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1 jmc 1.5 % $Header: /u/gcmpack/manual/part3/case_studies/held_suarez_cs/held_suarez_cs.tex,v 1.4 2005/08/09 18:46:39 jmc Exp $
2 jmc 1.1 % $Name: $
3    
4     \section[Held-Suarez Atmosphere MITgcm Example]{Held-Suarez atmospheric simulation
5     on cube-sphere grid with 32 square cube faces.}
6     \label{www:tutorials}
7     \label{sect:eg-hs}
8     \begin{rawhtml}
9     <!-- CMIREDIR:eg-hs: -->
10     \end{rawhtml}
11    
12     \bodytext{bgcolor="#FFFFFFFF"}
13    
14     %\begin{center}
15     %{\Large \bf Using MITgcm to Simulate Global Climatological Ocean Circulation
16     %At Four Degree Resolution with Asynchronous Time Stepping}
17     %
18     %\vspace*{4mm}
19     %
20     %\vspace*{3mm}
21     %{\large May 2001}
22     %\end{center}
23    
24     This example illustrates the use of the MITgcm as an Atmospheric GCM,
25 jmc 1.2 using simple \cite{held-suar:94} forcing
26 jmc 1.1 to simulate Atmospheric Dynamics on global scale.
27     The set-up use the rescaled pressure coordinate ($p^*$)\cite[]{adcroft:04a}
28     in the vertical direction, with 20 equaly-spaced levels, and
29     the conformal cube-sphere grid (C32) \cite[]{adcroft:04b}.
30    
31     \subsection{Overview}
32     \label{www:tutorials}
33    
34     This example demonstrates using the MITgcm to simulate
35     the planetary atmospheric circulation, with flat orography
36     and simplified forcing.
37     In particular, only dry air processes are considered and
38     radiation effects are represented by a simple newtownien cooling,
39 jmc 1.4 Thus this example does not rely on any particular atmospheric
40 jmc 1.1 physics package.
41     This kind of simplified atmospheric simulation has been widely
42     used in GFD-type experiments and in intercomparison projects of
43     AGCM dynamical cores \cite[]{held-suar:94}.
44    
45     The horizontal grid is obtain from the projection of a uniform gridded cube
46     to the sphere. Each of the 6 faces has the same resolution, with
47     $32 \times 32$ grid points. The equator line coincide with a grid line
48     and crosses, right in the midle, 4 of the 6 faces, leaving 2 faces
49     for the Northern and Southern polar regions.
50     This curvilinear grid requires the use of the 2nd generation exchange
51     topology ({\it pkg/exch2}) to connect tile and face edges,
52     but without any limitation on the number of processors.
53    
54     The use of the $p^*$ coordinate with 20 equally spaced levels
55     ($20 \times 50\,{\rm mb}$, from $p^*=1000,{\rm mb}$ to $0$ at the
56 jmc 1.2 top of the atmosphere) follows the choice of \cite{held-suar:94}.
57     Note that without topography, the $p^*$ coordinate and the normalized
58     pressure coordinate ($\sigma_p$) coincide exactly.
59     No viscosity and zero diffusion are used here, but
60     a $8^th$ order \cite{Shapiro_70} filter is applied to both momentum and
61     potential temperature, to remove selectively grid scale noise.
62     Apart from the horizontal grid, this experiment is made very similar to
63     the grid-point model case used in \cite{held-suar:94} study.
64 jmc 1.1
65     At this resolution, the configuration can be integrated forward
66     for many years on a single processor desktop computer.
67     \\
68    
69 jmc 1.2 \subsection{Forcing}
70     \label{www:tutorials}
71    
72 jmc 1.1 The model is forced by relaxation to a radiative equilibrium temperature from
73     \cite{held-suar:94}.
74     A linear frictional drag (Rayleigh damping) is applied in the lower
75     part of the atmosphere and account from surface friction and momentum
76     dissipation in the boundary layer.
