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\section{Example: Barotropic Ocean Gyre In Cartesian Coordinates} |
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\bodytext{bgcolor="#FFFFFFFF"} |
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%\begin{center} |
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%{\large May 2001} |
%{\large May 2001} |
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\subsection{Introduction} |
\section[Barotropic Gyre MITgcm Example]{Barotropic Ocean Gyre In Cartesian Coordinates} |
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\label{www:tutorials} |
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This document is the first in a series of documents describing |
\label{sect:eg-baro} |
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example MITgcm numerical experiments. The example experiments |
\begin{rawhtml} |
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include both straightforward examples of idealised geophysical |
<!-- CMIREDIR:eg-baro: --> |
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fluid simulations and more involved cases encompassing |
\end{rawhtml} |
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large scale modeling and |
\begin{center} |
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automatic differentiation. Both hydrostatic and non-hydrostatic |
(in directory: {\it verification/tutorial\_barotropic\_gyre/}) |
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experiements are presented, as well as experiments employing |
\end{center} |
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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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\subsection{Experiment Overview} |
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This example experiment demonstrates using the MITgcm to simulate |
This example experiment demonstrates using the MITgcm to simulate |
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a barotropic, wind-forced, ocean gyre circulation. The experiment |
a Barotropic, wind-forced, ocean gyre circulation. The files for this |
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is a numerical rendition of the gyre circulation problem simliar |
experiment can be found in the verification directory tutorial\_barotropic\_gyre. |
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The experiment is a numerical rendition of the gyre circulation problem similar |
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to the problems described analytically by Stommel in 1966 |
to the problems described analytically by Stommel in 1966 |
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\cite{Stommel66} and numerically in Holland et. al \cite{Holland75}. |
\cite{Stommel66} and numerically in Holland et. al \cite{Holland75}. |
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is configured to represent a rectangular enclosed box of fluid, |
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 |
$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 |
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 |
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 |
the coriolis parameter $f$ is defined according to a mid-latitude beta-plane |
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equation |
equation |
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\begin{equation} |
\begin{equation} |
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\label{EQ:fcori} |
\label{EQ:eg-baro-fcori} |
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f(y) = f_{0}+\beta y |
f(y) = f_{0}+\beta y |
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\end{equation} |
\end{equation} |
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\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 |
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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 |
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(\ref{EQ:fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$. |
(\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 |
The sinusoidal wind-stress variations are defined according to |
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\begin{equation} |
\begin{equation} |
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\label{EQ:taux} |
\label{EQ:eg-baro-taux} |
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\tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y}) |
\tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y}) |
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\end{equation} |
\end{equation} |
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$\tau_0$ is set to $0.1N m^{-2}$. |
$\tau_0$ is set to $0.1N m^{-2}$. |
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\\ |
\\ |
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\\ |
\\ |
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Figure \ref{FIG:simulation_config} |
Figure \ref{FIG:eg-baro-simulation_config} |
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summarises the configuration simulated. |
summarizes the configuration simulated. |
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%% === eh3 === |
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\begin{figure} |
\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{ |
\centerline{ |
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\resizebox{7.5in}{5.5in}{ |
\scalefig{.95} |
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\includegraphics*[0.2in,0.7in][10.5in,10.5in] |
\epsfbox{part3/case_studies/barotropic_gyre/simulation_config.eps} |
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{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 |
\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 |
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.} |
walls at $x=$~0,1200km and at $y=$~0,1200km.} |
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\label{FIG:simulation_config} |
\label{FIG:eg-baro-simulation_config} |
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\end{figure} |
\end{figure} |
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\subsection{Discrete Numerical Configuration} |
\subsection{Equations Solved} |
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\label{www:tutorials} |
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The model is configured in hydrostatic form. The domain is discretised with |
The model is configured in hydrostatic form. The implicit free surface form of the |
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a uniform grid spacing in the horizontal set to |
pressure equation described in Marshall et. al \cite{marshall:97a} is |
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$\Delta x=\Delta y=20$~km, so |
employed. |
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that there are sixty grid cells in the $x$ and $y$ directions. Vertically the |
A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous |
