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revision 1.9 by cnh, Thu Oct 25 18:36:55 2001 UTC revision 1.13 by adcroft, Thu May 16 15:54:37 2002 UTC
# Line 1  Line 1 
1  % $Header$  % $Header$
2  % $Name$  % $Name$
3    
4  \section{Example: Four layer Baroclinic Ocean Gyre In Spherical Coordinates}  \section{Four Layer Baroclinic Ocean Gyre In Spherical Coordinates}
5  \label{sec:eg-fourlayer}  \label{www:tutorials}
6    \label{sect:eg-fourlayer}
7    
8  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
9    
# Line 19  Line 20 
20  This document describes an example experiment using MITgcm  This document describes an example experiment using MITgcm
21  to simulate a baroclinic ocean gyre in spherical  to simulate a baroclinic ocean gyre in spherical
22  polar coordinates. The barotropic  polar coordinates. The barotropic
23  example experiment in section \ref{sec:eg-baro}  example experiment in section \ref{sect:eg-baro}
24  illustrated how to configure the code for a single layer  illustrated how to configure the code for a single layer
25  simulation in a Cartesian grid. In this example a similar physical problem  simulation in a Cartesian grid. In this example a similar physical problem
26  is simulated, but the code is now configured  is simulated, but the code is now configured
27  for four layers and in a spherical polar coordinate system.  for four layers and in a spherical polar coordinate system.
28    
29  \subsection{Overview}  \subsection{Overview}
30    \label{www:tutorials}
31    
32  This example experiment demonstrates using the MITgcm to simulate  This example experiment demonstrates using the MITgcm to simulate
33  a baroclinic, wind-forced, ocean gyre circulation. The experiment  a baroclinic, wind-forced, ocean gyre circulation. The experiment
# Line 43  domain is a sector on a sphere and the c Line 45  domain is a sector on a sphere and the c
45  according to latitude, $\varphi$  according to latitude, $\varphi$
46    
47  \begin{equation}  \begin{equation}
48  \label{EQ:fcori}  \label{EQ:eg-fourlayer-fcori}
49  f(\varphi) = 2 \Omega \sin( \varphi )  f(\varphi) = 2 \Omega \sin( \varphi )
50  \end{equation}  \end{equation}
51    
# Line 61  f(\varphi) = 2 \Omega \sin( \varphi ) Line 63  f(\varphi) = 2 \Omega \sin( \varphi )
63  $\tau_0$ is set to $0.1N m^{-2}$.  $\tau_0$ is set to $0.1N m^{-2}$.
64  \\  \\
65    
66  Figure \ref{FIG:simulation_config}  Figure \ref{FIG:eg-fourlayer-simulation_config}
67  summarizes the configuration simulated.  summarizes the configuration simulated.
68  In contrast to the example in section \ref{sec:eg-baro}, the  In contrast to the example in section \ref{sect:eg-baro}, the
69  current experiment simulates a spherical polar domain. As indicated  current experiment simulates a spherical polar domain. As indicated
70  by the axes in the lower left of the figure the model code works internally  by the axes in the lower left of the figure the model code works internally
71  in a locally orthogonal coordinate $(x,y,z)$. For this experiment description  in a locally orthogonal coordinate $(x,y,z)$. For this experiment description
# Line 82  $\theta_{1750}=6^{\circ}$~C. The equatio Line 84  $\theta_{1750}=6^{\circ}$~C. The equatio
84  linear  linear
85    
86  \begin{equation}  \begin{equation}
87  \label{EQ:linear1_eos}  \label{EQ:eg-fourlayer-linear1_eos}
88  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )
89  \end{equation}  \end{equation}
90    
91  \noindent which is implemented in the model as a density anomaly equation  \noindent which is implemented in the model as a density anomaly equation
92    
93  \begin{equation}  \begin{equation}
94  \label{EQ:linear1_eos_pert}  \label{EQ:eg-fourlayer-linear1_eos_pert}
95  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}
96  \end{equation}  \end{equation}
97    
# Line 114  An initial stratification is Line 116  An initial stratification is
116  imposed by setting the potential temperature, $\theta$, in each layer.  imposed by setting the potential temperature, $\theta$, in each layer.
