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# Line 1  Line 1 
1  % $Header$  % $Header$
2  % $Name$  % $Name$
3    
4  \section{Example: Centenial Time Scale Sensitivities}  \section{Centennial Time Scale Tracer Injection}
5    \label{www:tutorials}
6    \label{sect:eg-simple-tracer}
7    \begin{rawhtml}
8    <!-- CMIREDIR:eg-simple-tracer: -->
9    \end{rawhtml}
10    
11  \bodytext{bgcolor="#FFFFFFFF"}  \bodytext{bgcolor="#FFFFFFFF"}
12    
13  %\begin{center}  %\begin{center}
14  %{\Large \bf Using MITgcm to Look at Centenial Time Scale  %{\Large \bf Using MITgcm to Look at Centennial Time Scale
15  %Sensitivities}  %Sensitivities}
16  %  %
17  %\vspace*{4mm}  %\vspace*{4mm}
# Line 16  Line 21 
21  %\end{center}  %\end{center}
22    
23  \subsection{Introduction}  \subsection{Introduction}
24    \label{www:tutorials}
25    
26  This document describes the fourth example MITgcm experiment.  This document describes the fourth example MITgcm experiment.
27  This example iilustrates the use of  This example illustrates the use of
28  the MITgcm to perform sentivity analysis in a  the MITgcm to perform sensitivity analysis in a
29  large scale ocean circulation simulation.  large scale ocean circulation simulation.
30    
31  \subsection{Overview}  \subsection{Overview}
32    \label{www:tutorials}
33    
34  This example experiment demonstrates using the MITgcm to simulate  This example experiment demonstrates using the MITgcm to simulate
35  the planetary ocean circulation. The simulation is configured  the planetary ocean circulation. The simulation is configured
# Line 36  can be integrated forward for thousands Line 43  can be integrated forward for thousands
43  processor desktop computer.  processor desktop computer.
44  \\  \\
45    
46  The model is forced with climatalogical wind stress data and surface  The model is forced with climatological wind stress data and surface
47  flux data from Da Silva \cite{DaSilva94}. Climatalogical data  flux data from Da Silva \cite{DaSilva94}. Climatological data
48  from Levitus \cite{Levitus94} is used to initialise the model hydrography.  from Levitus \cite{Levitus94} is used to initialize the model hydrography.
49  Levitus data is also used throughout the calculation  Levitus data is also used throughout the calculation
50  to derive air-sea fluxes of heat at the ocean surface.  to derive air-sea fluxes of heat at the ocean surface.
51  These fluxes are combined with climatalogical estimates of  These fluxes are combined with climatological estimates of
52  surface heat flux and fresh water, resulting in a mixed boundary  surface heat flux and fresh water, resulting in a mixed boundary
53  condition of the style decribed in Haney \cite{Haney}.  condition of the style described in Haney \cite{Haney}.
54  Altogether, this yields the following forcing applied  Altogether, this yields the following forcing applied
55  in the model surface layer.  in the model surface layer.
56    
57  \begin{eqnarray}  \begin{eqnarray}
58  \label{EQ:global_forcing}  \label{EQ:eg-simple-tracer-global_forcing}
59  \label{EQ:global_forcing_fu}  \label{EQ:eg-simple-tracer-global_forcing_fu}
60  {\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}}  {\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}}
61  \\  \\
62  \label{EQ:global_forcing_fv}  \label{EQ:eg-simple-tracer-global_forcing_fv}
63  {\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}}  {\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}}
64  \\  \\
65  \label{EQ:global_forcing_ft}  \label{EQ:eg-simple-tracer-global_forcing_ft}
66  {\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} )  {\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} )
67   - \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q}   - \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q}
68  \\  \\
69  \label{EQ:global_forcing_fs}  \label{EQ:eg-simple-tracer-global_forcing_fs}
70  {\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} )  {\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} )
71   + \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R})   + \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R})
72  \end{eqnarray}  \end{eqnarray}
# Line 81  The configuration is illustrated in figu Line 88  The configuration is illustrated in figu
88    
89    
90  \subsection{Discrete Numerical Configuration}  \subsection{Discrete Numerical Configuration}
91    \label{www:tutorials}
92    
93    
94   The model is configured in hydrostatic form.  The domain is discretised with   The model is configured in hydrostatic form.  The domain is discretised with
# Line 118  $ Line 126  $
126   \Delta z_{20}=815\,{\rm m}   \Delta z_{20}=815\,{\rm m}
127  $ (here the numeric subscript indicates the model level index number, ${\tt k}$).  $ (here the numeric subscript indicates the model level index number, ${\tt k}$).
