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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{www:tutorials}
18  This document is the first in a series of documents describing  \label{sec:eg-baro}
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}
22  large scale modeling and  \begin{center}
23  automatic differentiation. Both hydrostatic and non-hydrostatic  (in directory: {\it verification/tutorial\_barotropic\_gyre/})
24  experiements are presented, as well as experiments employing  \end{center}
 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.  
   
 \subsection{Experiment Overview}  
25    
26  This example experiment demonstrates using the MITgcm to simulate  This example experiment demonstrates using the MITgcm to simulate
27  a barotropic, wind-forced, ocean gyre circulation. The experiment  a Barotropic, wind-forced, ocean gyre circulation. The files for this
28  is a numerical rendition of the gyre circulation problem simliar  experiment can be found in the verification directory tutorial\_barotropic\_gyre.
29    The experiment is a numerical rendition of the gyre circulation problem similar
30  to the problems described analytically by Stommel in 1966  to the problems described analytically by Stommel in 1966
31  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.
32    
# Line 41  In this experiment the model Line 34  In this experiment the model
34  is configured to represent a rectangular enclosed box of fluid,  is configured to represent a rectangular enclosed box of fluid,
35  $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
36  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
37  in the ``north-south'' direction. Topologically the grid is cartesian and  in the ``north-south'' direction. Topologically the grid is Cartesian and
38  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
39  equation  equation
40    
41  \begin{equation}  \begin{equation}
42  \label{EQ:fcori}  \label{eq:eg-baro-fcori}
43  f(y) = f_{0}+\beta y  f(y) = f_{0}+\beta y
44  \end{equation}  \end{equation}
45    
46  \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
47  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
48  (\ref{EQ:fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$.  (\ref{eq:eg-baro-fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$.
49  \\  \\
50  \\  \\
51   The sinusoidal wind-stress variations are defined according to   The sinusoidal wind-stress variations are defined according to
52    
53  \begin{equation}  \begin{equation}
54  \label{EQ:taux}  \label{eq:eg-baro-taux}
55  \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})  \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})
56  \end{equation}  \end{equation}
57    
# Line 66  simulated domain. For this experiment $f Line 59  simulated domain. For this experiment $f
59  $\tau_0$ is set to $0.1N m^{-2}$.  $\tau_0$ is set to $0.1N m^{-2}$.
60  \\  \\
61  \\  \\
62  Figure \ref{FIG:simulation_config}  Figure \ref{fig:eg-baro-simulation_config}
63  summarises the configuration simulated.  summarizes the configuration simulated.
64    
65    %% === eh3 ===
66  \begin{figure}  \begin{figure}
67    %% \begin{center}
68    %%  \resizebox{7.5in}{5.5in}{
69    %%    \includegraphics*[0.2in,0.7in][10.5in,10.5in]
70    %%     {s_examples/barotropic_gyre/simulation_config.eps} }
71    %% \end{center}
72  \centerline{  \centerline{
73   \resizebox{7.5in}{5.5in}{    \scalefig{.95}
74     \includegraphics*[0.2in,0.7in][10.5in,10.5in]    \epsfbox{s_examples/barotropic_gyre/simulation_config.eps}
     {part3/case_studies/barotropic_gyre/simulation_config.eps} }  
75  }  }
76  \caption{Schematic of simulation domain and wind-stress forcing function  \caption{Schematic of simulation domain and wind-stress forcing function
77  for barotropic gyre numerical experiment. The domain is enclosed bu solid  for barotropic gyre numerical experiment. The domain is enclosed bu solid
78  walls at $x=$~0,1200km and at $y=$~0,1200km.}  walls at $x=$~0,1200km and at $y=$~0,1200km.}
79  \label{FIG:simulation_config}  \label{fig:eg-baro-simulation_config}
80  \end{figure}  \end{figure}
81    
82  \subsection{Discrete Numerical Configuration}  \subsection{Equations Solved}
83    %\label{www:tutorials}
84   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
85  a uniform grid spacing in the horizontal set to  pressure equation described in Marshall et. al \cite{marshall:97a} is
86   $\Delta x=\Delta y=20$~km, so  employed.
