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revision 1.4 by afe, Mon Jul 26 17:52:43 2004 UTC revision 1.18 by jmc, Mon Aug 30 23:09:20 2010 UTC
# Line 14  Line 14 
14  %\end{center}  %\end{center}
15    
16  \section{A Rotating Tank in Cylindrical Coordinates}  \section{A Rotating Tank in Cylindrical Coordinates}
17  \label{sect:eg-tank}  \label{sec: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}
22  fluid dynamics.  \begin{center}
23    (in directory: {\it verification/rotating\_tank/})
24    \end{center}
25    
26  \subsection{Overview}  \subsection{Overview}
27  \label{www:tutorials}  %\label{www:tutorials}
28                                                                                                                                                              
29                                                                                    This example configuration demonstrates using the MITgcm to simulate a
30  This example experiment demonstrates using the MITgcm to simulate  laboratory demonstration using a differentially heated rotating
31  a laboratory experiment with a rotating tank of water with an ice  annulus of water.  The simulation is configured for a laboratory scale
32  bucket in the center. The simulation is configured for a laboratory  on a $3^{\circ}\times1\mathrm{cm}$ cyclindrical grid with twenty-nine
33  scale on a  vertical levels of 0.5cm each.  This is a typical laboratory setup for
34  $3^{\circ}$ $\times$ 20cm  illustration principles of GFD, as well as for a laboratory data
35  cyclindrical grid with twenty-nine vertical  assimilation project. The files for this experiment can be found in
36  levels.  the verification directory under rotating\_tank.
37  \\  \\
38    
39    example illustration from GFD lab here
40    \\
41    
42    
 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  
43    
 \begin{equation}  
 \label{EQ:eg-baro-taux}  
 \tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y})  
 \end{equation}  
44    
 \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}  
45    
46  \subsection{Equations Solved}  \subsection{Equations Solved}
47  \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 $\vec{u}$.  
 \\  
48    
49    
50  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
51  \label{www:tutorials}  %\label{www:tutorials}
   
  The domain is discretised with  
 a uniform grid spacing in the horizontal set to  
  $\Delta x=\Delta y=20$~km, so  
 that there are sixty grid cells in the $x$ and $y$ directions. Vertically the  
 model is configured with a single layer with depth, $\Delta z$, of $5000$~m.  
   
 \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}  
52    
53  \noindent evaluates to $0.0144$, which is well below the $0.5$ upper   The domain is discretised with a uniform cylindrical grid spacing in
54  limit for stability.  the horizontal set to $\Delta a=1$~cm and $\Delta \phi=3^{\circ}$, so
55    that there are 120 grid cells in the azimuthal direction and
56    thirty-one grid cells in the radial, representing a tank 62cm in
57    diameter.  The bathymetry file sets the depth=0 in the nine lowest
58    radial rows to represent the central of the annulus.  Vertically the
59    model is configured with twenty-nine layers of uniform 0.5cm
60    thickness.
61  \\  \\
62    something about heat flux
 \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 $1200s$.  
63    
64  \subsection{Code Configuration}  \subsection{Code Configuration}
65  \label{www:tutorials}  %\label{www:tutorials}
66  \label{SEC:eg-baro-code_config}  \label{sec:eg-tank-code_config}
67    
68  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
69  directory {\it verification/exp0/}.  The experiment files  directory {\it verification/rotatingi\_tank/}.  The experiment files
70  \begin{itemize}  \begin{itemize}
71  \item {\it input/data}  \item {\it input/data}
72  \item {\it input/data.pkg}  \item {\it input/data.pkg}
73  \item {\it input/eedata},  \item {\it input/eedata},
74  \item {\it input/windx.sin\_y},  \item {\it input/bathyPol.bin},
75  \item {\it input/topog.box},  \item {\it input/thetaPol.bin},
76  \item {\it code/CPP\_EEOPTIONS.h}  \item {\it code/CPP\_EEOPTIONS.h}
77  \item {\it code/CPP\_OPTIONS.h},  \item {\it code/CPP\_OPTIONS.h},
78  \item {\it code/SIZE.h}.  \item {\it code/SIZE.h}.
