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# Line 2  Line 2 
2  % $Name$  % $Name$
3    
4  \section[Baroclinic Gyre MITgcm Example]{Four Layer Baroclinic Ocean Gyre In Spherical Coordinates}  \section[Baroclinic Gyre MITgcm Example]{Four Layer Baroclinic Ocean Gyre In Spherical Coordinates}
5  \label{www:tutorials}  %\label{www:tutorials}
6  \label{sect:eg-fourlayer}  \label{sec:eg-fourlayer}
7  \begin{rawhtml}  \begin{rawhtml}
8  <!-- CMIREDIR:eg-fourlayer: -->  <!-- CMIREDIR:eg-fourlayer: -->
9  \end{rawhtml}  \end{rawhtml}
# Line 29  polar coordinates.  The files for this e Line 29  polar coordinates.  The files for this e
29  in the verification directory under tutorial\_baroclinic\_gyre.  in the verification directory under tutorial\_baroclinic\_gyre.
30    
31  \subsection{Overview}  \subsection{Overview}
32  \label{www:tutorials}  %\label{www:tutorials}
33    
34  This example experiment demonstrates using the MITgcm to simulate  This example experiment demonstrates using the MITgcm to simulate
35  a baroclinic, wind-forced, ocean gyre circulation. The experiment  a baroclinic, wind-forced, ocean gyre circulation. The experiment
# Line 47  domain is a sector on a sphere and the c Line 47  domain is a sector on a sphere and the c
47  according to latitude, $\varphi$  according to latitude, $\varphi$
48    
49  \begin{equation}  \begin{equation}
50  \label{EQ:eg-fourlayer-fcori}  \label{eq:eg-fourlayer-fcori}
51  f(\varphi) = 2 \Omega \sin( \varphi )  f(\varphi) = 2 \Omega \sin( \varphi )
52  \end{equation}  \end{equation}
53    
# Line 57  f(\varphi) = 2 \Omega \sin( \varphi ) Line 57  f(\varphi) = 2 \Omega \sin( \varphi )
57   The sinusoidal wind-stress variations are defined according to   The sinusoidal wind-stress variations are defined according to
58    
59  \begin{equation}  \begin{equation}
60  \label{EQ:taux}  \label{eq:taux}
61  \tau_{\lambda}(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}})  \tau_{\lambda}(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}})
62  \end{equation}  \end{equation}
63    
# Line 65  f(\varphi) = 2 \Omega \sin( \varphi ) Line 65  f(\varphi) = 2 \Omega \sin( \varphi )
65  $\tau_0$ is set to $0.1N m^{-2}$.  $\tau_0$ is set to $0.1N m^{-2}$.
66  \\  \\
67    
68  Figure \ref{FIG:eg-fourlayer-simulation_config}  Figure \ref{fig:eg-fourlayer-simulation_config}
69  summarizes the configuration simulated.  summarizes the configuration simulated.
70  In contrast to the example in section \ref{sect:eg-baro}, the  In contrast to the example in section \ref{sec:eg-baro}, the
71  current experiment simulates a spherical polar domain. As indicated  current experiment simulates a spherical polar domain. As indicated
72  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
73  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 86  $\theta_{1750}=6^{\circ}$~C. The equatio Line 86  $\theta_{1750}=6^{\circ}$~C. The equatio
86  linear  linear
87    
88  \begin{equation}  \begin{equation}
89  \label{EQ:eg-fourlayer-linear1_eos}  \label{eq:eg-fourlayer-linear1_eos}
90  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )
91  \end{equation}  \end{equation}
92    
93  \noindent which is implemented in the model as a density anomaly equation  \noindent which is implemented in the model as a density anomaly equation
94    
95  \begin{equation}  \begin{equation}
96  \label{EQ:eg-fourlayer-linear1_eos_pert}  \label{eq:eg-fourlayer-linear1_eos_pert}
97  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}
98  \end{equation}  \end{equation}
99    
# Line 122  An initial stratification is Line 122  An initial stratification is
122  imposed by setting the potential temperature, $\theta$, in each layer.  imposed by setting the potential temperature, $\theta$, in each layer.
