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revision 1.1 by adcroft, Wed Aug 8 16:15:49 2001 UTC revision 1.8 by cnh, Thu Feb 28 19:32:19 2002 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    
31    
32  This example experiment demonstrates using the MITgcm to simulate  This example experiment demonstrates using the MITgcm to simulate
33  a barotropic, wind-forced, ocean gyre circulation. The experiment  a Barotropic, wind-forced, ocean gyre circulation. The experiment
34  is a numerical rendition of the gyre circulation problem simliar  is a numerical rendition of the gyre circulation problem similar
35  to the problems described analytically by Stommel in 1966  to the problems described analytically by Stommel in 1966
36  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.
37    
# Line 41  In this experiment the model Line 39  In this experiment the model
39  is configured to represent a rectangular enclosed box of fluid,  is configured to represent a rectangular enclosed box of fluid,
40  $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
41  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
42  in the ``north-south'' direction. Topologically the grid is cartesian and  in the ``north-south'' direction. Topologically the grid is Cartesian and
43  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
44  equation  equation
45    
46  \begin{equation}  \begin{equation}
47  \label{EQ:fcori}  \label{EQ:eg-baro-fcori}
48  f(y) = f_{0}+\beta y  f(y) = f_{0}+\beta y
49  \end{equation}  \end{equation}
50    
51  \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
52  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
53  (\ref{EQ:fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$.  (\ref{EQ:eg-baro-fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$.
54  \\  \\
55  \\  \\
56   The sinusoidal wind-stress variations are defined according to   The sinusoidal wind-stress variations are defined according to
57    
58  \begin{equation}  \begin{equation}
59  \label{EQ:taux}  \label{EQ:eg-baro-taux}
60  \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})  \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})
61  \end{equation}  \end{equation}
62    
# Line 66  simulated domain. For this experiment $f Line 64  simulated domain. For this experiment $f
64  $\tau_0$ is set to $0.1N m^{-2}$.  $\tau_0$ is set to $0.1N m^{-2}$.
65  \\  \\
66  \\  \\
67  Figure \ref{FIG:simulation_config}  Figure \ref{FIG:eg-baro-simulation_config}
68  summarises the configuration simulated.  summarizes the configuration simulated.
69    
70  \begin{figure}  \begin{figure}
71  \centerline{  \begin{center}
72   \resizebox{7.5in}{5.5in}{   \resizebox{7.5in}{5.5in}{
73     \includegraphics*[0.2in,0.7in][10.5in,10.5in]     \includegraphics*[0.2in,0.7in][10.5in,10.5in]
74      {part3/case_studies/barotropic_gyre/simulation_config.eps} }      {part3/case_studies/barotropic_gyre/simulation_config.eps} }
75  }  \end{center}
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    The model is configured in hydrostatic form. The implicit free surface form of the
84   The model is configured in hydrostatic form.  The domain is discretised with  pressure equation described in Marshall et. al \cite{marshall:97a} is
85  a uniform grid spacing in the horizontal set to  employed.
86   $\Delta x=\Delta y=20$~km, so  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous
 that there are sixty grid cells in the $x$ and $y$ directions. Vertically the  
 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  
87  dissipation. The wind-stress momentum input is added to the momentum equation  dissipation. The wind-stress momentum input is added to the momentum equation
88  for the ``zonal flow'', $u$. Other terms in the model  for the ``zonal flow'', $u$. Other terms in the model
89  are explicitly switched off for this experiement configuration (see section  are explicitly switched off for this experiment configuration (see section
90  \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
91  configuration as follows  configuration as follows
92    
93  \begin{eqnarray}  \begin{eqnarray}
94  \label{EQ:model_equations}  \label{EQ:eg-baro-model_equations}
95  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
96                g\frac{\partial \eta}{\partial x} -                g\frac{\partial \eta}{\partial x} -
97                A_{h}\nabla_{h}^2u                A_{h}\nabla_{h}^2u
98  & = &  & = &
99  \frac{\tau_{x}}{\rho_{0}\Delta z}  \frac{\tau_{x}}{\rho_{0}\Delta z}
100  \\  \\
101  \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -  \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -
102                A_{h}\nabla_{h}^2v                A_{h}\nabla_{h}^2v
103  & = &  & = &
104  0  0
105  \\  \\
# Line 117  configuration as follows Line 109  configuration as follows
109  \end{eqnarray}  \end{eqnarray}
110    
111  \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
112  flow vector $\vec{u}$.  flow vector $\vec{u}$.
