| 36 |
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| 37 |
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| 38 |
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This example experiment demonstrates using the MITgcm to simulate |
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a Barotropic, wind-forced, ocean gyre circulation. The experiment |
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is a numerical rendition of the gyre circulation problem similar |
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to the problems described analytically by Stommel in 1966 |
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\cite{Stommel66} and numerically in Holland et. al \cite{Holland75}. |
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In this experiment the model |
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is configured to represent a rectangular enclosed box of fluid, |
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$1200 \times 1200 $~km in lateral extent. The fluid is $5$~km deep and is forced |
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by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally |
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in the ``north-south'' direction. Topologically the grid is Cartesian and |
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the coriolis parameter $f$ is defined according to a mid-latitude beta-plane |
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equation |
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\begin{equation} |
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\label{EQ:eg-baro-fcori} |
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f(y) = f_{0}+\beta y |
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\end{equation} |
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| 39 |
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\noindent where $y$ is the distance along the ``north-south'' axis of the |
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simulated domain. For this experiment $f_{0}$ is set to $10^{-4}s^{-1}$ in |
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(\ref{EQ:eg-baro-fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$. |
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\\ |
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\\ |
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The sinusoidal wind-stress variations are defined according to |
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\begin{equation} |
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\label{EQ:eg-baro-taux} |
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\tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y}) |
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\end{equation} |
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\noindent where $L_{y}$ is the lateral domain extent ($1200$~km) and |
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$\tau_0$ is set to $0.1N m^{-2}$. |
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\\ |
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\\ |
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Figure \ref{FIG:eg-baro-simulation_config} |
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summarizes the configuration simulated. |
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%% === eh3 === |
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\begin{figure} |
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%% \begin{center} |
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%% \resizebox{7.5in}{5.5in}{ |
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%% \includegraphics*[0.2in,0.7in][10.5in,10.5in] |
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%% {part3/case_studies/barotropic_gyre/simulation_config.eps} } |
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%% \end{center} |
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\centerline{ |
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\scalefig{.95} |
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\epsfbox{part3/case_studies/barotropic_gyre/simulation_config.eps} |
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} |
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\caption{Schematic of simulation domain and wind-stress forcing function |
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for barotropic gyre numerical experiment. The domain is enclosed bu solid |
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walls at $x=$~0,1200km and at $y=$~0,1200km.} |
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\label{FIG:eg-baro-simulation_config} |
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\end{figure} |
|
| 40 |
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| 41 |
\subsection{Equations Solved} |
\subsection{Equations Solved} |
| 42 |
\label{www:tutorials} |
\label{www:tutorials} |
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The model is configured in hydrostatic form. The implicit free surface form of the |
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pressure equation described in Marshall et. al \cite{marshall:97a} is |
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employed. |
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A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous |
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dissipation. The wind-stress momentum input is added to the momentum equation |
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for the ``zonal flow'', $u$. Other terms in the model |
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are explicitly switched off for this experiment configuration (see section |
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\ref{SEC:code_config} ), yielding an active set of equations solved in this |
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configuration as follows |
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\begin{eqnarray} |
