/[MITgcm]/manual/s_examples/tracer_adjsens/co2sens.tex
ViewVC logotype

Annotation of /manual/s_examples/tracer_adjsens/co2sens.tex

Parent Directory Parent Directory | Revision Log Revision Log | View Revision Graph Revision Graph


Revision 1.1 - (hide annotations) (download) (as text)
Wed Aug 8 16:16:16 2001 UTC (23 years, 11 months ago) by adcroft
Branch: MAIN
Branch point for: dummy
File MIME type: application/x-tex
Initial revision

1 adcroft 1.1 % $Header: $
2     % $Name: $
3    
4     \section{Example: Centenial Time Scale Sensitivities}
5    
6     \bodytext{bgcolor="#FFFFFFFF"}
7    
8     %\begin{center}
9     %{\Large \bf Using MITgcm to Look at Centenial Time Scale
10     %Sensitivities}
11     %
12     %\vspace*{4mm}
13     %
14     %\vspace*{3mm}
15     %{\large May 2001}
16     %\end{center}
17    
18     \subsection{Introduction}
19    
20     This document describes the fourth example MITgcm experiment.
21     This example iilustrates the use of
22     the MITgcm to perform sentivity analysis in a
23     large scale ocean circulation simulation.
24    
25     \subsection{Overview}
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     Twenty vertical layers are used in the vertical, ranging in thickness
32     from $50\,{\rm m}$ at the surface to $815\,{\rm m}$ at depth,
33     giving a maximum model depth of $6\,{\rm km}$.
34     At this resolution, the configuration
35     can be integrated forward for thousands of years on a single
36     processor desktop computer.
37     \\
38    
39     The model is forced with climatalogical wind stress data and surface
40     flux data from Da Silva \cite{DaSilva94}. Climatalogical data
41     from Levitus \cite{Levitus94} is used to initialise the model hydrography.
42     Levitus data is also used throughout the calculation
43     to derive air-sea fluxes of heat at the ocean surface.
44     These fluxes are combined with climatalogical estimates of
45     surface heat flux and fresh water, resulting in a mixed boundary
46     condition of the style decribed in Haney \cite{Haney}.
47     Altogether, this yields the following forcing applied
48     in the model surface layer.
49    
50     \begin{eqnarray}
51     \label{EQ:global_forcing}
52     \label{EQ:global_forcing_fu}
53     {\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}}
54     \\
55     \label{EQ:global_forcing_fv}
56     {\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}}
57     \\
58     \label{EQ:global_forcing_ft}
59     {\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} )
60     - \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q}
61     \\
62     \label{EQ:global_forcing_fs}
63     {\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} )
64     + \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R})
65     \end{eqnarray}
66    
67     \noindent where ${\cal F}_{u}$, ${\cal F}_{v}$, ${\cal F}_{\theta}$,
68     ${\cal F}_{s}$ are the forcing terms in the zonal and meridional
69     momentum and in the potential temperature and salinity
70     equations respectively.
