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revision 1.1 by cnh, Thu Sep 27 17:45:03 2001 UTC revision 1.2 by cnh, Tue Oct 9 10:48:03 2001 UTC
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3  %%%% %\usepackage{oldgerm}  %\usepackage{oldgerm}
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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 82  models - see fig.1% Line 81  models - see fig.1%
81  \marginpar{  \marginpar{
82  Fig.1 One model}\ref{fig:onemodel}  Fig.1 One model}\ref{fig:onemodel}
83    
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/onemodel.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:onemodel}  
 \end{figure}  
   
84  \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
85  small-scale and large scale processes - see fig.2%  small-scale and large scale processes - see fig.2%
86  \marginpar{  \marginpar{
87  Fig.2 All scales}\ref{fig:all-scales}  Fig.2 All scales}\ref{fig:all-scales}
88    
   
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/scales.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:scales}  
 \end{figure}  
   
   
89  \item finite volume techniques are employed yielding an intuitive  \item finite volume techniques are employed yielding an intuitive
90  discretization and support for the treatment of irregular geometries using  discretization and support for the treatment of irregular geometries using
91  orthogonal curvilinear grids and shaved cells - see fig.3%  orthogonal curvilinear grids and shaved cells - see fig.3%
# Line 147  atmospheric winds - see fig.2\ref{fig:al Line 118  atmospheric winds - see fig.2\ref{fig:al
118  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
119  of them here. A more detailed description of the underlying formulation,  of them here. A more detailed description of the underlying formulation,
120  numerical algorithm and implementation that lie behind these calculations is  numerical algorithm and implementation that lie behind these calculations is
121  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
122  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
123  with a FORTRAN\ 77 compiler) and follow the examples.  running linux, together with a FORTRAN\ 77 compiler) and follow the examples
124    described in detail in the documentation.
125    
126  \subsection{Global atmosphere: `Held-Suarez' benchmark}  \subsection{Global atmosphere: `Held-Suarez' benchmark}
127    
128  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
129  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.  
   
 Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal  
 wind and meridional overturning streamfunction from the 5-level model.  
130    
131    Fig.E1a.\ref{fig:Held-Suarez} shows an instantaneous plot of the 500$mb$
132    temperature field obtained using the atmospheric isomorph of MITgcm run at
133    2.8$^{\circ }$ resolution on the cubed sphere. We see cold air over the pole
134    (blue) and warm air along an equatorial band (red). Fully developed
135    baroclinic eddies spawned in the northern hemisphere storm track are
136    evident. There are no mountains or land-sea contrast in this calculation,
137    but you can easily put them in. The model is driven by relaxation to a
138    radiative-convective equilibrium profile, following the description set out
139    in Held and Suarez; 1994 designed to test atmospheric hydrodynamical cores -
140    there are no mountains or land-sea contrast.
141    
142    As described in Adcroft (2001), a `cubed sphere' is used to discretize the
143    globe permitting a uniform gridding and obviated the need to fourier filter.
144    The `vector-invariant' form of MITgcm supports any orthogonal curvilinear
145    grid, of which the cubed sphere is just one of many choices.
146    
147  \begin{figure}  Fig.E1b shows the 5-year mean, zonally averaged potential temperature, zonal
148  \begin{center}  wind and meridional overturning streamfunction from a 20-level version of
149  \resizebox{!}{4in}{  the model. It compares favorable with more conventional spatial
150   \rotatebox{90}{  discretization approaches.
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hscs.eps}  
  }  
 }  
 \end{center}  
 \label{fig:hscs}  
 \end{figure}  
   
151    
152  A regular spherical lat-lon grid can also be used.  A regular spherical lat-lon grid can also be used.
153    
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/hslatlon.eps}  
  }  
 }  
 \end{center}  
 \label{fig:hslatlon}  
 \end{figure}  
   
154  \subsection{Ocean gyres}  \subsection{Ocean gyres}
155    
156    Baroclinic instability is a ubiquitous process in the ocean, as well as the
157    atmosphere. Ocean eddies play an important role in modifying the
158    hydrographic structure and current systems of the oceans. Coarse resolution
159    models of the oceans cannot resolve the eddy field and yield rather broad,
160    diffusive patterns of ocean currents. But if the resolution of our models is
161    increased until the baroclinic instability process is resolved, numerical
162    solutions of a different and much more realistic kind, can be obtained.
