/[MITgcm]/manual/s_examples/rotating_tank/tank.tex

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-revision 1.1 by afe,
Tue Jun 22 15:07:37 2004 UTC
+revision 1.7 by afe,
Mon Jul 26 21:09:47 2004 UTC
 Line 4
  \bodytext{bgcolor="#FFFFFFFF"}
  %\begin{center}
  %{\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
+ \section{A Rotating Tank in Cylindrical Coordinates}
- example MITgcm numerical experiments. The example experiments
+ \label{sect:eg-tank}
- 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}
  \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
+ & = &
+ \\
+ \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}
+ &=&
+ \end{eqnarray}
+ \noindent where $u$ and $v$ and the $x$ and $y$ components of the
+ flow vector $\vec{u}$.
+ \\
  \subsection{Discrete Numerical Configuration}
-Line 48 
 model is configured with a single layer
+Line 138 
 model is configured with a single layer
  \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
+ 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/bathyPol.bin},
- \item {\it input/topog.box},
+ \item {\it input/thetaPol.bin},
  \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.
-Line 78 
 are
+Line 219 
 are
  \begin{itemize}
- \item Line 7, \begin{verbatim} viscAh=4.E2, \end{verbatim} this line sets
+ \item Line X, \begin{verbatim} viscAh=5.0E-6, \end{verbatim} this line sets
- the Laplacian friction coefficient to $400 m^2s^{-1}$
+ 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.,
+ rigidLid=.TRUE.,
- implicitFreeSurface=.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.
-Line 100 
 this line indicates that the experiment
+Line 247 
 this line indicates that the experiment
  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.
+ cylindrical coordinate system.
- \item Line 41,
+ \item Line qqq,
  \begin{verbatim}
- delX=60*20E3,
+ dXspacing=3,
  \end{verbatim}
- This line sets the horizontal grid spacing between each x-coordinate line
+ This line sets the azimuthal 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 $20 \times 10^{3}m$
+ should be comprise of $120$ grid lines each separated by $3^{\circ}$.
- ($20$~km).
- \item Line 42,
+ \item Line qqq,
  \begin{verbatim}
- delY=60*20E3,
+ dYspacing=0.01,
  \end{verbatim}
- This line sets the horizontal grid spacing between each y-coordinate line
+ This line sets the radial grid spacing between each $\rho$-coordinate line
- in the discrete grid to $20 \times 10^{3}m$ ($20$~km).
+ in the discrete grid to $1cm$.
  \item Line 43,
  \begin{verbatim}
- delZ=5000,
+ delZ=29*0.005,
  \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'
+ bathyFile='bathyPol.bin',
  \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
+ ``bathymetry'' (tank depth) 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
+ 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
+ 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
+ experiment, a depth of $0m$ indicates an area outside of the tank
- of $-5000m$ indicates open ocean. The matlab program
+ and a depth
- {\it input/gendata.m} shows an example of how to generate a
+ f $-0.145m$ indicates the tank itself.
- bathymetry file.
  \item Line 49,
  \begin{verbatim}
- zonalWindFile='windx.sin_y'
+ hydrogThetaFile='thetaPol.bin',
+ \end{verbatim}
+ This line specifies the name of the file from which the initial values
+ of $\theta$
+ are read. This file is a three-dimensional
+ ($x,y,z$) map and is enumerated and formatted in the same manner as the
+ bathymetry file.
+ \item Line qqq
+ \begin{verbatim}
+  tCyl  = 0
  \end{verbatim}
- This line specifies the name of the file from which the x-direction
+ This line specifies the temperature in degrees Celsius of the interior
- surface wind stress is read. This file is also a two-dimensional
+ wall of the tank -- usually a bucket of ice water.
- ($x,y$) 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}
-Line 177 
 code to generate a valid {\bf zonalWindF
+Line 326 
 code to generate a valid {\bf zonalWindF
  that are described in the MITgcm Getting Started and MITgcm Parameters
  notes.
- %%\begin{small}
+ \begin{small}
- %%\input{part3/case_studies/barotropic_gyre/input/data}
+ \input{part3/case_studies/rotating_tank/input/data}
- %%\end{small}
+ \end{small}
  \subsubsection{File {\it input/data.pkg}}
  \label{www:tutorials}
-Line 193 
 customizations for this experiment.
+Line 342 
 customizations for this experiment.
  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 ($x,y$)
+ The {\it input/thetaPol.bin} file specifies a three-dimensional ($x,y,z$)
- map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.
+ map of initial values of $\theta$ in degrees Celsius.
- 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}}
+ \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
+ $0m$ or {\bf -delZ}m, corresponding respectively to outside or inside of
- ocean. The file contains a raw binary stream of data that is enumerated
+ 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}
-Line 224 
 Two lines are customized in this file fo
+Line 366 
 Two lines are customized in this file fo
  \begin{itemize}
  \item Line 39,
- \begin{verbatim} sNx=60, \end{verbatim} this line sets
+ \begin{verbatim} sNx=120, \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
+ \begin{verbatim} sNy=31, \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}
+ \input{part3/case_studies/rotating_tank/code/SIZE.h}
  \end{small}
  \subsubsection{File {\it code/CPP\_OPTIONS.h}}

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