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1 jmc 1.20 % $Header: /u/gcmpack/manual/s_examples/global_oce_latlon/climatalogical_ogcm.tex,v 1.19 2010/08/30 23:09:20 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     %\begin{center}
17 cnh 1.3 %{\Large \bf Using MITgcm to Simulate Global Climatological Ocean Circulation
18 adcroft 1.1 %At Four Degree Resolution with Asynchronous Time Stepping}
19     %
20     %\vspace*{4mm}
21     %
22     %\vspace*{3mm}
23     %{\large May 2001}
24     %\end{center}
25    
26    
27     This example experiment demonstrates using the MITgcm to simulate
28     the planetary ocean circulation. The simulation is configured
29     with realistic geography and bathymetry on a
30     $4^{\circ} \times 4^{\circ}$ spherical polar grid.
31 molod 1.14 The files for this experiment are in the verification directory
32     under tutorial\_global\_oce\_latlon.
33 adcroft 1.1 Twenty levels are used in the vertical, ranging in thickness
34     from $50\,{\rm m}$ at the surface to $815\,{\rm m}$ at depth,
35     giving a maximum model depth of $6\,{\rm km}$.
36     At this resolution, the configuration
37     can be integrated forward for thousands of years on a single
38     processor desktop computer.
39     \\
40 cnh 1.8 \subsection{Overview}
41 jmc 1.19 %\label{www:tutorials}
42 adcroft 1.1
43 cnh 1.3 The model is forced with climatological wind stress data and surface
44     flux data from DaSilva \cite{DaSilva94}. Climatological data
45     from Levitus \cite{Levitus94} is used to initialize the model hydrography.
46     Levitus seasonal climatology data is also used throughout the calculation
47 adcroft 1.1 to provide additional air-sea fluxes.
48 cnh 1.3 These fluxes are combined with the DaSilva climatological estimates of
49 adcroft 1.1 surface heat flux and fresh water, resulting in a mixed boundary
50 cnh 1.3 condition of the style described in Haney \cite{Haney}.
51 adcroft 1.1 Altogether, this yields the following forcing applied
52     in the model surface layer.
53    
54     \begin{eqnarray}
55 jmc 1.19 \label{eq:eg-global-global_forcing}
56     \label{eq:eg-global-global_forcing_fu}
57 adcroft 1.1 {\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}}
58     \\
59 jmc 1.19 \label{eq:eg-global-global_forcing_fv}
60 adcroft 1.1 {\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}}
61     \\
62 jmc 1.19 \label{eq:eg-global-global_forcing_ft}
63 adcroft 1.1 {\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} )
64     - \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q}
65     \\
66 jmc 1.19 \label{eq:eg-global-global_forcing_fs}
67 adcroft 1.1 {\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} )
68     + \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R})
69     \end{eqnarray}
70    
71     \noindent where ${\cal F}_{u}$, ${\cal F}_{v}$, ${\cal F}_{\theta}$,
72     ${\cal F}_{s}$ are the forcing terms in the zonal and meridional
73     momentum and in the potential temperature and salinity
74     equations respectively.
75     The term $\Delta z_{s}$ represents the top ocean layer thickness in
76     meters.
77     It is used in conjunction with a reference density, $\rho_{0}$
78     (here set to $999.8\,{\rm kg\,m^{-3}}$), a
79     reference salinity, $S_{0}$ (here set to 35~ppt),
80     and a specific heat capacity, $C_{p}$ (here set to
81     $4000~{\rm J}~^{\circ}{\rm C}^{-1}~{\rm kg}^{-1}$), to convert
82     input dataset values into time tendencies of
83     potential temperature (with units of $^{\circ}{\rm C}~{\rm s}^{-1}$),
84     salinity (with units ${\rm ppt}~s^{-1}$) and
85     velocity (with units ${\rm m}~{\rm s}^{-2}$).
86     The externally supplied forcing fields used in this
87     experiment are $\tau_{x}$, $\tau_{y}$, $\theta^{\ast}$, $S^{\ast}$,
88     $\cal{Q}$ and $\cal{E}-\cal{P}-\cal{R}$. The wind stress fields ($\tau_x$, $\tau_y$)
89     have units of ${\rm N}~{\rm m}^{-2}$. The temperature forcing fields
90     ($\theta^{\ast}$ and $Q$) have units of $^{\circ}{\rm C}$ and ${\rm W}~{\rm m}^{-2}$
91     respectively. The salinity forcing fields ($S^{\ast}$ and
92     $\cal{E}-\cal{P}-\cal{R}$) have units of ${\rm ppt}$ and ${\rm m}~{\rm s}^{-1}$
93 cnh 1.8 respectively. The source files and procedures for ingesting this data into the
94     simulation are described in the experiment configuration discussion in section
95 jmc 1.19 \ref{sec:eg-global-clim_ocn_examp_exp_config}.
