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%\begin{center} |
%\begin{center} |
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%{\Large \bf Using MITgcm to Simulate a Rotating Tank in Cylindrical |
%{\Large \bf Using MITgcm to Simulate a Rotating Tank in Cylindrical |
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%Coordinates} |
%Coordinates} |
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% |
% |
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%\vspace*{4mm} |
%\vspace*{4mm} |
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% |
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%\vspace*{3mm} |
%\vspace*{3mm} |
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%{\large June 2004} |
%{\large May 2001} |
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%\end{center} |
%\end{center} |
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This is the first in a series of tutorials describing |
\section{A Rotating Tank in Cylindrical Coordinates} |
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example MITgcm numerical experiments. The example experiments |
\label{sect:eg-tank} |
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include both straightforward examples of idealized geophysical |
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fluid simulations and more involved cases encompassing |
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large scale modeling and |
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automatic differentiation. Both hydrostatic and non-hydrostatic |
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experiments are presented, as well as experiments employing |
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Cartesian, spherical-polar and cube-sphere coordinate systems. |
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These ``case study'' documents include information describing |
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the experimental configuration and detailed information on how to |
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configure the MITgcm code and input files for each experiment. |
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\section{Barotropic Ocean Gyre In Cartesian Coordinates} |
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\label{sect:eg-baro} |
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\label{www:tutorials} |
\label{www:tutorials} |
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This section illustrates an example of MITgcm simulating a laboratory |
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experiment on much smaller scales than those common to geophysical |
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fluid dynamics. |
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\subsection{Overview} |
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\label{www:tutorials} |
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This example experiment demonstrates using the MITgcm to simulate |
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a laboratory experiment with a rotating tank of water with an ice |
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bucket in the center. The simulation is configured for a laboratory |
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scale on a |
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$3^{\circ}$ $\times$ 20cm |
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cyclindrical grid with twenty-nine vertical |
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levels. |
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\\ |
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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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\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} |
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\subsection{Equations Solved} |
\subsection{Equations Solved} |
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\label{www:tutorials} |
\label{www:tutorials} |
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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 |
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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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\subsection{Discrete Numerical Configuration} |
\subsection{Discrete Numerical Configuration} |
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\subsubsection{Numerical Stability Criteria} |
\subsubsection{Numerical Stability Criteria} |
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\label{www:tutorials} |
\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 $1200s$. |
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\subsection{Code Configuration} |
\subsection{Code Configuration} |
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\label{www:tutorials} |
\label{www:tutorials} |
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\label{SEC:eg-baro-code_config} |
\label{SEC:eg-baro-code_config} |
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The model configuration for this experiment resides under the |
The model configuration for this experiment resides under the |
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directory {\it verification/exp0/}. The experiment files |
directory {\it verification/rotatingi\_tank/}. The experiment files |
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\begin{itemize} |
\begin{itemize} |
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\item {\it input/data} |
\item {\it input/data} |
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\item {\it input/data.pkg} |
\item {\it input/data.pkg} |
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\item {\it input/eedata}, |
\item {\it input/eedata}, |
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\item {\it input/windx.sin\_y}, |
\item {\it input/bathyPol.bin}, |
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\item {\it input/topog.box}, |
\item {\it input/thetaPol.bin}, |
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\item {\it code/CPP\_EEOPTIONS.h} |
\item {\it code/CPP\_EEOPTIONS.h} |
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\item {\it code/CPP\_OPTIONS.h}, |
\item {\it code/CPP\_OPTIONS.h}, |
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\item {\it code/SIZE.h}. |
\item {\it code/SIZE.h}. |
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\end{itemize} |
\end{itemize} |
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contain the code customizations and parameter settings for this |
contain the code customizations and parameter settings for this |
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experiments. Below we describe the customizations |
experiments. Below we describe the customizations |
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to these files associated with this experiment. |
to these files associated with this experiment. |
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that are described in the MITgcm Getting Started and MITgcm Parameters |
that are described in the MITgcm Getting Started and MITgcm Parameters |
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notes. |
notes. |
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%%\begin{small} |
\begin{small} |
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%%\input{part3/case_studies/barotropic_gyre/input/data} |
\input{part3/case_studies/rotating_tank/input/data} |
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%%\end{small} |
\end{small} |
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\subsubsection{File {\it input/data.pkg}} |
\subsubsection{File {\it input/data.pkg}} |
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\label{www:tutorials} |
\label{www:tutorials} |
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