123  The vertical spacing, $\Delta z$, is constant and equal to $500$m.  The vertical spacing, $\Delta z$, is constant and equal to $500$m.
124  }  }
125  \label{FIG:eg-fourlayer-simulation_config}  \label{fig:eg-fourlayer-simulation_config}
126  \end{figure}  \end{figure}
127    
128  \subsection{Equations solved}  \subsection{Equations solved}
129  \label{www:tutorials}  %\label{www:tutorials}
130  For this problem  For this problem
131  the implicit free surface, {\bf HPE} (see section \ref{sect:hydrostatic_and_quasi-hydrostatic_forms}) form of the  the implicit free surface, {\bf HPE} (see section \ref{sec:hydrostatic_and_quasi-hydrostatic_forms}) form of the
132  equations described in Marshall et. al \cite{marshall:97a} are  equations described in Marshall et. al \cite{marshall:97a} are
133  employed. The flow is three-dimensional with just temperature, $\theta$, as  employed. The flow is three-dimensional with just temperature, $\theta$, as
134  an active tracer.  The equation of state is linear.  an active tracer.  The equation of state is linear.
# Line 137  dissipation and provides a diffusive sub Line 137  dissipation and provides a diffusive sub
137  temperature equation. A wind-stress momentum forcing is added to the momentum  temperature equation. A wind-stress momentum forcing is added to the momentum
138  equation for the zonal flow, $u$. Other terms in the model  equation for the zonal flow, $u$. Other terms in the model
139  are explicitly switched off for this experiment configuration (see section  are explicitly switched off for this experiment configuration (see section
140  \ref{SEC:eg_fourl_code_config} ). This yields an active set of equations  \ref{sec:eg_fourl_code_config} ). This yields an active set of equations
141  solved in this configuration, written in spherical polar coordinates as  solved in this configuration, written in spherical polar coordinates as
142  follows  follows
143    
144  \begin{eqnarray}  \begin{eqnarray}
145  \label{EQ:eg-fourlayer-model_equations}  \label{eq:eg-fourlayer-model_equations}
146  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
147    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -
148    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 181  g\rho_{0} \eta + \int^{0}_{-z}\rho^{\pri Line 181  g\rho_{0} \eta + \int^{0}_{-z}\rho^{\pri
181  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).
182  The terms $H\widehat{u}$ and $H\widehat{v}$ are the components of the vertical  The terms $H\widehat{u}$ and $H\widehat{v}$ are the components of the vertical
183  integral term given in equation \ref{eq:free-surface} and  integral term given in equation \ref{eq:free-surface} and
184  explained in more detail in section \ref{sect:pressure-method-linear-backward}.  explained in more detail in section \ref{sec:pressure-method-linear-backward}.
185  However, for the problem presented here, the continuity relation (equation  However, for the problem presented here, the continuity relation (equation
186  \ref{eq:fourl_example_continuity}) differs from the general form given  \ref{eq:fourl_example_continuity}) differs from the general form given
187  in section \ref{sect:pressure-method-linear-backward},  in section \ref{sec:pressure-method-linear-backward},
188  equation \ref{eq:linear-free-surface=P-E+R}, because the source terms  equation \ref{eq:linear-free-surface=P-E}, because the source terms
189  ${\cal P}-{\cal E}+{\cal R}$  ${\cal P}-{\cal E}+{\cal R}$
190  are all $0$.  are all $0$.