96 adcroft 1.1
97    
98     \subsection{Discrete Numerical Configuration}
99 jmc 1.19 %\label{www:tutorials}
100 adcroft 1.1
101    
102     The model is configured in hydrostatic form. The domain is discretised with
103     a uniform grid spacing in latitude and longitude on the sphere
104     $\Delta \phi=\Delta \lambda=4^{\circ}$, so
105     that there are ninety grid cells in the zonal and forty in the
106     meridional direction. The internal model coordinate variables
107 cnh 1.3 $x$ and $y$ are initialized according to
108 adcroft 1.1 \begin{eqnarray}
109     x=r\cos(\phi),~\Delta x & = &r\cos(\Delta \phi) \\
110 cnh 1.8 y=r\lambda,~\Delta y &= &r\Delta \lambda
111 adcroft 1.1 \end{eqnarray}
112    
113     Arctic polar regions are not
114     included in this experiment. Meridionally the model extends from
115     $80^{\circ}{\rm S}$ to $80^{\circ}{\rm N}$.
116     Vertically the model is configured with twenty layers with the
117     following thicknesses
118     $\Delta z_{1} = 50\,{\rm m},\,
119     \Delta z_{2} = 50\,{\rm m},\,
120     \Delta z_{3} = 55\,{\rm m},\,
121     \Delta z_{4} = 60\,{\rm m},\,
122     \Delta z_{5} = 65\,{\rm m},\,
123     $
124     $
125     \Delta z_{6}~=~70\,{\rm m},\,
126     \Delta z_{7}~=~80\,{\rm m},\,
127     \Delta z_{8}~=95\,{\rm m},\,
128     \Delta z_{9}=120\,{\rm m},\,
129     \Delta z_{10}=155\,{\rm m},\,
130     $
131     $
132     \Delta z_{11}=200\,{\rm m},\,
133     \Delta z_{12}=260\,{\rm m},\,
134     \Delta z_{13}=320\,{\rm m},\,
135     \Delta z_{14}=400\,{\rm m},\,
136     \Delta z_{15}=480\,{\rm m},\,
137     $
138     $
139     \Delta z_{16}=570\,{\rm m},\,
140     \Delta z_{17}=655\,{\rm m},\,
141     \Delta z_{18}=725\,{\rm m},\,
142     \Delta z_{19}=775\,{\rm m},\,
143     \Delta z_{20}=815\,{\rm m}
144 cnh 1.8 $ (here the numeric subscript indicates the model level index number, ${\tt k}$) to
145     give a total depth, $H$, of $-5450{\rm m}$.
146 adcroft 1.1 The implicit free surface form of the pressure equation described in Marshall et. al
147 adcroft 1.6 \cite{marshall:97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous
148 cnh 1.3 dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.
149 adcroft 1.1
150 jmc 1.19 Wind-stress forcing is added to the momentum equations in (\ref{eq:eg-global-model_equations})
151 cnh 1.8 for both the zonal flow, $u$ and the meridional flow $v$, according to equations
152 jmc 1.19 (\ref{eq:eg-global-global_forcing_fu}) and (\ref{eq:eg-global-global_forcing_fv}).
153 cnh 1.8 Thermodynamic forcing inputs are added to the equations
154 jmc 1.19 in (\ref{eq:eg-global-model_equations}) for
155 adcroft 1.1 potential temperature, $\theta$, and salinity, $S$, according to equations
156 jmc 1.19 (\ref{eq:eg-global-global_forcing_ft}) and (\ref{eq:eg-global-global_forcing_fs}).