79  \end{itemize}  \end{itemize}
80    
81  contain the code customizations and parameter settings for this  contain the code customizations and parameter settings for this
82  experiments. Below we describe the customizations  experiments. Below we describe the customizations
83  to these files associated with this experiment.  to these files associated with this experiment.
84    
85  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
86  \label{www:tutorials}  %\label{www:tutorials}
87    
88  This file, reproduced completely below, specifies the main parameters  This file, reproduced completely below, specifies the main parameters
89  for the experiment. The parameters that are significant for this configuration  for the experiment. The parameters that are significant for this configuration
# Line 218  are Line 91  are
91    
92  \begin{itemize}  \begin{itemize}
93    
94  \item Line 7, \begin{verbatim} viscAh=4.E2, \end{verbatim} this line sets  \item Lines 9-10, \begin{verbatim}
95  the Laplacian friction coefficient to $400 m^2s^{-1}$  viscAh=5.0E-6,
96  \item Line 10, \begin{verbatim} beta=1.E-11, \end{verbatim} this line sets  viscAz=5.0E-6,
97  $\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$  \end{verbatim}
98    
99    
100    These lines set the Laplacian friction coefficient in the horizontal
101    and vertical, respectively.  Note that they are several orders of
102    magnitude smaller than the other examples due to the small scale of
103    this example.
104    
105    \item Lines 13-16, \begin{verbatim}
106     diffKhT=2.5E-6,
107     diffKzT=2.5E-6,
108     diffKhS=1.0E-6,
109     diffKzS=1.0E-6,
110    
111    \end{verbatim}
112    
113    
114    These lines set horizontal and vertical diffusion coefficients for
115    temperature and salinity.  Similarly to the friction coefficients, the
116    values are a couple of orders of magnitude less than most
117     configurations.
118    
119    
120  \item Lines 15 and 16  \item Line 17, \begin{verbatim}f0=0.5 , \end{verbatim} this line sets the
121    coriolis term, and represents a tank spinning at about 2.4 rpm.
122    
123    \item Lines 23 and 24
124  \begin{verbatim}  \begin{verbatim}
125  rigidLid=.FALSE.,  rigidLid=.TRUE.,
126  implicitFreeSurface=.TRUE.,  implicitFreeSurface=.FALSE.,
127  \end{verbatim}  \end{verbatim}
128  these lines suppress the rigid lid formulation of the surface  
129  pressure inverter and activate the implicit free surface form  These lines activate  the rigid lid formulation of the surface
130    pressure inverter and suppress the implicit free surface form
131  of the pressure inverter.  of the pressure inverter.
132    
133  \item Line 27,  \item Line 40,
134  \begin{verbatim}  \begin{verbatim}
135  startTime=0,  nIter=0,
136  \end{verbatim}  \end{verbatim}
137  this line indicates that the experiment should start from $t=0$  This line indicates that the experiment should start from $t=0$ and
138  and implicitly suppresses searching for checkpoint files associated  implicitly suppresses searching for checkpoint files associated with
139  with restarting an numerical integration from a previously saved state.  restarting an numerical integration from a previously saved state.
140    Instead, the file thetaPol.bin will be loaded to initialized the
141    temperature fields as indicated below, and other variables will be
142    initialized to their defaults.
143    
 \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.  
144    
145  \item Line 30,  \item Line 43,
146  \begin{verbatim}  \begin{verbatim}
147  deltaTmom=1200,  deltaT=0.1,
148  \end{verbatim}  \end{verbatim}
149  This line sets the momentum equation timestep to $1200s$.  This line sets the integration timestep to $0.1s$.  This is an
150    unsually small value among the examples due to the small physical
151    scale of the experiment.  Using the ensemble Kalman filter to produce
152    input fields can necessitate even shorter timesteps.
