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revision 1.1 by adcroft, Wed Aug 8 16:15:49 2001 UTC revision 1.10 by edhill, Thu Jan 29 15:11:39 2004 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}  This is the first in a series of tutorials describing
   
 This document is the first in a series of documents describing  
17  example MITgcm numerical experiments. The example experiments  example MITgcm numerical experiments. The example experiments
18  include both straightforward examples of idealised geophysical  include both straightforward examples of idealized geophysical
19  fluid simulations and more involved cases encompassing  fluid simulations and more involved cases encompassing
20  large scale modeling and  large scale modeling and
21  automatic differentiation. Both hydrostatic and non-hydrostatic  automatic differentiation. Both hydrostatic and non-hydrostatic
22  experiements are presented, as well as experiments employing  experiments are presented, as well as experiments employing
23  cartesian, spherical-polar and cube-sphere coordinate systems.  Cartesian, spherical-polar and cube-sphere coordinate systems.
24  These ``case study'' documents include information describing  These ``case study'' documents include information describing
25  the experimental configuration and detailed information on how to  the experimental configuration and detailed information on how to
26  configure the MITgcm code and input files for each experiment.  configure the MITgcm code and input files for each experiment.
27    
28  \subsection{Experiment Overview}  \section{Barotropic Ocean Gyre In Cartesian Coordinates}
29    \label{sect:eg-baro}
30    \label{www:tutorials}
31    
32    
33  This example experiment demonstrates using the MITgcm to simulate  This example experiment demonstrates using the MITgcm to simulate
34  a barotropic, wind-forced, ocean gyre circulation. The experiment  a Barotropic, wind-forced, ocean gyre circulation. The experiment
35  is a numerical rendition of the gyre circulation problem simliar  is a numerical rendition of the gyre circulation problem similar
36  to the problems described analytically by Stommel in 1966  to the problems described analytically by Stommel in 1966
37  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.
38    
# Line 41  In this experiment the model Line 40  In this experiment the model
40  is configured to represent a rectangular enclosed box of fluid,  is configured to represent a rectangular enclosed box of fluid,
41  $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
42  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
43  in the ``north-south'' direction. Topologically the grid is cartesian and  in the ``north-south'' direction. Topologically the grid is Cartesian and
44  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
45  equation  equation
46    
47  \begin{equation}  \begin{equation}
48  \label{EQ:fcori}  \label{EQ:eg-baro-fcori}
49  f(y) = f_{0}+\beta y  f(y) = f_{0}+\beta y
50  \end{equation}  \end{equation}
51    
52  \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
53  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
54  (\ref{EQ:fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$.  (\ref{EQ:eg-baro-fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$.
55  \\  \\
56  \\  \\
57   The sinusoidal wind-stress variations are defined according to   The sinusoidal wind-stress variations are defined according to
58    
59  \begin{equation}  \begin{equation}
60  \label{EQ:taux}  \label{EQ:eg-baro-taux}
61  \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})  \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})
62  \end{equation}  \end{equation}
63    
# Line 66  simulated domain. For this experiment $f Line 65  simulated domain. For this experiment $f
65  $\tau_0$ is set to $0.1N m^{-2}$.  $\tau_0$ is set to $0.1N m^{-2}$.
66  \\  \\
67  \\  \\
68  Figure \ref{FIG:simulation_config}  Figure \ref{FIG:eg-baro-simulation_config}
69  summarises the configuration simulated.  summarizes the configuration simulated.
70    
71    %% === eh3 ===
72  \begin{figure}  \begin{figure}
73    %% \begin{center}
74    %%  \resizebox{7.5in}{5.5in}{
75    %%    \includegraphics*[0.2in,0.7in][10.5in,10.5in]
76    %%     {part3/case_studies/barotropic_gyre/simulation_config.eps} }
77    %% \end{center}
78  \centerline{  \centerline{
79   \resizebox{7.5in}{5.5in}{    \scalefig{.95}
80     \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} }  
81  }  }
82  \caption{Schematic of simulation domain and wind-stress forcing function  \caption{Schematic of simulation domain and wind-stress forcing function
83  for barotropic gyre numerical experiment. The domain is enclosed bu solid  for barotropic gyre numerical experiment. The domain is enclosed bu solid
84  walls at $x=$~0,1200km and at $y=$~0,1200km.}  walls at $x=$~0,1200km and at $y=$~0,1200km.}
85  \label{FIG:simulation_config}  \label{FIG:eg-baro-simulation_config}
86  \end{figure}  \end{figure}
87    
88  \subsection{Discrete Numerical Configuration}  \subsection{Equations Solved}
89    \label{www:tutorials}
90   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
91  a uniform grid spacing in the horizontal set to  pressure equation described in Marshall et. al \cite{marshall:97a} is
92   $\Delta x=\Delta y=20$~km, so  employed.
