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revision 1.3 by cnh, Wed Oct 24 15:21:27 2001 UTC revision 1.13 by adcroft, Thu May 16 15:54:37 2002 UTC
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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  ilustrated 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
34  is a numerical rendition of the gyre circulation problem simliar  is a numerical rendition of the gyre circulation problem similar
35  to the problems described analytically by Stommel in 1966  to the problems described analytically by Stommel in 1966
36  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.  \cite{Stommel66} and numerically in Holland et. al \cite{Holland75}.
37  \\  \\
# Line 37  to the problems described analytically b Line 39  to the problems described analytically b
39  In this experiment the model is configured to represent a mid-latitude  In this experiment the model is configured to represent a mid-latitude
40  enclosed sector of fluid on a sphere, $60^{\circ} \times 60^{\circ}$ in  enclosed sector of fluid on a sphere, $60^{\circ} \times 60^{\circ}$ in
41  lateral extent. The fluid is $2$~km deep and is forced  lateral extent. The fluid is $2$~km deep and is forced
42  by a constant in time zonal wind stress, $\tau_x$, that varies sinusoidally  by a constant in time zonal wind stress, $\tau_{\lambda}$, that varies
43  in the north-south direction. Topologically the simulated  sinusoidally in the north-south direction. Topologically the simulated
44  domain is a sector on a sphere and the coriolis parameter, $f$, is defined  domain is a sector on a sphere and the coriolis parameter, $f$, is defined
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 54  f(\varphi) = 2 \Omega \sin( \varphi ) Line 56  f(\varphi) = 2 \Omega \sin( \varphi )
56    
57  \begin{equation}  \begin{equation}
58  \label{EQ:taux}  \label{EQ:taux}
59  \tau_x(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}})  \tau_{\lambda}(\varphi) = \tau_{0}\sin(\pi \frac{\varphi}{L_{\varphi}})
60  \end{equation}  \end{equation}
61    
62  \noindent where $L_{\varphi}$ is the lateral domain extent ($60^{\circ}$) and  \noindent where $L_{\varphi}$ is the lateral domain extent ($60^{\circ}$) and
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  summarises 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. However, 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 orthoganal coordinate $(x,y,z)$. For this experiment description  in a locally orthogonal coordinate $(x,y,z)$. For this experiment description
72  of this document the local orthogonal model coordinate $(x,y,z)$ is synonomous  the local orthogonal model coordinate $(x,y,z)$ is synonymous
73  with the spherical polar coordinate shown in figure  with the coordinates $(\lambda,\varphi,r)$ shown in figure
74  \ref{fig:spherical-polar-coord}  \ref{fig:spherical-polar-coord}
75  \\  \\
76    
# 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    
98  \noindent with $\rho_{0}=999.8\,{\rm kg\,m}^{-3}$ and  \noindent with $\rho_{0}=999.8\,{\rm kg\,m}^{-3}$ and
99  $\alpha_{\theta}=2\times10^{-4}\,{\rm degrees}^{-1} $. Integrated forward in  $\alpha_{\theta}=2\times10^{-4}\,{\rm degrees}^{-1} $. Integrated forward in
100  this configuration the model state variable {\bf theta} is synonomous with  this configuration the model state variable {\bf theta} is equivalent to
101  either in-situ temperature, $T$, or potential temperature, $\theta$. For  either in-situ temperature, $T$, or potential temperature, $\theta$. For
102  consistency with later examples, in which the equation of state is  consistency with later examples, in which the equation of state is
103  non-linear, we use $\theta$ to represent temperature here. This is  non-linear, we use $\theta$ to represent temperature here. This is
# Line 110  the quantity that is carried in the mode Line 112  the quantity that is carried in the mode
112  \caption{Schematic of simulation domain and wind-stress forcing function  \caption{Schematic of simulation domain and wind-stress forcing function
113  for the four-layer gyre numerical experiment. The domain is enclosed by solid  for the four-layer gyre numerical experiment. The domain is enclosed by solid
114  walls at $0^{\circ}$~E, $60^{\circ}$~E, $0^{\circ}$~N and $60^{\circ}$~N.  walls at $0^{\circ}$~E, $60^{\circ}$~E, $0^{\circ}$~N and $60^{\circ}$~N.
