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% $Header: $ |
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% $Name: $ |
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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 \bf Using MITgcm to Simulate a Barotropic Ocean Gyre In Cartesian |
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%Coordinates} |
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% |
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%\vspace*{4mm} |
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% |
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%\vspace*{3mm} |
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%{\large May 2001} |
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%\end{center} |
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\subsection{Introduction} |
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This document is the first in a series of documents describing |
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example MITgcm numerical experiments. The example experiments |
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include both straightforward examples of idealised 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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experiements 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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\subsection{Experiment Overview} |
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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 simliar |
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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: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: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: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:simulation_config} |
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summarises the configuration simulated. |
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\begin{figure} |
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\centerline{ |
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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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} |
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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:simulation_config} |
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\end{figure} |
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\subsection{Discrete Numerical Configuration} |
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The model is configured in hydrostatic form. 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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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 |
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for the ``zonal flow'', $u$. Other terms in the model |
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are explicitly switched off for this experiement 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: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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\subsubsection{Numerical Stability Criteria} |
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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_thesis}, |
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\begin{eqnarray} |
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\label{EQ: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_thesis} |
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\begin{eqnarray} |
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\label{EQ: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_thesis} |
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\begin{eqnarray} |
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\label{EQ: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_thesis} 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: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} |
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\label{SEC:code_config} |
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The model configuration for this experiment resides under the |
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directory {\it verification/exp0/}. The experiment files |
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\begin{itemize} |
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\item {\it input/data} |
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\item {\it input/data.pkg} |
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\item {\it input/eedata}, |
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\item {\it input/windx.sin\_y}, |
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\item {\it input/topog.box}, |
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\item {\it code/CPP\_EEOPTIONS.h} |
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\item {\it code/CPP\_OPTIONS.h}, |
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\item {\it code/SIZE.h}. |
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\end{itemize} |
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contain the code customisations and parameter settings for this |
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experiements. Below we describe the customisations |
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to these files associated with this experiment. |
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\subsubsection{File {\it input/data}} |
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This file, reproduced completely below, specifies the main parameters |
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for the experiment. The parameters that are significant for this configuration |
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are |
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\begin{itemize} |
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\item Line 7, \begin{verbatim} viscAh=4.E2, \end{verbatim} this line sets |
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the laplacian friction coefficient to $400 m^2s^{-1}$ |
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\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 $10^{-11} s^{-1}m^{-1}$ |
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\item Lines 15 and 16 |
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\begin{verbatim} |
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rigidLid=.FALSE., |
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implicitFreeSurface=.TRUE., |
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\end{verbatim} |
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these lines suppress the rigid lid formulation of the surface |
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pressure inverter and activate the implicit free surface form |
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of the pressure inverter. |
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\item Line 27, |
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\begin{verbatim} |
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startTime=0, |
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\end{verbatim} |
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this line indicates that the experiment should start from $t=0$ |
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and implicitly supresses searching for checkpoint files associated |
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with restarting an numerical integration from a previously saved state. |
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\item Line 29, |
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\begin{verbatim} |
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endTime=12000, |
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\end{verbatim} |
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this line indicates that the experiment should start finish at $t=12000s$. |
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A restart file will be written at this time that will enable the |
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simulation to be continued from this point. |
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\item Line 30, |
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\begin{verbatim} |
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deltaTmom=1200, |
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\end{verbatim} |
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This line sets the momentum equation timestep to $1200s$. |
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\item Line 39, |
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\begin{verbatim} |
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usingCartesianGrid=.TRUE., |
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\end{verbatim} |
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This line requests that the simulation be performed in a |
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cartesian coordinate system. |
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\item Line 41, |