163    
164    Fig. ?.? shows the surface temperature and velocity field obtained from
165    MITgcm run at $\frac{1}{6}^{\circ }$ horizontal resolution on a $lat-lon$
166    grid in which the pole has been rotated by 90$^{\circ }$ on to the equator
167    (to avoid the converging of meridian in northern latitudes). 21 vertical
168    levels are used in the vertical with a `lopped cell' representation of
169    topography. The development and propagation of anomalously warm and cold
170    eddies can be clearly been seen in the Gulf Stream region. The transport of
171    warm water northward by the mean flow of the Gulf Stream is also clearly
172    visible.
173    
174  \subsection{Global ocean circulation}  \subsection{Global ocean circulation}
175    
176  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$^{\circ }$
177  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
178  monthly-mean winds with mixed boundary conditions on temperature and  represent topography on a regular $lat-lon$ grid extending from 70$^{\circ
179  salinity at the surface. Fig.E2b shows the overturning (thermohaline)  }N $ to 70$^{\circ }S$. The model is driven using monthly-mean winds with
180  circulation. Lopped cells are used to represent topography on a regular $%  mixed boundary conditions on temperature and salinity at the surface. The
181  lat-lon$ grid extending from 70$^{\circ }N$ to 70$^{\circ }S$.  transfer properties of ocean eddies, convection and mixing is parameterized
182    in this model.
183    
184  \begin{figure}  Fig.E2b shows the meridional overturning circulation of the global ocean in
185  \begin{center}  Sverdrups.
186  \resizebox{!}{4in}{  
187  % \rotatebox{90}{  \subsection{Convection and mixing over topography}
188    \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/ocean_circ_455_2030.eps}  
189  % }  Dense plumes generated by localized cooling on the continental shelf of the
190  }  ocean may be influenced by rotation when the deformation radius is smaller
191  \end{center}  than the width of the cooling region. Rather than gravity plumes, the
192  \label{fig:horizcirc}  mechanism for moving dense fluid down the shelf is then through geostrophic
193  \end{figure}  eddies. The simulation shown in the figure (blue is cold dense fluid, red is
194    warmer, lighter fluid) employs the non-hydrostatic capability of MITgcm to
195  \begin{figure}  trigger convection by surface cooling. The cold, dense water falls down the
196  \begin{center}  slope but is deflected along the slope by rotation. It is found that
197  \resizebox{!}{4in}{  entrainment in the vertical plane is reduced when rotational control is
198   \rotatebox{90}{  strong, and replaced by lateral entrainment due to the baroclinic
199   \rotatebox{180}{  instability of the along-slope current.
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/moc.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:moc}  
 \end{figure}  
   
   
 \subsection{Flow over topography}  
   
 \subsection{Ocean convection}  
   
 Fig.E3 shows convection over a slope using the non-hydrostatic ocean  
 isomorph and lopped cells to respresent topography. .....The grid resolution  
 is  
200    
201  \subsection{Boundary forced internal waves}  \subsection{Boundary forced internal waves}
202    
203  \subsection{Carbon outgassing sensitivity}  The unique ability of MITgcm to treat non-hydrostatic dynamics in the
204    presence of complex geometry makes it an ideal tool to study internal wave
205  Fig.E4 shows....  dynamics and mixing in oceanic canyons and ridges driven by large amplitude
206    barotropic tidal currents imposed through open boundary conditions.