77     Altogether, this yields the following forcing
78     \cite[from][]{held-suar:94} that is applied to the fluid:
79    
80     \begin{eqnarray}
81     \label{EQ:eg-hs-global_forcing}
82 jmc 1.2 \label{EQ:eg-hs-global_forcing_fv}
83 jmc 1.4 \vec{{\cal F}_\mathbf{v}} & = & -k_\mathbf{v}(p)\vec{\mathbf{v}}_h
84 jmc 1.1 \\
85     \label{EQ:eg-hs-global_forcing_ft}
86 jmc 1.4 {\cal F}_{\theta} & = & -k_{\theta}(\varphi,p)[\theta-\theta_{eq}(\varphi,p)]
87 jmc 1.1 \end{eqnarray}
88    
89 jmc 1.4 \noindent where $\vec{\cal F}_\mathbf{v}$, ${\cal F}_{\theta}$,
90 jmc 1.1 are the forcing terms in the zonal and meridional
91     momentum and in the potential temperature equations respectively.
92 jmc 1.4 The term $k_\mathbf{v}$ in equation (\ref{EQ:eg-hs-global_forcing_fv}) applies a
93 jmc 1.2 Rayleigh damping that is active within the planetary boundary layer.
94     It is defined so as to decay as pressure decreases according to
95     \begin{eqnarray*}
96 jmc 1.1 \label{EQ:eg-hs-define_kv}
97 jmc 1.4 k_\mathbf{v} & = & k_{f}~\max[0,~(p^*/P^{0}_{s}-\sigma_{b})/(1-\sigma_{b})]
98 jmc 1.1 \\
99 jmc 1.2 \sigma_{b} & = & 0.7 ~~{\rm and}~~
100     k_{f} = 1/86400 ~{\rm s}^{-1}
101     \end{eqnarray*}
102 jmc 1.1
103     where $p^*$ is the pressure level of the cell center
104 jmc 1.2 and $P^{0}_{s}$ is the pressure at the base of the atmospheric column,
105 jmc 1.4 which is constant and uniform here ($= 10^5 {\rm Pa}$), in the absence
106 jmc 1.2 of topography.
107 jmc 1.1
108     The Equilibrium temperature $\theta_{eq}$ and relaxation time scale $k_{\theta}$
109     are set to:
110     \begin{eqnarray}
111 jmc 1.2 \label{EQ:eg-hs-define_Teq}
112 jmc 1.4 \theta_{eq}(\varphi,p^*) & = & \max \{ 200.K (P^{0}_{s}/p^*)^\kappa,\\
113 jmc 1.2 \nonumber
114 jmc 1.4 & & \hspace{8mm} 315.K - \Delta T_y~\sin^2(\varphi)
115     - \Delta \theta_z \cos^2(\varphi) \log(p^*/P^{0}_s) \}
116 jmc 1.2 \\
117 jmc 1.1 \label{EQ:eg-hs-define_kT}
118 jmc 1.4 k_{\theta}(\varphi,p^*) & = &
119     k_a + (k_s -k_a)~\cos^4(\varphi)~\max[0,(p^*/P^{0}_{s}-\sigma_{b})/(1-\sigma_{b})]
120 jmc 1.1 \end{eqnarray}
121 jmc 1.2 with:
122     \begin{eqnarray*}
123     \Delta T_y = 60.K & k_a = 1/(40 \cdot 86400) ~{\rm s}^{-1}\\
124     \Delta \theta_z = 10.K & k_s = 1/(4 \cdot 86400) ~{\rm s}^{-1}
125     \end{eqnarray*}
126 jmc 1.1
127     Initial conditions correspond to a resting state with horizontally uniform
128     stratified fluid. The initial temperature profile is simply the
129     horizontally average of the radiative equilibrium temperature.
130    
131 jmc 1.3 \subsection{Set-up description}
132     %\subsection{Discrete Numerical Configuration}
133 jmc 1.1 \label{www:tutorials}
134    
135 jmc 1.2 The model is configured in hydrostatic form, using non-boussinesq
136     $p^*$ coordinate.
137     The vertical resolution is uniform, $\Delta p^* = 50.10^2 Pa$,
138     with 20 levels, from $p^*=10^5 Pa$ to $0$ at the top.
139     The domain is discretised using C32 cube-sphere grid \cite[]{adcroft:04b}
140     that cover the whole sphere with a relatively uniform grid-spacing.
141     The resolution at the equator or along the Greenwitch meridian
142     is similar to the $128 \times 64$ equaly spaced longitude-latitude grid,
143     but requires $25\%$ less grid points.
144     Grid spacing and grid-point location are not computed by the model but
145     read from files.
146    
147     The vector-invariant form of the momentum equation (see section
148     \ref{sect:vect-inv_momentum_equations}) is used so that no explicit
149     metrics are necessary.
