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revision 1.8 by cnh, Thu Oct 25 12:06:56 2001 UTC revision 1.12 by cnh, Thu Feb 28 19:32:19 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{sect:eg-fourlayer}
6    
7  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
8    
# Line 19  Line 19 
19  This document describes an example experiment using MITgcm  This document describes an example experiment using MITgcm
20  to simulate a baroclinic ocean gyre in spherical  to simulate a baroclinic ocean gyre in spherical
21  polar coordinates. The barotropic  polar coordinates. The barotropic
22  example experiment in section \ref{sec:eg-baro}  example experiment in section \ref{sect:eg-baro}
23  ilustrated how to configure the code for a single layer  illustrated how to configure the code for a single layer
24  simulation in a cartesian grid. In this example a similar physical problem  simulation in a Cartesian grid. In this example a similar physical problem
25  is simulated, but the code is now configured  is simulated, but the code is now configured
26  for four layers and in a spherical polar coordinate system.  for four layers and in a spherical polar coordinate system.
27    
# Line 29  for four layers and in a spherical polar Line 29  for four layers and in a spherical polar
29    
30  This example experiment demonstrates using the MITgcm to simulate  This example experiment demonstrates using the MITgcm to simulate
31  a baroclinic, wind-forced, ocean gyre circulation. The experiment  a baroclinic, wind-forced, ocean gyre circulation. The experiment
32  is a numerical rendition of the gyre circulation problem simliar  is a numerical rendition of the gyre circulation problem similar
33  to the problems described analytically by Stommel in 1966  to the problems described analytically by Stommel in 1966
34  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.
35  \\  \\
# Line 43  domain is a sector on a sphere and the c Line 43  domain is a sector on a sphere and the c
43  according to latitude, $\varphi$  according to latitude, $\varphi$
44    
45  \begin{equation}  \begin{equation}
46  \label{EQ:fcori}  \label{EQ:eg-fourlayer-fcori}
47  f(\varphi) = 2 \Omega \sin( \varphi )  f(\varphi) = 2 \Omega \sin( \varphi )
48  \end{equation}  \end{equation}
49    
# Line 61  f(\varphi) = 2 \Omega \sin( \varphi ) Line 61  f(\varphi) = 2 \Omega \sin( \varphi )
61  $\tau_0$ is set to $0.1N m^{-2}$.  $\tau_0$ is set to $0.1N m^{-2}$.
62  \\  \\
63    
64  Figure \ref{FIG:simulation_config}  Figure \ref{FIG:eg-fourlayer-simulation_config}
65  summarises the configuration simulated.  summarizes the configuration simulated.
66  In contrast to the example in section \ref{sec:eg-baro}, the  In contrast to the example in section \ref{sect:eg-baro}, the
67  current experiment simulates a spherical polar domain. As indicated  current experiment simulates a spherical polar domain. As indicated
68  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
69  in a locally orthoganal coordinate $(x,y,z)$. For this experiment description  in a locally orthogonal coordinate $(x,y,z)$. For this experiment description
70  the local orthogonal model coordinate $(x,y,z)$ is synonomous  the local orthogonal model coordinate $(x,y,z)$ is synonymous
71  with the coordinates $(\lambda,\varphi,r)$ shown in figure  with the coordinates $(\lambda,\varphi,r)$ shown in figure
72  \ref{fig:spherical-polar-coord}  \ref{fig:spherical-polar-coord}
73  \\  \\
# Line 82  $\theta_{1750}=6^{\circ}$~C. The equatio Line 82  $\theta_{1750}=6^{\circ}$~C. The equatio
82  linear  linear
83    
84  \begin{equation}  \begin{equation}
85  \label{EQ:linear1_eos}  \label{EQ:eg-fourlayer-linear1_eos}
86  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )  \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} )
87  \end{equation}  \end{equation}
88    
89  \noindent which is implemented in the model as a density anomaly equation  \noindent which is implemented in the model as a density anomaly equation
90    
91  \begin{equation}  \begin{equation}
92  \label{EQ:linear1_eos_pert}  \label{EQ:eg-fourlayer-linear1_eos_pert}
93  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}  \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'}
94  \end{equation}  \end{equation}
95    
# Line 114  An initial stratification is Line 114  An initial stratification is
114  imposed by setting the potential temperature, $\theta$, in each layer.  imposed by setting the potential temperature, $\theta$, in each layer.