153    
154  \item Line 39,  \item Line 56,
155  \begin{verbatim}  \begin{verbatim}
156  usingCartesianGrid=.TRUE.,  usingCylindricalGrid=.TRUE.,
157  \end{verbatim}  \end{verbatim}
158  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
159  Cartesian coordinate system.  cylindrical coordinate system.
160    
161  \item Line 41,  \item Line 57,
162  \begin{verbatim}  \begin{verbatim}
163  delX=60*20E3,  dXspacing=3,
164  \end{verbatim}  \end{verbatim}
165  This line sets the horizontal grid spacing between each x-coordinate line  This line sets the azimuthal grid spacing between each $x$-coordinate line
166  in the discrete grid. The syntax indicates that the discrete grid  in the discrete grid. The syntax indicates that the discrete grid
167  should be comprise of $60$ grid lines each separated by $20 \times 10^{3}m$  should be comprised of $120$ grid lines each separated by $3^{\circ}$.
168  ($20$~km).                                                                                
169    
170  \item Line 42,  \item Line 58,
171  \begin{verbatim}  \begin{verbatim}
172  delY=60*20E3,  dYspacing=0.01,
173  \end{verbatim}  \end{verbatim}
 This line sets the horizontal grid spacing between each y-coordinate line  
 in the discrete grid to $20 \times 10^{3}m$ ($20$~km).  
174    
175  \item Line 43,  This line sets the radial cylindrical grid spacing between each
176    $a$-coordinate line in the discrete grid to $1cm$.
177    
178    \item Line 59,
179  \begin{verbatim}  \begin{verbatim}
180  delZ=5000,  delZ=29*0.005,
181  \end{verbatim}  \end{verbatim}
 This line sets the vertical grid spacing between each z-coordinate line  
 in the discrete grid to $5000m$ ($5$~km).  
182    
183  \item Line 46,  This line sets the vertical grid spacing between each of 29
184    z-coordinate lines in the discrete grid to $0.005m$ ($5$~mm).
185    
186    \item Line 64,
187  \begin{verbatim}  \begin{verbatim}
188  bathyFile='topog.box'  bathyFile='bathyPol.bin',
189  \end{verbatim}  \end{verbatim}
190  This line specifies the name of the file from which the domain  This line specifies the name of the file from which the domain
191  bathymetry is read. This file is a two-dimensional ($x,y$) map of  ``bathymetry'' (tank depth) is read. This file is a two-dimensional
192    ($a,\phi$) map of
193  depths. This file is assumed to contain 64-bit binary numbers  depths. This file is assumed to contain 64-bit binary numbers
194  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 $\phi$
195  coordinate varying fastest. The points are ordered from low coordinate  coordinate varying fastest. The points are ordered from low coordinate
196  to high coordinate for both axes. The units and orientation of the  to high coordinate for both axes.  The units and orientation of the
197  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
198  experiment, a depth of $0m$ indicates a solid wall and a depth  experiment, a depth of $0m$ indicates an area outside of the tank
199  of $-5000m$ indicates open ocean. The matlab program  and a depth
200  {\it input/gendata.m} shows an example of how to generate a  f $-0.145m$ indicates the tank itself.
 bathymetry file.  
201    
202    \item Line 65,
203    \begin{verbatim}
204    hydrogThetaFile='thetaPol.bin',
205    \end{verbatim}
206    This line specifies the name of the file from which the initial values
207    of temperature
208    are read. This file is a three-dimensional
209    ($x,y,z$) map and is enumerated and formatted in the same manner as the
210    bathymetry file.
211    
212  \item Line 49,  \item Lines 66 and 67
213  \begin{verbatim}  \begin{verbatim}
214  zonalWindFile='windx.sin_y'   tCylIn  = 0
215     tCylOut  = 20
216  \end{verbatim}  \end{verbatim}
217  This line specifies the name of the file from which the x-direction  These line specify the temperatures in degrees Celsius of the interior
218  surface wind stress is read. This file is also a two-dimensional  and exterior walls of the tank -- typically taken to be icewater on
219  ($x,y$) map and is enumerated and formatted in the same manner as the  the inside and room temperature on the outside.
220  bathymetry file. The matlab program {\it input/gendata.m} includes example  
 code to generate a valid {\bf zonalWindFile} file.    
221    
222  \end{itemize}  \end{itemize}
223    
224  \noindent other lines in the file {\it input/data} are standard values  \noindent Other lines in the file {\it input/data} are standard values
225  that are described in the MITgcm Getting Started and MITgcm Parameters  that are described in the MITgcm Getting Started and MITgcm Parameters
226  notes.  notes.
