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add a warning at the beginning of this section: Most of the parameter
 description corresponds to verification/exp2 whereas it's supposed
 to match tutorial_global_oce_latlon !

1 jmc 1.21 % $Header: /u/gcmpack/manual/s_examples/global_oce_latlon/climatalogical_ogcm.tex,v 1.20 2011/04/21 20:05:12 jmc Exp $
2 cnh 1.2 % $Name: $
3 adcroft 1.1
4 jmc 1.17 \section[Global Ocean MITgcm Example]{Global Ocean Simulation at $4^\circ$ Resolution}
5 jmc 1.19 %\label{www:tutorials}
6     \label{sec:eg-global}
7 edhill 1.12 \begin{rawhtml}
8     <!-- CMIREDIR:eg-global: -->
9     \end{rawhtml}
10 jmc 1.16 \begin{center}
11     (in directory: {\it verification/tutorial\_global\_oce\_latlon/})
12     \end{center}
13 adcroft 1.1
14     \bodytext{bgcolor="#FFFFFFFF"}
15    
16 jmc 1.21 \noindent {\bf WARNING: the description of this experiment is not up-to-date.
17     In particular, most of the parameters description corresponds to an older
18     version of {\it verification/exp2} instead of the current tutorial}\\
19    
20 adcroft 1.1 %\begin{center}
21 cnh 1.3 %{\Large \bf Using MITgcm to Simulate Global Climatological Ocean Circulation
22 adcroft 1.1 %At Four Degree Resolution with Asynchronous Time Stepping}
23     %
24     %\vspace*{4mm}
25     %
26     %\vspace*{3mm}
27     %{\large May 2001}
28     %\end{center}
29    
30    
31     This example experiment demonstrates using the MITgcm to simulate
32     the planetary ocean circulation. The simulation is configured
33     with realistic geography and bathymetry on a
34     $4^{\circ} \times 4^{\circ}$ spherical polar grid.
35 molod 1.14 The files for this experiment are in the verification directory
36     under tutorial\_global\_oce\_latlon.
37 adcroft 1.1 Twenty levels are used in the vertical, ranging in thickness
38     from $50\,{\rm m}$ at the surface to $815\,{\rm m}$ at depth,
39     giving a maximum model depth of $6\,{\rm km}$.
40     At this resolution, the configuration
41     can be integrated forward for thousands of years on a single
42     processor desktop computer.
43     \\
44 cnh 1.8 \subsection{Overview}
45 jmc 1.19 %\label{www:tutorials}
46 adcroft 1.1
47 cnh 1.3 The model is forced with climatological wind stress data and surface
48     flux data from DaSilva \cite{DaSilva94}. Climatological data
49     from Levitus \cite{Levitus94} is used to initialize the model hydrography.
50     Levitus seasonal climatology data is also used throughout the calculation
51 adcroft 1.1 to provide additional air-sea fluxes.
52 cnh 1.3 These fluxes are combined with the DaSilva climatological estimates of
53 adcroft 1.1 surface heat flux and fresh water, resulting in a mixed boundary
54 cnh 1.3 condition of the style described in Haney \cite{Haney}.
55 adcroft 1.1 Altogether, this yields the following forcing applied
56     in the model surface layer.
57    
58     \begin{eqnarray}
59 jmc 1.19 \label{eq:eg-global-global_forcing}
60     \label{eq:eg-global-global_forcing_fu}
61 adcroft 1.1 {\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}}
62     \\
63 jmc 1.19 \label{eq:eg-global-global_forcing_fv}
64 adcroft 1.1 {\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}}
65     \\
66 jmc 1.19 \label{eq:eg-global-global_forcing_ft}
67 adcroft 1.1 {\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} )
68     - \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q}
69     \\
70 jmc 1.19 \label{eq:eg-global-global_forcing_fs}
71 adcroft 1.1 {\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} )
72     + \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R})
73     \end{eqnarray}
74    
75     \noindent where ${\cal F}_{u}$, ${\cal F}_{v}$, ${\cal F}_{\theta}$,
76     ${\cal F}_{s}$ are the forcing terms in the zonal and meridional
77     momentum and in the potential temperature and salinity
78     equations respectively.