207  \begin{figure}  
208  \begin{center}  Fig. ?.? shows the influence of cross-slope topographic variations on
209  \resizebox{!}{4in}{  internal wave breaking - the cross-slope velocity is in color, the density
210    \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/co209.eps}  contoured. The internal waves are excited by application of open boundary
211  }  conditions on the left.\ They propagate to the sloping boundary (represented
212  \end{center}  using MITgcm's finite volume spatial discretization) where they break under
213  \label{fig:co2mrt}  nonhydrostatic dynamics.
214  \end{figure}  
215    \subsection{Parameter sensitivity using the adjoint of MITgcm}
216    
217    Forward and tangent linear counterparts of MITgcm are supported using an
218    `automatic adjoint compiler'. These can be used in parameter sensitivity and
219    data assimilation studies.
220    
221    As one example of application of the MITgcm adjoint, Fig.E4 maps the
222    gradient $\frac{\partial J}{\partial \mathcal{H}}$where $J$ is the magnitude
223    of the overturning streamfunction shown in fig?.? at 40$^{\circ }$N and $%
224    \mathcal{H}$ is the air-sea heat flux 100 years before. We see that $J$ is
225    sensitive to heat fluxes over the Labrador Sea, one of the important sources
226    of deep water for the thermohaline circulations. This calculation also
227    yields sensitivities to all other model parameters.
228    
229    \subsection{Global state estimation of the ocean}
230    
231    An important application of MITgcm is in state estimation of the global
232    ocean circulation. An appropriately defined `cost function', which measures
233    the departure of the model from observations (both remotely sensed and
234    insitu) over an interval of time, is minimized by adjusting `control
235    parameters' such as air-sea fluxes, the wind field, the initial conditions
236    etc. Figure ?.? shows an estimate of the time-mean surface elevation of the
237    ocean obtained by bringing the model in to consistency with altimetric and
238    in-situ observations over the period 1992-1997.
239    
240    \subsection{Ocean biogeochemical cycles}
241    
242    MITgcm is being used to study global biogeochemical cycles in the ocean. For
243    example one can study the effects of interannual changes in meteorological
244    forcing and upper ocean circulation on the fluxes of carbon dioxide and
245    oxygen between the ocean and atmosphere. The figure shows the annual air-sea
246    flux of oxygen and its relation to density outcrops in the southern oceans
247    from a single year of a global, interannually varying simulation.
248    
249    Chris - get figure here: http://puddle.mit.edu/\symbol{126}%
250    mick/biogeochem.html
251    
252    \subsection{Simulations of laboratory experiments}
253    
254    Figure ?.? shows MITgcm being used to simulate a laboratory experiment
255    enquiring in to the dynamics of the Antarctic Circumpolar Current (ACC). An
256    initially homogeneous tank of water ($1m$ in diameter) is driven from its
257    free surface by a rotating heated disk. The combined action of mechanical
258    and thermal forcing creates a lens of fluid which becomes baroclinically
259    unstable. The stratification and depth of penetration of the lens is
260    arrested by its instability in a process analogous to that whic sets the
261    stratification of the ACC.