117  The vertical spacing, $\Delta z$, is constant and equal to $500$m.  The vertical spacing, $\Delta z$, is constant and equal to $500$m.
118  }  }
119  \label{FIG:simulation_config}  \label{FIG:eg-fourlayer-simulation_config}
120  \end{figure}  \end{figure}
121    
122  \subsection{Equations solved}  \subsection{Equations solved}
123    \label{www:tutorials}
124  For this problem  For this problem
125  the implicit free surface, {\bf HPE} (see section \ref{sec:hydrostatic_and_quasi-hydrostatic_forms}) form of the  the implicit free surface, {\bf HPE} (see section \ref{sect:hydrostatic_and_quasi-hydrostatic_forms}) form of the
126  equations described in Marshall et. al \cite{Marshall97a} are  equations described in Marshall et. al \cite{marshall:97a} are
127  employed. The flow is three-dimensional with just temperature, $\theta$, as  employed. The flow is three-dimensional with just temperature, $\theta$, as
128  an active tracer.  The equation of state is linear.  an active tracer.  The equation of state is linear.
129  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous
# Line 133  solved in this configuration, written in Line 136  solved in this configuration, written in
136  follows  follows
137    
138  \begin{eqnarray}  \begin{eqnarray}
139  \label{EQ:model_equations}  \label{EQ:eg-fourlayer-model_equations}
140  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
141    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -
142    A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}    A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}
# Line 202  e.g. $\frac{\partial \theta}{\partial \v Line 205  e.g. $\frac{\partial \theta}{\partial \v
205    
206    
207  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
208    \label{www:tutorials}
209    
210   The domain is discretised with   The domain is discretised with
211  a uniform grid spacing in latitude and longitude  a uniform grid spacing in latitude and longitude
# Line 221  y=r\varphi,~\Delta y &= &r\Delta \varphi Line 225  y=r\varphi,~\Delta y &= &r\Delta \varphi
225    
226  The procedure for generating a set of internal grid variables from a  The procedure for generating a set of internal grid variables from a
227  spherical polar grid specification is discussed in section  spherical polar grid specification is discussed in section
228  \ref{sec:spatial_discrete_horizontal_grid}.  \ref{sect:spatial_discrete_horizontal_grid}.
229    
230  \noindent\fbox{ \begin{minipage}{5.5in}  \noindent\fbox{ \begin{minipage}{5.5in}
231  {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em  {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em
# Line 242  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, { Line 246  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, {
246    
247    
248    
249  As described in \ref{sec:tracer_equations}, the time evolution of potential  As described in \ref{sect:tracer_equations}, the time evolution of potential
250  temperature,  temperature,
251  $\theta$, (equation \ref{eq:eg_fourl_theta})  $\theta$, (equation \ref{eq:eg_fourl_theta})
252  is evaluated prognostically. The centered second-order scheme with  is evaluated prognostically. The centered second-order scheme with
253  Adams-Bashforth time stepping described in section  Adams-Bashforth time stepping described in section
254  \ref{sec:tracer_equations_abII} is used to step forward the temperature  \ref{sect:tracer_equations_abII} is used to step forward the temperature
255  equation. Prognostic terms in  equation. Prognostic terms in
256  the momentum equations are solved using flux form as  the momentum equations are solved using flux form as
257  described in section \ref{sec:flux-form_momentum_eqautions}.  described in section \ref{sect:flux-form_momentum_eqautions}.
258  The pressure forces that drive the fluid motions, (  The pressure forces that drive the fluid motions, (
259  $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface  $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$), are found by summing pressure due to surface
260  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the
# Line 258  pressure is diagnosed explicitly by inte Line 262  pressure is diagnosed explicitly by inte
262  height, $\eta$, is diagnosed using an implicit scheme. The pressure  height, $\eta$, is diagnosed using an implicit scheme. The pressure
263  field solution method is described in sections  field solution method is described in sections
264  \ref{sect:pressure-method-linear-backward} and  \ref{sect:pressure-method-linear-backward} and
265  \ref{sec:finding_the_pressure_field}.  \ref{sect:finding_the_pressure_field}.