79     The term $\Delta z_{s}$ represents the top ocean layer thickness in
80     meters.
81     It is used in conjunction with a reference density, $\rho_{0}$
82     (here set to $999.8\,{\rm kg\,m^{-3}}$), a
83     reference salinity, $S_{0}$ (here set to 35~ppt),
84     and a specific heat capacity, $C_{p}$ (here set to
85     $4000~{\rm J}~^{\circ}{\rm C}^{-1}~{\rm kg}^{-1}$), to convert
86     input dataset values into time tendencies of
87     potential temperature (with units of $^{\circ}{\rm C}~{\rm s}^{-1}$),
88     salinity (with units ${\rm ppt}~s^{-1}$) and
89     velocity (with units ${\rm m}~{\rm s}^{-2}$).
90     The externally supplied forcing fields used in this
91     experiment are $\tau_{x}$, $\tau_{y}$, $\theta^{\ast}$, $S^{\ast}$,
92     $\cal{Q}$ and $\cal{E}-\cal{P}-\cal{R}$. The wind stress fields ($\tau_x$, $\tau_y$)
93     have units of ${\rm N}~{\rm m}^{-2}$. The temperature forcing fields
94     ($\theta^{\ast}$ and $Q$) have units of $^{\circ}{\rm C}$ and ${\rm W}~{\rm m}^{-2}$
95     respectively. The salinity forcing fields ($S^{\ast}$ and
96     $\cal{E}-\cal{P}-\cal{R}$) have units of ${\rm ppt}$ and ${\rm m}~{\rm s}^{-1}$
97 cnh 1.8 respectively. The source files and procedures for ingesting this data into the
98     simulation are described in the experiment configuration discussion in section
99 jmc 1.19 \ref{sec:eg-global-clim_ocn_examp_exp_config}.
100 adcroft 1.1
101    
102     \subsection{Discrete Numerical Configuration}
103 jmc 1.19 %\label{www:tutorials}
104 adcroft 1.1
105    
106     The model is configured in hydrostatic form. The domain is discretised with
107     a uniform grid spacing in latitude and longitude on the sphere
108     $\Delta \phi=\Delta \lambda=4^{\circ}$, so
109     that there are ninety grid cells in the zonal and forty in the
110     meridional direction. The internal model coordinate variables
111 cnh 1.3 $x$ and $y$ are initialized according to
112 adcroft 1.1 \begin{eqnarray}
113     x=r\cos(\phi),~\Delta x & = &r\cos(\Delta \phi) \\
114 cnh 1.8 y=r\lambda,~\Delta y &= &r\Delta \lambda
115 adcroft 1.1 \end{eqnarray}
116    
117     Arctic polar regions are not
118     included in this experiment. Meridionally the model extends from
119     $80^{\circ}{\rm S}$ to $80^{\circ}{\rm N}$.
120     Vertically the model is configured with twenty layers with the
121     following thicknesses
122     $\Delta z_{1} = 50\,{\rm m},\,
123     \Delta z_{2} = 50\,{\rm m},\,
124     \Delta z_{3} = 55\,{\rm m},\,
125     \Delta z_{4} = 60\,{\rm m},\,
126     \Delta z_{5} = 65\,{\rm m},\,
127     $
128     $
129     \Delta z_{6}~=~70\,{\rm m},\,
130     \Delta z_{7}~=~80\,{\rm m},\,
131     \Delta z_{8}~=95\,{\rm m},\,
132     \Delta z_{9}=120\,{\rm m},\,
133     \Delta z_{10}=155\,{\rm m},\,
134     $
135     $
136     \Delta z_{11}=200\,{\rm m},\,
137     \Delta z_{12}=260\,{\rm m},\,
138     \Delta z_{13}=320\,{\rm m},\,
139     \Delta z_{14}=400\,{\rm m},\,
140     \Delta z_{15}=480\,{\rm m},\,
141     $
142     $
143     \Delta z_{16}=570\,{\rm m},\,
144     \Delta z_{17}=655\,{\rm m},\,
145     \Delta z_{18}=725\,{\rm m},\,
146     \Delta z_{19}=775\,{\rm m},\,
147     \Delta z_{20}=815\,{\rm m}
148 cnh 1.8 $ (here the numeric subscript indicates the model level index number, ${\tt k}$) to
149     give a total depth, $H$, of $-5450{\rm m}$.
