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revision 1.1 by adcroft, Wed Aug 8 16:15:49 2001 UTC revision 1.14 by molod, Tue Jun 27 19:08:22 2006 UTC
# Line 1  Line 1 
1  % $Header$  % $Header$
2  % $Name$  % $Name$
3    
 \section{Example: Barotropic Ocean Gyre In Cartesian Coordinates}  
   
4  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
5    
6  %\begin{center}  %\begin{center}
# Line 15  Line 13 
13  %{\large May 2001}  %{\large May 2001}
14  %\end{center}  %\end{center}
15    
16  \subsection{Introduction}  \section[Barotropic Gyre MITgcm Example]{Barotropic Ocean Gyre In Cartesian Coordinates}
17    \label{sect:eg-baro}
18  This document is the first in a series of documents describing  \label{www:tutorials}
19  example MITgcm numerical experiments. The example experiments  \begin{rawhtml}
20  include both straightforward examples of idealised geophysical  <!-- CMIREDIR:eg-baro: -->
21  fluid simulations and more involved cases encompassing  \end{rawhtml}
 large scale modeling and  
 automatic differentiation. Both hydrostatic and non-hydrostatic  
 experiements are presented, as well as experiments employing  
 cartesian, spherical-polar and cube-sphere coordinate systems.  
 These ``case study'' documents include information describing  
 the experimental configuration and detailed information on how to  
 configure the MITgcm code and input files for each experiment.  
22    
 \subsection{Experiment Overview}  
23    
24  This example experiment demonstrates using the MITgcm to simulate  This example experiment demonstrates using the MITgcm to simulate
25  a barotropic, wind-forced, ocean gyre circulation. The experiment  a Barotropic, wind-forced, ocean gyre circulation. The files for this
26  is a numerical rendition of the gyre circulation problem simliar  experiment can be found in the verification directory tutorial\_barotropic\_gyre.
27    The experiment is a numerical rendition of the gyre circulation problem similar
28  to the problems described analytically by Stommel in 1966  to the problems described analytically by Stommel in 1966
29  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.
30    
# Line 41  In this experiment the model Line 32  In this experiment the model
32  is configured to represent a rectangular enclosed box of fluid,  is configured to represent a rectangular enclosed box of fluid,
33  $1200 \times 1200 $~km in lateral extent. The fluid is $5$~km deep and is forced  $1200 \times 1200 $~km in lateral extent. The fluid is $5$~km deep and is forced
34  by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally  by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally
35  in the ``north-south'' direction. Topologically the grid is cartesian and  in the ``north-south'' direction. Topologically the grid is Cartesian and
36  the coriolis parameter $f$ is defined according to a mid-latitude beta-plane  the coriolis parameter $f$ is defined according to a mid-latitude beta-plane
37  equation  equation
38    
39  \begin{equation}  \begin{equation}
40  \label{EQ:fcori}  \label{EQ:eg-baro-fcori}
41  f(y) = f_{0}+\beta y  f(y) = f_{0}+\beta y
42  \end{equation}  \end{equation}
43    
44  \noindent where $y$ is the distance along the ``north-south'' axis of the  \noindent where $y$ is the distance along the ``north-south'' axis of the
45  simulated domain. For this experiment $f_{0}$ is set to $10^{-4}s^{-1}$ in  simulated domain. For this experiment $f_{0}$ is set to $10^{-4}s^{-1}$ in
46  (\ref{EQ:fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$.  (\ref{EQ:eg-baro-fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$.
47  \\  \\
48  \\  \\
49   The sinusoidal wind-stress variations are defined according to   The sinusoidal wind-stress variations are defined according to
50    
51  \begin{equation}  \begin{equation}
52  \label{EQ:taux}  \label{EQ:eg-baro-taux}
53  \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})  \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})
54  \end{equation}  \end{equation}
55    
# Line 66  simulated domain. For this experiment $f Line 57  simulated domain. For this experiment $f
57  $\tau_0$ is set to $0.1N m^{-2}$.  $\tau_0$ is set to $0.1N m^{-2}$.