266    
267  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
268    \label{www:tutorials}
269    
270  The Laplacian viscosity coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian viscosity coefficient, $A_{h}$, is set to $400 m s^{-1}$.
271  This value is chosen to yield a Munk layer width,  This value is chosen to yield a Munk layer width,
272    
273  \begin{eqnarray}  \begin{eqnarray}
274  \label{EQ:munk_layer}  \label{EQ:eg-fourlayer-munk_layer}
275  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
276  \end{eqnarray}  \end{eqnarray}
277    
# Line 282  time step $\delta t=1200$secs. With this Line 287  time step $\delta t=1200$secs. With this
287  parameter to the horizontal Laplacian friction  parameter to the horizontal Laplacian friction
288    
289  \begin{eqnarray}  \begin{eqnarray}
290  \label{EQ:laplacian_stability}  \label{EQ:eg-fourlayer-laplacian_stability}
291  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
292  \end{eqnarray}  \end{eqnarray}
293    
# Line 294  for stability for this term under ABII t Line 299  for stability for this term under ABII t
299  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
300    
301  \begin{eqnarray}  \begin{eqnarray}
302  \label{EQ:laplacian_stability_z}  \label{EQ:eg-fourlayer-laplacian_stability_z}
303  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}
304  \end{eqnarray}  \end{eqnarray}
305    
# Line 307  and vertical ($K_{z}$) diffusion coeffic Line 312  and vertical ($K_{z}$) diffusion coeffic
312  \noindent The numerical stability for inertial oscillations  \noindent The numerical stability for inertial oscillations
313    
314  \begin{eqnarray}  \begin{eqnarray}
315  \label{EQ:inertial_stability}  \label{EQ:eg-fourlayer-inertial_stability}
316  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
317  \end{eqnarray}  \end{eqnarray}
318    
# Line 320  horizontal flow Line 325  horizontal flow
325  speed of $ | \vec{u} | = 2 ms^{-1}$  speed of $ | \vec{u} | = 2 ms^{-1}$
326    
327  \begin{eqnarray}  \begin{eqnarray}
328  \label{EQ:cfl_stability}  \label{EQ:eg-fourlayer-cfl_stability}
329  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
330  \end{eqnarray}  \end{eqnarray}
331    
# Line 332  limit of 0.5. Line 337  limit of 0.5.
337  propagating at $2~{\rm m}~{\rm s}^{-1}$  propagating at $2~{\rm m}~{\rm s}^{-1}$
338    
339  \begin{eqnarray}  \begin{eqnarray}
340  \label{EQ:igw_stability}  \label{EQ:eg-fourlayer-igw_stability}
341  S_{c} = \frac{c_{g} \delta t}{ \Delta x}  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
342  \end{eqnarray}  \end{eqnarray}
343    
# Line 340  S_{c} = \frac{c_{g} \delta t}{ \Delta x} Line 345  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
345  stability limit of 0.25.  stability limit of 0.25.
346        
347  \subsection{Code Configuration}  \subsection{Code Configuration}
348    \label{www:tutorials}
349  \label{SEC:eg_fourl_code_config}  \label{SEC:eg_fourl_code_config}
350    
351  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
# Line 359  experiments. Below we describe the custo Line 365  experiments. Below we describe the custo
365  to these files associated with this experiment.  to these files associated with this experiment.
366    
367  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
368    \label{www:tutorials}
369    
370  This file, reproduced completely below, specifies the main parameters  This file, reproduced completely below, specifies the main parameters
371  for the experiment. The parameters that are significant for this configuration  for the experiment. The parameters that are significant for this configuration
# Line 946  goto code Line 953  goto code
953  \begin{rawhtml}</PRE>\end{rawhtml}  \begin{rawhtml}</PRE>\end{rawhtml}
954    
955  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
956    \label{www:tutorials}
957    
958  This file uses standard default values and does not contain  This file uses standard default values and does not contain
959  customisations for this experiment.  customisations for this experiment.