157 adcroft 1.1 This produces a set of equations solved in this configuration as follows:
158    
159     \begin{eqnarray}
160 jmc 1.19 \label{eq:eg-global-model_equations}
161 adcroft 1.1 \frac{Du}{Dt} - fv +
162     \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -
163     \nabla_{h}\cdot A_{h}\nabla_{h}u -
164     \frac{\partial}{\partial z}A_{z}\frac{\partial u}{\partial z}
165     & = &
166     \begin{cases}
167     {\cal F}_u & \text{(surface)} \\
168     0 & \text{(interior)}
169     \end{cases}
170     \\
171     \frac{Dv}{Dt} + fu +
172     \frac{1}{\rho}\frac{\partial p^{'}}{\partial y} -
173     \nabla_{h}\cdot A_{h}\nabla_{h}v -
174     \frac{\partial}{\partial z}A_{z}\frac{\partial v}{\partial z}
175     & = &
176     \begin{cases}
177     {\cal F}_v & \text{(surface)} \\
178     0 & \text{(interior)}
179     \end{cases}
180     \\
181     \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}
182     &=&
183     0
184     \\
185     \frac{D\theta}{Dt} -
186     \nabla_{h}\cdot K_{h}\nabla_{h}\theta
187     - \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial\theta}{\partial z}
188     & = &
189     \begin{cases}
190     {\cal F}_\theta & \text{(surface)} \\
191     0 & \text{(interior)}
192     \end{cases}
193     \\
194     \frac{D s}{Dt} -
195     \nabla_{h}\cdot K_{h}\nabla_{h}s
196     - \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial s}{\partial z}
197     & = &
198     \begin{cases}
199     {\cal F}_s & \text{(surface)} \\
200     0 & \text{(interior)}
201     \end{cases}
202     \\
203     g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz & = & p^{'}
204     \end{eqnarray}
205    
206     \noindent where $u=\frac{Dx}{Dt}=r \cos(\phi)\frac{D \lambda}{Dt}$ and
207     $v=\frac{Dy}{Dt}=r \frac{D \phi}{Dt}$
208     are the zonal and meridional components of the
209     flow vector, $\vec{u}$, on the sphere. As described in
210 adcroft 1.5 MITgcm Numerical Solution Procedure \ref{chap:discretization}, the time
211 adcroft 1.1 evolution of potential temperature, $\theta$, equation is solved prognostically.
212     The total pressure, $p$, is diagnosed by summing pressure due to surface
213     elevation $\eta$ and the hydrostatic pressure.
214     \\
215    
216     \subsubsection{Numerical Stability Criteria}
217 jmc 1.19 %\label{www:tutorials}
218 adcroft 1.1
219 cnh 1.3 The Laplacian dissipation coefficient, $A_{h}$, is set to $5 \times 10^5 m s^{-1}$.
220 adcroft 1.4 This value is chosen to yield a Munk layer width \cite{adcroft:95},
221 adcroft 1.1 \begin{eqnarray}
222 jmc 1.19 \label{eq:eg-global-munk_layer}
223 adcroft 1.10 && M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
224 adcroft 1.1 \end{eqnarray}
225    
226     \noindent of $\approx 600$km. This is greater than the model
227     resolution in low-latitudes, $\Delta x \approx 400{\rm km}$, ensuring that the frictional
228     boundary layer is adequately resolved.
229     \\
230    
231     \noindent The model is stepped forward with a
232     time step $\delta t_{\theta}=30~{\rm hours}$ for thermodynamic variables and
233     $\delta t_{v}=40~{\rm minutes}$ for momentum terms. With this time step, the stability
234 adcroft 1.4 parameter to the horizontal Laplacian friction \cite{adcroft:95}
235 adcroft 1.1 \begin{eqnarray}
236 jmc 1.19 \label{eq:eg-global-laplacian_stability}
237 adcroft 1.10 && S_{l} = 4 \frac{A_{h} \delta t_{v}}{{\Delta x}^2}
238 adcroft 1.1 \end{eqnarray}
239    
240     \noindent evaluates to 0.16 at a latitude of $\phi=80^{\circ}$, which is below the
241     0.3 upper limit for stability. The zonal grid spacing $\Delta x$ is smallest at
242     $\phi=80^{\circ}$ where $\Delta x=r\cos(\phi)\Delta \phi\approx 77{\rm km}$.
243     \\
244    
245     \noindent The vertical dissipation coefficient, $A_{z}$, is set to
246     $1\times10^{-3} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
247     \begin{eqnarray}
248 jmc 1.19 \label{eq:eg-global-laplacian_stability_z}
249 adcroft 1.1 S_{l} = 4 \frac{A_{z} \delta t_{v}}{{\Delta z}^2}
250     \end{eqnarray}
251    
252     \noindent evaluates to $0.015$ for the smallest model
253 cnh 1.3 level spacing ($\Delta z_{1}=50{\rm m}$) which is again well below
254 adcroft 1.1 the upper stability limit.