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\label{EQ:eg-baro-model_equations} |
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\frac{Du}{Dt} - fv + |
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g\frac{\partial \eta}{\partial x} - |
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A_{h}\nabla_{h}^2u |
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& = & |
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\frac{\tau_{x}}{\rho_{0}\Delta z} |
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\\ |
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\frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} - |
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A_{h}\nabla_{h}^2v |
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& = & |
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0 |
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\\ |
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\frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u} |
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&=& |
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0 |
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\end{eqnarray} |
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\noindent where $u$ and $v$ and the $x$ and $y$ components of the |
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flow vector $\vec{u}$. |
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\\ |
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| 43 |
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| 44 |
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| 45 |
\subsection{Discrete Numerical Configuration} |
\subsection{Discrete Numerical Configuration} |
| 51 |
that there are sixty grid cells in the $x$ and $y$ directions. Vertically the |
that there are sixty grid cells in the $x$ and $y$ directions. Vertically the |
| 52 |
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. |
| 53 |
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\subsubsection{Numerical Stability Criteria} |
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\label{www:tutorials} |
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The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$. |
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This value is chosen to yield a Munk layer width \cite{adcroft:95}, |
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\begin{eqnarray} |
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\label{EQ:eg-baro-munk_layer} |
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M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}} |
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\end{eqnarray} |
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\noindent of $\approx 100$km. This is greater than the model |
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resolution $\Delta x$, ensuring that the frictional boundary |
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layer is well resolved. |
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\\ |
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\noindent The model is stepped forward with a |
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time step $\delta t=1200$secs. With this time step the stability |
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parameter to the horizontal Laplacian friction \cite{adcroft:95} |
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\begin{eqnarray} |
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\label{EQ:eg-baro-laplacian_stability} |
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S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2} |
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\end{eqnarray} |
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\noindent evaluates to 0.012, which is well below the 0.3 upper limit |
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for stability. |
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\\ |
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\noindent The numerical stability for inertial oscillations |
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\cite{adcroft:95} |
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\begin{eqnarray} |
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\label{EQ:eg-baro-inertial_stability} |
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S_{i} = f^{2} {\delta t}^2 |
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\end{eqnarray} |
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\noindent evaluates to $0.0144$, which is well below the $0.5$ upper |
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limit for stability. |
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\\ |
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\noindent The advective CFL \cite{adcroft:95} for an extreme maximum |
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horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$ |
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\begin{eqnarray} |
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\label{EQ:eg-baro-cfl_stability} |
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S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x} |
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\end{eqnarray} |
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\noindent evaluates to 0.12. This is approaching the stability limit |
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of 0.5 and limits $\delta t$ to $1200s$. |
|
| 54 |
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| 55 |
\subsection{Code Configuration} |
\subsection{Code Configuration} |
| 56 |
\label{www:tutorials} |
\label{www:tutorials} |
| 123 |
usingCylindricalGrid=.TRUE., |
usingCylindricalGrid=.TRUE., |
| 124 |
\end{verbatim} |
\end{verbatim} |
| 125 |
This line requests that the simulation be performed in a |
This line requests that the simulation be performed in a |
| 126 |
Cartesian coordinate system. |
