/[MITgcm]/manual/s_examples/rotating_tank/tank.tex
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revision 1.1 by afe, Tue Jun 22 15:07:37 2004 UTC revision 1.3 by afe, Mon Jul 26 16:21:15 2004 UTC
# Line 4  Line 4 
4  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
5    
6  %\begin{center}  %\begin{center}
7  %{\Large \bf Using MITgcm to Simulate a Rotating Tank in Cylindrical  %{\Large \bf Using MITgcm to Simulate a Rotating Tank in Cylindrical
8  %Coordinates}  %Coordinates}
9  %  %
10  %\vspace*{4mm}  %\vspace*{4mm}
11  %  %
12  %\vspace*{3mm}  %\vspace*{3mm}
13  %{\large June 2004}  %{\large May 2001}
14  %\end{center}  %\end{center}
15    
16  This is the first in a series of tutorials describing  This is the first in a series of tutorials describing
# Line 25  These ``case study'' documents include i Line 25  These ``case study'' documents include i
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  \section{Barotropic Ocean Gyre In Cartesian Coordinates}  \section{A Rotating Tank in Cylindrical Coordinates}
29  \label{sect:eg-baro}  \label{sect:eg-tank}
30  \label{www:tutorials}  \label{www:tutorials}
31    
32    
33    This example experiment demonstrates using the MITgcm to simulate
34    a Barotropic, wind-forced, ocean gyre circulation. The experiment
35    is a numerical rendition of the gyre circulation problem similar
36    to the problems described analytically by Stommel in 1966
37    \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.
38    
39    In this experiment the model
40    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
42    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
44    the coriolis parameter $f$ is defined according to a mid-latitude beta-plane
45    equation
46    
47    \begin{equation}
48    \label{EQ:eg-baro-fcori}
49    f(y) = f_{0}+\beta y
50    \end{equation}
51    
52    \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
54    (\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
58    
59    \begin{equation}
60    \label{EQ:eg-baro-taux}
61    \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})
62    \end{equation}
63    
64    \noindent where $L_{y}$ is the lateral domain extent ($1200$~km) and
65    $\tau_0$ is set to $0.1N m^{-2}$.
66    \\
67    \\
68    Figure \ref{FIG:eg-baro-simulation_config}
69    summarizes the configuration simulated.
70    
71    %% === eh3 ===
72    \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{
79      \scalefig{.95}
80      \epsfbox{part3/case_studies/barotropic_gyre/simulation_config.eps}
81    }
82    \caption{Schematic of simulation domain and wind-stress forcing function
83    for barotropic gyre numerical experiment. The domain is enclosed bu solid
84    walls at $x=$~0,1200km and at $y=$~0,1200km.}
85    \label{FIG:eg-baro-simulation_config}
86    \end{figure}
87    
88  \subsection{Equations Solved}  \subsection{Equations Solved}
89  \label{www:tutorials}  \label{www:tutorials}
90  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
91    pressure equation described in Marshall et. al \cite{marshall:97a} is
92    employed.
93    A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous
94    dissipation. The wind-stress momentum input is added to the momentum equation
95    for the ``zonal flow'', $u$. Other terms in the model
96    are explicitly switched off for this experiment configuration (see section
97    \ref{SEC:code_config} ), yielding an active set of equations solved in this
98    configuration as follows
99    
100    \begin{eqnarray}
101    \label{EQ:eg-baro-model_equations}
102    \frac{Du}{Dt} - fv +
103                  g\frac{\partial \eta}{\partial x} -
104                  A_{h}\nabla_{h}^2u
105    & = &
106    \frac{\tau_{x}}{\rho_{0}\Delta z}
107    \\
108    \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} -
109                  A_{h}\nabla_{h}^2v
110    & = &
111    0
112    \\
113    \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}
114    &=&
115    0
116    \end{eqnarray}
117    
118    \noindent where $u$ and $v$ and the $x$ and $y$ components of the
119    flow vector $\vec{u}$.
120    \\
121    
122    
123  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
# Line 48  model is configured with a single layer Line 132  model is configured with a single layer
132  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
133  \label{www:tutorials}  \label{www:tutorials}
134    
135    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:95},
137    
138    \begin{eqnarray}
139    \label{EQ:eg-baro-munk_layer}
140    M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
141    \end{eqnarray}
142    
143    \noindent  of $\approx 100$km. This is greater than the model
144    resolution $\Delta x$, ensuring that the frictional boundary
145    layer is well resolved.
146    \\
147    
148    \noindent The model is stepped forward with a
149    time step $\delta t=1200$secs. With this time step the stability
150    parameter to the horizontal Laplacian friction \cite{adcroft:95}
151    
152    
153    
154    \begin{eqnarray}
155    \label{EQ:eg-baro-laplacian_stability}
156    S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
157    \end{eqnarray}
158    
159    \noindent evaluates to 0.012, which is well below the 0.3 upper limit
160    for stability.
161    \\
162    
163    \noindent The numerical stability for inertial oscillations  
164    \cite{adcroft:95}
165    
166    \begin{eqnarray}
167    \label{EQ:eg-baro-inertial_stability}
168    S_{i} = f^{2} {\delta t}^2
169    \end{eqnarray}
170    
171    \noindent evaluates to $0.0144$, which is well below the $0.5$ upper
172    limit for stability.
173    \\
174    
175    \noindent The advective CFL \cite{adcroft:95} for an extreme maximum
176    horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$
177    
178    \begin{eqnarray}
179    \label{EQ:eg-baro-cfl_stability}
180    S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
181    \end{eqnarray}
182    
183    \noindent evaluates to 0.12. This is approaching the stability limit
184    of 0.5 and limits $\delta t$ to $1200s$.
185    
186  \subsection{Code Configuration}  \subsection{Code Configuration}
187  \label{www:tutorials}  \label{www:tutorials}
# Line 177  code to generate a valid {\bf zonalWindF Line 311  code to generate a valid {\bf zonalWindF
311  that are described in the MITgcm Getting Started and MITgcm Parameters  that are described in the MITgcm Getting Started and MITgcm Parameters
312  notes.  notes.
313    
314  %%\begin{small}  \begin{small}
315  %%\input{part3/case_studies/barotropic_gyre/input/data}  \input{part3/case_studies/barotropic_gyre/input/data}
316  %%\end{small}  \end{small}
317    
318  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
319  \label{www:tutorials}  \label{www:tutorials}
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