150 adcroft 1.1 The implicit free surface form of the pressure equation described in Marshall et. al
151 adcroft 1.6 \cite{marshall:97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous
152 cnh 1.3 dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.
153 adcroft 1.1
154 jmc 1.19 Wind-stress forcing is added to the momentum equations in (\ref{eq:eg-global-model_equations})
155 cnh 1.8 for both the zonal flow, $u$ and the meridional flow $v$, according to equations
156 jmc 1.19 (\ref{eq:eg-global-global_forcing_fu}) and (\ref{eq:eg-global-global_forcing_fv}).
157 cnh 1.8 Thermodynamic forcing inputs are added to the equations
158 jmc 1.19 in (\ref{eq:eg-global-model_equations}) for
159 adcroft 1.1 potential temperature, $\theta$, and salinity, $S$, according to equations
160 jmc 1.19 (\ref{eq:eg-global-global_forcing_ft}) and (\ref{eq:eg-global-global_forcing_fs}).
161 adcroft 1.1 This produces a set of equations solved in this configuration as follows:
162    
163     \begin{eqnarray}
164 jmc 1.19 \label{eq:eg-global-model_equations}
165 adcroft 1.1 \frac{Du}{Dt} - fv +
166     \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -
167     \nabla_{h}\cdot A_{h}\nabla_{h}u -
168     \frac{\partial}{\partial z}A_{z}\frac{\partial u}{\partial z}
169     & = &
170     \begin{cases}
171     {\cal F}_u & \text{(surface)} \\
172     0 & \text{(interior)}
173     \end{cases}
174     \\
175     \frac{Dv}{Dt} + fu +
176     \frac{1}{\rho}\frac{\partial p^{'}}{\partial y} -
177     \nabla_{h}\cdot A_{h}\nabla_{h}v -
178     \frac{\partial}{\partial z}A_{z}\frac{\partial v}{\partial z}
179     & = &
180     \begin{cases}
181     {\cal F}_v & \text{(surface)} \\
182     0 & \text{(interior)}
183     \end{cases}
184     \\
185     \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}
186     &=&
187     0
188     \\
189     \frac{D\theta}{Dt} -
190     \nabla_{h}\cdot K_{h}\nabla_{h}\theta
191     - \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial\theta}{\partial z}
192     & = &
193     \begin{cases}
194     {\cal F}_\theta & \text{(surface)} \\
195     0 & \text{(interior)}
196     \end{cases}
197     \\
198     \frac{D s}{Dt} -
199     \nabla_{h}\cdot K_{h}\nabla_{h}s
200     - \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial s}{\partial z}
201     & = &
202     \begin{cases}
203     {\cal F}_s & \text{(surface)} \\
204     0 & \text{(interior)}
205     \end{cases}
206     \\
207     g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz & = & p^{'}
208     \end{eqnarray}
209    
210     \noindent where $u=\frac{Dx}{Dt}=r \cos(\phi)\frac{D \lambda}{Dt}$ and
211     $v=\frac{Dy}{Dt}=r \frac{D \phi}{Dt}$
212     are the zonal and meridional components of the
213     flow vector, $\vec{u}$, on the sphere. As described in
214 adcroft 1.5 MITgcm Numerical Solution Procedure \ref{chap:discretization}, the time
215 adcroft 1.1 evolution of potential temperature, $\theta$, equation is solved prognostically.
