| 1 |
% $Header: /u/gcmpack/manual/part3/case_studies/barotropic_gyre/baro.tex,v 1.10 2004/01/29 15:11:39 edhill Exp $ |
| 2 |
% $Name: $ |
| 3 |
|
| 4 |
\bodytext{bgcolor="#FFFFFFFF"} |
| 5 |
|
| 6 |
%\begin{center} |
| 7 |
%{\Large \bf Using MITgcm to Simulate a Rotating Tank in Cylindrical |
| 8 |
%Coordinates} |
| 9 |
% |
| 10 |
%\vspace*{4mm} |
| 11 |
% |
| 12 |
%\vspace*{3mm} |
| 13 |
%{\large May 2001} |
| 14 |
%\end{center} |
| 15 |
|
| 16 |
This is the first in a series of tutorials describing |
| 17 |
example MITgcm numerical experiments. The example experiments |
| 18 |
include both straightforward examples of idealized geophysical |
| 19 |
fluid simulations and more involved cases encompassing |
| 20 |
large scale modeling and |
| 21 |
automatic differentiation. Both hydrostatic and non-hydrostatic |
| 22 |
experiments are presented, as well as experiments employing |
| 23 |
Cartesian, spherical-polar and cube-sphere coordinate systems. |
| 24 |
These ``case study'' documents include information describing |
| 25 |
the experimental configuration and detailed information on how to |
| 26 |
configure the MITgcm code and input files for each experiment. |
| 27 |
|
| 28 |
\section{A Rotating Tank in Cylindrical Coordinates} |
| 29 |
\label{sect:eg-tank} |
| 30 |
\label{www:tutorials} |
| 31 |
|
| 32 |
|
| 33 |
This example experiment demonstrates using the MITgcm to simulate |
| 34 |
a Barotropic, wind-forced, ocean gyre circulation. The experiment |
| 35 |
is a numerical rendition of the gyre circulation problem similar |
| 36 |
to the problems described analytically by Stommel in 1966 |
| 37 |
\cite{Stommel66} and numerically in Holland et. al \cite{Holland75}. |
| 38 |
|
| 39 |
In this experiment the model |
| 40 |
is configured to represent a rectangular enclosed box of fluid, |
| 41 |
$1200 \times 1200 $~km in lateral extent. The fluid is $5$~km deep and is forced |
| 42 |
by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally |
| 43 |
in the ``north-south'' direction. Topologically the grid is Cartesian and |
| 44 |
the coriolis parameter $f$ is defined according to a mid-latitude beta-plane |
| 45 |
equation |
| 46 |
|
| 47 |
\begin{equation} |
| 48 |
\label{EQ:eg-baro-fcori} |
| 49 |
f(y) = f_{0}+\beta y |
| 50 |
\end{equation} |
| 51 |
|
| 52 |
\noindent where $y$ is the distance along the ``north-south'' axis of the |
| 53 |
simulated domain. For this experiment $f_{0}$ is set to $10^{-4}s^{-1}$ in |
| 54 |
(\ref{EQ:eg-baro-fcori}) and $\beta = 10^{-11}s^{-1}m^{-1}$. |
| 55 |
\\ |
| 56 |
\\ |
| 57 |
The sinusoidal wind-stress variations are defined according to |
| 58 |
|
| 59 |
\begin{equation} |
| 60 |
\label{EQ:eg-baro-taux} |
| 61 |
\tau_x(y) = \tau_{0}\sin(\pi \frac{y}{L_y}) |
| 62 |
\end{equation} |
| 63 |
|
| 64 |
\noindent where $L_{y}$ is the lateral domain extent ($1200$~km) and |
| 65 |
$\tau_0$ is set to $0.1N m^{-2}$. |
| 66 |
\\ |
| 67 |
\\ |
| 68 |
Figure \ref{FIG:eg-baro-simulation_config} |
| 69 |
summarizes the configuration simulated. |
| 70 |
|
| 71 |
%% === eh3 === |
| 72 |
\begin{figure} |
| 73 |
%% \begin{center} |
| 74 |
%% \resizebox{7.5in}{5.5in}{ |
| 75 |
%% \includegraphics*[0.2in,0.7in][10.5in,10.5in] |