cylindrical coordinate system. |
| 127 |
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|
| 128 |
\item Line 41, |
\item Line qqq, |
| 129 |
\begin{verbatim} |
\begin{verbatim} |
| 130 |
delX=60*20E3, |
dXspacing=3, |
| 131 |
\end{verbatim} |
\end{verbatim} |
| 132 |
This line sets the horizontal grid spacing between each x-coordinate line |
This line sets the azimuthal grid spacing between each x-coordinate line |
| 133 |
in the discrete grid. The syntax indicates that the discrete grid |
in the discrete grid. The syntax indicates that the discrete grid |
| 134 |
should be comprise of $60$ grid lines each separated by $20 \times 10^{3}m$ |
should be comprise of $120$ grid lines each separated by $3^{\circ}$. |
| 135 |
($20$~km). |
|
| 136 |
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| 137 |
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| 138 |
\item Line 42, |
\item Line qqq, |
| 139 |
\begin{verbatim} |
\begin{verbatim} |
| 140 |
delY=60*20E3, |
dYspacing=0.01, |
| 141 |
\end{verbatim} |
\end{verbatim} |
| 142 |
This line sets the horizontal grid spacing between each y-coordinate line |
This line sets the radial grid spacing between each $\rho$-coordinate line |
| 143 |
in the discrete grid to $20 \times 10^{3}m$ ($20$~km). |
in the discrete grid to $1cm$. |
| 144 |
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|
| 145 |
\item Line 43, |
\item Line 43, |
| 146 |
\begin{verbatim} |
\begin{verbatim} |
| 147 |
delZ=5000, |
delZ=29*0.005, |
| 148 |
\end{verbatim} |
\end{verbatim} |
| 149 |
This line sets the vertical grid spacing between each z-coordinate line |
This line sets the vertical grid spacing between each z-coordinate line |
| 150 |
in the discrete grid to $5000m$ ($5$~km). |
in the discrete grid to $5000m$ ($5$~km). |
| 151 |
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|
| 152 |
\item Line 46, |
\item Line 46, |
| 153 |
\begin{verbatim} |
\begin{verbatim} |
| 154 |
bathyFile='topog.box' |
bathyFile='bathyPol.bin', |
| 155 |
\end{verbatim} |
\end{verbatim} |
| 156 |
This line specifies the name of the file from which the domain |
This line specifies the name of the file from which the domain |
| 157 |
bathymetry is read. This file is a two-dimensional ($x,y$) map of |
``bathymetry'' (tank depth) is read. This file is a two-dimensional |
| 158 |
|
($x,y$) map of |
| 159 |
depths. This file is assumed to contain 64-bit binary numbers |
depths. This file is assumed to contain 64-bit binary numbers |
| 160 |
giving the depth of the model at each grid cell, ordered with the x |
giving the depth of the model at each grid cell, ordered with the $x$ |
| 161 |
coordinate varying fastest. The points are ordered from low coordinate |
coordinate varying fastest. The points are ordered from low coordinate |
| 162 |
to high coordinate for both axes. The units and orientation of the |
to high coordinate for both axes. The units and orientation of the |
| 163 |
depths in this file are the same as used in the MITgcm code. In this |
depths in this file are the same as used in the MITgcm code. In this |
| 164 |
experiment, a depth of $0m$ indicates a solid wall and a depth |
experiment, a depth of $0m$ indicates an area outside of the tank |
| 165 |
of $-5000m$ indicates open ocean. The matlab program |
and a depth |
| 166 |
{\it input/gendata.m} shows an example of how to generate a |
f $-0.145m$ indicates the tank itself. |
|
bathymetry file. |
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|
| 167 |
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| 168 |
\item Line 49, |
\item Line 49, |
| 169 |
\begin{verbatim} |
\begin{verbatim} |
| 170 |
zonalWindFile='windx.sin_y' |
hydrogThetaFile='thetaPol.bin', |
| 171 |
\end{verbatim} |
\end{verbatim} |
| 172 |
This line specifies the name of the file from which the x-direction |
This line specifies the name of the file from which the initial values |
| 173 |
surface wind stress is read. This file is also a two-dimensional |
of $\theta$ |
| 174 |
($x,y$) map and is enumerated and formatted in the same manner as the |
are read. This file is a three-dimensional |
| 175 |
bathymetry file. The matlab program {\it input/gendata.m} includes example |
($x,y,z$) map and is enumerated and formatted in the same manner as the |
| 176 |
code to generate a valid {\bf zonalWindFile} file. |
bathymetry file. |
| 177 |
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|
| 178 |
|
\item Line qqq |
| 179 |
|
\begin{verbatim} |
| 180 |
|
tCyl = 0 |
| 181 |
|
\end{verbatim} |
| 182 |
|
This line specifies the temperature in degrees Celsius of the interior |
| 183 |
|
wall of the tank -- usually a bucket of ice water. |
| 184 |
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|
| 185 |
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|
| 186 |
\end{itemize} |
\end{itemize} |
| 187 |
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|
| 229 |
\begin{itemize} |
\begin{itemize} |
| 230 |
|
|
| 231 |
\item Line 39, |
\item Line 39, |
| 232 |
\begin{verbatim} sNx=60, \end{verbatim} this line sets |
\begin{verbatim} sNx=120, \end{verbatim} this line sets |
| 233 |
the lateral domain extent in grid points for the |
the lateral domain extent in grid points for the |
| 234 |
axis aligned with the x-coordinate. |
axis aligned with the x-coordinate. |
| 235 |
|
|
| 236 |
\item Line 40, |
\item Line 40, |
| 237 |
\begin{verbatim} sNy=60, \end{verbatim} this line sets |
\begin{verbatim} sNy=31, \end{verbatim} this line sets |
| 238 |
the lateral domain extent in grid points for the |
the lateral domain extent in grid points for the |
| 239 |
axis aligned with the y-coordinate. |
axis aligned with the y-coordinate. |
| 240 |
|
|
| 241 |
\end{itemize} |
\end{itemize} |
| 242 |
|
|
| 243 |
\begin{small} |
\begin{small} |
| 244 |
\input{part3/case_studies/barotropic_gyre/code/SIZE.h} |
\input{part3/case_studies/rotating_tank/code/SIZE.h} |
| 245 |
\end{small} |
\end{small} |
| 246 |
|
|
| 247 |
\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
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