71     The term $\Delta z_{s}$ represents the top ocean layer thickness.
72     It is used in conjunction with the reference density, $\rho_{0}$
73     (here set to $999.8\,{\rm kg\,m^{-3}}$), the
74     reference salinity, $S_{0}$ (here set to 35ppt),
75     and a specific heat capacity $C_{p}$ to convert
76     wind-stress fluxes given in ${\rm N}\,m^{-2}$,
77     \\
78    
79    
80     The configuration is illustrated in figure \ref{simulation_config}.
81    
82    
83     \subsection{Discrete Numerical Configuration}
84    
85    
86     The model is configured in hydrostatic form. The domain is discretised with
87     a uniform grid spacing in latitude and longitude of
88     $\Delta x=\Delta y=4^{\circ}$, so
89     that there are ninety grid cells in the $x$ and forty in the
90     $y$ direction (Arctic polar regions are not
91     included in this experiment). Vertically the
92     model is configured with twenty layers with the following thicknesses
93     $\Delta z_{1} = 50\,{\rm m},\,
94     \Delta z_{2} = 50\,{\rm m},\,
95     \Delta z_{3} = 55\,{\rm m},\,
96     \Delta z_{4} = 60\,{\rm m},\,
97     \Delta z_{5} = 65\,{\rm m},\,
98     $
99     $
100     \Delta z_{6}~=~70\,{\rm m},\,
101     \Delta z_{7}~=~80\,{\rm m},\,
102     \Delta z_{8}~=95\,{\rm m},\,
103     \Delta z_{9}=120\,{\rm m},\,
104     \Delta z_{10}=155\,{\rm m},\,
105     $
106     $
107     \Delta z_{11}=200\,{\rm m},\,
108     \Delta z_{12}=260\,{\rm m},\,
109     \Delta z_{13}=320\,{\rm m},\,
110     \Delta z_{14}=400\,{\rm m},\,
111     \Delta z_{15}=480\,{\rm m},\,
112     $
113     $
114     \Delta z_{16}=570\,{\rm m},\,
115     \Delta z_{17}=655\,{\rm m},\,
116     \Delta z_{18}=725\,{\rm m},\,
117     \Delta z_{19}=775\,{\rm m},\,
118     \Delta z_{20}=815\,{\rm m}
119     $ (here the numeric subscript indicates the model level index number, ${\tt k}$).
120     The implicit free surface form of the pressure equation described in Marshall et. al
121     \cite{Marshall97a} is employed. A laplacian operator, $\nabla^2$, provides viscous
122     dissipation. Thermal and haline diffusion is also represented by a laplacian operator.
123     \\
124    
125     Wind-stress momentum inputs are added to the momentum equations for both
126     the zonal flow, $u$ and the merdional flow $v$, according to equations
127     (\ref{EQ:global_forcing_fu}) and (\ref{EQ:global_forcing_fv}).
128     Thermodynamic forcing inputs are added to the equations for
129     potential temperature, $\theta$, and salinity, $S$, according to equations
130     (\ref{EQ:global_forcing_ft}) and (\ref{EQ:global_forcing_fs}).
131     This produces a set of equations solved in this configuration as follows:
132     % {\fracktur}
133    
134    
135     \begin{eqnarray}
136     \label{EQ:model_equations}
137     \frac{Du}{Dt} - fv +
138     \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -
139     A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}
140     & = &
141     {\cal F}_{u}
142     \\
143     \frac{Dv}{Dt} + fu +
144     \frac{1}{\rho}\frac{\partial p^{'}}{\partial y} -
145     A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}
146     & = &
147     {\cal F}_{v}
148     \\
149     \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}
150     &=&
151     0
152     \\
153     \frac{D\theta}{Dt} -
154     K_{h}\nabla_{h}^2\theta - \Gamma(K_{z})\frac{\partial^{2}\theta}{\partial z^{2}}
155     & = &
156     {\cal F}_{\theta}
157     \\
158     \frac{D s}{Dt} -
159     K_{h}\nabla_{h}^2 s - \Gamma(K_{z})\frac{\partial^{2} s}{\partial z^{2}}
160     & = &
161     {\cal F}_{s}
162     \\
163     g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz & = & p^{'}
164     \\
165     \end{eqnarray}
166    
167     \noindent where $u$ and $v$ are the $x$ and $y$ components of the
168     flow vector $\vec{u}$. The suffices ${s},{i}$ indicate surface and
169     interior model levels respectively. As described in
170     MITgcm Numerical Solution Procedure \cite{MITgcm_Numerical_Scheme}, the time
171     evolution of potential temperature, $\theta$, equation is solved prognostically.
172     The total pressure, $p$, is diagnosed by summing pressure due to surface
173     elevation $\eta$ and the hydrostatic pressure.
174     \\
175    
176     \subsubsection{Numerical Stability Criteria}
177    
178     The laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.
179     This value is chosen to yield a Munk layer width \cite{Adcroft_thesis},
180    
181     \begin{eqnarray}
182     \label{EQ:munk_layer}
183     M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
184     \end{eqnarray}
185    
186     \noindent of $\approx 100$km. This is greater than the model
187     resolution in mid-latitudes $\Delta x$, ensuring that the frictional
188     boundary layer is well resolved.
189     \\
190    
191     \noindent The model is stepped forward with a
192     time step $\delta t=1200$secs. With this time step the stability
193     parameter to the horizontal laplacian friction \cite{Adcroft_thesis}
194    
195     \begin{eqnarray}
196     \label{EQ:laplacian_stability}
197     S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
198     \end{eqnarray}
199    
200     \noindent evaluates to 0.012, which is well below the 0.3 upper limit
201     for stability.