115  In the four-layer case an initial temperature stratification is  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  The implicit free surface form of the  For this problem
125  pressure equation described in Marshall et. al \cite{Marshall97a} is  the implicit free surface, {\bf HPE} (see section \ref{sect:hydrostatic_and_quasi-hydrostatic_forms}) form of the
126  employed.  equations described in Marshall et. al \cite{marshall:97a} are
127  A horizontal laplacian operator $\nabla_{h}^2$ provides viscous  employed. The flow is three-dimensional with just temperature, $\theta$, as
128  dissipation. The wind-stress momentum input is added to the momentum equation  an active tracer.  The equation of state is linear.
129  for the ``zonal flow'', $u$. Other terms in the model  A horizontal Laplacian operator $\nabla_{h}^2$ provides viscous
130  are explicitly switched off for this experiement configuration (see section  dissipation and provides a diffusive sub-grid scale closure for the
131  \ref{SEC:code_config} ). This yields an active set of equations in  temperature equation. A wind-stress momentum forcing is added to the momentum
132    equation for the zonal flow, $u$. Other terms in the model
133    are explicitly switched off for this experiment configuration (see section
134    \ref{SEC:eg_fourl_code_config} ). This yields an active set of equations
135  solved in this configuration, written in spherical polar coordinates as  solved in this configuration, written in spherical polar coordinates as
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^{'}}{\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}}
143  & = &  & = &
144  \cal{F}  \cal{F}_{\lambda}
145  \\  \\
146  \frac{Dv}{Dt} + fu +  \frac{Dv}{Dt} + fu +
147    \frac{1}{\rho}\frac{\partial p^{'}}{\partial \varphi} -    \frac{1}{\rho}\frac{\partial p^{\prime}}{\partial \varphi} -
148    A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}    A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}
149  & = &  & = &
150  0  0
151  \\  \\
152  \frac{\partial \eta}{\partial t} + \frac{\partial H \hat{u}}{\partial \lambda} +  \frac{\partial \eta}{\partial t} + \frac{\partial H \widehat{u}}{\partial \lambda} +
153  \frac{\partial H \hat{v}}{\partial \varphi}  \frac{\partial H \widehat{v}}{\partial \varphi}
154  &=&  &=&
155  0  0
156    \label{eq:fourl_example_continuity}
157  \\  \\
158  \frac{D\theta}{Dt} -  \frac{D\theta}{Dt} -
159   K_{h}\nabla_{h}^2\theta  - K_{z}\frac{\partial^{2}\theta}{\partial z^{2}}   K_{h}\nabla_{h}^2\theta  - K_{z}\frac{\partial^{2}\theta}{\partial z^{2}}
160  & = &  & = &
161  0  0
162    \label{eq:eg_fourl_theta}
163  \\  \\
164  p^{'} & = &  p^{\prime} & = &
165  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz  g\rho_{0} \eta + \int^{0}_{-z}\rho^{\prime} dz
166  \\  \\
167  \rho^{'} & = &- \alpha_{\theta}\rho_{0}\theta^{'}  \rho^{\prime} & = &- \alpha_{\theta}\rho_{0}\theta^{\prime}
168  \\  \\
169  {\cal F} |_{s} & = & \frac{\tau_{x}}{\rho_{0}\Delta z_{s}}  {\cal F}_{\lambda} |_{s} & = & \frac{\tau_{\lambda}}{\rho_{0}\Delta z_{s}}
170  \\  \\
171  {\cal F} |_{i} & = & 0  {\cal F}_{\lambda} |_{i} & = & 0
172  \end{eqnarray}  \end{eqnarray}
173    
174  \noindent where $u$ and $v$ are the components of the horizontal  \noindent where $u$ and $v$ are the components of the horizontal
175  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).  flow vector $\vec{u}$ on the sphere ($u=\dot{\lambda},v=\dot{\varphi}$).
176  The suffices ${s},{i}$ indicate surface and interior of the domain.  The terms $H\widehat{u}$ and $H\widehat{v}$ are the components of the vertical
177  The forcing $\cal F$ is only applied at the surface.  integral term given in equation \ref{eq:free-surface} and
178  The pressure field $p^{'}$ is separated into a barotropic part  explained in more detail in section \ref{sect:pressure-method-linear-backward}.
179    However, for the problem presented here, the continuity relation (equation
180    \ref{eq:fourl_example_continuity}) differs from the general form given
181    in section \ref{sect:pressure-method-linear-backward},
182    equation \ref{eq:linear-free-surface=P-E+R}, because the source terms
183    ${\cal P}-{\cal E}+{\cal R}$
184    are all $0$.