216     The total pressure, $p$, is diagnosed by summing pressure due to surface
217     elevation $\eta$ and the hydrostatic pressure.
218     \\
219    
220     \subsubsection{Numerical Stability Criteria}
221 jmc 1.19 %\label{www:tutorials}
222 adcroft 1.1
223 cnh 1.3 The Laplacian dissipation coefficient, $A_{h}$, is set to $5 \times 10^5 m s^{-1}$.
224 adcroft 1.4 This value is chosen to yield a Munk layer width \cite{adcroft:95},
225 adcroft 1.1 \begin{eqnarray}
226 jmc 1.19 \label{eq:eg-global-munk_layer}
227 adcroft 1.10 && M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
228 adcroft 1.1 \end{eqnarray}
229    
230     \noindent of $\approx 600$km. This is greater than the model
231     resolution in low-latitudes, $\Delta x \approx 400{\rm km}$, ensuring that the frictional
232     boundary layer is adequately resolved.
233     \\
234    
235     \noindent The model is stepped forward with a
236     time step $\delta t_{\theta}=30~{\rm hours}$ for thermodynamic variables and
237     $\delta t_{v}=40~{\rm minutes}$ for momentum terms. With this time step, the stability
238 adcroft 1.4 parameter to the horizontal Laplacian friction \cite{adcroft:95}
239 adcroft 1.1 \begin{eqnarray}
240 jmc 1.19 \label{eq:eg-global-laplacian_stability}
241 adcroft 1.10 && S_{l} = 4 \frac{A_{h} \delta t_{v}}{{\Delta x}^2}
242 adcroft 1.1 \end{eqnarray}
243    
244     \noindent evaluates to 0.16 at a latitude of $\phi=80^{\circ}$, which is below the
245     0.3 upper limit for stability. The zonal grid spacing $\Delta x$ is smallest at
246     $\phi=80^{\circ}$ where $\Delta x=r\cos(\phi)\Delta \phi\approx 77{\rm km}$.
247     \\
248    
249     \noindent The vertical dissipation coefficient, $A_{z}$, is set to
250     $1\times10^{-3} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
251     \begin{eqnarray}
252 jmc 1.19 \label{eq:eg-global-laplacian_stability_z}
253 adcroft 1.1 S_{l} = 4 \frac{A_{z} \delta t_{v}}{{\Delta z}^2}
254     \end{eqnarray}
255    
256     \noindent evaluates to $0.015$ for the smallest model
257 cnh 1.3 level spacing ($\Delta z_{1}=50{\rm m}$) which is again well below
258 adcroft 1.1 the upper stability limit.
259     \\
260    
261     The values of the horizontal ($K_{h}$) and vertical ($K_{z}$) diffusion coefficients
262     for both temperature and salinity are set to $1 \times 10^{3}~{\rm m}^{2}{\rm s}^{-1}$
263     and $3 \times 10^{-5}~{\rm m}^{2}{\rm s}^{-1}$ respectively. The stability limit
264     related to $K_{h}$ will be at $\phi=80^{\circ}$ where $\Delta x \approx 77 {\rm km}$.
265     Here the stability parameter
266     \begin{eqnarray}
267 jmc 1.19 \label{eq:eg-global-laplacian_stability_xtheta}
268 adcroft 1.1 S_{l} = \frac{4 K_{h} \delta t_{\theta}}{{\Delta x}^2}
269     \end{eqnarray}
270 cnh 1.3 evaluates to $0.07$, well below the stability limit of $S_{l} \approx 0.5$. The
271 adcroft 1.1 stability parameter related to $K_{z}$
272     \begin{eqnarray}
273 jmc 1.19 \label{eq:eg-global-laplacian_stability_ztheta}
274 adcroft 1.1 S_{l} = \frac{4 K_{z} \delta t_{\theta}}{{\Delta z}^2}
275     \end{eqnarray}
276     evaluates to $0.005$ for $\min(\Delta z)=50{\rm m}$, well below the stability limit
277     of $S_{l} \approx 0.5$.