| 76 |
%% {part3/case_studies/barotropic_gyre/simulation_config.eps} } |
| 77 |
%% \end{center} |
| 78 |
\centerline{ |
| 79 |
\scalefig{.95} |
| 80 |
\epsfbox{part3/case_studies/barotropic_gyre/simulation_config.eps} |
| 81 |
} |
| 82 |
\caption{Schematic of simulation domain and wind-stress forcing function |
| 83 |
for barotropic gyre numerical experiment. The domain is enclosed bu solid |
| 84 |
walls at $x=$~0,1200km and at $y=$~0,1200km.} |
| 85 |
\label{FIG:eg-baro-simulation_config} |
| 86 |
\end{figure} |
| 87 |
|
| 88 |
\subsection{Equations Solved} |
| 89 |
\label{www:tutorials} |
| 90 |
The model is configured in hydrostatic form. The implicit free surface form of the |
| 91 |
pressure equation described in Marshall et. al \cite{marshall:97a} is |
| 92 |
employed. |
| 93 |
A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous |
| 94 |
dissipation. The wind-stress momentum input is added to the momentum equation |
| 95 |
for the ``zonal flow'', $u$. Other terms in the model |
| 96 |
are explicitly switched off for this experiment configuration (see section |
| 97 |
\ref{SEC:code_config} ), yielding an active set of equations solved in this |
| 98 |
configuration as follows |
| 99 |
|
| 100 |
\begin{eqnarray} |
| 101 |
\label{EQ:eg-baro-model_equations} |
| 102 |
\frac{Du}{Dt} - fv + |
| 103 |
g\frac{\partial \eta}{\partial x} - |
| 104 |
A_{h}\nabla_{h}^2u |
| 105 |
& = & |
| 106 |
\frac{\tau_{x}}{\rho_{0}\Delta z} |
| 107 |
\\ |
| 108 |
\frac{Dv}{Dt} + fu + g\frac{\partial \eta}{\partial y} - |
| 109 |
A_{h}\nabla_{h}^2v |
| 110 |
& = & |
| 111 |
0 |
| 112 |
\\ |
| 113 |
\frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u} |
| 114 |
&=& |
| 115 |
0 |
| 116 |
\end{eqnarray} |
| 117 |
|
| 118 |
\noindent where $u$ and $v$ and the $x$ and $y$ components of the |
| 119 |
flow vector $\vec{u}$. |
| 120 |
\\ |
| 121 |
|
| 122 |
|
| 123 |
\subsection{Discrete Numerical Configuration} |
| 124 |
\label{www:tutorials} |
| 125 |
|
| 126 |
The domain is discretised with |
| 127 |
a uniform grid spacing in the horizontal set to |
| 128 |
$\Delta x=\Delta y=20$~km, so |
| 129 |
that there are sixty grid cells in the $x$ and $y$ directions. Vertically the |
| 130 |
model is configured with a single layer with depth, $\Delta z$, of $5000$~m. |
| 131 |
|
| 132 |
\subsubsection{Numerical Stability Criteria} |
| 133 |
\label{www:tutorials} |
| 134 |
|
| 135 |
The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$. |
| 136 |
This value is chosen to yield a Munk layer width \cite{adcroft:95}, |
| 137 |
|
| 138 |
\begin{eqnarray} |
| 139 |
\label{EQ:eg-baro-munk_layer} |
| 140 |
M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}} |
| 141 |
\end{eqnarray} |
| 142 |
|
| 143 |
\noindent of $\approx 100$km. This is greater than the model |
| 144 |
resolution $\Delta x$, ensuring that the frictional boundary |
| 145 |
layer is well resolved. |
| 146 |
\\ |
| 147 |
|
| 148 |
\noindent The model is stepped forward with a |
| 149 |
time step $\delta t=1200$secs. With this time step the stability |
| 150 |
parameter to the horizontal Laplacian friction \cite{adcroft:95} |
| 151 |
|
| 152 |
|
| 153 |
|
| 154 |
\begin{eqnarray} |
| 155 |