113  \\  \\
114    
115    
116    \subsection{Discrete Numerical Configuration}
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    
126  The laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.
127  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},
128    
129  \begin{eqnarray}  \begin{eqnarray}
130  \label{EQ:munk_layer}  \label{EQ:eg-baro-munk_layer}
131  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
132  \end{eqnarray}  \end{eqnarray}
133    
# Line 137  layer is well resolved. Line 138  layer is well resolved.
138    
139  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
140  time step $\delta t=1200$secs. With this time step the stability  time step $\delta t=1200$secs. With this time step the stability
141  parameter to the horizontal laplacian friction \cite{Adcroft_thesis}  parameter to the horizontal Laplacian friction \cite{adcroft:95}
142    
143    
144    
145  \begin{eqnarray}  \begin{eqnarray}
146  \label{EQ:laplacian_stability}  \label{EQ:eg-baro-laplacian_stability}
147  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
148  \end{eqnarray}  \end{eqnarray}
149    
# Line 151  for stability. Line 152  for stability.
152  \\  \\
153    
154  \noindent The numerical stability for inertial oscillations    \noindent The numerical stability for inertial oscillations  
155  \cite{Adcroft_thesis}  \cite{adcroft:95}
156    
157  \begin{eqnarray}  \begin{eqnarray}
158  \label{EQ:inertial_stability}  \label{EQ:eg-baro-inertial_stability}
159  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
160  \end{eqnarray}  \end{eqnarray}
161    
# Line 162  S_{i} = f^{2} {\delta t}^2 Line 163  S_{i} = f^{2} {\delta t}^2
163  limit for stability.  limit for stability.
164  \\  \\
165    
166  \noindent The advective CFL \cite{Adcroft_thesis} for an extreme maximum  \noindent The advective CFL \cite{adcroft:95} for an extreme maximum
167  horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$  horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$
168    
169  \begin{eqnarray}  \begin{eqnarray}
170  \label{EQ:cfl_stability}  \label{EQ:eg-baro-cfl_stability}
171  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
172  \end{eqnarray}  \end{eqnarray}
173    
# Line 174  S_{a} = \frac{| \vec{u} | \delta t}{ \De Line 175  S_{a} = \frac{| \vec{u} | \delta t}{ \De
175  of 0.5 and limits $\delta t$ to $1200s$.  of 0.5 and limits $\delta t$ to $1200s$.
176    
177  \subsection{Code Configuration}  \subsection{Code Configuration}
178  \label{SEC:code_config}  \label{SEC:eg-baro-code_config}
179    
180  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
181  directory {\it verification/exp0/}.  The experiment files  directory {\it verification/exp0/}.  The experiment files
# Line 188  directory {\it verification/exp0/}.  The Line 189  directory {\it verification/exp0/}.  The
189  \item {\it code/CPP\_OPTIONS.h},  \item {\it code/CPP\_OPTIONS.h},
190  \item {\it code/SIZE.h}.  \item {\it code/SIZE.h}.
191  \end{itemize}  \end{itemize}
192  contain the code customisations and parameter settings for this  contain the code customizations and parameter settings for this
193  experiements. Below we describe the customisations  experiments. Below we describe the customizations
194  to these files associated with this experiment.  to these files associated with this experiment.
195    
196  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
# Line 201  are Line 202  are
202  \begin{itemize}  \begin{itemize}
203    
204  \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
205  the laplacian friction coefficient to $400 m^2s^{-1}$  the Laplacian friction coefficient to $400 m^2s^{-1}$
206  \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
207  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$
208    
# Line 219  of the pressure inverter. Line 220  of the pressure inverter.
220  startTime=0,  startTime=0,
221  \end{verbatim}  \end{verbatim}
222  this line indicates that the experiment should start from $t=0$  this line indicates that the experiment should start from $t=0$
223  and implicitly supresses searching for checkpoint files associated  and implicitly suppresses searching for checkpoint files associated
224  with restarting an numerical integration from a previously saved state.  with restarting an numerical integration from a previously saved state.
225    
226  \item Line 29,  \item Line 29,
# Line 241  This line sets the momentum equation tim Line 242  This line sets the momentum equation tim
242  usingCartesianGrid=.TRUE.,  usingCartesianGrid=.TRUE.,
243  \end{verbatim}  \end{verbatim}
244  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
245  cartesian coordinate system.  Cartesian coordinate system.
246    
247  \item Line 41,  \item Line 41,
248  \begin{verbatim}  \begin{verbatim}
# Line 306  notes. Line 307  notes.
307  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
308    
309  This file uses standard default values and does not contain  This file uses standard default values and does not contain
310  customisations for this experiment.  customizations for this experiment.
311    
312  \subsubsection{File {\it input/eedata}}  \subsubsection{File {\it input/eedata}}
313    
314  This file uses standard default values and does not contain  This file uses standard default values and does not contain
315  customisations for this experiment.  customizations for this experiment.
316    
317  \subsubsection{File {\it input/windx.sin\_y}}  \subsubsection{File {\it input/windx.sin\_y}}
318    
# Line 359  axis aligned with the y-coordinate. Line 360  axis aligned with the y-coordinate.
360  \subsubsection{File {\it code/CPP\_OPTIONS.h}}  \subsubsection{File {\it code/CPP\_OPTIONS.h}}
361    
362  This file uses standard default values and does not contain  This file uses standard default values and does not contain
363  customisations for this experiment.  customizations for this experiment.
364    
365    
366  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
367    
368  This file uses standard default values and does not contain  This file uses standard default values and does not contain
369  customisations for this experiment.  customizations for this experiment.
370    
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