191    
# Line 203  In the momentum equations Line 203  In the momentum equations
203  lateral and vertical boundary conditions for the $\nabla_{h}^{2}$  lateral and vertical boundary conditions for the $\nabla_{h}^{2}$
204  and $\frac{\partial^{2}}{\partial z^{2}}$ operators are specified  and $\frac{\partial^{2}}{\partial z^{2}}$ operators are specified
205  when the numerical simulation is run - see section  when the numerical simulation is run - see section
206  \ref{SEC:eg_fourl_code_config}. For temperature  \ref{sec:eg_fourl_code_config}. For temperature
207  the boundary condition is ``zero-flux''  the boundary condition is ``zero-flux''
208  e.g. $\frac{\partial \theta}{\partial \varphi}=  e.g. $\frac{\partial \theta}{\partial \varphi}=
209  \frac{\partial \theta}{\partial \lambda}=\frac{\partial \theta}{\partial z}=0$.  \frac{\partial \theta}{\partial \lambda}=\frac{\partial \theta}{\partial z}=0$.
# Line 211  e.g. $\frac{\partial \theta}{\partial \v Line 211  e.g. $\frac{\partial \theta}{\partial \v
211    
212    
213  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
214  \label{www:tutorials}  %\label{www:tutorials}
215    
216   The domain is discretised with   The domain is discretised with
217  a uniform grid spacing in latitude and longitude  a uniform grid spacing in latitude and longitude
# Line 231  y=r\varphi,~\Delta y &= &r\Delta \varphi Line 231  y=r\varphi,~\Delta y &= &r\Delta \varphi
231    
232  The procedure for generating a set of internal grid variables from a  The procedure for generating a set of internal grid variables from a
233  spherical polar grid specification is discussed in section  spherical polar grid specification is discussed in section
234  \ref{sect:spatial_discrete_horizontal_grid}.  \ref{sec:spatial_discrete_horizontal_grid}.
235    
236  \noindent\fbox{ \begin{minipage}{5.5in}  \noindent\fbox{ \begin{minipage}{5.5in}
237  {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em  {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em
# Line 252  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, { Line 252  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, {
252    
253    
254    
255  As described in \ref{sect:tracer_equations}, the time evolution of potential  As described in \ref{sec:tracer_equations}, the time evolution of potential
256  temperature,  temperature,
257  $\theta$, (equation \ref{eq:eg_fourl_theta})  $\theta$, (equation \ref{eq:eg_fourl_theta})
258  is evaluated prognostically. The centered second-order scheme with  is evaluated prognostically. The centered second-order scheme with
259  Adams-Bashforth time stepping described in section  Adams-Bashforth time stepping described in section
260  \ref{sect:tracer_equations_abII} is used to step forward the temperature  \ref{sec:tracer_equations_abII} is used to step forward the temperature
261  equation. Prognostic terms in  equation. Prognostic terms in
262  the momentum equations are solved using flux form as  the momentum equations are solved using flux form as
263  described in section \ref{sect:flux-form_momentum_eqautions}.  described in section \ref{sec:flux-form_momentum_equations}.
264  The pressure forces that drive the fluid motions, (  The pressure forces that drive the fluid motions, (
265  $\frac{\partial p^{'}}{\partial \lambda}$ 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
266  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the
267  pressure is diagnosed explicitly by integrating density. The sea-surface  pressure is diagnosed explicitly by integrating density. The sea-surface
268  height, $\eta$, is diagnosed using an implicit scheme. The pressure  height, $\eta$, is diagnosed using an implicit scheme. The pressure
269  field solution method is described in sections  field solution method is described in sections
270  \ref{sect:pressure-method-linear-backward} and  \ref{sec:pressure-method-linear-backward} and
271  \ref{sect:finding_the_pressure_field}.  \ref{sec:finding_the_pressure_field}.
272    
273  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
274  \label{www:tutorials}  %\label{www:tutorials}
275    
276  The Laplacian viscosity coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian viscosity coefficient, $A_{h}$, is set to $400 m s^{-1}$.