185    
186    The pressure field, $p^{\prime}$, is separated into a barotropic part
187  due to variations in sea-surface height, $\eta$, and a hydrostatic  due to variations in sea-surface height, $\eta$, and a hydrostatic
188  part due to variations in density, $\rho^{'}$, over the water column.  part due to variations in density, $\rho^{\prime}$, integrated
189    through the water column.
190    
191    The suffices ${s},{i}$ indicate surface layer and the interior of the domain.
192    The windstress forcing, ${\cal F}_{\lambda}$, is applied in the surface layer
193    by a source term in the zonal momentum equation. In the ocean interior
194    this term is zero.
195    
196    In the momentum equations
197    lateral and vertical boundary conditions for the $\nabla_{h}^{2}$
198    and $\frac{\partial^{2}}{\partial z^{2}}$ operators are specified
199    when the numerical simulation is run - see section
200    \ref{SEC:eg_fourl_code_config}. For temperature
201    the boundary condition is ``zero-flux''
202    e.g. $\frac{\partial \theta}{\partial \varphi}=
203    \frac{\partial \theta}{\partial \lambda}=\frac{\partial \theta}{\partial z}=0$.
204    
205    
206    
207  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
208    \label{www:tutorials}
209    
210   The model is configured in hydrostatic form.  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
212   $\Delta \lambda=\Delta \varphi=1^{\circ}$, so   $\Delta \lambda=\Delta \varphi=1^{\circ}$, so
213  that there are sixty grid cells in the zonal and meridional directions.  that there are sixty grid cells in the zonal and meridional directions.
214  Vertically the  Vertically the
215  model is configured with a four layers with constant depth,  model is configured with four layers with constant depth,
216  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate  $\Delta z$, of $500$~m. The internal, locally orthogonal, model coordinate
217  variables $x$ and $y$ are initialised from the values of  variables $x$ and $y$ are initialized from the values of
218  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in  $\lambda$, $\varphi$, $\Delta \lambda$ and $\Delta \varphi$ in
219  radians according to  radians according to
220    
# Line 192  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 213  $\Delta x_v$, $\Delta y_u$: {\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 is solved prognostically.  $\theta$, (equation \ref{eq:eg_fourl_theta})
252    is evaluated prognostically. The centered second-order scheme with
253    Adams-Bashforth time stepping described in section
254    \ref{sect:tracer_equations_abII} is used to step forward the temperature
255    equation. Prognostic terms in
256    the momentum equations are solved using flux form as
257    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.  elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the
261    pressure is diagnosed explicitly by integrating density. The sea-surface
262    height, $\eta$, is diagnosed using an implicit scheme. The pressure
263    field solution method is described in sections
264    \ref{sect:pressure-method-linear-backward} and
265    \ref{sect:finding_the_pressure_field}.
266    
267  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
268    \label{www:tutorials}
269    
270  The laplacian dissipation 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 \cite{Adcroft_thesis},  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    
278  \noindent  of $\approx 100$km. This is greater than the model  \noindent  of $\approx 100$km. This is greater than the model
279  resolution in mid-latitudes $\Delta x$, ensuring that the frictional  resolution in mid-latitudes
280    $\Delta x=r \cos(\varphi) \Delta \lambda \approx 80~{\rm km}$ at
281    $\varphi=45^{\circ}$, ensuring that the frictional
282  boundary layer is well resolved.  boundary layer is well resolved.
283  \\  \\
284    
285  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
286  time step $\delta t=1200$secs. With this time step the stability  time step $\delta t=1200$secs. With this time step the stability
287  parameter to the horizontal laplacian friction \cite{Adcroft_thesis}  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    
294  \noindent evaluates to 0.012, which is well below the 0.3 upper limit  \noindent evaluates to 0.012, which is well below the 0.3 upper limit
295  for stability.  for stability for this term under ABII time-stepping.
296  \\  \\
297    
298  \noindent The vertical dissipation coefficient, $A_{z}$, is set to  \noindent The vertical dissipation coefficient, $A_{z}$, is set to
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 263  and vertical ($K_{z}$) diffusion coeffic Line 310  and vertical ($K_{z}$) diffusion coeffic
310  \\  \\
311    
312  \noindent The numerical stability for inertial oscillations  \noindent The numerical stability for inertial oscillations
 \cite{Adcroft_thesis}  
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 274  S_{i} = f^{2} {\delta t}^2 Line 320  S_{i} = f^{2} {\delta t}^2
320  limit for stability.  limit for stability.