93  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  
94  dissipation. The wind-stress momentum input is added to the momentum equation  dissipation. The wind-stress momentum input is added to the momentum equation
95  for the ``zonal flow'', $u$. Other terms in the model  for the ``zonal flow'', $u$. Other terms in the model
96  are explicitly switched off for this experiement configuration (see section  are explicitly switched off for this experiment configuration (see section
97  \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
98  configuration as follows  configuration as follows
99    
100  \begin{eqnarray}  \begin{eqnarray}
101  \label{EQ:model_equations}  \label{EQ:eg-baro-model_equations}
102  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
103                g\frac{\partial \eta}{\partial x} -                g\frac{\partial \eta}{\partial x} -
104                A_{h}\nabla_{h}^2u                A_{h}\nabla_{h}^2u
105  & = &  & = &
106  \frac{\tau_{x}}{\rho_{0}\Delta z}  \frac{\tau_{x}}{\rho_{0}\Delta z}
107  \\  \\
108  \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -  \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -
109                A_{h}\nabla_{h}^2v                A_{h}\nabla_{h}^2v
110  & = &  & = &
111  0  0
112  \\  \\
# Line 117  configuration as follows Line 116  configuration as follows
116  \end{eqnarray}  \end{eqnarray}
117    
118  \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
119  flow vector $\vec{u}$.  flow vector $\vec{u}$.
120  \\  \\
121    
122    
123    \subsection{Discrete Numerical Configuration}
124    \label{www:tutorials}
125    
126     The domain is discretised with
127    a uniform grid spacing in the horizontal set to
128     $\Delta x=\Delta y=20$~km, so
129    that there are sixty grid cells in the $x$ and $y$ directions. Vertically the
130    model is configured with a single layer with depth, $\Delta z$, of $5000$~m.
131    
132  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
133    \label{www:tutorials}
134    
135  The laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.
136  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},
137    
138  \begin{eqnarray}  \begin{eqnarray}
139  \label{EQ:munk_layer}  \label{EQ:eg-baro-munk_layer}
140  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
141  \end{eqnarray}  \end{eqnarray}
142    
# Line 137  layer is well resolved. Line 147  layer is well resolved.
147    
148  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
149  time step $\delta t=1200$secs. With this time step the stability  time step $\delta t=1200$secs. With this time step the stability
150  parameter to the horizontal laplacian friction \cite{Adcroft_thesis}  parameter to the horizontal Laplacian friction \cite{adcroft:95}
151    
152    
153    
154  \begin{eqnarray}  \begin{eqnarray}
155  \label{EQ:laplacian_stability}  \label{EQ:eg-baro-laplacian_stability}
156  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
157  \end{eqnarray}  \end{eqnarray}
158    
# Line 151  for stability. Line 161  for stability.
161  \\  \\
162    
163  \noindent The numerical stability for inertial oscillations    \noindent The numerical stability for inertial oscillations  
164  \cite{Adcroft_thesis}  \cite{adcroft:95}
165    
166  \begin{eqnarray}  \begin{eqnarray}
167  \label{EQ:inertial_stability}  \label{EQ:eg-baro-inertial_stability}
168  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
169  \end{eqnarray}  \end{eqnarray}
170    
# Line 162  S_{i} = f^{2} {\delta t}^2 Line 172  S_{i} = f^{2} {\delta t}^2
172  limit for stability.  limit for stability.
173  \\  \\
174    
175  \noindent The advective CFL \cite{Adcroft_thesis} for an extreme maximum  \noindent The advective CFL \cite{adcroft:95} for an extreme maximum
176  horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$  horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$
177    
178  \begin{eqnarray}  \begin{eqnarray}
179  \label{EQ:cfl_stability}  \label{EQ:eg-baro-cfl_stability}
180  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
181  \end{eqnarray}  \end{eqnarray}
182    
# Line 174  S_{a} = \frac{| \vec{u} | \delta t}{ \De Line 184  S_{a} = \frac{| \vec{u} | \delta t}{ \De
184  of 0.5 and limits $\delta t$ to $1200s$.  of 0.5 and limits $\delta t$ to $1200s$.