58  \\  \\
59  \\  \\
60  Figure \ref{FIG:simulation_config}  Figure \ref{FIG:eg-baro-simulation_config}
61  summarises the configuration simulated.  summarizes the configuration simulated.
62    
63    %% === eh3 ===
64  \begin{figure}  \begin{figure}
65    %% \begin{center}
66    %%  \resizebox{7.5in}{5.5in}{
67    %%    \includegraphics*[0.2in,0.7in][10.5in,10.5in]
68    %%     {part3/case_studies/barotropic_gyre/simulation_config.eps} }
69    %% \end{center}
70  \centerline{  \centerline{
71   \resizebox{7.5in}{5.5in}{    \scalefig{.95}
72     \includegraphics*[0.2in,0.7in][10.5in,10.5in]    \epsfbox{part3/case_studies/barotropic_gyre/simulation_config.eps}
     {part3/case_studies/barotropic_gyre/simulation_config.eps} }  
73  }  }
74  \caption{Schematic of simulation domain and wind-stress forcing function  \caption{Schematic of simulation domain and wind-stress forcing function
75  for barotropic gyre numerical experiment. The domain is enclosed bu solid  for barotropic gyre numerical experiment. The domain is enclosed bu solid
76  walls at $x=$~0,1200km and at $y=$~0,1200km.}  walls at $x=$~0,1200km and at $y=$~0,1200km.}
77  \label{FIG:simulation_config}  \label{FIG:eg-baro-simulation_config}
78  \end{figure}  \end{figure}
79    
80  \subsection{Discrete Numerical Configuration}  \subsection{Equations Solved}
81    \label{www:tutorials}
82   The model is configured in hydrostatic form.  The domain is discretised with  The model is configured in hydrostatic form. The implicit free surface form of the
83  a uniform grid spacing in the horizontal set to  pressure equation described in Marshall et. al \cite{marshall:97a} is
84   $\Delta x=\Delta y=20$~km, so  employed.
85  that there are sixty grid cells in the $x$ and $y$ directions. Vertically the  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous
 model is configured with a single layer with depth, $\Delta z$, of $5000$~m.  
 The implicit free surface form of the  
 pressure equation described in Marshall et. al \cite{Marshall97a} is  
 employed.  
 A horizontal laplacian operator $\nabla_{h}^2$ provides viscous  
86  dissipation. The wind-stress momentum input is added to the momentum equation  dissipation. The wind-stress momentum input is added to the momentum equation
87  for the ``zonal flow'', $u$. Other terms in the model  for the ``zonal flow'', $u$. Other terms in the model
88  are explicitly switched off for this experiement configuration (see section  are explicitly switched off for this experiment configuration (see section
89  \ref{SEC:code_config} ), yielding an active set of equations solved in this  \ref{SEC:code_config} ), yielding an active set of equations solved in this
90  configuration as follows  configuration as follows
91    
92  \begin{eqnarray}  \begin{eqnarray}
93  \label{EQ:model_equations}  \label{EQ:eg-baro-model_equations}
94  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
95                g\frac{\partial \eta}{\partial x} -                g\frac{\partial \eta}{\partial x} -
96                A_{h}\nabla_{h}^2u                A_{h}\nabla_{h}^2u
97  & = &  & = &
98  \frac{\tau_{x}}{\rho_{0}\Delta z}  \frac{\tau_{x}}{\rho_{0}\Delta z}
99  \\  \\
100  \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -  \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -
101                A_{h}\nabla_{h}^2v                A_{h}\nabla_{h}^2v
102  & = &  & = &
103  0  0
104  \\  \\
# Line 117  configuration as follows Line 108  configuration as follows
108  \end{eqnarray}  \end{eqnarray}
109    
110  \noindent where $u$ and $v$ and the $x$ and $y$ components of the  \noindent where $u$ and $v$ and the $x$ and $y$ components of the
111  flow vector $\vec{u}$.  flow vector $\vec{u}$.