321  \\  \\
322    
323  \noindent The advective CFL \cite{Adcroft_thesis} for a extreme maximum  \noindent The advective CFL for a extreme maximum
324  horizontal flow  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  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  C_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
330  \end{eqnarray}  \end{eqnarray}
331    
332  \noindent evaluates to $5 \times 10^{-2}$. This is well below the stability  \noindent evaluates to $5 \times 10^{-2}$. This is well below the stability
333  limit of 0.5.  limit of 0.5.
334  \\  \\
335    
336  \noindent The stability parameter for internal gravity waves  \noindent The stability parameter for internal gravity waves
337  \cite{Adcroft_thesis}  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    
344  \noindent evaluates to $5 \times 10^{-2}$. This is well below the linear  \noindent evaluates to $\approx 5 \times 10^{-2}$. This is well below the linear
345  stability limit of 0.25.  stability limit of 0.25.
346        
347  \subsection{Code Configuration}  \subsection{Code Configuration}
348  \label{SEC:code_config}  \label{www:tutorials}
349    \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
352  directory {\it verification/exp1/}.  The experiment files  directory {\it verification/exp2/}.  The experiment files
353  \begin{itemize}  \begin{itemize}
354  \item {\it input/data}  \item {\it input/data}
355  \item {\it input/data.pkg}  \item {\it input/data.pkg}
# Line 314  directory {\it verification/exp1/}.  The Line 361  directory {\it verification/exp1/}.  The
361  \item {\it code/SIZE.h}.  \item {\it code/SIZE.h}.
362  \end{itemize}  \end{itemize}
363  contain the code customisations and parameter settings for this  contain the code customisations and parameter settings for this
364  experiements. Below we describe the customisations  experiments. Below we describe the customisations
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 331  this line sets Line 379  this line sets
379  the initial and reference values of potential temperature at each model  the initial and reference values of potential temperature at each model
380  level in units of $^{\circ}$C.  level in units of $^{\circ}$C.
381  The entries are ordered from surface to depth. For each  The entries are ordered from surface to depth. For each
382  depth level the inital and reference profiles will be uniform in  depth level the initial and reference profiles will be uniform in
383  $x$ and $y$. The values specified here are read into the  $x$ and $y$. The values specified here are read into the
384  variable  variable
385  {\bf  {\bf
# Line 377  goto code Line 425  goto code
425    
426  \item Line 6,  \item Line 6,
427  \begin{verbatim} viscAz=1.E-2, \end{verbatim}  \begin{verbatim} viscAz=1.E-2, \end{verbatim}
428  this line sets the vertical laplacian dissipation coefficient to  this line sets the vertical Laplacian dissipation coefficient to
429  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions
430  for this operator are specified later.  for this operator are specified later.
431  The variable  The variable
# Line 397  and is copied into model general vertica Line 445  and is copied into model general vertica
445  \begin{rawhtml} <A href=../../../code_reference/vdb/names/PF.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/names/PF.htm> \end{rawhtml}
446  viscAr  viscAr
447  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
448  }.  }. At each time step, the viscous term contribution to the momentum equations
449    is calculated in routine
450    {\it S/R CALC\_DIFFUSIVITY}.
451    
452  \fbox{  \fbox{
453  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 428  is read in the routine Line 478  is read in the routine
478  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
479  INI\_PARMS  INI\_PARMS
480  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
481  }.  } and applied in routines {\it CALC\_MOM\_RHS} and {\it CALC\_GW}.
482    
483  \fbox{  \fbox{
484  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 471  is read in the routine Line 521  is read in the routine
521  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
522  INI\_PARMS  INI\_PARMS
523  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
524  }.  } and the boundary condition is evaluated in routine
525    {\it S/R CALC\_MOM\_RHS}.
526    
527    
528  \fbox{  \fbox{
# Line 503  is read in the routine Line 554  is read in the routine
554  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
555  INI\_PARMS  INI\_PARMS
556  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
557  }.  } and is applied in the routine {\it S/R CALC\_MOM\_RHS}.
558    
559  \fbox{  \fbox{
560  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 535  is read in the routine Line 586  is read in the routine
586  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
587  INI\_PARMS  INI\_PARMS
588  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
589  }.  } and used in routine {\it S/R CALC\_GT}.