262    
263  % $Header$  % $Header$
264  % $Name$  % $Name$
# Line 280  a generic vertical coordinate, $r$, see Line 285  a generic vertical coordinate, $r$, see
285  \marginpar{  \marginpar{
286  Fig.5 The vertical coordinate of model}:  Fig.5 The vertical coordinate of model}:
287    
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/vertcoord.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:vertcoord}  
 \end{figure}  
   
288  \begin{equation*}  \begin{equation*}
289  \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}}%
290  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}%  \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}%
# Line 311  vertical mtm} Line 303  vertical mtm}
303  \end{equation}  \end{equation}
304    
305  \begin{equation*}  \begin{equation*}
306  b=b(\theta ,S,r)\text{ equation of state}  b=b(\theta ,S,r)\text{ equation of state}
307  \end{equation*}  \end{equation*}
308    
309  \begin{equation*}  \begin{equation*}
310  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{  potential temperature}  \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature}
311  \end{equation*}  \end{equation*}
312    
313  \begin{equation*}  \begin{equation*}
314  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{  humidity/salinity}  \frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity}
315  \end{equation*}  \end{equation*}
316    
317  Here:  Here:
318    
319  \begin{equation*}  \begin{equation*}
320  r\text{ is the vertical coordinate}  r\text{ is the vertical coordinate}
321  \end{equation*}  \end{equation*}
322    
323  \begin{equation*}  \begin{equation*}
324  \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{
325  is the total derivative}  is the total derivative}
326  \end{equation*}  \end{equation*}
327    
328  \begin{equation*}  \begin{equation*}
329  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%  \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%
330  \text{ is the `grad' operator}  \text{ is the `grad' operator}
331  \end{equation*}  \end{equation*}
332  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%  with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%
333  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$  \frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$
334  is a unit vector in the vertical  is a unit vector in the vertical
335    
336  \begin{equation*}  \begin{equation*}
337  t\text{ is time}  t\text{ is time}
338  \end{equation*}  \end{equation*}
339    
340  \begin{equation*}  \begin{equation*}
341  \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
342  velocity}  velocity}
343  \end{equation*}  \end{equation*}
344    
345  \begin{equation*}  \begin{equation*}
346  \phi \text{ is the `pressure'/`geopotential'}  \phi \text{ is the `pressure'/`geopotential'}
347  \end{equation*}  \end{equation*}
348    
349  \begin{equation*}  \begin{equation*}
350  \vec{\Omega}\text{ is the Earth's rotation}  \vec{\Omega}\text{ is the Earth's rotation}
351  \end{equation*}  \end{equation*}
352    
353  \begin{equation*}  \begin{equation*}
354  b\text{ is the `buoyancy'}  b\text{ is the `buoyancy'}
355  \end{equation*}  \end{equation*}
356    
357  \begin{equation*}  \begin{equation*}
358  \theta \text{ is potential temperature}  \theta \text{ is potential temperature}
359  \end{equation*}  \end{equation*}
360    
361  \begin{equation*}  \begin{equation*}
362  S\text{ is specific humidity in the atmosphere; salinity in the ocean}  S\text{ is specific humidity in the atmosphere; salinity in the ocean}
363  \end{equation*}  \end{equation*}
364    
365  \begin{equation*}  \begin{equation*}
# Line 376  S\text{ is specific humidity in the atmo Line 368  S\text{ is specific humidity in the atmo
368  \end{equation*}  \end{equation*}
369    
370  \begin{equation*}  \begin{equation*}
371  \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }%  \mathcal{Q}_{\theta }\mathcal{\ }\text{are forcing and dissipation of }\theta
 \theta  
372  \end{equation*}  \end{equation*}
373    
374  \begin{equation*}  \begin{equation*}
# Line 406  at fixed and moving $r$ surfaces we set Line 397  at fixed and moving $r$ surfaces we set
397  Here  Here
398    
399  \begin{equation*}  \begin{equation*}
400  R_{moving}=R_{o}+\eta  R_{moving}=R_{o}+\eta
401  \end{equation*}  \end{equation*}
402  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
403  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 465  constant and $c_{p}$ the specific heat o
465  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):
466    
467  \begin{equation*}  \begin{equation*}
468  R_{fixed}=p_{top}=0  R_{fixed}=p_{top}=0
469  \end{equation*}  \end{equation*}
470  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
471  given by  given by
472  \begin{equation*}  \begin{equation*}
473  R_{moving}=R_{o}(x,y)=p_{o}(x,y)  R_{moving}=R_{o}(x,y)=p_{o}(x,y)
474  \end{equation*}  \end{equation*}
475  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
476  atmosphere.  atmosphere.
