% $Header: /home/ubuntu/mnt/e9_copy/manual/s_examples/rotating_tank/tank.tex,v 1.4 2004/07/26 17:52:43 afe Exp $ % $Name: $ \bodytext{bgcolor="#FFFFFFFF"} %\begin{center} %{\Large \bf Using MITgcm to Simulate a Rotating Tank in Cylindrical %Coordinates} % %\vspace*{4mm} % %\vspace*{3mm} %{\large May 2001} %\end{center} \section{A Rotating Tank in Cylindrical Coordinates} \label{sect:eg-tank} \label{www:tutorials} This section illustrates an example of MITgcm simulating a laboratory experiment on much smaller scales than those common to geophysical fluid dynamics. \subsection{Overview} \label{www:tutorials} This example experiment demonstrates using the MITgcm to simulate a laboratory experiment with a rotating tank of water with an ice bucket in the center. The simulation is configured for a laboratory scale on a $3^{\circ}$ $\times$ 20cm cyclindrical grid with twenty-nine vertical levels. \\ This example experiment demonstrates using the MITgcm to simulate a Barotropic, wind-forced, ocean gyre circulation. The experiment is a numerical rendition of the gyre circulation problem similar to the problems described analytically by Stommel in 1966 \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}. In this experiment the model is configured to represent a rectangular enclosed box of fluid, $1200 \times 1200 $~km in lateral extent. The fluid is $5$~km deep and is forced by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally in the ``north-south'' direction. Topologically the grid is Cartesian and the coriolis parameter $f$ is defined according to a mid-latitude beta-plane equation \begin{equation} \label{EQ:eg-baro-fcori} f(y) = f_{0}+\beta y \end{equation} \noindent where $y$ is the distance along the ``north-south'' axis of the simulated domain. For this experiment $f_{0}$ is set to $10^{-4}s^{-1}$ in (\ref{EQ:eg-baro-fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$. \\ \\ The sinusoidal wind-stress variations are defined according to \begin{equation} \label{EQ:eg-baro-taux} \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y}) \end{equation} \noindent where $L_{y}$ is the lateral domain extent ($1200$~km) and $\tau_0$ is set to $0.1N m^{-2}$. \\ \\ Figure \ref{FIG:eg-baro-simulation_config} summarizes the configuration simulated. %% === eh3 === \begin{figure} %% \begin{center} %% \resizebox{7.5in}{5.5in}{ %% \includegraphics*[0.2in,0.7in][10.5in,10.5in] %% {part3/case_studies/barotropic_gyre/simulation_config.eps} } %% \end{center} \centerline{ \scalefig{.95} \epsfbox{part3/case_studies/barotropic_gyre/simulation_config.eps} } \caption{Schematic of simulation domain and wind-stress forcing function for barotropic gyre numerical experiment. The domain is enclosed bu solid walls at $x=$~0,1200km and at $y=$~0,1200km.} \label{FIG:eg-baro-simulation_config} \end{figure} \subsection{Equations Solved} \label{www:tutorials} The model is configured in hydrostatic form. The implicit free surface form of the pressure equation described in Marshall et. al \cite{marshall:97a} is employed. A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous dissipation. The wind-stress momentum input is added to the momentum equation for the ``zonal flow'', $u$. Other terms in the model are explicitly switched off for this experiment configuration (see section \ref{SEC:code_config} ), yielding an active set of equations solved in this configuration as follows \begin{eqnarray} \label{EQ:eg-baro-model_equations} \frac{Du}{Dt} - fv + g\frac{\partial \eta}{\partial x} - A_{h}\nabla_{h}^2u & = & \frac{\tau_{x}}{\rho_{0}\Delta z} \\ \frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} - A_{h}\nabla_{h}^2v & = & 0 \\ \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u} &=& 0 \end{eqnarray} \noindent where $u$ and $v$ and the $x$ and $y$ components of the flow vector $\vec{u}$. \\ \subsection{Discrete Numerical Configuration} \label{www:tutorials} The domain is discretised with a uniform grid spacing in the horizontal set to $\Delta x=\Delta y=20$~km, so