590    
591  \fbox{ \begin{minipage}{5.0in}  \fbox{ \begin{minipage}{5.0in}
592  {\it S/R CALC\_GT}({\it calc\_gt.F})  {\it S/R CALC\_GT}({\it calc\_gt.F})
# Line 571  It is copied into model general vertical Line 622  It is copied into model general vertical
622  \begin{rawhtml} <A href=../../../code_reference/vdb/names/PD.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/names/PD.htm> \end{rawhtml}
623  diffKrT  diffKrT
624  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
625  }.  } which is used in routine {\it S/R CALC\_DIFFUSIVITY}.
626    
627  \fbox{ \begin{minipage}{5.0in}  \fbox{ \begin{minipage}{5.0in}
628  {\it S/R CALC\_DIFFUSIVITY}({\it calc\_diffusivity.F})  {\it S/R CALC\_DIFFUSIVITY}({\it calc\_diffusivity.F})
# Line 602  is read in the routine Line 653  is read in the routine
653  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
654  INI\_PARMS  INI\_PARMS
655  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
656  }.  }. The routine {\it S/R FIND\_RHO} makes use of {\bf tAlpha}.
657    
658  \fbox{  \fbox{
659  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 631  is read in the routine Line 682  is read in the routine
682  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
683  INI\_PARMS  INI\_PARMS
684  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
685  }.  }. The values of {\bf eosType} sets which formula in routine
686    {\it FIND\_RHO} is used to calculate density.
687    
688  \fbox{  \fbox{
689  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 652  usingSphericalPolarGrid=.TRUE., Line 704  usingSphericalPolarGrid=.TRUE.,
704  \end{verbatim}  \end{verbatim}
705  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
706  spherical polar coordinate system. It affects the interpretation of  spherical polar coordinate system. It affects the interpretation of
707  grid inoput parameters, for exampl {\bf delX} and {\bf delY} and  grid input parameters, for example {\bf delX} and {\bf delY} and
708  causes the grid generation routines to initialise an internal grid based  causes the grid generation routines to initialize an internal grid based
709  on spherical polar geometry.  on spherical polar geometry.
710  The variable  The variable
711  {\bf  {\bf
# Line 666  is read in the routine Line 718  is read in the routine
718  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
719  INI\_PARMS  INI\_PARMS
720  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
721  }.  }. When set to {\bf .TRUE.} the settings of {\bf delX} and {\bf delY} are
722    taken to be in degrees. These values are used in the
723    routine {\it INI\_SPEHRICAL\_POLAR\_GRID}.
724    
725  \fbox{  \fbox{
726  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 686  phiMin=0., Line 740  phiMin=0.,
740  This line sets the southern boundary of the modeled  This line sets the southern boundary of the modeled
741  domain to $0^{\circ}$ latitude. This value affects both the  domain to $0^{\circ}$ latitude. This value affects both the
742  generation of the locally orthogonal grid that the model  generation of the locally orthogonal grid that the model
743  uses internally and affects the initialisation of the coriolis force.  uses internally and affects the initialization of the coriolis force.
744  Note - it is not required to set  Note - it is not required to set
745  a longitude boundary, since the absolute longitude does  a longitude boundary, since the absolute longitude does
746  not alter the kernel equation discretisation.  not alter the kernel equation discretisation.
# Line 701  is read in the routine Line 755  is read in the routine
755  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
756  INI\_PARMS  INI\_PARMS
757  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
758  }.  } and is used in routine {\it INI\_SPEHRICAL\_POLAR\_GRID}.
759    
760  \fbox{  \fbox{
761  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 731  is read in the routine Line 785  is read in the routine
785  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
786  INI\_PARMS  INI\_PARMS
787  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
788  }.  } and is used in routine {\it INI\_SPEHRICAL\_POLAR\_GRID}.
789    
790  \fbox{  \fbox{
791  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 761  is read in the routine Line 815  is read in the routine
815  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
816  INI\_PARMS  INI\_PARMS
817  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
818  }.  } and is used in routine {\it INI\_SPEHRICAL\_POLAR\_GRID}.
819    
820  \fbox{  \fbox{
821  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 799  model coordinate variable Line 853  model coordinate variable
853  \begin{rawhtml} <A href=../../../code_reference/vdb/names/10Y.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/names/10Y.htm> \end{rawhtml}
854  delR  delR
855  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
856  }.  } which is used in routine {\it INI\_VERTICAL\_GRID}.