115  The vertical spacing, $\Delta z$, is constant and equal to $500$m.  The vertical spacing, $\Delta z$, is constant and equal to $500$m.
116  }  }
117  \label{FIG:simulation_config}  \label{FIG:eg-fourlayer-simulation_config}
118  \end{figure}  \end{figure}
119    
120  \subsection{Equations solved}  \subsection{Equations solved}
121  For this problem  For this problem
122  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
123  equations described in Marshall et. al \cite{Marshall97a} are  equations described in Marshall et. al \cite{marshall:97a} are
124  employed. The flow is three-dimensional with just temperature, $\theta$, as  employed. The flow is three-dimensional with just temperature, $\theta$, as
125  an active tracer.  The equation of state is linear.  an active tracer.  The equation of state is linear.
126  A horizontal laplacian operator $\nabla_{h}^2$ provides viscous  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous
127  dissipation and provides a diffusive sub-grid scale closure for the  dissipation and provides a diffusive sub-grid scale closure for the
128  temperature equation. A wind-stress momentum forcing is added to the momentum  temperature equation. A wind-stress momentum forcing is added to the momentum
129  equation for the zonal flow, $u$. Other terms in the model  equation for the zonal flow, $u$. Other terms in the model
130  are explicitly switched off for this experiement configuration (see section  are explicitly switched off for this experiment configuration (see section
131  \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
132  solved in this configuration, written in spherical polar coordinates as  solved in this configuration, written in spherical polar coordinates as
133  follows  follows
134    
135  \begin{eqnarray}  \begin{eqnarray}
136  \label{EQ:model_equations}  \label{EQ:eg-fourlayer-model_equations}
137  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
138    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \lambda} -
139    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 210  that there are sixty grid cells in the z Line 210  that there are sixty grid cells in the z
210  Vertically the  Vertically the
211  model is configured with four layers with constant depth,  model is configured with four layers with constant depth,
212  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate
213  variables $x$ and $y$ are initialised from the values of  variables $x$ and $y$ are initialized from the values of
214  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in
215  radians according to  radians according to
216    
# Line 221  y=r\varphi,~\Delta y &= &r\Delta \varphi Line 221  y=r\varphi,~\Delta y &= &r\Delta \varphi
221    
222  The procedure for generating a set of internal grid variables from a  The procedure for generating a set of internal grid variables from a
223  spherical polar grid specification is discussed in section  spherical polar grid specification is discussed in section
224  \ref{sec:spatial_discrete_horizontal_grid}.  \ref{sect:spatial_discrete_horizontal_grid}.
225    
226  \noindent\fbox{ \begin{minipage}{5.5in}  \noindent\fbox{ \begin{minipage}{5.5in}
227  {\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 242  $\Delta x_v$, $\Delta y_u$: {\bf DXv}, {
242    
243    
244    
245  As described in \ref{sec:tracer_equations}, the time evolution of potential  As described in \ref{sect:tracer_equations}, the time evolution of potential
246  temperature,  temperature,
247  $\theta$, (equation \ref{eq:eg_fourl_theta})  $\theta$, (equation \ref{eq:eg_fourl_theta})
248  is evaluated prognostically. The centered second-order scheme with  is evaluated prognostically. The centered second-order scheme with
249  Adams-Bashforth time stepping described in section  Adams-Bashforth time stepping described in section
250  \ref{sec:tracer_equations_abII} is used to step forward the temperature  \ref{sect:tracer_equations_abII} is used to step forward the temperature
251  equation. Prognostic terms in  equation. Prognostic terms in
252  the momentum equations are solved using flux form as  the momentum equations are solved using flux form as
253  described in section \ref{sec:flux-form_momentum_eqautions}.  described in section \ref{sect:flux-form_momentum_eqautions}.