112  \\  \\
113    
114    
115    \subsection{Discrete Numerical Configuration}
116    \label{www:tutorials}
117    
118     The domain is discretised with
119    a uniform grid spacing in the horizontal set to
120     $\Delta x=\Delta y=20$~km, so
121    that there are sixty grid cells in the $x$ and $y$ directions. Vertically the
122    model is configured with a single layer with depth, $\Delta z$, of $5000$~m.
123    
124  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
125    \label{www:tutorials}
126    
127  The laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.
128  This value is chosen to yield a Munk layer width \cite{Adcroft_thesis},  This value is chosen to yield a Munk layer width \cite{adcroft:95},
129    
130  \begin{eqnarray}  \begin{eqnarray}
131  \label{EQ:munk_layer}  \label{EQ:eg-baro-munk_layer}
132  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
133  \end{eqnarray}  \end{eqnarray}
134    
# Line 137  layer is well resolved. Line 139  layer is well resolved.
139    
140  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
141  time step $\delta t=1200$secs. With this time step the stability  time step $\delta t=1200$secs. With this time step the stability
142  parameter to the horizontal laplacian friction \cite{Adcroft_thesis}  parameter to the horizontal Laplacian friction \cite{adcroft:95}
143    
144    
145    
146  \begin{eqnarray}  \begin{eqnarray}
147  \label{EQ:laplacian_stability}  \label{EQ:eg-baro-laplacian_stability}
148  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
149  \end{eqnarray}  \end{eqnarray}
150    
# Line 151  for stability. Line 153  for stability.
153  \\  \\
154    
155  \noindent The numerical stability for inertial oscillations    \noindent The numerical stability for inertial oscillations  
156  \cite{Adcroft_thesis}  \cite{adcroft:95}
157    
158  \begin{eqnarray}  \begin{eqnarray}
159  \label{EQ:inertial_stability}  \label{EQ:eg-baro-inertial_stability}
160  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
161  \end{eqnarray}  \end{eqnarray}
162    
# Line 162  S_{i} = f^{2} {\delta t}^2 Line 164  S_{i} = f^{2} {\delta t}^2
164  limit for stability.  limit for stability.
165  \\  \\
166    
167  \noindent The advective CFL \cite{Adcroft_thesis} for an extreme maximum  \noindent The advective CFL \cite{adcroft:95} for an extreme maximum
168  horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$  horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$
169    
170  \begin{eqnarray}  \begin{eqnarray}
171  \label{EQ:cfl_stability}  \label{EQ:eg-baro-cfl_stability}
172  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
173  \end{eqnarray}  \end{eqnarray}
174    
# Line 174  S_{a} = \frac{| \vec{u} | \delta t}{ \De Line 176  S_{a} = \frac{| \vec{u} | \delta t}{ \De
176  of 0.5 and limits $\delta t$ to $1200s$.  of 0.5 and limits $\delta t$ to $1200s$.
177    
178  \subsection{Code Configuration}  \subsection{Code Configuration}
179  \label{SEC:code_config}  \label{www:tutorials}
180    \label{SEC:eg-baro-code_config}
181    
182  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
183  directory {\it verification/exp0/}.  The experiment files  directory {\it verification/exp0/}.  The experiment files
# Line 188  directory {\it verification/exp0/}.  The Line 191  directory {\it verification/exp0/}.  The
191  \item {\it code/CPP\_OPTIONS.h},  \item {\it code/CPP\_OPTIONS.h},
192  \item {\it code/SIZE.h}.  \item {\it code/SIZE.h}.
193  \end{itemize}  \end{itemize}
194  contain the code customisations and parameter settings for this  contain the code customizations and parameter settings for this
195  experiements. Below we describe the customisations  experiments. Below we describe the customizations
196  to these files associated with this experiment.  to these files associated with this experiment.