that there are sixty grid cells in the $x$ and $y$ directions. Vertically the model is configured with a single layer with depth, $\Delta z$, of $5000$~m. \subsubsection{Numerical Stability Criteria} \label{www:tutorials} The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$. This value is chosen to yield a Munk layer width \cite{adcroft:95}, \begin{eqnarray} \label{EQ:eg-baro-munk_layer} M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}} \end{eqnarray} \noindent of $\approx 100$km. This is greater than the model resolution $\Delta x$, ensuring that the frictional boundary layer is well resolved. \\ \noindent The model is stepped forward with a time step $\delta t=1200$secs. With this time step the stability parameter to the horizontal Laplacian friction \cite{adcroft:95} \begin{eqnarray} \label{EQ:eg-baro-laplacian_stability} S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2} \end{eqnarray} \noindent evaluates to 0.012, which is well below the 0.3 upper limit for stability. \\ \noindent The numerical stability for inertial oscillations \cite{adcroft:95} \begin{eqnarray} \label{EQ:eg-baro-inertial_stability} S_{i} = f^{2} {\delta t}^2 \end{eqnarray} \noindent evaluates to $0.0144$, which is well below the $0.5$ upper limit for stability. \\ \noindent The advective CFL \cite{adcroft:95} for an extreme maximum horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$ \begin{eqnarray} \label{EQ:eg-baro-cfl_stability} S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x} \end{eqnarray} \noindent evaluates to 0.12. This is approaching the stability limit of 0.5 and limits $\delta t$ to $1200s$. \subsection{Code Configuration} \label{www:tutorials} \label{SEC:eg-baro-code_config} The model configuration for this experiment resides under the directory {\it verification/exp0/}. The experiment files \begin{itemize} \item {\it input/data} \item {\it input/data.pkg} \item {\it input/eedata}, \item {\it input/windx.sin\_y}, \item {\it input/topog.box}, \item {\it code/CPP\_EEOPTIONS.h} \item {\it code/CPP\_OPTIONS.h}, \item {\it code/SIZE.h}. \end{itemize} contain the code customizations and parameter settings for this experiments. Below we describe the customizations to these files associated with this experiment. \subsubsection{File {\it input/data}} \label{www:tutorials} This file, reproduced completely below, specifies the main parameters for the experiment. The parameters that are significant for this configuration are \begin{itemize} \item Line 7, \begin{verbatim} viscAh=4.E2, \end{verbatim} this line sets the Laplacian friction coefficient to $400 m^2s^{-1}$ \item Line 10, \begin{verbatim} beta=1.E-11, \end{verbatim} this line sets $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$ \item Lines 15 and 16 \begin{verbatim} rigidLid=.FALSE., implicitFreeSurface=.TRUE., \end{verbatim} these lines suppress the rigid lid formulation of the surface pressure inverter and activate the implicit free surface form of the pressure inverter. \item Line 27, \begin{verbatim} startTime=0, \end{verbatim} this line indicates that the experiment should start from $t=0$ and implicitly suppresses searching for checkpoint files associated with restarting an numerical integration from a previously saved state. \item Line 29, \begin{verbatim} endTime=12000, \end{verbatim} this line indicates that the experiment should start finish at $t=12000s$. A restart file will be written at this time that will enable the simulation to be continued from this point. \item Line 30, \begin{verbatim} deltaTmom=1200, \end{verbatim} This line sets the momentum equation timestep to $1200s$. \item Line 39, \begin{verbatim} usingCartesianGrid=.TRUE., \end{verbatim} This line requests that the simulation be performed in a Cartesian coordinate system. \item Line 41, \begin{verbatim} delX=60*20E3, \end{verbatim} This line sets the horizontal grid spacing between each x-coordinate line in the discrete grid. The syntax indicates that the discrete grid should be comprise of $60$ grid