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model is configured with a single layer with depth, $\Delta z$, of $5000$~m. |
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The implicit free surface form of the |
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pressure equation described in Marshall et. al \cite{Marshall97a} 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 |
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 |
for the ``zonal flow'', $u$. Other terms in the model |
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are explicitly switched off for this experiement configuration (see section |
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 |
\ref{SEC:code_config} ), yielding an active set of equations solved in this |
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configuration as follows |
configuration as follows |
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\begin{eqnarray} |
\begin{eqnarray} |
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\label{EQ:model_equations} |
\label{EQ:eg-baro-model_equations} |
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\frac{Du}{Dt} - fv + |
\frac{Du}{Dt} - fv + |
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g\frac{\partial \eta}{\partial x} - |
g\frac{\partial \eta}{\partial x} - |
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A_{h}\nabla_{h}^2u |
A_{h}\nabla_{h}^2u |
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& = & |
& = & |
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\frac{\tau_{x}}{\rho_{0}\Delta z} |
\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} - |
\frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} - |
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A_{h}\nabla_{h}^2v |
A_{h}\nabla_{h}^2v |
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& = & |
& = & |
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0 |
0 |
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\\ |
\\ |
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\end{eqnarray} |
\end{eqnarray} |
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\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 |
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flow vector $\vec{u}$. |
flow vector $\vec{u}$. |
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\\ |
\\ |
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\subsection{Discrete Numerical Configuration} |
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\label{www:tutorials} |
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The domain is discretised with |
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a uniform grid spacing in the horizontal set to |
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$\Delta x=\Delta y=20$~km, so |
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that there are sixty grid cells in the $x$ and $y$ directions. Vertically the |
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model is configured with a single layer with depth, $\Delta z$, of $5000$~m. |
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\subsubsection{Numerical Stability Criteria} |
\subsubsection{Numerical Stability Criteria} |
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\label{www:tutorials} |
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The laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$. |
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_thesis}, |
This value is chosen to yield a Munk layer width \cite{adcroft:95}, |
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\begin{eqnarray} |
\begin{eqnarray} |
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\label{EQ:munk_layer} |
\label{EQ:eg-baro-munk_layer} |
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M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}} |
M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}} |
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\end{eqnarray} |
\end{eqnarray} |
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\noindent The model is stepped forward with a |
\noindent The model is stepped forward with a |
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time step $\delta t=1200$secs. With this time step the stability |
time step $\delta t=1200$secs. With this time step the stability |
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parameter to the horizontal laplacian friction \cite{Adcroft_thesis} |
parameter to the horizontal Laplacian friction \cite{adcroft:95} |
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\begin{eqnarray} |
\begin{eqnarray} |
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\label{EQ:laplacian_stability} |
\label{EQ:eg-baro-laplacian_stability} |
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S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2} |
S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2} |
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\end{eqnarray} |
\end{eqnarray} |
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\\ |
\\ |
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\noindent The numerical stability for inertial oscillations |
\noindent The numerical stability for inertial oscillations |
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\cite{Adcroft_thesis} |
\cite{adcroft:95} |
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\begin{eqnarray} |
\begin{eqnarray} |
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\label{EQ:inertial_stability} |
\label{EQ:eg-baro-inertial_stability} |
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S_{i} = f^{2} {\delta t}^2 |
S_{i} = f^{2} {\delta t}^2 |
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\end{eqnarray} |
\end{eqnarray} |
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limit for stability. |
limit for stability. |
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\\ |
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\noindent The advective CFL \cite{Adcroft_thesis} for an extreme maximum |
\noindent The advective CFL \cite{adcroft:95} for an extreme maximum |
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horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$ |
horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$ |
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\begin{eqnarray} |
\begin{eqnarray} |
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\label{EQ:cfl_stability} |
\label{EQ:eg-baro-cfl_stability} |
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S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x} |
S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x} |
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\end{eqnarray} |
\end{eqnarray} |
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of 0.5 and limits $\delta t$ to $1200s$. |
of 0.5 and limits $\delta t$ to $1200s$. |
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\subsection{Code Configuration} |
\subsection{Code Configuration} |
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\label{SEC:code_config} |
\label{www:tutorials} |
