--- manual/s_examples/rotating_tank/tank.tex 2004/06/22 15:07:37 1.1 +++ manual/s_examples/rotating_tank/tank.tex 2004/07/26 18:41:32 1.5 @@ -1,39 +1,129 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_examples/rotating_tank/tank.tex,v 1.1 2004/06/22 15:07:37 afe Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_examples/rotating_tank/tank.tex,v 1.5 2004/07/26 18:41:32 afe Exp $ % $Name: $ \bodytext{bgcolor="#FFFFFFFF"} %\begin{center} -%{\Large \bf Using MITgcm to Simulate a Rotating Tank in Cylindrical +%{\Large \bf Using MITgcm to Simulate a Rotating Tank in Cylindrical %Coordinates} % %\vspace*{4mm} % %\vspace*{3mm} -%{\large June 2004} +%{\large May 2001} %\end{center} -This is the first in a series of tutorials describing -example MITgcm numerical experiments. The example experiments -include both straightforward examples of idealized geophysical -fluid simulations and more involved cases encompassing -large scale modeling and -automatic differentiation. Both hydrostatic and non-hydrostatic -experiments are presented, as well as experiments employing -Cartesian, spherical-polar and cube-sphere coordinate systems. -These ``case study'' documents include information describing -the experimental configuration and detailed information on how to -configure the MITgcm code and input files for each experiment. - -\section{Barotropic Ocean Gyre In Cartesian Coordinates} -\label{sect:eg-baro} +\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} @@ -48,23 +138,74 @@ \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 +The model configuration for this experiment resides under the +directory {\it verification/rotatingi\_tank/}. 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 input/bathyPol.bin}, +\item {\it input/thetaPol.bin}, \item {\it code/CPP\_EEOPTIONS.h} \item {\it code/CPP\_OPTIONS.h}, -\item {\it code/SIZE.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. @@ -177,9 +318,9 @@ that are described in the MITgcm Getting Started and MITgcm Parameters notes. -%%\begin{small} -%%\input{part3/case_studies/barotropic_gyre/input/data} -%%\end{small} +\begin{small} +\input{part3/case_studies/rotating_tank/input/data} +\end{small} \subsubsection{File {\it input/data.pkg}} \label{www:tutorials}