227    
228  \begin{small}  \begin{small}
229  \input{part3/case_studies/barotropic_gyre/input/data}  \input{s_examples/rotating_tank/input/data}
230  \end{small}  \end{small}
231    
232  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
233  \label{www:tutorials}  %\label{www:tutorials}
234    
235  This file uses standard default values and does not contain  This file uses standard default values and does not contain
236  customizations for this experiment.  customizations for this experiment.
237    
238  \subsubsection{File {\it input/eedata}}  \subsubsection{File {\it input/eedata}}
239  \label{www:tutorials}  %\label{www:tutorials}
240    
241  This file uses standard default values and does not contain  This file uses standard default values and does not contain
242  customizations for this experiment.  customizations for this experiment.
243    
244  \subsubsection{File {\it input/windx.sin\_y}}  \subsubsection{File {\it input/thetaPol.bin}}
245  \label{www:tutorials}  %\label{www:tutorials}
246    
247  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)  The {\it input/thetaPol.bin} file specifies a three-dimensional ($x,y,z$)
248  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.  map of initial values of $\theta$ in degrees Celsius.  This particular
249  Although $\tau_{x}$ is only a function of $y$n in this experiment  experiment is set to random values x around 20C to provide initial
250  this file must still define a complete two-dimensional map in order  perturbations.
 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.  
251    
252  \subsubsection{File {\it input/topog.box}}  \subsubsection{File {\it input/bathyPol.bin}}
253  \label{www:tutorials}  %\label{www:tutorials}
254    
255    
256  The {\it input/topog.box} file specifies a two-dimensional ($x,y$)  The {\it input/bathyPol.bin} file specifies a two-dimensional ($x,y$)
257  map of depth values. For this experiment values are either  map of depth values. For this experiment values are either
258  $0m$ or {\bf -delZ}m, corresponding respectively to a wall or to deep  $0m$ or {\bf -delZ}m, corresponding respectively to outside or inside of
259  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
260  in the same way as standard MITgcm two-dimensional, horizontal arrays.  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.  
261    
262  \subsubsection{File {\it code/SIZE.h}}  \subsubsection{File {\it code/SIZE.h}}
263  \label{www:tutorials}  %\label{www:tutorials}
264    
265  Two lines are customized in this file for the current experiment  Two lines are customized in this file for the current experiment
266    
267  \begin{itemize}  \begin{itemize}
268    
269  \item Line 39,  \item Line 39,
270  \begin{verbatim} sNx=60, \end{verbatim} this line sets  \begin{verbatim} sNx=120, \end{verbatim} this line sets
271  the lateral domain extent in grid points for the  the lateral domain extent in grid points for the
272  axis aligned with the x-coordinate.  axis aligned with the x-coordinate.
273    
274  \item Line 40,  \item Line 40,
275  \begin{verbatim} sNy=60, \end{verbatim} this line sets  \begin{verbatim} sNy=31, \end{verbatim} this line sets
276  the lateral domain extent in grid points for the  the lateral domain extent in grid points for the
277  axis aligned with the y-coordinate.  axis aligned with the y-coordinate.
278    
279  \end{itemize}  \end{itemize}
280    
281  \begin{small}  \begin{small}
282  \input{part3/case_studies/barotropic_gyre/code/SIZE.h}  \input{s_examples/rotating_tank/code/SIZE.h}
283  \end{small}  \end{small}
284    
285  \subsubsection{File {\it code/CPP\_OPTIONS.h}}  \subsubsection{File {\it code/CPP\_OPTIONS.h}}
286  \label{www:tutorials}  %\label{www:tutorials}
287    
288  This file uses standard default values and does not contain  This file uses standard default values and does not contain
289  customizations for this experiment.  customizations for this experiment.
290    
291    
292  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
293  \label{www:tutorials}  %\label{www:tutorials}
294    
295  This file uses standard default values and does not contain  This file uses standard default values and does not contain
296  customizations for this experiment.  customizations for this experiment.
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