255     \\
256    
257     The values of the horizontal ($K_{h}$) and vertical ($K_{z}$) diffusion coefficients
258     for both temperature and salinity are set to $1 \times 10^{3}~{\rm m}^{2}{\rm s}^{-1}$
259     and $3 \times 10^{-5}~{\rm m}^{2}{\rm s}^{-1}$ respectively. The stability limit
260     related to $K_{h}$ will be at $\phi=80^{\circ}$ where $\Delta x \approx 77 {\rm km}$.
261     Here the stability parameter
262     \begin{eqnarray}
263 jmc 1.19 \label{eq:eg-global-laplacian_stability_xtheta}
264 adcroft 1.1 S_{l} = \frac{4 K_{h} \delta t_{\theta}}{{\Delta x}^2}
265     \end{eqnarray}
266 cnh 1.3 evaluates to $0.07$, well below the stability limit of $S_{l} \approx 0.5$. The
267 adcroft 1.1 stability parameter related to $K_{z}$
268     \begin{eqnarray}
269 jmc 1.19 \label{eq:eg-global-laplacian_stability_ztheta}
270 adcroft 1.1 S_{l} = \frac{4 K_{z} \delta t_{\theta}}{{\Delta z}^2}
271     \end{eqnarray}
272     evaluates to $0.005$ for $\min(\Delta z)=50{\rm m}$, well below the stability limit
273     of $S_{l} \approx 0.5$.
274     \\
275    
276     \noindent The numerical stability for inertial oscillations
277 adcroft 1.4 \cite{adcroft:95}
278 adcroft 1.1
279     \begin{eqnarray}
280 jmc 1.19 \label{eq:eg-global-inertial_stability}
281 adcroft 1.1 S_{i} = f^{2} {\delta t_v}^2
282     \end{eqnarray}
283    
284     \noindent evaluates to $0.24$ for $f=2\omega\sin(80^{\circ})=1.43\times10^{-4}~{\rm s}^{-1}$, which is close to
285     the $S_{i} < 1$ upper limit for stability.
286     \\
287    
288 adcroft 1.4 \noindent The advective CFL \cite{adcroft:95} for a extreme maximum
289 adcroft 1.1 horizontal flow
290     speed of $ | \vec{u} | = 2 ms^{-1}$
291    
292     \begin{eqnarray}
293 jmc 1.19 \label{eq:eg-global-cfl_stability}
294 adcroft 1.1 S_{a} = \frac{| \vec{u} | \delta t_{v}}{ \Delta x}
295     \end{eqnarray}
296    
297     \noindent evaluates to $6 \times 10^{-2}$. This is well below the stability
298     limit of 0.5.
299     \\
300    
301 cnh 1.3 \noindent The stability parameter for internal gravity waves propagating
302 adcroft 1.1 with a maximum speed of $c_{g}=10~{\rm ms}^{-1}$
303 adcroft 1.4 \cite{adcroft:95}
304 adcroft 1.1
305     \begin{eqnarray}
306 jmc 1.19 \label{eq:eg-global-gfl_stability}
307 adcroft 1.1 S_{c} = \frac{c_{g} \delta t_{v}}{ \Delta x}
308     \end{eqnarray}
309    
310     \noindent evaluates to $3 \times 10^{-1}$. This is close to the linear
311     stability limit of 0.5.
312    
313     \subsection{Experiment Configuration}
314 jmc 1.19 %\label{www:tutorials}
315     \label{sec:eg-global-clim_ocn_examp_exp_config}
316 adcroft 1.1
317     The model configuration for this experiment resides under the
318 cnh 1.8 directory {\it tutorial\_examples/global\_ocean\_circulation/}.
319     The experiment files
320    
321 adcroft 1.1 \begin{itemize}
322     \item {\it input/data}
323     \item {\it input/data.pkg}
324     \item {\it input/eedata},
325     \item {\it input/windx.bin},
326     \item {\it input/windy.bin},
327     \item {\it input/salt.bin},
328     \item {\it input/theta.bin},
329     \item {\it input/SSS.bin},
330     \item {\it input/SST.bin},
331     \item {\it input/topog.bin},
332     \item {\it code/CPP\_EEOPTIONS.h}
333     \item {\it code/CPP\_OPTIONS.h},
334     \item {\it code/SIZE.h}.
335     \end{itemize}
336 cnh 1.3 contain the code customizations and parameter settings for these
337     experiments. Below we describe the customizations
338 adcroft 1.1 to these files associated with this experiment.