128  The implicit free surface form of the pressure equation described in Marshall et. al  The implicit free surface form of the pressure equation described in Marshall et. al
129  \cite{Marshall97a} is employed. A laplacian operator, $\nabla^2$, provides viscous  \cite{marshall:97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous
130  dissipation. Thermal and haline diffusion is also represented by a laplacian operator.  dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.
131  \\  \\
132    
133  Wind-stress momentum inputs are added to the momentum equations for both  Wind-stress momentum inputs are added to the momentum equations for both
134  the zonal flow, $u$ and the merdional flow $v$, according to equations  the zonal flow, $u$ and the meridional flow $v$, according to equations
135  (\ref{EQ:global_forcing_fu}) and (\ref{EQ:global_forcing_fv}).  (\ref{EQ:eg-simple-tracer-global_forcing_fu}) and (\ref{EQ:eg-simple-tracer-global_forcing_fv}).
136  Thermodynamic forcing inputs are added to the equations for  Thermodynamic forcing inputs are added to the equations for
137  potential temperature, $\theta$, and salinity, $S$, according to equations  potential temperature, $\theta$, and salinity, $S$, according to equations
138  (\ref{EQ:global_forcing_ft}) and (\ref{EQ:global_forcing_fs}).  (\ref{EQ:eg-simple-tracer-global_forcing_ft}) and (\ref{EQ:eg-simple-tracer-global_forcing_fs}).
139  This produces a set of equations solved in this configuration as follows:  This produces a set of equations solved in this configuration as follows:
140  % {\fracktur}  % {\fracktur}
141    
142    
143  \begin{eqnarray}  \begin{eqnarray}
144  \label{EQ:model_equations}  \label{EQ:eg-simple-tracer-model_equations}
145  \frac{Du}{Dt} - fv +  \frac{Du}{Dt} - fv +
146    \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -    \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} -
147    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}}
# Line 167  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} d Line 175  g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} d
175  \noindent where $u$ and $v$ are the $x$ and $y$ components of the  \noindent where $u$ and $v$ are the $x$ and $y$ components of the
176  flow vector $\vec{u}$. The suffices ${s},{i}$ indicate surface and  flow vector $\vec{u}$. The suffices ${s},{i}$ indicate surface and
177  interior model levels respectively. As described in  interior model levels respectively. As described in
178  MITgcm Numerical Solution Procedure \cite{MITgcm_Numerical_Scheme}, the time  MITgcm Numerical Solution Procedure \ref{chap:discretization}, the time
179  evolution of potential temperature, $\theta$, equation is solved prognostically.  evolution of potential temperature, $\theta$, equation is solved prognostically.
180  The total pressure, $p$, is diagnosed by summing pressure due to surface  The total pressure, $p$, is diagnosed by summing pressure due to surface
181  elevation $\eta$ and the hydrostatic pressure.  elevation $\eta$ and the hydrostatic pressure.
182  \\  \\
183    
184  \subsubsection{Numerical Stability Criteria}  \subsubsection{Numerical Stability Criteria}
185    \label{www:tutorials}
186    
187  The laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.  The Laplacian dissipation coefficient, $A_{h}$, is set to $400 m s^{-1}$.
188  This value is chosen to yield a Munk layer width \cite{Adcroft_thesis},  This value is chosen to yield a Munk layer width \cite{adcroft:95},
189    
190  \begin{eqnarray}  \begin{eqnarray}
191  \label{EQ:munk_layer}  \label{EQ:eg-simple-tracer-munk_layer}
192  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}  M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}
193  \end{eqnarray}  \end{eqnarray}
194    
# Line 190  boundary layer is well resolved. Line 199  boundary layer is well resolved.
199    
200  \noindent The model is stepped forward with a  \noindent The model is stepped forward with a
201  time step $\delta t=1200$secs. With this time step the stability  time step $\delta t=1200$secs. With this time step the stability
202  parameter to the horizontal laplacian friction \cite{Adcroft_thesis}  parameter to the horizontal Laplacian friction \cite{adcroft:95}
203    
204  \begin{eqnarray}  \begin{eqnarray}
205  \label{EQ:laplacian_stability}  \label{EQ:eg-simple-tracer-laplacian_stability}
206  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}  S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}