254  The pressure forces that drive the fluid motions, (  The pressure forces that drive the fluid motions, (
255  $\frac{\partial p^{'}}{\partial \lambda}$ 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
256  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 258  pressure is diagnosed explicitly by inte
258  height, $\eta$, is diagnosed using an implicit scheme. The pressure  height, $\eta$, is diagnosed using an implicit scheme. The pressure
259  field solution method is described in sections  field solution method is described in sections
260  \ref{sect:pressure-method-linear-backward} and  \ref{sect:pressure-method-linear-backward} and
261  \ref{sec:finding_the_pressure_field}.  \ref{sect:finding_the_pressure_field}.
262    
263  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
264    
265  The laplacian viscosity coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian viscosity coefficient, $A_{h}$, is set to $400 m s^{-1}$.
266  This value is chosen to yield a Munk layer width,  This value is chosen to yield a Munk layer width,
267    
268  \begin{eqnarray}  \begin{eqnarray}
269  \label{EQ:munk_layer}  \label{EQ:eg-fourlayer-munk_layer}
270  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
271  \end{eqnarray}  \end{eqnarray}
272    
# Line 279  boundary layer is well resolved. Line 279  boundary layer is well resolved.
279    
280  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
281  time step $\delta t=1200$secs. With this time step the stability  time step $\delta t=1200$secs. With this time step the stability
282  parameter to the horizontal laplacian friction  parameter to the horizontal Laplacian friction
283    
284  \begin{eqnarray}  \begin{eqnarray}
285  \label{EQ:laplacian_stability}  \label{EQ:eg-fourlayer-laplacian_stability}
286  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
287  \end{eqnarray}  \end{eqnarray}
288    
# Line 294  for stability for this term under ABII t Line 294  for stability for this term under ABII t
294  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
295    
296  \begin{eqnarray}  \begin{eqnarray}
297  \label{EQ:laplacian_stability_z}  \label{EQ:eg-fourlayer-laplacian_stability_z}
298  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}
299  \end{eqnarray}  \end{eqnarray}
300    
# Line 307  and vertical ($K_{z}$) diffusion coeffic Line 307  and vertical ($K_{z}$) diffusion coeffic
307  \noindent The numerical stability for inertial oscillations  \noindent The numerical stability for inertial oscillations
308    
309  \begin{eqnarray}  \begin{eqnarray}
310  \label{EQ:inertial_stability}  \label{EQ:eg-fourlayer-inertial_stability}
311  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
312  \end{eqnarray}  \end{eqnarray}
313    
# Line 320  horizontal flow Line 320  horizontal flow
320  speed of $ | \vec{u} | = 2 ms^{-1}$  speed of $ | \vec{u} | = 2 ms^{-1}$
321    
322  \begin{eqnarray}  \begin{eqnarray}
323  \label{EQ:cfl_stability}  \label{EQ:eg-fourlayer-cfl_stability}
324  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
325  \end{eqnarray}  \end{eqnarray}
326    
# Line 329  limit of 0.5. Line 329  limit of 0.5.
329  \\  \\
330    
331  \noindent The stability parameter for internal gravity waves  \noindent The stability parameter for internal gravity waves
332  propogating at $2~{\rm m}~{\rm s}^{-1}$  propagating at $2~{\rm m}~{\rm s}^{-1}$
333    
334  \begin{eqnarray}  \begin{eqnarray}
335  \label{EQ:igw_stability}  \label{EQ:eg-fourlayer-igw_stability}
336  S_{c} = \frac{c_{g} \delta t}{ \Delta x}  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
337  \end{eqnarray}  \end{eqnarray}
338    
# Line 355  directory {\it verification/exp2/}.  The Line 355  directory {\it verification/exp2/}.  The
355  \item {\it code/SIZE.h}.  \item {\it code/SIZE.h}.
