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revision 1.4 by cnh, Thu Oct 25 18:36:54 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}  
 \label{sec:eg-baro}  
   
4  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
5    
6  %\begin{center}  %\begin{center}
# Line 16  Line 13 
13  %{\large May 2001}  %{\large May 2001}
14  %\end{center}  %\end{center}
15    
16  This is the first in a series of sections describing  \section[Barotropic Gyre MITgcm Example]{Barotropic Ocean Gyre In Cartesian Coordinates}
17  example MITgcm numerical experiments. The example experiments  \label{sect:eg-baro}
18  include both straightforward examples of idealized geophysical  \label{www:tutorials}
19  fluid simulations and more involved cases encompassing  \begin{rawhtml}
20  large scale modeling and  <!-- CMIREDIR:eg-baro: -->
21  automatic differentiation. Both hydrostatic and non-hydrostatic  \end{rawhtml}
 experiments 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 similar  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 45  the coriolis parameter $f$ is defined ac Line 37  the coriolis parameter $f$ is defined ac
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 65  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  summarizes the configuration simulated.  summarizes the configuration simulated.
62    
63    %% === eh3 ===
64  \begin{figure}  \begin{figure}
65  \begin{center}  %% \begin{center}
66   \resizebox{7.5in}{5.5in}{  %%  \resizebox{7.5in}{5.5in}{
67     \includegraphics*[0.2in,0.7in][10.5in,10.5in]  %%    \includegraphics*[0.2in,0.7in][10.5in,10.5in]
68      {part3/case_studies/barotropic_gyre/simulation_config.eps} }  %%     {part3/case_studies/barotropic_gyre/simulation_config.eps} }
69  \end{center}  %% \end{center}
70    \centerline{
71      \scalefig{.95}
72      \epsfbox{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{Equations Solved}  \subsection{Equations Solved}
81    \label{www:tutorials}
82  The model is configured in hydrostatic form. The implicit free surface form of the  The model is configured in hydrostatic form. The implicit free surface form of the
83  pressure equation described in Marshall et. al \cite{Marshall97a} is  pressure equation described in Marshall et. al \cite{marshall:97a} is
84  employed.  employed.
85  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous  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
# Line 92  are explicitly switched off for this exp Line 90  are explicitly switched off for this exp
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
# Line 115  flow vector $\vec{u}$. Line 113  flow vector $\vec{u}$.
113    
114    
115  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
116    \label{www:tutorials}
117    
118   The domain is discretised with   The domain is discretised with
119  a uniform grid spacing in the horizontal set to  a uniform grid spacing in the horizontal set to
# Line 123  that there are sixty grid cells in the $ Line 122  that there are sixty grid cells in the $
122  model is configured with a single layer with depth, $\Delta z$, of $5000$~m.  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 139  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 153  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 164  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 176  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 195  experiments. Below we describe the custo Line 196  experiments. Below we describe the custo
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 306  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  customizations 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  customizations 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 326  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 337  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 359  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  customizations 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  customizations for this experiment.  customizations for this experiment.
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