--- 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 19:13:08 1.6 @@ -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 1.6 2004/07/26 19:13:08 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. @@ -78,17 +219,23 @@ \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 X, \begin{verbatim} viscAh=5.0E-6, \end{verbatim} this line sets +the Laplacian friction coefficient to $0.000006 m^2s^{-1}$, which is ususally +low because of the small scale, presumably.... qqq + +\item Line X, \begin{verbatim}f0=0.5 , \end{verbatim} this line sets the +coriolis term, and represents a tank spinning at qqq \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., +rigidLid=.TRUE., +implicitFreeSurface=.FALSE., \end{verbatim} -these lines suppress the rigid lid formulation of the surface + +these lines do the opposite of the following: +suppress the rigid lid formulation of the surface pressure inverter and activate the implicit free surface form of the pressure inverter. @@ -100,23 +247,17 @@ 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, +deltaT=0.1, \end{verbatim} -This line sets the momentum equation timestep to $1200s$. +This line sets the integration timestep to $0.1s$. This is an unsually +small value among the examples due to the small physical scale of the +experiment. \item Line 39, \begin{verbatim} -usingCartesianGrid=.TRUE., +usingCylindricalGrid=.TRUE., \end{verbatim} This line requests that the simulation be performed in a Cartesian coordinate system. @@ -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} @@ -193,28 +334,21 @@ This file uses standard default values and does not contain customizations for this experiment. -\subsubsection{File {\it input/windx.sin\_y}} +\subsubsection{File {\it input/thetaPol.bin}} \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. +The {\it input/thetaPol.bin} file specifies a three-dimensional ($x,y,z$) +map of initial values of $\theta$ in degrees Celsius. -\subsubsection{File {\it input/topog.box}} +\subsubsection{File {\it input/bathyPol.bin}} \label{www:tutorials} -The {\it input/topog.box} file specifies a two-dimensional ($x,y$) +The {\it input/bathyPol.bin} 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 +$0m$ or {\bf -delZ}m, corresponding respectively to outside or inside of +the tank. 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}