# Line 589  $+\mathcal{F}_{u}$% Line 580  $+\mathcal{F}_{u}$%
580  \textit{Coriolis} \\  \textit{Coriolis} \\
581  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}%
582  \end{tabular}%  \end{tabular}%
583  \ \right. \qquad   \label{eq:gu-speherical}  \ \right. \qquad  \label{eq:gu-speherical}
584  \end{equation}  \end{equation}
585    
586  \begin{equation}  \begin{equation}
# Line 608  $+\mathcal{F}_{v}$% Line 599  $+\mathcal{F}_{v}$%
599  \textit{Coriolis} \\  \textit{Coriolis} \\
600  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}%
601  \end{tabular}%  \end{tabular}%
602  \ \right. \qquad   \label{eq:gv-spherical}  \ \right. \qquad  \label{eq:gv-spherical}
603  \end{equation}%  \end{equation}%
604  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
605    
606  \begin{equation}  \begin{equation}
607  \left.  \left.
# Line 627  $\underline{\underline{\mathcal{F}_{\dot Line 618  $\underline{\underline{\mathcal{F}_{\dot
618  \textit{Coriolis} \\  \textit{Coriolis} \\
619  \textit{\ Forcing/Dissipation}%  \textit{\ Forcing/Dissipation}%
620  \end{tabular}%  \end{tabular}%
621  \ \right.   \label{eq:gw-spherical}  \ \right.  \label{eq:gw-spherical}
622  \end{equation}%  \end{equation}%
623  \qquad \qquad \qquad \qquad \qquad  \qquad \qquad \qquad \qquad \qquad
624    
625  In the above `${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$%
626  ' is latitude.  ' is latitude.
# Line 639  OPERATORS.% Line 630  OPERATORS.%
630  \marginpar{  \marginpar{
631  Fig.6 Spherical polar coordinate system.}  Fig.6 Spherical polar coordinate system.}
632    
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/spherical-polar.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:spcoord}  
 \end{figure}  
   
   
633  \subsubsection{Shallow atmosphere approximation}  \subsubsection{Shallow atmosphere approximation}
634    
635  Most models are based on the `hydrostatic primitive equations' (HPE's) in  Most models are based on the `hydrostatic primitive equations' (HPE's) in
# Line 661  hydrostatic balance and the `traditional Line 638  hydrostatic balance and the `traditional
638  Coriolis force is treated approximately and the shallow atmosphere  Coriolis force is treated approximately and the shallow atmosphere
639  approximation is made.\ The MITgcm need not make the `traditional  approximation is made.\ The MITgcm need not make the `traditional
640  approximation'. To be able to support consistent non-hydrostatic forms the  approximation'. To be able to support consistent non-hydrostatic forms the
641  shallow atmosphere approximation can be relaxed - when dividing through by $r  shallow atmosphere approximation can be relaxed - when dividing through by $%
642  $ 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 $a$,
643  the radius of the earth.  the radius of the earth.
644    
645  \subsubsection{Hydrostatic and quasi-hydrostatic forms}  \subsubsection{Hydrostatic and quasi-hydrostatic forms}
# Line 688  variation of the radial position of a pa Line 665  variation of the radial position of a pa
665  vertical momentum equation (\ref{eq:mom-w}) becomes:  vertical momentum equation (\ref{eq:mom-w}) becomes:
666    
667  \begin{equation*}  \begin{equation*}
668  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat  \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat
669  \end{equation*}  \end{equation*}
670  making a small correction to the hydrostatic pressure.  making a small correction to the hydrostatic pressure.
671    
# Line 769  forward and $\dot{r}$ found from continu Line 746  forward and $\dot{r}$ found from continu
746  stepping forward the horizontal momentum equations; $\dot{r}$ is found by  stepping forward the horizontal momentum equations; $\dot{r}$ is found by
747  stepping forward the vertical momentum equation.  stepping forward the vertical momentum equation.
748    
 \begin{figure}  
 \begin{center}  
 \resizebox{!}{4in}{  
  \rotatebox{90}{  
  \rotatebox{180}{  
   \includegraphics*[0.2in,0.7in][10.5in,10.5in]{part1/soln_strategy.eps}  
  }  
  }  
 }  
 \end{center}  
 \label{fig:solnstart}  
 \end{figure}  
   
   
749  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
750  course, some complication that goes with the inclusion of $\cos \phi \ $%  course, some complication that goes with the inclusion of $\cos \phi \ $%
751  Coriolis terms and the relaxation of the shallow atmosphere approximation.  Coriolis terms and the relaxation of the shallow atmosphere approximation.