lines each separated by $20 \times 10^{3}m$ ($20$~km). \item Line 42, \begin{verbatim} delY=60*20E3, \end{verbatim} This line sets the horizontal grid spacing between each y-coordinate line in the discrete grid to $20 \times 10^{3}m$ ($20$~km). \item Line 43, \begin{verbatim} delZ=5000, \end{verbatim} This line sets the vertical grid spacing between each z-coordinate line in the discrete grid to $5000m$ ($5$~km). \item Line 46, \begin{verbatim} bathyFile='topog.box' \end{verbatim} This line specifies the name of the file from which the domain bathymetry is read. This file is a two-dimensional ($x,y$) map of depths. This file is assumed to contain 64-bit binary numbers giving the depth of the model at each grid cell, ordered with the x coordinate varying fastest. The points are ordered from low coordinate to high coordinate for both axes. The units and orientation of the depths in this file are the same as used in the MITgcm code. In this experiment, a depth of $0m$ indicates a solid wall and a depth of $-5000m$ indicates open ocean. The matlab program {\it input/gendata.m} shows an example of how to generate a bathymetry file. \item Line 49, \begin{verbatim} zonalWindFile='windx.sin_y' \end{verbatim} This line specifies the name of the file from which the x-direction surface wind stress is read. This file is also a two-dimensional ($x,y$) map and is enumerated and formatted in the same manner as the bathymetry file. The matlab program {\it input/gendata.m} includes example code to generate a valid {\bf zonalWindFile} file. \end{itemize} \noindent other lines in the file {\it input/data} are standard values that are described in the MITgcm Getting Started and MITgcm Parameters notes. \begin{small} \input{part3/case_studies/barotropic_gyre/input/data} \end{small} \subsubsection{File {\it input/data.pkg}} \label{www:tutorials} This file uses standard default values and does not contain customizations for this experiment. \subsubsection{File {\it input/eedata}} \label{www:tutorials} This file uses standard default values and does not contain customizations for this experiment. \subsubsection{File {\it input/windx.sin\_y}} \label{www:tutorials} The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$) map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$. Although $\tau_{x}$ is only a function of $y$n in this experiment this file must still define a complete two-dimensional map in order to be compatible with the standard code for loading forcing fields in MITgcm. The included matlab program {\it input/gendata.m} gives a complete code for creating the {\it input/windx.sin\_y} file. \subsubsection{File {\it input/topog.box}} \label{www:tutorials} The {\it input/topog.box} file specifies a two-dimensional ($x,y$) map of depth values. For this experiment values are either $0m$ or {\bf -delZ}m, corresponding respectively to a wall or to deep ocean. The file contains a raw binary stream of data that is enumerated in the same way as standard MITgcm two-dimensional, horizontal arrays. The included matlab program {\it input/gendata.m} gives a complete code for creating the {\it input/topog.box} file. \subsubsection{File {\it code/SIZE.h}} \label{www:tutorials} Two lines are customized in this file for the current experiment \begin{itemize} \item Line 39, \begin{verbatim} sNx=60, \end{verbatim} this line sets the lateral domain extent in grid points for the axis aligned with the x-coordinate. \item Line 40, \begin{verbatim} sNy=60, \end{verbatim} this line sets the lateral domain extent in grid points for the axis aligned with the y-coordinate. \end{itemize} \begin{small} \input{part3/case_studies/barotropic_gyre/code/SIZE.h} \end{small} \subsubsection{File {\it code/CPP\_OPTIONS.h}} \label{www:tutorials} This file uses standard default values and does not contain customizations for this experiment. \subsubsection{File {\it code/CPP\_EEOPTIONS.h}} \label{www:tutorials} This file uses standard default values and does not contain customizations for this experiment.