150    
151     Applying the vector-invariant discretization to the
152     atmospheric equations \ref{eq:atmos-prime}, and adding the
153     forcing term
154     (\ref{EQ:eg-hs-global_forcing_fv}, \ref{EQ:eg-hs-global_forcing_ft})
155     on the right-hand-side,
156     leads to the set of equations that are solved in this configuration:
157    
158     %the The set of equations solved here is der
159     %Wind-stress forcing is added to the momentum equations for both
160     %the zonal flow, $u$ and the meridional flow $v$, according to equations
161     %(\ref{EQ:eg-hs-global_forcing_fv}) and (\ref{EQ:eg-hs-global_forcing_fv}).
162     %Thermodynamic forcing inputs are added to the equations for
163     %potential temperature, $\theta$, and salinity, $S$, according to equations
164     %(\ref{EQ:eg-hs-global_forcing_ft}) and (\ref{EQ:eg-hs-global_forcing_fs}).
165 jmc 1.1
166     \begin{eqnarray}
167     \label{EQ:eg-hs-model_equations}
168 jmc 1.2 \frac{\partial \vec{\mathbf{v}}_h}{\partial t}
169     +(f + \zeta)\hat{\mathbf{k}} \times \vec{\mathbf{v}}_h
170     %+\mathbf{\nabla }_{p} ({\rm KE})
171     +\mathbf{\nabla }_{p} (\mbox{\sc ke})
172     + \omega \frac{\partial \vec{\mathbf{v}}_h }{\partial p}
173 jmc 1.5 +\mathbf{\nabla }_p \Phi ^{\prime }
174 jmc 1.2 &=&
175     % \vec{{\cal F}_v} =
176 jmc 1.4 -k_\mathbf{v}\vec{\mathbf{v}}_h
177 jmc 1.2 \\
178 jmc 1.5 \frac{\partial \Phi ^{\prime }}{\partial p}
179 jmc 1.2 +\frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&0
180     \\
181     \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_h+\frac{\partial \omega }{
182     \partial p} &=&0
183     \\
184     \frac{\partial \theta }{\partial t}
185     + \mathbf{\nabla }_{p}\cdot (\theta \vec{\mathbf{v}}_h)
186     + \frac{\partial (\theta \omega)}{\partial p}
187     %= \frac{\mathcal{Q}}{\Pi }
188     &=& -k_{\theta}[\theta-\theta_{eq}]
189     \end{eqnarray}
190    
191     %\begin{equation}
192     %\partial_t \vec{v} + ( 2\vec{\Omega} + \vec{\zeta}) \wedge \vec{v}
193     %- b \hat{r}
194     %+ \vec{\nabla} B = \vec{\nabla} \cdot \vec{\bf \tau}
195     %\end{equation}
196 jmc 1.4 %{\cal F}_{\theta} & = & -k_{\theta}(\varphi,p)[\theta-\theta_{eq}(\varphi,p)]
197 jmc 1.2
198 jmc 1.4 \noindent where $\vec{\mathbf{v}}_h$ and $\omega = \frac{Dp}{Dt}$
199     are the horizontal velocity vector and the vertical velocity in pressure coordinate,
200 jmc 1.2 $\zeta$ is the relative vorticity and $f$ the Coriolis parameter,
201     $\hat{\mathbf{k}}$ is the vertical unity vector,
202     {\sc ke} is the kinetic energy, $\Phi$ is the geopotential
203     and $\Pi$ the Exner function
204     ($\Pi = C_p (p/p_c)^\kappa ~{\rm with}~ p_c = 10^5 Pa$).
205 jmc 1.4 Variables marked with ' corresponds to anomaly from
206 jmc 1.2 the resting, uniformly stratified state.
207    
208     As described in MITgcm Numerical Solution Procedure \ref{chap:discretization},
209     the continuity equation is integrated vertically, to give a prognostic
210     equation for the surface pressure $p_s$:
211     \begin{equation}
212 jmc 1.4 \frac{\partial p_s}{\partial t} + \nabla_{h}\cdot \int_{0}^{p_s} \vec{\mathbf{v}}_h dp
213 jmc 1.2 = 0
214     \end{equation}
215    
216     The implicit free surface form of the pressure equation described in
217     \cite{marshall:97a} is employed to solve for $p_s$;
218     Integrating vertically the hydrostatic balance
219 jmc 1.4 gives the geopotential $\Phi'$ and allow to step forward the momentum equation
220 jmc 1.2 \ref{EQ:eg-hs-model_equations}.