185    
186  \subsection{Code Configuration}  \subsection{Code Configuration}
187  \label{SEC:code_config}  \label{www:tutorials}
188    \label{SEC:eg-baro-code_config}
189    
190  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
191  directory {\it verification/exp0/}.  The experiment files  directory {\it verification/exp0/}.  The experiment files
# Line 188  directory {\it verification/exp0/}.  The Line 199  directory {\it verification/exp0/}.  The
199  \item {\it code/CPP\_OPTIONS.h},  \item {\it code/CPP\_OPTIONS.h},
200  \item {\it code/SIZE.h}.  \item {\it code/SIZE.h}.
201  \end{itemize}  \end{itemize}
202  contain the code customisations and parameter settings for this  contain the code customizations and parameter settings for this
203  experiements. Below we describe the customisations  experiments. Below we describe the customizations
204  to these files associated with this experiment.  to these files associated with this experiment.
205    
206  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
207    \label{www:tutorials}
208    
209  This file, reproduced completely below, specifies the main parameters  This file, reproduced completely below, specifies the main parameters
210  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 213  are
213  \begin{itemize}  \begin{itemize}
214    
215  \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
216  the laplacian friction coefficient to $400 m^2s^{-1}$  the Laplacian friction coefficient to $400 m^2s^{-1}$
217  \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
218  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$
219    
# Line 219  of the pressure inverter. Line 231  of the pressure inverter.
231  startTime=0,  startTime=0,
232  \end{verbatim}  \end{verbatim}
233  this line indicates that the experiment should start from $t=0$  this line indicates that the experiment should start from $t=0$
234  and implicitly supresses searching for checkpoint files associated  and implicitly suppresses searching for checkpoint files associated
235  with restarting an numerical integration from a previously saved state.  with restarting an numerical integration from a previously saved state.
236    
237  \item Line 29,  \item Line 29,
# Line 241  This line sets the momentum equation tim Line 253  This line sets the momentum equation tim
253  usingCartesianGrid=.TRUE.,  usingCartesianGrid=.TRUE.,
254  \end{verbatim}  \end{verbatim}
255  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
256  cartesian coordinate system.  Cartesian coordinate system.
257    
258  \item Line 41,  \item Line 41,
259  \begin{verbatim}  \begin{verbatim}
# Line 304  notes. Line 316  notes.
316  \end{small}  \end{small}
317    
318  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
319    \label{www:tutorials}
320    
321  This file uses standard default values and does not contain  This file uses standard default values and does not contain
322  customisations for this experiment.  customizations for this experiment.
323    
324  \subsubsection{File {\it input/eedata}}  \subsubsection{File {\it input/eedata}}
325    \label{www:tutorials}
326    
327  This file uses standard default values and does not contain  This file uses standard default values and does not contain
328  customisations for this experiment.  customizations for this experiment.
329    
330  \subsubsection{File {\it input/windx.sin\_y}}  \subsubsection{File {\it input/windx.sin\_y}}
331    \label{www:tutorials}
332    
333  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
334  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 339  in MITgcm. The included matlab program {
339  code for creating the {\it input/windx.sin\_y} file.  code for creating the {\it input/windx.sin\_y} file.
340    
341  \subsubsection{File {\it input/topog.box}}  \subsubsection{File {\it input/topog.box}}
342    \label{www:tutorials}
343    
344    
345  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 351  The included matlab program {\it input/g
351  code for creating the {\it input/topog.box} file.  code for creating the {\it input/topog.box} file.
352    
353  \subsubsection{File {\it code/SIZE.h}}  \subsubsection{File {\it code/SIZE.h}}
354    \label{www:tutorials}
355    
356  Two lines are customized in this file for the current experiment  Two lines are customized in this file for the current experiment
357    
# Line 357  axis aligned with the y-coordinate. Line 374  axis aligned with the y-coordinate.
374  \end{small}  \end{small}
375    
376  \subsubsection{File {\it code/CPP\_OPTIONS.h}}  \subsubsection{File {\it code/CPP\_OPTIONS.h}}
377    \label{www:tutorials}
378    
379  This file uses standard default values and does not contain  This file uses standard default values and does not contain
380  customisations for this experiment.  customizations for this experiment.
381    
382    
383  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
384    \label{www:tutorials}
385    
386  This file uses standard default values and does not contain  This file uses standard default values and does not contain
387  customisations for this experiment.  customizations for this experiment.
388    
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