277  This value is chosen to yield a Munk layer width,  This value is chosen to yield a Munk layer width,
278    
279  \begin{eqnarray}  \begin{eqnarray}
280  \label{EQ:eg-fourlayer-munk_layer}  \label{eq:eg-fourlayer-munk_layer}
281  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
282  \end{eqnarray}  \end{eqnarray}
283    
# Line 293  time step $\delta t=1200$secs. With this Line 293  time step $\delta t=1200$secs. With this
293  parameter to the horizontal Laplacian friction  parameter to the horizontal Laplacian friction
294    
295  \begin{eqnarray}  \begin{eqnarray}
296  \label{EQ:eg-fourlayer-laplacian_stability}  \label{eq:eg-fourlayer-laplacian_stability}
297  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
298  \end{eqnarray}  \end{eqnarray}
299    
# Line 305  for stability for this term under ABII t Line 305  for stability for this term under ABII t
305  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
306    
307  \begin{eqnarray}  \begin{eqnarray}
308  \label{EQ:eg-fourlayer-laplacian_stability_z}  \label{eq:eg-fourlayer-laplacian_stability_z}
309  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}
310  \end{eqnarray}  \end{eqnarray}
311    
# Line 318  and vertical ($K_{z}$) diffusion coeffic Line 318  and vertical ($K_{z}$) diffusion coeffic
318  \noindent The numerical stability for inertial oscillations  \noindent The numerical stability for inertial oscillations
319    
320  \begin{eqnarray}  \begin{eqnarray}
321  \label{EQ:eg-fourlayer-inertial_stability}  \label{eq:eg-fourlayer-inertial_stability}
322  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
323  \end{eqnarray}  \end{eqnarray}
324    
# Line 331  horizontal flow Line 331  horizontal flow
331  speed of $ | \vec{u} | = 2 ms^{-1}$  speed of $ | \vec{u} | = 2 ms^{-1}$
332    
333  \begin{eqnarray}  \begin{eqnarray}
334  \label{EQ:eg-fourlayer-cfl_stability}  \label{eq:eg-fourlayer-cfl_stability}
335  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
336  \end{eqnarray}  \end{eqnarray}
337    
# Line 343  limit of 0.5. Line 343  limit of 0.5.
343  propagating at $2~{\rm m}~{\rm s}^{-1}$  propagating at $2~{\rm m}~{\rm s}^{-1}$
344    
345  \begin{eqnarray}  \begin{eqnarray}
346  \label{EQ:eg-fourlayer-igw_stability}  \label{eq:eg-fourlayer-igw_stability}
347  S_{c} = \frac{c_{g} \delta t}{ \Delta x}  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
348  \end{eqnarray}  \end{eqnarray}
349    
# Line 351  S_{c} = \frac{c_{g} \delta t}{ \Delta x} Line 351  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
351  stability limit of 0.25.  stability limit of 0.25.
352        
353  \subsection{Code Configuration}  \subsection{Code Configuration}
354  \label{www:tutorials}  %\label{www:tutorials}
355  \label{SEC:eg_fourl_code_config}  \label{sec:eg_fourl_code_config}
356    
357  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
358  directory {\it verification/tutorial\_barotropic\_gyre/}.  directory {\it verification/tutorial\_barotropic\_gyre/}.
# Line 372  experiment. Below we describe the custom Line 372  experiment. Below we describe the custom
372  associated with this experiment.  associated with this experiment.
373    
374  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
375  \label{www:tutorials}  %\label{www:tutorials}
376    
377  This file, reproduced completely below, specifies the main parameters  This file, reproduced completely below, specifies the main parameters
378  for the experiment. The parameters that are significant for this configuration  for the experiment. The parameters that are significant for this configuration
# Line 684  zonalWindFile='windx.sin_y' Line 684  zonalWindFile='windx.sin_y'
684  \begin{rawhtml}</PRE>\end{rawhtml}  \begin{rawhtml}</PRE>\end{rawhtml}
685    
686  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
687  \label{www:tutorials}  %\label{www:tutorials}
688    
689  This file uses standard default values and does not contain  This file uses standard default values and does not contain
690  customisations for this experiment.  customisations for this experiment.