# Line 809  vertically from $r=R_{o}$ where $\phi _{ Line 772  vertically from $r=R_{o}$ where $\phi _{
772    
773  \begin{equation*}  \begin{equation*}
774  \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}%
775  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr  \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr
776  \end{equation*}  \end{equation*}
777  and so  and so
778    
# Line 831  The surface pressure equation can be obt Line 794  The surface pressure equation can be obt
794    
795  \begin{equation*}  \begin{equation*}
796  \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}%
797  }_{h}+\partial _{r}\dot{r}\right) dr=0  }_{h}+\partial _{r}\dot{r}\right) dr=0
798  \end{equation*}  \end{equation*}
799    
800  Thus:  Thus:
# Line 839  Thus: Line 802  Thus:
802  \begin{equation*}  \begin{equation*}
803  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
804  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%  +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%
805  _{h}dr=0  _{h}dr=0
806  \end{equation*}  \end{equation*}
807  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 $%
808  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 818  Whether $\phi $ is pressure (ocean model
818  (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can  (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can
819  be written  be written
820  \begin{equation}  \begin{equation}
821  \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)
822  \label{eq:phi-surf}  \label{eq:phi-surf}
823  \end{equation}%  \end{equation}%
824  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 877  _{s}+\mathbf{\nabla }\phi _{hyd}\right)
877  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
878  (\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
879  exploit classical 3D potential theory and, by introducing an appropriately  exploit classical 3D potential theory and, by introducing an appropriately
880  chosen $\delta $-function sheet of `source-charge', replace the inhomogenous  chosen $\delta $-function sheet of `source-charge', replace the
881  boundary condition on pressure by a homogeneous one. The source term $rhs$  inhomogeneous boundary condition on pressure by a homogeneous one. The
882  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 $%
883  By simultaneously setting $%  \vec{\mathbf{F}}.$ By simultaneously setting $%
884  \begin{array}{l}  \begin{array}{l}
885  \widehat{n}.\vec{\mathbf{F}}%  \widehat{n}.\vec{\mathbf{F}}%
886  \end{array}%  \end{array}%
887  =0$\ 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
888  self-consistent but simpler homogenised Elliptic problem is obtained:  self-consistent but simpler homogenized Elliptic problem is obtained:
889    
890  \begin{equation*}  \begin{equation*}
891  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad  \nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad
892  \end{equation*}%  \end{equation*}%
893  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such  where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such
894  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 963  For some purposes it is advantageous to
963  \begin{equation}  \begin{equation}
964  \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}%
965  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %  +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla %
966  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]  \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
967  \label{eq:vi-identity}  \label{eq:vi-identity}
968  \end{equation}%  \end{equation}%
969  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 976  to discretize the model.
976    
977  \subsection{Adjoint}  \subsection{Adjoint}
978    
979  Tangent linear and adoint counterparts of the forward model and described in  Tangent linear and adjoint counterparts of the forward model and described