960    
961  \subsubsection{File {\it input/eedata}}  \subsubsection{File {\it input/eedata}}
962    \label{www:tutorials}
963    
964  This file uses standard default values and does not contain  This file uses standard default values and does not contain
965  customisations for this experiment.  customisations for this experiment.
966    
967  \subsubsection{File {\it input/windx.sin\_y}}  \subsubsection{File {\it input/windx.sin\_y}}
968    \label{www:tutorials}
969    
970  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
971  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$ (the  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$ (the
# Line 969  The included matlab program {\it input/g Line 979  The included matlab program {\it input/g
979  code for creating the {\it input/windx.sin\_y} file.  code for creating the {\it input/windx.sin\_y} file.
980    
981  \subsubsection{File {\it input/topog.box}}  \subsubsection{File {\it input/topog.box}}
982    \label{www:tutorials}
983    
984    
985  The {\it input/topog.box} file specifies a two-dimensional ($x,y$)  The {\it input/topog.box} file specifies a two-dimensional ($x,y$)
# Line 980  The included matlab program {\it input/g Line 991  The included matlab program {\it input/g
991  code for creating the {\it input/topog.box} file.  code for creating the {\it input/topog.box} file.
992    
993  \subsubsection{File {\it code/SIZE.h}}  \subsubsection{File {\it code/SIZE.h}}
994    \label{www:tutorials}
995    
996  Two lines are customized in this file for the current experiment  Two lines are customized in this file for the current experiment
997    
# Line 1006  the vertical domain extent in grid point Line 1018  the vertical domain extent in grid point
1018  \end{small}  \end{small}
1019    
1020  \subsubsection{File {\it code/CPP\_OPTIONS.h}}  \subsubsection{File {\it code/CPP\_OPTIONS.h}}
1021    \label{www:tutorials}
1022    
1023  This file uses standard default values and does not contain  This file uses standard default values and does not contain
1024  customisations for this experiment.  customisations for this experiment.
1025    
1026    
1027  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
1028    \label{www:tutorials}
1029    
1030  This file uses standard default values and does not contain  This file uses standard default values and does not contain
1031  customisations for this experiment.  customisations for this experiment.
1032    
1033  \subsubsection{Other Files }  \subsubsection{Other Files }
1034    \label{www:tutorials}
1035    
1036  Other files relevant to this experiment are  Other files relevant to this experiment are
1037  \begin{itemize}  \begin{itemize}
# Line 1029  dxF, dyF, dxG, dyG, dxC, dyC}. Line 1044  dxF, dyF, dxG, dyG, dxC, dyC}.
1044  \end{itemize}  \end{itemize}
1045    
1046  \subsection{Running The Example}  \subsection{Running The Example}
1047    \label{www:tutorials}
1048  \label{SEC:running_the_example}  \label{SEC:running_the_example}
1049    
1050  \subsubsection{Code Download}  \subsubsection{Code Download}
1051    \label{www:tutorials}
1052    
1053   In order to run the examples you must first download the code distribution.   In order to run the examples you must first download the code distribution.
1054  Instructions for downloading the code can be found in section  Instructions for downloading the code can be found in section
1055  \ref{sect:obtainingCode}.  \ref{sect:obtainingCode}.
1056    
1057  \subsubsection{Experiment Location}  \subsubsection{Experiment Location}
1058    \label{www:tutorials}
1059    
1060   This example experiments is located under the release sub-directory   This example experiments is located under the release sub-directory
1061    
# Line 1045  Instructions for downloading the code ca Line 1063  Instructions for downloading the code ca
1063  {\it verification/exp2/ }  {\it verification/exp2/ }
1064    
1065  \subsubsection{Running the Experiment}  \subsubsection{Running the Experiment}
1066    \label{www:tutorials}
1067    
1068   To run the experiment   To run the experiment
1069    
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