339 cnh 1.8
340     \subsubsection{Driving Datasets}
341 jmc 1.19 %\label{www:tutorials}
342 cnh 1.8
343 jmc 1.19 Figures ({\it --- missing figures ---})
344     %(\ref{fig:sim_config_tclim}-\ref{fig:sim_config_empmr})
345     show the relaxation temperature ($\theta^{\ast}$) and salinity ($S^{\ast}$)
346     fields, the wind stress components ($\tau_x$ and $\tau_y$), the heat flux ($Q$)
347 cnh 1.8 and the net fresh water flux (${\cal E} - {\cal P} - {\cal R}$) used
348 jmc 1.19 in equations
349     (\ref{eq:eg-global-global_forcing_fu}-\ref{eq:eg-global-global_forcing_fs}).
350     The figures also indicate the lateral extent and coastline used in the
351     experiment. Figure ({\it --- missing figure --- }) %ref{fig:model_bathymetry})
352     shows the depth contours of the model domain.
353 adcroft 1.1
354     \subsubsection{File {\it input/data}}
355 jmc 1.19 %\label{www:tutorials}
356 adcroft 1.1
357 jmc 1.20 \input{s_examples/global_oce_latlon/inp_data}
358 adcroft 1.1
359     \subsubsection{File {\it input/data.pkg}}
360 jmc 1.19 %\label{www:tutorials}
361 adcroft 1.1
362     This file uses standard default values and does not contain
363     customisations for this experiment.
364    
365     \subsubsection{File {\it input/eedata}}
366 jmc 1.19 %\label{www:tutorials}
367 adcroft 1.1
368     This file uses standard default values and does not contain
369     customisations for this experiment.
370    
371     \subsubsection{File {\it input/windx.sin\_y}}
372 jmc 1.19 %\label{www:tutorials}
373 adcroft 1.1
374     The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
375     map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.
376     Although $\tau_{x}$ is only a function of $y$n in this experiment
377     this file must still define a complete two-dimensional map in order
378     to be compatible with the standard code for loading forcing fields
379     in MITgcm. The included matlab program {\it input/gendata.m} gives a complete
380     code for creating the {\it input/windx.sin\_y} file.
381    
382     \subsubsection{File {\it input/topog.box}}
383 jmc 1.19 %\label{www:tutorials}
384 adcroft 1.1
385    
386     The {\it input/topog.box} file specifies a two-dimensional ($x,y$)
387     map of depth values. For this experiment values are either
388     $0m$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep
389     ocean. The file contains a raw binary stream of data that is enumerated
390     in the same way as standard MITgcm two-dimensional, horizontal arrays.
391     The included matlab program {\it input/gendata.m} gives a complete
392     code for creating the {\it input/topog.box} file.
393    
394     \subsubsection{File {\it code/SIZE.h}}
395 jmc 1.19 %\label{www:tutorials}
396 adcroft 1.1
397     Two lines are customized in this file for the current experiment
398    
399     \begin{itemize}
400    
401     \item Line 39,
402     \begin{verbatim} sNx=60, \end{verbatim} this line sets
403     the lateral domain extent in grid points for the
404     axis aligned with the x-coordinate.
405    
406     \item Line 40,
407     \begin{verbatim} sNy=60, \end{verbatim} this line sets
408     the lateral domain extent in grid points for the
409     axis aligned with the y-coordinate.
410    
411     \item Line 49,
412     \begin{verbatim} Nr=4, \end{verbatim} this line sets
413     the vertical domain extent in grid points.
414    
415     \end{itemize}
416    
417     \begin{small}
418 jmc 1.18 \input{s_examples/global_oce_latlon/code/SIZE.h}
419 adcroft 1.1 \end{small}
420    
421     \subsubsection{File {\it code/CPP\_OPTIONS.h}}
422 jmc 1.19 %\label{www:tutorials}
423 adcroft 1.1
424     This file uses standard default values and does not contain
425     customisations for this experiment.
426    
427    
428     \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
429 jmc 1.19 %\label{www:tutorials}
430 adcroft 1.1
431     This file uses standard default values and does not contain
432     customisations for this experiment.
433    
434     \subsubsection{Other Files }
435 jmc 1.19 %\label{www:tutorials}
436 adcroft 1.1
437     Other files relevant to this experiment are
438     \begin{itemize}
439     \item {\it model/src/ini\_cori.F}. This file initializes the model
440     coriolis variables {\bf fCorU}.
441     \item {\it model/src/ini\_spherical\_polar\_grid.F}
442     \item {\it model/src/ini\_parms.F},
443     \item {\it input/windx.sin\_y},
444     \end{itemize}
445     contain the code customisations and parameter settings for this
446 cnh 1.3 experiments. Below we describe the customisations
447 adcroft 1.1 to these files associated with this experiment.

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