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\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/tutorial\_barotropic\_gyre/}. |
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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 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 customisations and parameter settings for this |
contain the code customizations and parameter settings for this |
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experiements. Below we describe the customisations |
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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\subsubsection{File {\it input/data}} |
\subsubsection{File {\it input/data}} |
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\label{www:tutorials} |
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This file, reproduced completely below, specifies the main parameters |
This file, reproduced completely below, specifies the main parameters |
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for the experiment. The parameters that are significant for this configuration |
for the experiment. The parameters that are significant for this configuration |
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\begin{itemize} |
\begin{itemize} |
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\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 |
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the laplacian friction coefficient to $400 m^2s^{-1}$ |
the Laplacian friction coefficient to $400 m^2s^{-1}$ |
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\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 |
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$\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$ |
$\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$ |
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startTime=0, |
startTime=0, |
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\end{verbatim} |
\end{verbatim} |
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this line indicates that the experiment should start from $t=0$ |
this line indicates that the experiment should start from $t=0$ |
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and implicitly supresses searching for checkpoint files associated |
and implicitly suppresses searching for checkpoint files associated |
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with restarting an numerical integration from a previously saved state. |
with restarting an numerical integration from a previously saved state. |
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\item Line 29, |
\item Line 29, |
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usingCartesianGrid=.TRUE., |
usingCartesianGrid=.TRUE., |
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\end{verbatim} |
\end{verbatim} |
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This line requests that the simulation be performed in a |
This line requests that the simulation be performed in a |
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cartesian coordinate system. |
Cartesian coordinate system. |
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\item Line 41, |
\item Line 41, |
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\begin{verbatim} |
\begin{verbatim} |
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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} |
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This file uses standard default values and does not contain |
This file uses standard default values and does not contain |
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customisations for this experiment. |
customizations for this experiment. |
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\subsubsection{File {\it input/eedata}} |
\subsubsection{File {\it input/eedata}} |
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\label{www:tutorials} |
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This file uses standard default values and does not contain |
This file uses standard default values and does not contain |
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customisations for this experiment. |
customizations for this experiment. |
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\subsubsection{File {\it input/windx.sin\_y}} |
\subsubsection{File {\it input/windx.sin\_y}} |
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\label{www:tutorials} |
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The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$) |
The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$) |
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map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$. |
map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$. |
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code for creating the {\it input/windx.sin\_y} file. |
code for creating the {\it input/windx.sin\_y} file. |
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\subsubsection{File {\it input/topog.box}} |
\subsubsection{File {\it input/topog.box}} |
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\label{www:tutorials} |
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The {\it input/topog.box} file specifies a two-dimensional ($x,y$) |
The {\it input/topog.box} file specifies a two-dimensional ($x,y$) |
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code for creating the {\it input/topog.box} file. |
code for creating the {\it input/topog.box} file. |
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\subsubsection{File {\it code/SIZE.h}} |
\subsubsection{File {\it code/SIZE.h}} |
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\label{www:tutorials} |
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Two lines are customized in this file for the current experiment |
Two lines are customized in this file for the current experiment |
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\end{small} |
\end{small} |
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\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
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\label{www:tutorials} |
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This file uses standard default values and does not contain |
This file uses standard default values and does not contain |
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customisations for this experiment. |
customizations for this experiment. |
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\subsubsection{File {\it code/CPP\_EEOPTIONS.h}} |
\subsubsection{File {\it code/CPP\_EEOPTIONS.h}} |
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\label{www:tutorials} |
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This file uses standard default values and does not contain |
This file uses standard default values and does not contain |
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customisations for this experiment. |
customizations for this experiment. |
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