980  Chapter 5.  in Chapter 5.
981    
982  % $Header$  % $Header$
983  % $Name$  % $Name$
# Line 1029  coordinates} Line 992  coordinates}
992  The hydrostatic primitive equations (HPEs) in p-coordinates are:  The hydrostatic primitive equations (HPEs) in p-coordinates are:
993  \begin{eqnarray}  \begin{eqnarray}
994  \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}}%
995  _{h}+\mathbf{\nabla }_{p}\phi  &=&\vec{\mathbf{\mathcal{F}}}  _{h}+\mathbf{\nabla }_{p}\phi &=&\vec{\mathbf{\mathcal{F}}}
996  \label{eq:atmos-mom} \\  \label{eq:atmos-mom} \\
997  \frac{\partial \phi }{\partial p}+\alpha  &=&0  \label{eq-p-hydro-start} \\  \frac{\partial \phi }{\partial p}+\alpha &=&0  \label{eq-p-hydro-start} \\
998  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%  \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{%
999  \partial p} &=&0  \label{eq:atmos-cont} \\  \partial p} &=&0  \label{eq:atmos-cont} \\
1000  p\alpha  &=&RT  \label{eq:atmos-eos} \\  p\alpha &=&RT  \label{eq:atmos-eos} \\
1001  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}
1002  \end{eqnarray}%  \end{eqnarray}%
1003  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 1044  where $b=\frac{\partial \ \Pi }{\partial
1044  The heat equation is obtained by noting that  The heat equation is obtained by noting that
1045  \begin{equation*}  \begin{equation*}
1046  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
1047  \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}
1048  \end{equation*}  \end{equation*}
1049  and on substituting into (\ref{eq-p-heat-interim}) gives:  and on substituting into (\ref{eq-p-heat-interim}) gives:
1050  \begin{equation}  \begin{equation}
# Line 1160  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z} Line 1123  _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}
1123  &=&\epsilon _{nh}\mathcal{F}_{w} \\  &=&\epsilon _{nh}\mathcal{F}_{w} \\
1124  \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}}%
1125  _{h}+\frac{\partial w}{\partial z} &=&0 \\  _{h}+\frac{\partial w}{\partial z} &=&0 \\
1126  \rho  &=&\rho (\theta ,S,p) \\  \rho &=&\rho (\theta ,S,p) \\
1127  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\  \frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\
1128  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}  \frac{DS}{Dt} &=&\mathcal{Q}_{s}  \label{eq:non-boussinesq}
1129  \end{eqnarray}%  \end{eqnarray}%
# Line 1242  continuity and EOS. A better solution is Line 1205  continuity and EOS. A better solution is
1205  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
1206  height and a perturbation:  height and a perturbation:
1207  \begin{equation*}  \begin{equation*}
1208  \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })  \rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime })
1209  \end{equation*}  \end{equation*}
1210  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from  Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from
1211  differentiating the EOS, the continuity equation then becomes:  differentiating the EOS, the continuity equation then becomes:
1212  \begin{equation*}  \begin{equation*}
1213  \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{%
1214  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%  Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+%
1215  \frac{\partial w}{\partial z}=0  \frac{\partial w}{\partial z}=0
1216  \end{equation*}  \end{equation*}
1217  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
1218  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 1335  In spherical coordinates, the velocity c
1335  and vertical direction respectively, are given by (see Fig.2) :  and vertical direction respectively, are given by (see Fig.2) :
1336    
1337  \begin{equation*}  \begin{equation*}
1338  u=r\cos \phi \frac{D\lambda }{Dt}  u=r\cos \phi \frac{D\lambda }{Dt}
1339  \end{equation*}  \end{equation*}
1340    
1341  \begin{equation*}  \begin{equation*}
1342  v=r\frac{D\phi }{Dt}\qquad  v=r\frac{D\phi }{Dt}\qquad
1343  \end{equation*}  \end{equation*}
1344  $\qquad \qquad \qquad \qquad $  $\qquad \qquad \qquad \qquad $
1345    
1346  \begin{equation*}  \begin{equation*}
1347  \dot{r}=\frac{Dr}{Dt}  \dot{r}=\frac{Dr}{Dt}
1348  \end{equation*}  \end{equation*}
1349    
1350  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 1357  spherical coordinates:
1357  \begin{equation*}  \begin{equation*}
1358  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%  \nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }%
1359  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%  ,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}%
1360  \right)  \right)
1361  \end{equation*}  \end{equation*}
1362    
1363  \begin{equation*}  \begin{equation*}
1364  \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
1365  \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}  \lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}
1366  +\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}
1367  \end{equation*}  \end{equation*}
1368    
1369  %%%% \end{document}  %%%% \end{document}
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