221     The potential temperature, $\theta$, is stepped forward using the
222     new velocity field ({\it staggered time-step}, section
223     \ref{sect:adams-bashforth-staggered}).
224 jmc 1.1 \\
225    
226     \subsubsection{Numerical Stability Criteria}
227     \label{www:tutorials}
228    
229     \noindent The numerical stability for inertial oscillations
230     \cite{adcroft:95}
231    
232     \begin{eqnarray}
233     \label{EQ:eg-hs-inertial_stability}
234 jmc 1.3 S_{i} = f^{2} {\Delta t}^2
235 jmc 1.1 \end{eqnarray}
236    
237 jmc 1.3 \noindent evaluates to $4.\times10^{-3}$ at the poles,
238     for $f=2\Omega\sin(\pi / 2) =1.45\times10^{-4}~{\rm s}^{-1}$,
239     which is well below the $S_{i} < 1$ upper limit for stability.
240 jmc 1.1 \\
241    
242 jmc 1.3 \noindent The advective CFL \cite{adcroft:95}
243     for a extreme maximum horizontal flow speed of $ | \vec{u} | = 90. {\rm m/s}$~
244     and the smallest horizontal grid spacing $ \Delta x = 1.1\times10^5 {\rm m}$~:
245 jmc 1.1
246     \begin{eqnarray}
247     \label{EQ:eg-hs-cfl_stability}
248 jmc 1.3 S_{a} = \frac{| \vec{u} | \Delta t}{ \Delta x}
249 jmc 1.1 \end{eqnarray}
250    
251 jmc 1.3 \noindent evaluates to $0.37$, which is close to the stability
252 jmc 1.1 limit of 0.5.
253     \\
254    
255     \noindent The stability parameter for internal gravity waves propagating
256 jmc 1.3 with a maximum speed of $c_{g}=100~{\rm m/s}$
257 jmc 1.1 \cite{adcroft:95}
258    
259     \begin{eqnarray}
260     \label{EQ:eg-hs-gfl_stability}
261 jmc 1.3 S_{c} = \frac{c_{g} \Delta t}{ \Delta x}
262 jmc 1.1 \end{eqnarray}
263    
264 jmc 1.3 \noindent evaluates to $4 \times 10^{-1}$. This is close to the linear
265 jmc 1.1 stability limit of 0.5.
266    
267     \subsection{Experiment Configuration}
268     \label{www:tutorials}
269     \label{SEC:eg-hs_examp_exp_config}
270    
271     The model configuration for this experiment resides under the
272 jmc 1.4 directory {\it verification/tutorial\_held\_suarez\_cs}. The experiment files
273 jmc 1.1 \begin{itemize}
274     \item {\it input/data}
275     \item {\it input/data.pkg}
276     \item {\it input/eedata},
277 jmc 1.3 \item {\it input/data.shap},
278     \item {\it code/packages.conf},
279 jmc 1.1 \item {\it code/CPP\_OPTIONS.h},
280 jmc 1.3 \item {\it code/SIZE.h},
281     \item {\it code/DIAGNOSTICS\_SIZE.h},
282     \item {\it code/external\_forcing.F},
283 jmc 1.1 \end{itemize}
284     contain the code customizations and parameter settings for these
285     experiments. Below we describe the customizations
286     to these files associated with this experiment.
287    
288     \subsubsection{File {\it input/data}}
289     \label{www:tutorials}
290    
291 jmc 1.3 \input{part3/case_studies/held_suarez_cs/inp_data}
292 jmc 1.1
293     \subsubsection{File {\it input/data.pkg}}
294     \label{www:tutorials}
295    
296 jmc 1.3 \input{part3/case_studies/held_suarez_cs/inp_data.pkg}
297    
298 jmc 1.1 \subsubsection{File {\it input/eedata}}
299     \label{www:tutorials}
300    
301 jmc 1.3 This file uses standard default values except line 6:
302     \begin{verbatim}
303     useCubedSphereExchange=.TRUE.,
304     \end{verbatim}
305     This line selects the cubed-sphere specific exchanges to
306     to connect tiles and faces edges.