202     \\
203    
204     \noindent The vertical dissipation coefficient, $A_{z}$, is set to
205     $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
206    
207     \begin{eqnarray}
208     \label{EQ:laplacian_stability_z}
209     S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}
210     \end{eqnarray}
211    
212     \noindent evaluates to $4.8 \times 10^{-5}$ which is again well below
213     the upper limit.
214     The values of $A_{h}$ and $A_{z}$ are also used for the horizontal ($K_{h}$)
215     and vertical ($K_{z}$) diffusion coefficients for temperature respectively.
216     \\
217    
218     \noindent The numerical stability for inertial oscillations
219     \cite{Adcroft_thesis}
220    
221     \begin{eqnarray}
222     \label{EQ:inertial_stability}
223     S_{i} = f^{2} {\delta t}^2
224     \end{eqnarray}
225    
226     \noindent evaluates to $0.0144$, which is well below the $0.5$ upper
227     limit for stability.
228     \\
229    
230     \noindent The advective CFL \cite{Adcroft_thesis} for a extreme maximum
231     horizontal flow
232     speed of $ | \vec{u} | = 2 ms^{-1}$
233    
234     \begin{eqnarray}
235     \label{EQ:cfl_stability}
236     S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
237     \end{eqnarray}
238    
239     \noindent evaluates to $5 \times 10^{-2}$. This is well below the stability
240     limit of 0.5.
241     \\
242    
243     \noindent The stbility parameter for internal gravity waves
244     \cite{Adcroft_thesis}
245    
246     \begin{eqnarray}
247     \label{EQ:cfl_stability}
248     S_{c} = \frac{c_{g} \delta t}{ \Delta x}
249     \end{eqnarray}
250    
251     \noindent evaluates to $5 \times 10^{-2}$. This is well below the linear
252     stability limit of 0.25.
253    
254     \subsection{Code Configuration}
255     \label{SEC:code_config}
256    
257     The model configuration for this experiment resides under the
258     directory {\it verification/exp1/}. The experiment files
259     \begin{itemize}
260     \item {\it input/data}
261     \item {\it input/data.pkg}
262     \item {\it input/eedata},
263     \item {\it input/windx.sin\_y},
264     \item {\it input/topog.box},
265     \item {\it code/CPP\_EEOPTIONS.h}
266     \item {\it code/CPP\_OPTIONS.h},
267     \item {\it code/SIZE.h}.
268     \end{itemize}
269     contain the code customisations and parameter settings for this
270     experiements. Below we describe the customisations
271     to these files associated with this experiment.
272    
273     \subsubsection{File {\it input/data}}
274    
275     This file, reproduced completely below, specifies the main parameters
276     for the experiment. The parameters that are significant for this configuration
277     are
278    
279     \begin{itemize}
280    
281     \item Line 4,
282     \begin{verbatim} tRef=20.,10.,8.,6., \end{verbatim}
283     this line sets
284     the initial and reference values of potential temperature at each model
285     level in units of $^{\circ}$C.
286     The entries are ordered from surface to depth. For each
287     depth level the inital and reference profiles will be uniform in
288     $x$ and $y$.
289    
290     \fbox{
291     \begin{minipage}{5.0in}
292     {\it S/R INI\_THETA}({\it ini\_theta.F})
293     \end{minipage}
294     }
295    
296    
297     \item Line 6,
298     \begin{verbatim} viscAz=1.E-2, \end{verbatim}
299     this line sets the vertical laplacian dissipation coefficient to
300     $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions
301     for this operator are specified later. This variable is copied into
302     model general vertical coordinate variable {\bf viscAr}.
303    
304     \fbox{
305     \begin{minipage}{5.0in}
306     {\it S/R CALC\_DIFFUSIVITY}({\it calc\_diffusivity.F})
307     \end{minipage}
308     }
309    
310     \item Line 7,
311     \begin{verbatim}
312     viscAh=4.E2,
313     \end{verbatim}
314     this line sets the horizontal laplacian frictional dissipation coefficient to
315     $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions
316     for this operator are specified later.
317    
318     \item Lines 8,
319     \begin{verbatim}
320     no_slip_sides=.FALSE.
321     \end{verbatim}
322     this line selects a free-slip lateral boundary condition for
323     the horizontal laplacian friction operator
324     e.g. $\frac{\partial u}{\partial y}$=0 along boundaries in $y$ and
325     $\frac{\partial v}{\partial x}$=0 along boundaries in $x$.