\label{EQ:eg-baro-laplacian_stability} |
| 156 |
S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2} |
| 157 |
\end{eqnarray} |
| 158 |
|
| 159 |
\noindent evaluates to 0.012, which is well below the 0.3 upper limit |
| 160 |
for stability. |
| 161 |
\\ |
| 162 |
|
| 163 |
\noindent The numerical stability for inertial oscillations |
| 164 |
\cite{adcroft:95} |
| 165 |
|
| 166 |
\begin{eqnarray} |
| 167 |
\label{EQ:eg-baro-inertial_stability} |
| 168 |
S_{i} = f^{2} {\delta t}^2 |
| 169 |
\end{eqnarray} |
| 170 |
|
| 171 |
\noindent evaluates to $0.0144$, which is well below the $0.5$ upper |
| 172 |
limit for stability. |
| 173 |
\\ |
| 174 |
|
| 175 |
\noindent The advective CFL \cite{adcroft:95} for an extreme maximum |
| 176 |
horizontal flow speed of $ | \vec{u} | = 2 ms^{-1}$ |
| 177 |
|
| 178 |
\begin{eqnarray} |
| 179 |
\label{EQ:eg-baro-cfl_stability} |
| 180 |
S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x} |
| 181 |
\end{eqnarray} |
| 182 |
|
| 183 |
\noindent evaluates to 0.12. This is approaching the stability limit |
| 184 |
of 0.5 and limits $\delta t$ to $1200s$. |
| 185 |
|
| 186 |
\subsection{Code Configuration} |
| 187 |
\label{www:tutorials} |
| 188 |
\label{SEC:eg-baro-code_config} |
| 189 |
|
| 190 |
The model configuration for this experiment resides under the |
| 191 |
directory {\it verification/exp0/}. The experiment files |
| 192 |
\begin{itemize} |
| 193 |
\item {\it input/data} |
| 194 |
\item {\it input/data.pkg} |
| 195 |
\item {\it input/eedata}, |
| 196 |
\item {\it input/windx.sin\_y}, |
| 197 |
\item {\it input/topog.box}, |
| 198 |
\item {\it code/CPP\_EEOPTIONS.h} |
| 199 |
\item {\it code/CPP\_OPTIONS.h}, |
| 200 |
\item {\it code/SIZE.h}. |
| 201 |
\end{itemize} |
| 202 |
contain the code customizations and parameter settings for this |
| 203 |
experiments. Below we describe the customizations |
| 204 |
to these files associated with this experiment. |
| 205 |
|
| 206 |
\subsubsection{File {\it input/data}} |
| 207 |
\label{www:tutorials} |
| 208 |
|
| 209 |
This file, reproduced completely below, specifies the main parameters |
| 210 |
for the experiment. The parameters that are significant for this configuration |
| 211 |
are |
| 212 |
|
| 213 |
\begin{itemize} |
| 214 |
|
| 215 |
\item Line 7, \begin{verbatim} viscAh=4.E2, \end{verbatim} this line sets |
| 216 |
the Laplacian friction coefficient to $400 m^2s^{-1}$ |
| 217 |
\item Line 10, \begin{verbatim} beta=1.E-11, \end{verbatim} this line sets |
| 218 |
$\beta$ (the gradient of the coriolis parameter, $f$) to $10^{-11} s^{-1}m^{-1}$ |
| 219 |
|
| 220 |
\item Lines 15 and 16 |
| 221 |
\begin{verbatim} |
| 222 |
rigidLid=.FALSE., |
| 223 |
implicitFreeSurface=.TRUE., |
| 224 |
\end{verbatim} |
| 225 |
these lines suppress the rigid lid formulation of the surface |
| 226 |
pressure inverter and activate the implicit free surface form |
| 227 |
of the pressure inverter. |
| 228 |
|
| 229 |
\item Line 27, |
| 230 |
\begin{verbatim} |
| 231 |
startTime=0, |
| 232 |
\end{verbatim} |
| 233 |
this line indicates that the experiment should start from $t=0$ |
| 234 |
and implicitly suppresses searching for checkpoint files associated |
| 235 |