278     \\
279    
280     \noindent The numerical stability for inertial oscillations
281 adcroft 1.4 \cite{adcroft:95}
282 adcroft 1.1
283     \begin{eqnarray}
284 jmc 1.19 \label{eq:eg-global-inertial_stability}
285 adcroft 1.1 S_{i} = f^{2} {\delta t_v}^2
286     \end{eqnarray}
287    
288     \noindent evaluates to $0.24$ for $f=2\omega\sin(80^{\circ})=1.43\times10^{-4}~{\rm s}^{-1}$, which is close to
289     the $S_{i} < 1$ upper limit for stability.
290     \\
291    
292 adcroft 1.4 \noindent The advective CFL \cite{adcroft:95} for a extreme maximum
293 adcroft 1.1 horizontal flow
294     speed of $ | \vec{u} | = 2 ms^{-1}$
295    
296     \begin{eqnarray}
297 jmc 1.19 \label{eq:eg-global-cfl_stability}
298 adcroft 1.1 S_{a} = \frac{| \vec{u} | \delta t_{v}}{ \Delta x}
299     \end{eqnarray}
300    
301     \noindent evaluates to $6 \times 10^{-2}$. This is well below the stability
302     limit of 0.5.
303     \\
304    
305 cnh 1.3 \noindent The stability parameter for internal gravity waves propagating
306 adcroft 1.1 with a maximum speed of $c_{g}=10~{\rm ms}^{-1}$
307 adcroft 1.4 \cite{adcroft:95}
308 adcroft 1.1
309     \begin{eqnarray}
310 jmc 1.19 \label{eq:eg-global-gfl_stability}
311 adcroft 1.1 S_{c} = \frac{c_{g} \delta t_{v}}{ \Delta x}
312     \end{eqnarray}
313    
314     \noindent evaluates to $3 \times 10^{-1}$. This is close to the linear
315     stability limit of 0.5.
316    
317     \subsection{Experiment Configuration}
318 jmc 1.19 %\label{www:tutorials}
319     \label{sec:eg-global-clim_ocn_examp_exp_config}
320 adcroft 1.1
321     The model configuration for this experiment resides under the
322 cnh 1.8 directory {\it tutorial\_examples/global\_ocean\_circulation/}.
323     The experiment files
324    
325 adcroft 1.1 \begin{itemize}
326     \item {\it input/data}
327     \item {\it input/data.pkg}
328     \item {\it input/eedata},
329     \item {\it input/windx.bin},
330     \item {\it input/windy.bin},
331     \item {\it input/salt.bin},
332     \item {\it input/theta.bin},
333     \item {\it input/SSS.bin},
334     \item {\it input/SST.bin},
335     \item {\it input/topog.bin},
336     \item {\it code/CPP\_EEOPTIONS.h}
337     \item {\it code/CPP\_OPTIONS.h},
338     \item {\it code/SIZE.h}.
339     \end{itemize}
340 cnh 1.3 contain the code customizations and parameter settings for these
341     experiments. Below we describe the customizations
342 adcroft 1.1 to these files associated with this experiment.
343 cnh 1.8
344     \subsubsection{Driving Datasets}
345 jmc 1.19 %\label{www:tutorials}
346 cnh 1.8
347 jmc 1.19 Figures ({\it --- missing figures ---})
348     %(\ref{fig:sim_config_tclim}-\ref{fig:sim_config_empmr})
349     show the relaxation temperature ($\theta^{\ast}$) and salinity ($S^{\ast}$)
350     fields, the wind stress components ($\tau_x$ and $\tau_y$), the heat flux ($Q$)
351 cnh 1.8 and the net fresh water flux (${\cal E} - {\cal P} - {\cal R}$) used
352 jmc 1.19 in equations
353     (\ref{eq:eg-global-global_forcing_fu}-\ref{eq:eg-global-global_forcing_fs}).