326    
327     \item Lines 9,
328     \begin{verbatim}
329     no_slip_bottom=.TRUE.
330     \end{verbatim}
331     this line selects a no-slip boundary condition for bottom
332     boundary condition in the vertical laplacian friction operator
333     e.g. $u=v=0$ at $z=-H$, where $H$ is the local depth of the domain.
334    
335     \item Line 10,
336     \begin{verbatim}
337     diffKhT=4.E2,
338     \end{verbatim}
339     this line sets the horizontal diffusion coefficient for temperature
340     to $400\,{\rm m^{2}s^{-1}}$. The boundary condition on this
341     operator is $\frac{\partial}{\partial x}=\frac{\partial}{\partial y}=0$ at
342     all boundaries.
343    
344     \item Line 11,
345     \begin{verbatim}
346     diffKzT=1.E-2,
347     \end{verbatim}
348     this line sets the vertical diffusion coefficient for temperature
349     to $10^{-2}\,{\rm m^{2}s^{-1}}$. The boundary condition on this
350     operator is $\frac{\partial}{\partial z}$ = 0 on all boundaries.
351    
352     \item Line 13,
353     \begin{verbatim}
354     tAlpha=2.E-4,
355     \end{verbatim}
356     This line sets the thermal expansion coefficient for the fluid
357     to $2 \times 10^{-4}\,{\rm degrees}^{-1}$
358    
359     \fbox{
360     \begin{minipage}{5.0in}
361     {\it S/R FIND\_RHO}({\it find\_rho.F})
362     \end{minipage}
363     }
364    
365     \item Line 18,
366     \begin{verbatim}
367     eosType='LINEAR'
368     \end{verbatim}
369     This line selects the linear form of the equation of state.
370    
371     \fbox{
372     \begin{minipage}{5.0in}
373     {\it S/R FIND\_RHO}({\it find\_rho.F})
374     \end{minipage}
375     }
376    
377    
378    
379     \item Line 40,
380     \begin{verbatim}
381     usingSphericalPolarGrid=.TRUE.,
382     \end{verbatim}
383     This line requests that the simulation be performed in a
384     spherical polar coordinate system. It affects the interpretation of
385     grid inoput parameters, for exampl {\bf delX} and {\bf delY} and
386     causes the grid generation routines to initialise an internal grid based
387     on spherical polar geometry.
388    
389     \fbox{
390     \begin{minipage}{5.0in}
391     {\it S/R INI\_SPEHRICAL\_POLAR\_GRID}({\it ini\_spherical\_polar\_grid.F})
392     \end{minipage}
393     }
394    
395     \item Line 41,
396     \begin{verbatim}
397     phiMin=0.,
398     \end{verbatim}
399     This line sets the southern boundary of the modeled
400     domain to $0^{\circ}$ latitude. This value affects both the
401     generation of the locally orthogonal grid that the model
402     uses internally and affects the initialisation of the coriolis force.
403     Note - it is not required to set
404     a longitude boundary, since the absolute longitude does
405     not alter the kernel equation discretisation.
406    
407     \item Line 42,
408     \begin{verbatim}
409     delX=60*1.,
410     \end{verbatim}
411     This line sets the horizontal grid spacing between each y-coordinate line
412     in the discrete grid to $1^{\circ}$ in longitude.
413    
414     \item Line 43,
415     \begin{verbatim}
416     delY=60*1.,
417     \end{verbatim}
418     This line sets the horizontal grid spacing between each y-coordinate line
419     in the discrete grid to $1^{\circ}$ in latitude.
420    
421     \item Line 44,
422     \begin{verbatim}
423     delZ=500.,500.,500.,500.,
424     \end{verbatim}
425     This line sets the vertical grid spacing between each z-coordinate line
426     in the discrete grid to $500\,{\rm m}$, so that the total model depth
427     is $2\,{\rm km}$. The variable {\bf delZ} is copied into the internal
428     model coordinate variable {\bf delR}
429    
430     \fbox{
431     \begin{minipage}{5.0in}
432     {\it S/R INI\_VERTICAL\_GRID}({\it ini\_vertical\_grid.F})
433     \end{minipage}
434     }
435    
436     \item Line 47,
437     \begin{verbatim}
438     bathyFile='topog.box'
439     \end{verbatim}
440     This line specifies the name of the file from which the domain
441     bathymetry is read. This file is a two-dimensional ($x,y$) map of
442     depths. This file is assumed to contain 64-bit binary numbers
443     giving the depth of the model at each grid cell, ordered with the x
444     coordinate varying fastest. The points are ordered from low coordinate
445     to high coordinate for both axes. The units and orientation of the
446     depths in this file are the same as used in the MITgcm code. In this
447     experiment, a depth of $0m$ indicates a solid wall and a depth
448     of $-2000m$ indicates open ocean. The matlab program
449     {\it input/gendata.m} shows an example of how to generate a
450     bathymetry file.