857    
858  \fbox{  \fbox{
859  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 838  is read in the routine Line 892  is read in the routine
892  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
893  INI\_PARMS  INI\_PARMS
894  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
895  }.  }. The bathymetry file is read in the routine {\it INI\_DEPTHS}.
896    
897  \fbox{  \fbox{
898  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 857  goto code Line 911  goto code
911  zonalWindFile='windx.sin_y'  zonalWindFile='windx.sin_y'
912  \end{verbatim}  \end{verbatim}
913  This line specifies the name of the file from which the x-direction  This line specifies the name of the file from which the x-direction
914  surface wind stress is read. This file is also a two-dimensional  (zonal) surface wind stress is read. This file is also a two-dimensional
915  ($x,y$) map and is enumerated and formatted in the same manner as the  ($x,y$) map and is enumerated and formatted in the same manner as the
916  bathymetry file. The matlab program {\it input/gendata.m} includes example  bathymetry file. The matlab program {\it input/gendata.m} includes example
917  code to generate a valid  code to generate a valid
# Line 874  is read in the routine Line 928  is read in the routine
928  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}  \begin{rawhtml} <A href=../../../code_reference/vdb/code/94.htm> \end{rawhtml}
929  INI\_PARMS  INI\_PARMS
930  \begin{rawhtml} </A>\end{rawhtml}  \begin{rawhtml} </A>\end{rawhtml}
931  }.  }.  The wind-stress file is read in the routine
932    {\it EXTERNAL\_FIELDS\_LOAD}.
933    
934  \fbox{  \fbox{
935  \begin{minipage}{5.0in}  \begin{minipage}{5.0in}
# Line 889  goto code Line 944  goto code
944    
945  \end{itemize}  \end{itemize}
946    
947  \noindent other lines in the file {\it input/data} are standard values  \noindent other lines in the file {\it input/data} are standard values.
 that are described in the MITgcm Getting Started and MITgcm Parameters  
 notes.  
948    
949  \begin{rawhtml}<PRE>\end{rawhtml}  \begin{rawhtml}<PRE>\end{rawhtml}
950  \begin{small}  \begin{small}
# Line 900  notes. Line 953  notes.
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 $Nm^{-2}$.  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$ (the
972  Although $\tau_{x}$ is only a function of $y$n in this experiment  default for MITgcm).
973    Although $\tau_{x}$ is only a function of latitude, $y$,
974    in this experiment
975  this file must still define a complete two-dimensional map in order  this file must still define a complete two-dimensional map in order
976  to be compatible with the standard code for loading forcing fields  to be compatible with the standard code for loading forcing fields
977  in MITgcm. The included matlab program {\it input/gendata.m} gives a complete  in MITgcm (routine {\it EXTERNAL\_FIELDS\_LOAD}.
978    The included matlab program {\it input/gendata.m} gives a complete
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 ($x,y$)
986  map of depth values. For this experiment values are either  map of depth values. For this experiment values are either
987  $0m$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep  $0~{\rm m}$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep
988  ocean. The file contains a raw binary stream of data that is enumerated  ocean. The file contains a raw binary stream of data that is enumerated
989  in the same way as standard MITgcm two-dimensional, horizontal arrays.  in the same way as standard MITgcm two-dimensional, horizontal arrays.
990  The included matlab program {\it input/gendata.m} gives a complete  The included matlab program {\it input/gendata.m} gives a complete
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 957  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 980  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 the Getting Started  Instructions for downloading the code can be found in section
1055  Guide \cite{MITgcm_Getting_Started}.  \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    
1062  \vspace{5mm}  \vspace{5mm}
1063  {\it verification/exp1/ }  {\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    
# Line 1012  Guide \cite{MITgcm_Getting_Started}. Line 1080  Guide \cite{MITgcm_Getting_Started}.
1080  % pwd  % pwd
1081  \end{verbatim}  \end{verbatim}
1082    
1083   You shold see a response on the screen ending in   You should see a response on the screen ending in
1084    
1085  {\it verification/exp1/input }  {\it verification/exp2/input }
1086    
1087    
1088  \item Run the genmake script to create the experiment {\it Makefile}  \item Run the genmake script to create the experiment {\it Makefile}
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