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\begin{verbatim} |
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delX=60*20E3, |
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\end{verbatim} |
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This line sets the horizontal grid spacing between each x-coordinate line |
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in the discrete grid. The syntax indicates that the discrete grid |
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should be comprise of $60$ grid lines each separated by $20 \times 10^{3}m$ |
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($20$~km). |
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\item Line 42, |
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\begin{verbatim} |
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delY=60*20E3, |
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\end{verbatim} |
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This line sets the horizontal grid spacing between each y-coordinate line |
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in the discrete grid to $20 \times 10^{3}m$ ($20$~km). |
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\item Line 43, |
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\begin{verbatim} |
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delZ=5000, |
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\end{verbatim} |
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This line sets the vertical grid spacing between each z-coordinate line |
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in the discrete grid to $5000m$ ($5$~km). |
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\item Line 46, |
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\begin{verbatim} |
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bathyFile='topog.box' |
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\end{verbatim} |
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This line specifies the name of the file from which the domain |
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bathymetry is read. This file is a two-dimensional ($x,y$) map of |
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depths. This file is assumed to contain 64-bit binary numbers |
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giving the depth of the model at each grid cell, ordered with the x |
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coordinate varying fastest. The points are ordered from low coordinate |
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to high coordinate for both axes. The units and orientation of the |
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depths in this file are the same as used in the MITgcm code. In this |
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experiment, a depth of $0m$ indicates a solid wall and a depth |
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of $-5000m$ indicates open ocean. The matlab program |
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{\it input/gendata.m} shows an example of how to generate a |
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bathymetry file. |
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\item Line 49, |
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\begin{verbatim} |
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zonalWindFile='windx.sin_y' |
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\end{verbatim} |
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This line specifies the name of the file from which the x-direction |
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surface wind stress is read. This file is also a two-dimensional |
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($x,y$) map and is enumerated and formatted in the same manner as the |
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bathymetry file. The matlab program {\it input/gendata.m} includes example |
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code to generate a valid {\bf zonalWindFile} file. |
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\end{itemize} |
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\noindent other lines in the file {\it input/data} are standard values |
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that are described in the MITgcm Getting Started and MITgcm Parameters |
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notes. |
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\begin{small} |
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\input{part3/case_studies/barotropic_gyre/input/data} |
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\end{small} |
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\subsubsection{File {\it input/data.pkg}} |
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This file uses standard default values and does not contain |
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customisations for this experiment. |
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\subsubsection{File {\it input/eedata}} |
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This file uses standard default values and does not contain |
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customisations for this experiment. |
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\subsubsection{File {\it input/windx.sin\_y}} |
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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 $Nm^{-2}$. |
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Although $\tau_{x}$ is only a function of $y$n in this experiment |
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this file must still define a complete two-dimensional map in order |
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to be compatible with the standard code for loading forcing fields |
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in MITgcm. The included matlab program {\it input/gendata.m} gives a complete |
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code for creating the {\it input/windx.sin\_y} file. |
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\subsubsection{File {\it input/topog.box}} |
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The {\it input/topog.box} file specifies a two-dimensional ($x,y$) |
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map of depth values. For this experiment values are either |
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$0m$ or {\bf -delZ}m, corresponding respectively to a wall or to deep |
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ocean. The file contains a raw binary stream of data that is enumerated |
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in the same way as standard MITgcm two-dimensional, horizontal arrays. |
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The included matlab program {\it input/gendata.m} gives a complete |
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code for creating the {\it input/topog.box} file. |
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\subsubsection{File {\it code/SIZE.h}} |
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Two lines are customized in this file for the current experiment |
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\begin{itemize} |
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\item Line 39, |
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\begin{verbatim} sNx=60, \end{verbatim} this line sets |
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the lateral domain extent in grid points for the |
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axis aligned with the x-coordinate. |
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\item Line 40, |
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\begin{verbatim} sNy=60, \end{verbatim} this line sets |
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the lateral domain extent in grid points for the |
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axis aligned with the y-coordinate. |
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\end{itemize} |
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\begin{small} |
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\input{part3/case_studies/barotropic_gyre/code/SIZE.h} |
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\end{small} |
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\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
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| 361 |
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This file uses standard default values and does not contain |
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customisations for this experiment. |
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\subsubsection{File {\it code/CPP\_EEOPTIONS.h}} |
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This file uses standard default values and does not contain |
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customisations for this experiment. |
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