--- 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}