197    
198  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
199    \label{www:tutorials}
200    
201  This file, reproduced completely below, specifies the main parameters  This file, reproduced completely below, specifies the main parameters
202  for the experiment. The parameters that are significant for this configuration  for the experiment. The parameters that are significant for this configuration
# Line 201  are Line 205  are
205  \begin{itemize}  \begin{itemize}
206    
207  \item Line 7, \begin{verbatim} viscAh=4.E2, \end{verbatim} this line sets  \item Line 7, \begin{verbatim} viscAh=4.E2, \end{verbatim} this line sets
208  the laplacian friction coefficient to $400 m^2s^{-1}$  the Laplacian friction coefficient to $400 m^2s^{-1}$
209  \item Line 10, \begin{verbatim} beta=1.E-11, \end{verbatim} this line sets  \item Line 10, \begin{verbatim} beta=1.E-11, \end{verbatim} this line sets
210  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$
211    
# Line 219  of the pressure inverter. Line 223  of the pressure inverter.
223  startTime=0,  startTime=0,
224  \end{verbatim}  \end{verbatim}
225  this line indicates that the experiment should start from $t=0$  this line indicates that the experiment should start from $t=0$
226  and implicitly supresses searching for checkpoint files associated  and implicitly suppresses searching for checkpoint files associated
227  with restarting an numerical integration from a previously saved state.  with restarting an numerical integration from a previously saved state.
228    
229  \item Line 29,  \item Line 29,
# Line 241  This line sets the momentum equation tim Line 245  This line sets the momentum equation tim
245  usingCartesianGrid=.TRUE.,  usingCartesianGrid=.TRUE.,
246  \end{verbatim}  \end{verbatim}
247  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
248  cartesian coordinate system.  Cartesian coordinate system.
249    
250  \item Line 41,  \item Line 41,
251  \begin{verbatim}  \begin{verbatim}
# Line 304  notes. Line 308  notes.
308  \end{small}  \end{small}
309    
310  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
311    \label{www:tutorials}
312    
313  This file uses standard default values and does not contain  This file uses standard default values and does not contain
314  customisations for this experiment.  customizations for this experiment.
315    
316  \subsubsection{File {\it input/eedata}}  \subsubsection{File {\it input/eedata}}
317    \label{www:tutorials}
318    
319  This file uses standard default values and does not contain  This file uses standard default values and does not contain
320  customisations for this experiment.  customizations for this experiment.
321    
322  \subsubsection{File {\it input/windx.sin\_y}}  \subsubsection{File {\it input/windx.sin\_y}}
323    \label{www:tutorials}
324    
325  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
326  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.
# Line 324  in MITgcm. The included matlab program { Line 331  in MITgcm. The included matlab program {
331  code for creating the {\it input/windx.sin\_y} file.  code for creating the {\it input/windx.sin\_y} file.
332    
333  \subsubsection{File {\it input/topog.box}}  \subsubsection{File {\it input/topog.box}}
334    \label{www:tutorials}
335    
336    
337  The {\it input/topog.box} file specifies a two-dimensional ($x,y$)  The {\it input/topog.box} file specifies a two-dimensional ($x,y$)
# Line 335  The included matlab program {\it input/g Line 343  The included matlab program {\it input/g
343  code for creating the {\it input/topog.box} file.  code for creating the {\it input/topog.box} file.
344    
345  \subsubsection{File {\it code/SIZE.h}}  \subsubsection{File {\it code/SIZE.h}}
346    \label{www:tutorials}
347    
348  Two lines are customized in this file for the current experiment  Two lines are customized in this file for the current experiment
349    
# Line 357  axis aligned with the y-coordinate. Line 366  axis aligned with the y-coordinate.
366  \end{small}  \end{small}
367    
368  \subsubsection{File {\it code/CPP\_OPTIONS.h}}  \subsubsection{File {\it code/CPP\_OPTIONS.h}}
369    \label{www:tutorials}
370    
371  This file uses standard default values and does not contain  This file uses standard default values and does not contain
372  customisations for this experiment.  customizations for this experiment.
373    
374    
375  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
376    \label{www:tutorials}
377    
378  This file uses standard default values and does not contain  This file uses standard default values and does not contain
379  customisations for this experiment.  customizations for this experiment.
380    
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