307 jmc 1.1
308 jmc 1.3 \subsubsection{File {\it input/data.shap}}
309 jmc 1.1 \label{www:tutorials}
310    
311 jmc 1.3 \input{part3/case_studies/held_suarez_cs/inp_data.shap}
312 jmc 1.1
313     \subsubsection{File {\it code/SIZE.h}}
314     \label{www:tutorials}
315    
316 jmc 1.3 Four lines are customized in this file for the current experiment
317 jmc 1.1
318     \begin{itemize}
319    
320     \item Line 39,
321 jmc 1.3 \begin{verbatim} sNx=32, \end{verbatim}
322 jmc 1.4 sets the lateral domain extent in grid points along the x-direction,
323 jmc 1.3 for 1 face.
324    
325     \item Line 40,
326     \begin{verbatim} sNy=32, \end{verbatim}
327 jmc 1.4 sets the lateral domain extent in grid points along the y-direction,
328 jmc 1.3 for 1 face.
329    
330     \item Line 43,
331     \begin{verbatim} nSx=6, \end{verbatim}
332 jmc 1.4 sets the number of tiles in the x-direction, for the model domain
333 jmc 1.3 decomposition. In this simple case (one processor and 1 tile per
334 jmc 1.4 face), this number corresponds to the total number of faces.
335 jmc 1.1
336     \item Line 49,
337 jmc 1.3 \begin{verbatim} Nr=20, \end{verbatim}
338     sets the vertical domain extent in grid points.
339 jmc 1.1
340     \end{itemize}
341    
342 jmc 1.3 %\begin{small}
343     %\input{part3/case_studies/held_suarez_cs/code/SIZE.h}
344     %\end{small}
345 jmc 1.1
346 jmc 1.3 \subsubsection{File {\it code/packages.conf}}
347 jmc 1.1 \label{www:tutorials}
348    
349 jmc 1.3 \input{part3/case_studies/held_suarez_cs/cod_packages.conf}
350 jmc 1.1
351 jmc 1.3 \subsubsection{File {\it code/CPP\_OPTIONS.h}}
352 jmc 1.1 \label{www:tutorials}
353    
354 jmc 1.3 This file uses standard default except for Line 40\\
355     ({\it diff CPP\_OPTIONS.h ../../../model/inc}):
356     \begin{verbatim}
357     #define NONLIN_FRSURF
358     \end{verbatim}
359     This line allow to use the non-linear free-surface part of the code,
360     which is required for the $p^*$ coordinate formulation.
361 jmc 1.1
362     \subsubsection{Other Files }
363     \label{www:tutorials}
364    
365     Other files relevant to this experiment are
366     \begin{itemize}
367 jmc 1.3 \item {\it code/external\_forcing.F}
368     \item {\it input/grid\_cs32.face00[n].bin}, with $n=1,2,3,4,5,6$
369 jmc 1.1 \end{itemize}
370 jmc 1.3 contain the code customisations and binary input files for this
371 jmc 1.1 experiments. Below we describe the customisations
372 jmc 1.3 to these files associated with this experiment.\\
373    
374     The file {\it code/external\_forcing.F} contains 4 subroutines
375     that calculate the forcing terms (Right-Hand side term) in the
376     momentum equation (\ref{EQ:eg-hs-global_forcing_fv},
377     {\it S/R EXTERNAL\_FORCING\_U} and {\it EXTERNAL\_FORCING\_V})
378     and in the potential temperature equation
379     (\ref{EQ:eg-hs-global_forcing_ft}, {\it S/R EXTERNAL\_FORCING\_T}).
380     The water-vapour forcing subroutine ({\it S/R EXTERNAL\_FORCING\_S})
381     is left empty for this experiment.\\
382    
383     The grid-files {\it input/grid\_cs32.face00[n].bin}, with $n=1,2,3,4,5,6$,
384     are binary files (direct-access, big-endian 64.bits real) that
385     contains all the cubed-sphere grid lengths, areas and grid-point
386     positions, with one file per face.
387 jmc 1.4 Each file contains 18 2-D arrays (dimension $33 \times 33$) that correspond
388 jmc 1.3 to the model variables:
389     {\it
390     XC YC DXF DYF RA XG YG DXV DYU RAZ DXC DYC RAW RAS DXG DYG AngleCS AngleSN
391     }
392     (see {\it GRID.h} file)
393    

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