354     The figures also indicate the lateral extent and coastline used in the
355     experiment. Figure ({\it --- missing figure --- }) %ref{fig:model_bathymetry})
356     shows the depth contours of the model domain.
357 adcroft 1.1
358     \subsubsection{File {\it input/data}}
359 jmc 1.19 %\label{www:tutorials}
360 adcroft 1.1
361 jmc 1.20 \input{s_examples/global_oce_latlon/inp_data}
362 adcroft 1.1
363     \subsubsection{File {\it input/data.pkg}}
364 jmc 1.19 %\label{www:tutorials}
365 adcroft 1.1
366     This file uses standard default values and does not contain
367     customisations for this experiment.
368    
369     \subsubsection{File {\it input/eedata}}
370 jmc 1.19 %\label{www:tutorials}
371 adcroft 1.1
372     This file uses standard default values and does not contain
373     customisations for this experiment.
374    
375     \subsubsection{File {\it input/windx.sin\_y}}
376 jmc 1.19 %\label{www:tutorials}
377 adcroft 1.1
378     The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
379     map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.
380     Although $\tau_{x}$ is only a function of $y$n in this experiment
381     this file must still define a complete two-dimensional map in order
382     to be compatible with the standard code for loading forcing fields
383     in MITgcm. The included matlab program {\it input/gendata.m} gives a complete
384     code for creating the {\it input/windx.sin\_y} file.
385    
386     \subsubsection{File {\it input/topog.box}}
387 jmc 1.19 %\label{www:tutorials}
388 adcroft 1.1
389    
390     The {\it input/topog.box} file specifies a two-dimensional ($x,y$)
391     map of depth values. For this experiment values are either
392     $0m$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep
393     ocean. The file contains a raw binary stream of data that is enumerated
394     in the same way as standard MITgcm two-dimensional, horizontal arrays.
395     The included matlab program {\it input/gendata.m} gives a complete
396     code for creating the {\it input/topog.box} file.
397    
398     \subsubsection{File {\it code/SIZE.h}}
399 jmc 1.19 %\label{www:tutorials}
400 adcroft 1.1
401     Two lines are customized in this file for the current experiment
402    
403     \begin{itemize}
404    
405     \item Line 39,
406     \begin{verbatim} sNx=60, \end{verbatim} this line sets
407     the lateral domain extent in grid points for the
408     axis aligned with the x-coordinate.
409    
410     \item Line 40,
411     \begin{verbatim} sNy=60, \end{verbatim} this line sets
412     the lateral domain extent in grid points for the
413     axis aligned with the y-coordinate.
414    
415     \item Line 49,
416     \begin{verbatim} Nr=4, \end{verbatim} this line sets
417     the vertical domain extent in grid points.
418    
419     \end{itemize}
420    
421     \begin{small}
422 jmc 1.18 \input{s_examples/global_oce_latlon/code/SIZE.h}
423 adcroft 1.1 \end{small}
424    
425     \subsubsection{File {\it code/CPP\_OPTIONS.h}}
426 jmc 1.19 %\label{www:tutorials}
427 adcroft 1.1
428     This file uses standard default values and does not contain
429     customisations for this experiment.
430    
431    
432     \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
433 jmc 1.19 %\label{www:tutorials}
434 adcroft 1.1
435     This file uses standard default values and does not contain
436     customisations for this experiment.
437    
438     \subsubsection{Other Files }
439 jmc 1.19 %\label{www:tutorials}
440 adcroft 1.1
441     Other files relevant to this experiment are
442     \begin{itemize}
443     \item {\it model/src/ini\_cori.F}. This file initializes the model
444     coriolis variables {\bf fCorU}.
445     \item {\it model/src/ini\_spherical\_polar\_grid.F}
446     \item {\it model/src/ini\_parms.F},
447     \item {\it input/windx.sin\_y},
448     \end{itemize}
449     contain the code customisations and parameter settings for this
450 cnh 1.3 experiments. Below we describe the customisations
451 adcroft 1.1 to these files associated with this experiment.

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