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revision 1.7 by afe, Mon Jul 26 21:09:47 2004 UTC revision 1.15 by molod, Tue Jun 27 19:08:23 2006 UTC
# Line 16  Line 16 
16  \section{A Rotating Tank in Cylindrical Coordinates}  \section{A Rotating Tank in Cylindrical Coordinates}
17  \label{sect:eg-tank}  \label{sect:eg-tank}
18  \label{www:tutorials}  \label{www:tutorials}
19    \begin{rawhtml}
20  This section illustrates an example of MITgcm simulating a laboratory  <!-- CMIREDIR:eg-tank: -->
21  experiment on much smaller scales than those common to geophysical  \end{rawhtml}
 fluid dynamics.  
22    
23  \subsection{Overview}  \subsection{Overview}
24  \label{www:tutorials}  \label{www:tutorials}
25                                                                                                                                                              
26                                                                                    This example configuration demonstrates using the MITgcm to simulate a
27  This example experiment demonstrates using the MITgcm to simulate  laboratory demonstration using a differentially heated rotating
28  a laboratory experiment with a rotating tank of water with an ice  annulus of water.  The simulation is configured for a laboratory scale
29  bucket in the center. The simulation is configured for a laboratory  on a $3^{\circ}\times1\mathrm{cm}$ cyclindrical grid with twenty-nine
30  scale on a  vertical levels of 0.5cm each.  This is a typical laboratory setup for
31  $3^{\circ}$ $\times$ 20cm  illustration principles of GFD, as well as for a laboratory data
32  cyclindrical grid with twenty-nine vertical  assimilation project. The files for this experiment can be found in
33  levels.  the verification directory under rotating\_tank.
34  \\  \\
35    
36    example illustration from GFD lab here
37    \\
38    
39    
 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  
40    
 \begin{equation}  
 \label{EQ:eg-baro-taux}  
 \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})  
 \end{equation}  
41    
 \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}  
42    
43  \subsection{Equations Solved}  \subsection{Equations Solved}
44  \label{www:tutorials}  \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 $\vec{u}$.  
 \\  
45    
46    
47  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
48  \label{www:tutorials}  \label{www:tutorials}
49    
50   The domain is discretised with   The domain is discretised with a uniform cylindrical grid spacing in
51  a uniform grid spacing in the horizontal set to  the horizontal set to $\Delta a=1$~cm and $\Delta \phi=3^{\circ}$, so
52   $\Delta x=\Delta y=20$~km, so  that there are 120 grid cells in the azimuthal direction and
53  that there are sixty grid cells in the $x$ and $y$ directions. Vertically the  thirty-one grid cells in the radial, representing a tank 62cm in
54  model is configured with a single layer with depth, $\Delta z$, of $5000$~m.  diameter.  The bathymetry file sets the depth=0 in the nine lowest
55    radial rows to represent the central of the annulus.  Vertically the
56  \subsubsection{Numerical Stability Criteria}  model is configured with twenty-nine layers of uniform 0.5cm
57  \label{www:tutorials}  thickness.
   
 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.  
58  \\  \\
59    something about heat flux
 \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 $1200s$.  
60    
61  \subsection{Code Configuration}  \subsection{Code Configuration}
62  \label{www:tutorials}  \label{www:tutorials}
# Line 219  are Line 88  are
88    
89  \begin{itemize}  \begin{itemize}
90    
91  \item Line X, \begin{verbatim} viscAh=5.0E-6, \end{verbatim} this line sets  \item Lines 9-10, \begin{verbatim}
92  the Laplacian friction coefficient to $0.000006 m^2s^{-1}$, which is ususally  viscAh=5.0E-6,
93  low because of the small scale, presumably.... qqq  viscAz=5.0E-6,
94    \end{verbatim}
95  \item Line X, \begin{verbatim}f0=0.5 , \end{verbatim} this line sets the  
96  coriolis term, and represents a tank spinning at qqq  
97  \item Line 10, \begin{verbatim} beta=1.E-11, \end{verbatim} this line sets  These lines set the Laplacian friction coefficient in the horizontal
98  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$  and vertical, respectively.  Note that they are several orders of
99    magnitude smaller than the other examples due to the small scale of
100    this example.