87  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  
88  dissipation. The wind-stress momentum input is added to the momentum equation  dissipation. The wind-stress momentum input is added to the momentum equation
89  for the ``zonal flow'', $u$. Other terms in the model  for the ``zonal flow'', $u$. Other terms in the model
90  are explicitly switched off for this experiement configuration (see section  are explicitly switched off for this experiment configuration (see section
91  \ref{SEC:code_config} ), yielding an active set of equations solved in this  \ref{sec:eg-baro-code_config} ), yielding an active set of equations solved
92  configuration as follows  in this configuration as follows
93    
94  \begin{eqnarray}  \begin{eqnarray}
95  \label{EQ:model_equations}  \label{eq:eg-baro-model_equations}
96  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
97                g\frac{\partial \eta}{\partial x} -                g\frac{\partial \eta}{\partial x} -
98                A_{h}\nabla_{h}^2u                A_{h}\nabla_{h}^2u
99  & = &  & = &
100  \frac{\tau_{x}}{\rho_{0}\Delta z}  \frac{\tau_{x}}{\rho_{0}\Delta z}
101  \\  \\
102  \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -  \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -
103                A_{h}\nabla_{h}^2v                A_{h}\nabla_{h}^2v
104  & = &  & = &
105  0  0
106  \\  \\
# Line 117  configuration as follows Line 110  configuration as follows
110  \end{eqnarray}  \end{eqnarray}
111    
112  \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
113  flow vector $\vec{u}$.  flow vector $\vec{u}$.
114  \\  \\
115    
116    
117    \subsection{Discrete Numerical Configuration}
118    %\label{www:tutorials}
119    
120     The domain is discretised with
121    a uniform grid spacing in the horizontal set to
122     $\Delta x=\Delta y=20$~km, so
123    that there are sixty grid cells in the $x$ and $y$ directions. Vertically the
124    model is configured with a single layer with depth, $\Delta z$, of $5000$~m.
125    
126  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
127    %\label{www:tutorials}
128    
129  The laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.
130  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},
131    
132  \begin{eqnarray}  \begin{eqnarray}
133  \label{EQ:munk_layer}  \label{eq:eg-baro-munk_layer}
134  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
135  \end{eqnarray}  \end{eqnarray}
136    
# Line 137  layer is well resolved. Line 141  layer is well resolved.
141    
142  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
143  time step $\delta t=1200$secs. With this time step the stability  time step $\delta t=1200$secs. With this time step the stability
144  parameter to the horizontal laplacian friction \cite{Adcroft_thesis}  parameter to the horizontal Laplacian friction \cite{adcroft:95}
145    
146    
147    
148  \begin{eqnarray}  \begin{eqnarray}
149  \label{EQ:laplacian_stability}  \label{eq:eg-baro-laplacian_stability}
150  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
151  \end{eqnarray}  \end{eqnarray}
152    
# Line 151  for stability. Line 155  for stability.
155  \\  \\
156    
157  \noindent The numerical stability for inertial oscillations    \noindent The numerical stability for inertial oscillations  
158  \cite{Adcroft_thesis}  \cite{adcroft:95}
159    
160  \begin{eqnarray}  \begin{eqnarray}
161  \label{EQ:inertial_stability}  \label{eq:eg-baro-inertial_stability}
162  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
163  \end{eqnarray}  \end{eqnarray}
164    
# Line 162  S_{i} = f^{2} {\delta t}^2 Line 166  S_{i} = f^{2} {\delta t}^2
166  limit for stability.  limit for stability.
167  \\  \\
168    
169  \noindent The advective CFL \cite{Adcroft_thesis} for an extreme maximum  \noindent The advective CFL \cite{adcroft:95} for an extreme maximum
170  horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$  horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$
171    
172  \begin{eqnarray}  \begin{eqnarray}
173  \label{EQ:cfl_stability}  \label{eq:eg-baro-cfl_stability}
174  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
175  \end{eqnarray}  \end{eqnarray}
176    
# Line 174  S_{a} = \frac{| \vec{u} | \delta t}{ \De Line 178  S_{a} = \frac{| \vec{u} | \delta t}{ \De
178  of 0.5 and limits $\delta t$ to $1200s$.  of 0.5 and limits $\delta t$ to $1200s$.
179    
180  \subsection{Code Configuration}  \subsection{Code Configuration}
181  \label{SEC:code_config}  %\label{www:tutorials}
182    \label{sec:eg-baro-code_config}
183    
184  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
185  directory {\it verification/exp0/}.  The experiment files  directory {\it verification/tutorial\_barotropic\_gyre/}.  
186    The experiment files
187  \begin{itemize}  \begin{itemize}
188  \item {\it input/data}  \item {\it input/data}
189  \item {\it input/data.pkg}  \item {\it input/data.pkg}
# Line 188  directory {\it verification/exp0/}.  The Line 194  directory {\it verification/exp0/}.  The
194  \item {\it code/CPP\_OPTIONS.h},  \item {\it code/CPP\_OPTIONS.h},
195  \item {\it code/SIZE.h}.  \item {\it code/SIZE.h}.