451    
452    
453     \item Line 50,
454     \begin{verbatim}
455     zonalWindFile='windx.sin_y'
456     \end{verbatim}
457     This line specifies the name of the file from which the x-direction
458     surface wind stress is read. This file is also a two-dimensional
459     ($x,y$) map and is enumerated and formatted in the same manner as the
460     bathymetry file. The matlab program {\it input/gendata.m} includes example
461     code to generate a valid
462     {\bf zonalWindFile}
463     file.
464    
465     \end{itemize}
466    
467     \noindent other lines in the file {\it input/data} are standard values
468     that are described in the MITgcm Getting Started and MITgcm Parameters
469     notes.
470    
471     \begin{small}
472     % \input{part3/case_studies/carbon_outgassing_sensitivity/input/data}
473     \end{small}
474    
475     \subsubsection{File {\it input/data.pkg}}
476    
477     This file uses standard default values and does not contain
478     customisations for this experiment.
479    
480     \subsubsection{File {\it input/eedata}}
481    
482     This file uses standard default values and does not contain
483     customisations for this experiment.
484    
485     \subsubsection{File {\it input/windx.sin\_y}}
486    
487     The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
488     map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.
489     Although $\tau_{x}$ is only a function of $y$n in this experiment
490     this file must still define a complete two-dimensional map in order
491     to be compatible with the standard code for loading forcing fields
492     in MITgcm. The included matlab program {\it input/gendata.m} gives a complete
493     code for creating the {\it input/windx.sin\_y} file.
494    
495     \subsubsection{File {\it input/topog.box}}
496    
497    
498     The {\it input/topog.box} file specifies a two-dimensional ($x,y$)
499     map of depth values. For this experiment values are either
500     $0m$ or $-2000\,{\rm m}$, corresponding respectively to a wall or to deep
501     ocean. The file contains a raw binary stream of data that is enumerated
502     in the same way as standard MITgcm two-dimensional, horizontal arrays.
503     The included matlab program {\it input/gendata.m} gives a complete
504     code for creating the {\it input/topog.box} file.
505    
506     \subsubsection{File {\it code/SIZE.h}}
507    
508     Two lines are customized in this file for the current experiment
509    
510     \begin{itemize}
511    
512     \item Line 39,
513     \begin{verbatim} sNx=60, \end{verbatim} this line sets
514     the lateral domain extent in grid points for the
515     axis aligned with the x-coordinate.
516    
517     \item Line 40,
518     \begin{verbatim} sNy=60, \end{verbatim} this line sets
519     the lateral domain extent in grid points for the
520     axis aligned with the y-coordinate.
521    
522     \item Line 49,
523     \begin{verbatim} Nr=4, \end{verbatim} this line sets
524     the vertical domain extent in grid points.
525    
526     \end{itemize}
527    
528     \begin{small}
529     % \include{code/SIZE.h}
530     \end{small}
531    
532     \subsubsection{File {\it code/CPP\_OPTIONS.h}}
533    
534     This file uses standard default values and does not contain
535     customisations for this experiment.
536    
537    
538     \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
539    
540     This file uses standard default values and does not contain
541     customisations for this experiment.
542    
543     \subsubsection{Other Files }
544    
545     Other files relevant to this experiment are
546     \begin{itemize}
547     \item {\it model/src/ini\_cori.F}. This file initializes the model
548     coriolis variables {\bf fCorU}.
549     \item {\it model/src/ini\_spherical\_polar\_grid.F}
550     \item {\it model/src/ini\_parms.F},
551     \item {\it input/windx.sin\_y},
552     \end{itemize}
553     contain the code customisations and parameter settings for this
554     experiements. Below we describe the customisations
555     to these files associated with this experiment.

  ViewVC Help
Powered by ViewVC 1.1.22