101    
102    \item Lines 13-16, \begin{verbatim}
103     diffKhT=2.5E-6,
104     diffKzT=2.5E-6,
105     diffKhS=1.0E-6,
106     diffKzS=1.0E-6,
107    
108    \end{verbatim}
109    
110  \item Lines 15 and 16  
111    These lines set horizontal and vertical diffusion coefficients for
112    temperature and salinity.  Similarly to the friction coefficients, the
113    values are a couple of orders of magnitude less than most
114     configurations.
115    
116    
117    \item Line 17, \begin{verbatim}f0=0.5 , \end{verbatim} this line sets the
118    coriolis term, and represents a tank spinning at about 2.4 rpm.
119    
120    \item Lines 23 and 24
121  \begin{verbatim}  \begin{verbatim}
122  rigidLid=.TRUE.,  rigidLid=.TRUE.,
123  implicitFreeSurface=.FALSE.,  implicitFreeSurface=.FALSE.,
124  \end{verbatim}  \end{verbatim}
125    
126  these lines do the opposite of the following:  These lines activate  the rigid lid formulation of the surface
127  suppress the rigid lid formulation of the surface  pressure inverter and suppress the implicit free surface form
 pressure inverter and activate the implicit free surface form  
128  of the pressure inverter.  of the pressure inverter.
129    
130  \item Line 27,  \item Line 40,
131  \begin{verbatim}  \begin{verbatim}
132  startTime=0,  nIter=0,
133  \end{verbatim}  \end{verbatim}
134  this line indicates that the experiment should start from $t=0$  This line indicates that the experiment should start from $t=0$ and
135  and implicitly suppresses searching for checkpoint files associated  implicitly suppresses searching for checkpoint files associated with
136  with restarting an numerical integration from a previously saved state.  restarting an numerical integration from a previously saved state.
137    Instead, the file thetaPol.bin will be loaded to initialized the
138    temperature fields as indicated below, and other variables will be
139    initialized to their defaults.
140    
141    
142  \item Line 30,  \item Line 43,
143  \begin{verbatim}  \begin{verbatim}
144  deltaT=0.1,  deltaT=0.1,
145  \end{verbatim}  \end{verbatim}
146  This line sets the integration timestep to $0.1s$.  This is an unsually  This line sets the integration timestep to $0.1s$.  This is an
147  small value among the examples due to the small physical scale of the  unsually small value among the examples due to the small physical
148  experiment.  scale of the experiment.  Using the ensemble Kalman filter to produce
149    input fields can necessitate even shorter timesteps.
150    
151  \item Line 39,  \item Line 56,
152  \begin{verbatim}  \begin{verbatim}
153  usingCylindricalGrid=.TRUE.,  usingCylindricalGrid=.TRUE.,
154  \end{verbatim}  \end{verbatim}
155  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
156  cylindrical coordinate system.  cylindrical coordinate system.
157    
158  \item Line qqq,  \item Line 57,
159  \begin{verbatim}  \begin{verbatim}
160  dXspacing=3,  dXspacing=3,
161  \end{verbatim}  \end{verbatim}
162  This line sets the azimuthal grid spacing between each x-coordinate line  This line sets the azimuthal grid spacing between each $x$-coordinate line
163  in the discrete grid. The syntax indicates that the discrete grid  in the discrete grid. The syntax indicates that the discrete grid
164  should be comprise of $120$ grid lines each separated by $3^{\circ}$.  should be comprised of $120$ grid lines each separated by $3^{\circ}$.
165                                                                                                                                                                  
   
166    
167  \item Line qqq,  \item Line 58,
168  \begin{verbatim}  \begin{verbatim}
169  dYspacing=0.01,  dYspacing=0.01,
170  \end{verbatim}  \end{verbatim}
 This line sets the radial grid spacing between each $\rho$-coordinate line  
 in the discrete grid to $1cm$.  
171    
172  \item Line 43,  This line sets the radial cylindrical grid spacing between each
173    $a$-coordinate line in the discrete grid to $1cm$.