with restarting an numerical integration from a previously saved state. |
| 236 |
|
| 237 |
\item Line 29, |
| 238 |
\begin{verbatim} |
| 239 |
endTime=12000, |
| 240 |
\end{verbatim} |
| 241 |
this line indicates that the experiment should start finish at $t=12000s$. |
| 242 |
A restart file will be written at this time that will enable the |
| 243 |
simulation to be continued from this point. |
| 244 |
|
| 245 |
\item Line 30, |
| 246 |
\begin{verbatim} |
| 247 |
deltaTmom=1200, |
| 248 |
\end{verbatim} |
| 249 |
This line sets the momentum equation timestep to $1200s$. |
| 250 |
|
| 251 |
\item Line 39, |
| 252 |
\begin{verbatim} |
| 253 |
usingCartesianGrid=.TRUE., |
| 254 |
\end{verbatim} |
| 255 |
This line requests that the simulation be performed in a |
| 256 |
Cartesian coordinate system. |
| 257 |
|
| 258 |
\item Line 41, |
| 259 |
\begin{verbatim} |
| 260 |
delX=60*20E3, |
| 261 |
\end{verbatim} |
| 262 |
This line sets the horizontal grid spacing between each x-coordinate line |
| 263 |
in the discrete grid. The syntax indicates that the discrete grid |
| 264 |
should be comprise of $60$ grid lines each separated by $20 \times 10^{3}m$ |
| 265 |
($20$~km). |
| 266 |
|
| 267 |
\item Line 42, |
| 268 |
\begin{verbatim} |
| 269 |
delY=60*20E3, |
| 270 |
\end{verbatim} |
| 271 |
This line sets the horizontal grid spacing between each y-coordinate line |
| 272 |
in the discrete grid to $20 \times 10^{3}m$ ($20$~km). |
| 273 |
|
| 274 |
\item Line 43, |
| 275 |
\begin{verbatim} |
| 276 |
delZ=5000, |
| 277 |
\end{verbatim} |
| 278 |
This line sets the vertical grid spacing between each z-coordinate line |
| 279 |
in the discrete grid to $5000m$ ($5$~km). |
| 280 |
|
| 281 |
\item Line 46, |
| 282 |
\begin{verbatim} |
| 283 |
bathyFile='topog.box' |
| 284 |
\end{verbatim} |
| 285 |
This line specifies the name of the file from which the domain |
| 286 |
bathymetry is read. This file is a two-dimensional ($x,y$) map of |
| 287 |
depths. This file is assumed to contain 64-bit binary numbers |
| 288 |
giving the depth of the model at each grid cell, ordered with the x |
| 289 |
coordinate varying fastest. The points are ordered from low coordinate |
| 290 |
to high coordinate for both axes. The units and orientation of the |
| 291 |
depths in this file are the same as used in the MITgcm code. In this |
| 292 |
experiment, a depth of $0m$ indicates a solid wall and a depth |
| 293 |
of $-5000m$ indicates open ocean. The matlab program |
| 294 |
{\it input/gendata.m} shows an example of how to generate a |
| 295 |
bathymetry file. |
| 296 |
|
| 297 |
|
| 298 |
\item Line 49, |
| 299 |
\begin{verbatim} |
| 300 |
zonalWindFile='windx.sin_y' |
| 301 |
\end{verbatim} |
| 302 |
This line specifies the name of the file from which the x-direction |
| 303 |
surface wind stress is read. This file is also a two-dimensional |
| 304 |
($x,y$) map and is enumerated and formatted in the same manner as the |
| 305 |
bathymetry file. The matlab program {\it input/gendata.m} includes example |
| 306 |
code to generate a valid {\bf zonalWindFile} file. |
| 307 |
|
| 308 |
\end{itemize} |
| 309 |
|
| 310 |
\noindent other lines in the file {\it input/data} are standard values |
| 311 |
