{equation*}  \end{equation*}
1276  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
1277  differentiating the EOS, the continuity equation then becomes:  differentiating the EOS, the continuity equation then becomes:
1278  \begin{equation*}  \begin{equation*}
1279  \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%  \frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{%
1280  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%
1281  \frac{\partial w}{\partial z}=0  \frac{\partial w}{\partial z}=0
1282  \end{equation*}  \end{equation*}
1283  If the time- and space-scales of the motions of interest are longer than  If the time- and space-scales of the motions of interest are longer than
1284  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%  those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},%
# Line 1372  In spherical coordinates, the velocity c Line 1401  In spherical coordinates, the velocity c
1401  and vertical direction respectively, are given by (see Fig.2) :  and vertical direction respectively, are given by (see Fig.2) :
1402    
1403  \begin{equation*}  \begin{equation*}
1404  u=r\cos \phi \frac{D\lambda }{Dt}  u=r\cos \phi \frac{D\lambda }{Dt}
1405  \end{equation*}  \end{equation*}
1406    
1407  \begin{equation*}  \begin{equation*}
1408  v=r\frac{D\phi }{Dt}\qquad  v=r\frac{D\phi }{Dt}\qquad
1409  \end{equation*}  \end{equation*}
1410  $\qquad \qquad \qquad \qquad $  $\qquad \qquad \qquad \qquad $
1411    
1412  \begin{equation*}  \begin{equation*}
1413  \dot{r}=\frac{Dr}{Dt}  \dot{r}=\frac{Dr}{Dt}
1414  \end{equation*}  \end{equation*}
1415    
1416  Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial  Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial
# Line 1394  spherical coordinates: Line 1423  spherical coordinates:
1423  \begin{equation*}  \begin{equation*}
1424  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%
1425  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%
1426  \right)  \right)
1427  \end{equation*}  \end{equation*}
1428    
1429  \begin{equation*}  \begin{equation*}
1430  \nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial  \nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial
1431  \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}  \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}
1432  +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}  +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
1433  \end{equation*}  \end{equation*}
1434    
1435  %%%% \end{document}  %%%% \end{document}
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