691    
692  \subsubsection{File {\it input/eedata}}  \subsubsection{File {\it input/eedata}}
693  \label{www:tutorials}  %\label{www:tutorials}
694    
695  This file uses standard default values and does not contain  This file uses standard default values and does not contain
696  customisations for this experiment.  customisations for this experiment.
697    
698  \subsubsection{File {\it input/windx.sin\_y}}  \subsubsection{File {\it input/windx.sin\_y}}
699  \label{www:tutorials}  %\label{www:tutorials}
700    
701  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
702  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$
# Line 709  standard code for loading forcing fields Line 709  standard code for loading forcing fields
709    input/windx.sin\_y} file.    input/windx.sin\_y} file.
710    
711  \subsubsection{File {\it input/topog.box}}  \subsubsection{File {\it input/topog.box}}
712  \label{www:tutorials}  %\label{www:tutorials}
713    
714    
715  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 721  The included matlab program {\it input/g Line 721  The included matlab program {\it input/g
721  code for creating the {\it input/topog.box} file.  code for creating the {\it input/topog.box} file.
722    
723  \subsubsection{File {\it code/SIZE.h}}  \subsubsection{File {\it code/SIZE.h}}
724  \label{www:tutorials}  %\label{www:tutorials}
725    
726  Two lines are customized in this file for the current experiment  Two lines are customized in this file for the current experiment
727    
# Line 748  the vertical domain extent in grid point Line 748  the vertical domain extent in grid point
748  \end{small}  \end{small}
749    
750  \subsubsection{File {\it code/CPP\_OPTIONS.h}}  \subsubsection{File {\it code/CPP\_OPTIONS.h}}
751  \label{www:tutorials}  %\label{www:tutorials}
752    
753  This file uses standard default values and does not contain  This file uses standard default values and does not contain
754  customisations for this experiment.  customisations for this experiment.
755    
756    
757  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
758  \label{www:tutorials}  %\label{www:tutorials}
759    
760  This file uses standard default values and does not contain  This file uses standard default values and does not contain
761  customisations for this experiment.  customisations for this experiment.
762    
763  \subsubsection{Other Files }  \subsubsection{Other Files }
764  \label{www:tutorials}  %\label{www:tutorials}
765    
766  Other files relevant to this experiment are  Other files relevant to this experiment are
767  \begin{itemize}  \begin{itemize}
# Line 774  dxF, dyF, dxG, dyG, dxC, dyC}. Line 774  dxF, dyF, dxG, dyG, dxC, dyC}.
774  \end{itemize}  \end{itemize}
775    
776  \subsection{Running The Example}  \subsection{Running The Example}
777  \label{www:tutorials}  %\label{www:tutorials}
778  \label{SEC:running_the_example}  %\label{sec:running_the_example}
779    
780  \subsubsection{Code Download}  \subsubsection{Code Download}
781  \label{www:tutorials}  %\label{www:tutorials}
782    
783   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.
784  Instructions for downloading the code can be found in section  Instructions for downloading the code can be found in section
785  \ref{sect:obtainingCode}.  \ref{sec:obtainingCode}.
786    
787  \subsubsection{Experiment Location}  \subsubsection{Experiment Location}
788  \label{www:tutorials}  %\label{www:tutorials}
789    
790   This example experiments is located under the release sub-directory   This example experiments is located under the release sub-directory
791    
# Line 793  Instructions for downloading the code ca Line 793  Instructions for downloading the code ca
793  {\it verification/exp2/ }  {\it verification/exp2/ }
794    
795  \subsubsection{Running the Experiment}  \subsubsection{Running the Experiment}
796  \label{www:tutorials}  %\label{www:tutorials}
797    
798   To run the experiment   To run the experiment
799    
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