196  \end{itemize}  \end{itemize}
197  contain the code customisations and parameter settings for this  contain the code customizations and parameter settings for this
198  experiements. Below we describe the customisations  experiments. Below we describe the customizations
199  to these files associated with this experiment.  to these files associated with this experiment.
200    
201  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
202    %\label{www:tutorials}
203    
204  This file, reproduced completely below, specifies the main parameters  This file, reproduced completely below, specifies the main parameters
205  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 208  are
208  \begin{itemize}  \begin{itemize}
209    
210  \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
211  the laplacian friction coefficient to $400 m^2s^{-1}$  the Laplacian friction coefficient to $400 m^2s^{-1}$
212  \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
213  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$
214    
# Line 219  of the pressure inverter. Line 226  of the pressure inverter.
226  startTime=0,  startTime=0,
227  \end{verbatim}  \end{verbatim}
228  this line indicates that the experiment should start from $t=0$  this line indicates that the experiment should start from $t=0$
229  and implicitly supresses searching for checkpoint files associated  and implicitly suppresses searching for checkpoint files associated
230  with restarting an numerical integration from a previously saved state.  with restarting an numerical integration from a previously saved state.
231    
232  \item Line 29,  \item Line 29,
# Line 241  This line sets the momentum equation tim Line 248  This line sets the momentum equation tim
248  usingCartesianGrid=.TRUE.,  usingCartesianGrid=.TRUE.,
249  \end{verbatim}  \end{verbatim}
250  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
251  cartesian coordinate system.  Cartesian coordinate system.
252    
253  \item Line 41,  \item Line 41,
254  \begin{verbatim}  \begin{verbatim}
# Line 300  that are described in the MITgcm Getting Line 307  that are described in the MITgcm Getting
307  notes.  notes.
308    
309  \begin{small}  \begin{small}
310  \input{part3/case_studies/barotropic_gyre/input/data}  \input{s_examples/barotropic_gyre/input/data}
311  \end{small}  \end{small}
312    
313  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
314    %\label{www:tutorials}
315    
316  This file uses standard default values and does not contain  This file uses standard default values and does not contain
317  customisations for this experiment.  customizations for this experiment.
318    
319  \subsubsection{File {\it input/eedata}}  \subsubsection{File {\it input/eedata}}
320    %\label{www:tutorials}
321    
322  This file uses standard default values and does not contain  This file uses standard default values and does not contain
323  customisations for this experiment.  customizations for this experiment.
324    
325  \subsubsection{File {\it input/windx.sin\_y}}  \subsubsection{File {\it input/windx.sin\_y}}
326    %\label{www:tutorials}
327    
328  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
329  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 334  in MITgcm. The included matlab program {
334  code for creating the {\it input/windx.sin\_y} file.  code for creating the {\it input/windx.sin\_y} file.
335    
336  \subsubsection{File {\it input/topog.box}}  \subsubsection{File {\it input/topog.box}}
337    %\label{www:tutorials}
338    
339    
340  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 346  The included matlab program {\it input/g
346  code for creating the {\it input/topog.box} file.  code for creating the {\it input/topog.box} file.
347    
348  \subsubsection{File {\it code/SIZE.h}}  \subsubsection{File {\it code/SIZE.h}}
349    %\label{www:tutorials}
350    
351  Two lines are customized in this file for the current experiment  Two lines are customized in this file for the current experiment
352    
# Line 353  axis aligned with the y-coordinate. Line 365  axis aligned with the y-coordinate.
365  \end{itemize}  \end{itemize}
366    
367  \begin{small}  \begin{small}
368  \input{part3/case_studies/barotropic_gyre/code/SIZE.h}  \input{s_examples/barotropic_gyre/code/SIZE.h}
369  \end{small}  \end{small}
370    
371  \subsubsection{File {\it code/CPP\_OPTIONS.h}}  \subsubsection{File {\it code/CPP\_OPTIONS.h}}
372    %\label{www:tutorials}
373    
374  This file uses standard default values and does not contain  This file uses standard default values and does not contain
375  customisations for this experiment.  customizations for this experiment.
376    
377    
378  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
379    %\label{www:tutorials}
380    
381  This file uses standard default values and does not contain  This file uses standard default values and does not contain
382  customisations for this experiment.  customizations for this experiment.
383    
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