174    
175    \item Line 59,
176  \begin{verbatim}  \begin{verbatim}
177  delZ=29*0.005,  delZ=29*0.005,
178  \end{verbatim}  \end{verbatim}
 This line sets the vertical grid spacing between each z-coordinate line  
 in the discrete grid to $5000m$ ($5$~km).  
179    
180  \item Line 46,  This line sets the vertical grid spacing between each of 29
181    z-coordinate lines in the discrete grid to $0.005m$ ($5$~mm).
182    
183    \item Line 64,
184  \begin{verbatim}  \begin{verbatim}
185  bathyFile='bathyPol.bin',  bathyFile='bathyPol.bin',
186  \end{verbatim}  \end{verbatim}
187  This line specifies the name of the file from which the domain  This line specifies the name of the file from which the domain
188  ``bathymetry'' (tank depth) is read. This file is a two-dimensional  ``bathymetry'' (tank depth) is read. This file is a two-dimensional
189  ($x,y$) map of  ($a,\phi$) map of
190  depths. This file is assumed to contain 64-bit binary numbers  depths. This file is assumed to contain 64-bit binary numbers
191  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 $\phi$
192  coordinate varying fastest. The points are ordered from low coordinate  coordinate varying fastest. The points are ordered from low coordinate
193  to high coordinate for both axes.  The units and orientation of the  to high coordinate for both axes.  The units and orientation of the
194  depths in this file are the same as used in the MITgcm code. In this  depths in this file are the same as used in the MITgcm code. In this
# Line 302  experiment, a depth of $0m$ indicates an Line 196  experiment, a depth of $0m$ indicates an
196  and a depth  and a depth
197  f $-0.145m$ indicates the tank itself.  f $-0.145m$ indicates the tank itself.
198    
199  \item Line 49,  \item Line 65,
200  \begin{verbatim}  \begin{verbatim}
201  hydrogThetaFile='thetaPol.bin',  hydrogThetaFile='thetaPol.bin',
202  \end{verbatim}  \end{verbatim}
203  This line specifies the name of the file from which the initial values  This line specifies the name of the file from which the initial values
204  of $\theta$  of temperature
205  are read. This file is a three-dimensional  are read. This file is a three-dimensional
206  ($x,y,z$) map and is enumerated and formatted in the same manner as the  ($x,y,z$) map and is enumerated and formatted in the same manner as the
207  bathymetry file.  bathymetry file.
208    
209  \item Line qqq  \item Lines 66 and 67
210  \begin{verbatim}  \begin{verbatim}
211   tCyl  = 0   tCylIn  = 0
212     tCylOut  = 20
213  \end{verbatim}  \end{verbatim}
214  This line specifies the temperature in degrees Celsius of the interior  These line specify the temperatures in degrees Celsius of the interior
215  wall of the tank -- usually a bucket of ice water.  and exterior walls of the tank -- typically taken to be icewater on
216    the inside and room temperature on the outside.
217    
218    
219  \end{itemize}  \end{itemize}
220    
221  \noindent other lines in the file {\it input/data} are standard values  \noindent Other lines in the file {\it input/data} are standard values
222  that are described in the MITgcm Getting Started and MITgcm Parameters  that are described in the MITgcm Getting Started and MITgcm Parameters
223  notes.  notes.
224    
# Line 346  customizations for this experiment. Line 242  customizations for this experiment.
242  \label{www:tutorials}  \label{www:tutorials}
243    
244  The {\it input/thetaPol.bin} file specifies a three-dimensional ($x,y,z$)  The {\it input/thetaPol.bin} file specifies a three-dimensional ($x,y,z$)
245  map of initial values of $\theta$ in degrees Celsius.  map of initial values of $\theta$ in degrees Celsius.  This particular
246    experiment is set to random values x around 20C to provide initial
247    perturbations.
248    
249  \subsubsection{File {\it input/bathyPol.bin}}  \subsubsection{File {\it input/bathyPol.bin}}
250  \label{www:tutorials}  \label{www:tutorials}
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