that are described in the MITgcm Getting Started and MITgcm Parameters |
| 312 |
notes. |
| 313 |
|
| 314 |
\begin{small} |
| 315 |
\input{part3/case_studies/barotropic_gyre/input/data} |
| 316 |
\end{small} |
| 317 |
|
| 318 |
\subsubsection{File {\it input/data.pkg}} |
| 319 |
\label{www:tutorials} |
| 320 |
|
| 321 |
This file uses standard default values and does not contain |
| 322 |
customizations for this experiment. |
| 323 |
|
| 324 |
\subsubsection{File {\it input/eedata}} |
| 325 |
\label{www:tutorials} |
| 326 |
|
| 327 |
This file uses standard default values and does not contain |
| 328 |
customizations for this experiment. |
| 329 |
|
| 330 |
\subsubsection{File {\it input/windx.sin\_y}} |
| 331 |
\label{www:tutorials} |
| 332 |
|
| 333 |
The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$) |
| 334 |
map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$. |
| 335 |
Although $\tau_{x}$ is only a function of $y$n in this experiment |
| 336 |
this file must still define a complete two-dimensional map in order |
| 337 |
to be compatible with the standard code for loading forcing fields |
| 338 |
in MITgcm. The included matlab program {\it input/gendata.m} gives a complete |
| 339 |
code for creating the {\it input/windx.sin\_y} file. |
| 340 |
|
| 341 |
\subsubsection{File {\it input/topog.box}} |
| 342 |
\label{www:tutorials} |
| 343 |
|
| 344 |
|
| 345 |
The {\it input/topog.box} file specifies a two-dimensional ($x,y$) |
| 346 |
map of depth values. For this experiment values are either |
| 347 |
$0m$ or {\bf -delZ}m, corresponding respectively to a wall or to deep |
| 348 |
ocean. The file contains a raw binary stream of data that is enumerated |
| 349 |
in the same way as standard MITgcm two-dimensional, horizontal arrays. |
| 350 |
The included matlab program {\it input/gendata.m} gives a complete |
| 351 |
code for creating the {\it input/topog.box} file. |
| 352 |
|
| 353 |
\subsubsection{File {\it code/SIZE.h}} |
| 354 |
\label{www:tutorials} |
| 355 |
|
| 356 |
Two lines are customized in this file for the current experiment |
| 357 |
|
| 358 |
\begin{itemize} |
| 359 |
|
| 360 |
\item Line 39, |
| 361 |
\begin{verbatim} sNx=60, \end{verbatim} this line sets |
| 362 |
the lateral domain extent in grid points for the |
| 363 |
axis aligned with the x-coordinate. |
| 364 |
|
| 365 |
\item Line 40, |
| 366 |
\begin{verbatim} sNy=60, \end{verbatim} this line sets |
| 367 |
the lateral domain extent in grid points for the |
| 368 |
axis aligned with the y-coordinate. |
| 369 |
|
| 370 |
\end{itemize} |
| 371 |
|
| 372 |
\begin{small} |
| 373 |
\input{part3/case_studies/barotropic_gyre/code/SIZE.h} |
| 374 |
\end{small} |
| 375 |
|
| 376 |
\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
| 377 |
\label{www:tutorials} |
| 378 |
|
| 379 |
This file uses standard default values and does not contain |
| 380 |
customizations for this experiment. |
| 381 |
|
| 382 |
|
| 383 |
\subsubsection{File {\it code/CPP\_EEOPTIONS.h}} |
| 384 |
\label{www:tutorials} |
| 385 |
|
| 386 |
This file uses standard default values and does not contain |
| 387 |
customizations for this experiment. |
| 388 |
|
Parent Directory
| 
Revision Log
| 
Revision Graph