356  \end{itemize}  \end{itemize}
357  contain the code customisations and parameter settings for this  contain the code customisations and parameter settings for this
358  experiements. Below we describe the customisations  experiments. Below we describe the customisations
359  to these files associated with this experiment.  to these files associated with this experiment.
360    
361  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
# Line 372  this line sets Line 372  this line sets
372  the initial and reference values of potential temperature at each model  the initial and reference values of potential temperature at each model
373  level in units of $^{\circ}$C.  level in units of $^{\circ}$C.
374  The entries are ordered from surface to depth. For each  The entries are ordered from surface to depth. For each
375  depth level the inital and reference profiles will be uniform in  depth level the initial and reference profiles will be uniform in
376  $x$ and $y$. The values specified here are read into the  $x$ and $y$. The values specified here are read into the
377  variable  variable
378  {\bf  {\bf
# Line 418  goto code Line 418  goto code
418    
419  \item Line 6,  \item Line 6,
420  \begin{verbatim} viscAz=1.E-2, \end{verbatim}  \begin{verbatim} viscAz=1.E-2, \end{verbatim}
421  this line sets the vertical laplacian dissipation coefficient to  this line sets the vertical Laplacian dissipation coefficient to
422  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions
423  for this operator are specified later.  for this operator are specified later.
424  The variable  The variable
# Line 438  and is copied into model general vertica Line 438  and is copied into model general vertica
438  \begin{rawhtml} <A href=../../../code_reference/vdb/names/PF.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/names/PF.htm> \end{rawhtml}
439  viscAr  viscAr
440  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
441  }. At each time step, the viscous term contribution to the momentum eqautions  }. At each time step, the viscous term contribution to the momentum equations
442  is calculated in routine  is calculated in routine
443  {\it S/R CALC\_DIFFUSIVITY}.  {\it S/R CALC\_DIFFUSIVITY}.
444    
# Line 697  usingSphericalPolarGrid=.TRUE., Line 697  usingSphericalPolarGrid=.TRUE.,
697  \end{verbatim}  \end{verbatim}
698  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
699  spherical polar coordinate system. It affects the interpretation of  spherical polar coordinate system. It affects the interpretation of
700  grid inoput parameters, for exampl {\bf delX} and {\bf delY} and  grid input parameters, for example {\bf delX} and {\bf delY} and
701  causes the grid generation routines to initialise an internal grid based  causes the grid generation routines to initialize an internal grid based
702  on spherical polar geometry.  on spherical polar geometry.
703  The variable  The variable
704  {\bf  {\bf
# Line 733  phiMin=0., Line 733  phiMin=0.,
733  This line sets the southern boundary of the modeled  This line sets the southern boundary of the modeled
734  domain to $0^{\circ}$ latitude. This value affects both the  domain to $0^{\circ}$ latitude. This value affects both the
735  generation of the locally orthogonal grid that the model  generation of the locally orthogonal grid that the model
736  uses internally and affects the initialisation of the coriolis force.  uses internally and affects the initialization of the coriolis force.
737  Note - it is not required to set  Note - it is not required to set
738  a longitude boundary, since the absolute longitude does  a longitude boundary, since the absolute longitude does
739  not alter the kernel equation discretisation.  not alter the kernel equation discretisation.
# Line 960  customisations for this experiment. Line 960  customisations for this experiment.
960  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
961  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
962  default for MITgcm).  default for MITgcm).
963  Although $\tau_{x}$ is only a function of latituted, $y$,  Although $\tau_{x}$ is only a function of latitude, $y$,
964  in this experiment  in this experiment
965  this file must still define a complete two-dimensional map in order  this file must still define a complete two-dimensional map in order
966  to be compatible with the standard code for loading forcing fields  to be compatible with the standard code for loading forcing fields
# Line 1061  Instructions for downloading the code ca Line 1061  Instructions for downloading the code ca
1061  % pwd  % pwd
1062  \end{verbatim}  \end{verbatim}
1063    
1064   You shold see a response on the screen ending in   You should see a response on the screen ending in
1065    
1066  {\it verification/exp2/input }  {\it verification/exp2/input }
1067    
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