207  \end{eqnarray}  \end{eqnarray}
208    
# Line 205  for stability. Line 214  for stability.
214  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit  $1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit
215    
216  \begin{eqnarray}  \begin{eqnarray}
217  \label{EQ:laplacian_stability_z}  \label{EQ:eg-simple-tracer-laplacian_stability_z}
218  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}  S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}
219  \end{eqnarray}  \end{eqnarray}
220    
# Line 216  and vertical ($K_{z}$) diffusion coeffic Line 225  and vertical ($K_{z}$) diffusion coeffic
225  \\  \\
226    
227  \noindent The numerical stability for inertial oscillations  \noindent The numerical stability for inertial oscillations
228  \cite{Adcroft_thesis}  \cite{adcroft:95}
229    
230  \begin{eqnarray}  \begin{eqnarray}
231  \label{EQ:inertial_stability}  \label{EQ:eg-simple-tracer-inertial_stability}
232  S_{i} = f^{2} {\delta t}^2  S_{i} = f^{2} {\delta t}^2
233  \end{eqnarray}  \end{eqnarray}
234    
# Line 227  S_{i} = f^{2} {\delta t}^2 Line 236  S_{i} = f^{2} {\delta t}^2
236  limit for stability.  limit for stability.
237  \\  \\
238    
239  \noindent The advective CFL \cite{Adcroft_thesis} for a extreme maximum  \noindent The advective CFL \cite{adcroft:95} for a extreme maximum
240  horizontal flow  horizontal flow
241  speed of $ | \vec{u} | = 2 ms^{-1}$  speed of $ | \vec{u} | = 2 ms^{-1}$
242    
243  \begin{eqnarray}  \begin{eqnarray}
244  \label{EQ:cfl_stability}  \label{EQ:eg-simple-tracer-cfl_stability}
245  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}  S_{a} = \frac{| \vec{u} | \delta t}{ \Delta x}
246  \end{eqnarray}  \end{eqnarray}
247    
# Line 240  S_{a} = \frac{| \vec{u} | \delta t}{ \De Line 249  S_{a} = \frac{| \vec{u} | \delta t}{ \De
249  limit of 0.5.  limit of 0.5.
250  \\  \\
251    
252  \noindent The stbility parameter for internal gravity waves  \noindent The stability parameter for internal gravity waves
253  \cite{Adcroft_thesis}  \cite{adcroft:95}
254    
255  \begin{eqnarray}  \begin{eqnarray}
256  \label{EQ:cfl_stability}  \label{EQ:eg-simple-tracer-igw_stability}
257  S_{c} = \frac{c_{g} \delta t}{ \Delta x}  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
258  \end{eqnarray}  \end{eqnarray}
259    
# Line 252  S_{c} = \frac{c_{g} \delta t}{ \Delta x} Line 261  S_{c} = \frac{c_{g} \delta t}{ \Delta x}
261  stability limit of 0.25.  stability limit of 0.25.
262        
263  \subsection{Code Configuration}  \subsection{Code Configuration}
264    \label{www:tutorials}
265  \label{SEC:code_config}  \label{SEC:code_config}
266    
267  The model configuration for this experiment resides under the  The model configuration for this experiment resides under the
# Line 266  directory {\it verification/exp1/}.  The Line 276  directory {\it verification/exp1/}.  The
276  \item {\it code/CPP\_OPTIONS.h},  \item {\it code/CPP\_OPTIONS.h},
277  \item {\it code/SIZE.h}.  \item {\it code/SIZE.h}.
278  \end{itemize}  \end{itemize}
279  contain the code customisations and parameter settings for this  contain the code customizations and parameter settings for this
280  experiements. Below we describe the customisations  experiments. Below we describe the customizations
281  to these files associated with this experiment.  to these files associated with this experiment.
282    
283  \subsubsection{File {\it input/data}}  \subsubsection{File {\it input/data}}
284    \label{www:tutorials}
285    
286  This file, reproduced completely below, specifies the main parameters  This file, reproduced completely below, specifies the main parameters
287  for the experiment. The parameters that are significant for this configuration  for the experiment. The parameters that are significant for this configuration
# Line 282  are Line 293  are
293  \begin{verbatim} tRef=20.,10.,8.,6., \end{verbatim}  \begin{verbatim} tRef=20.,10.,8.,6., \end{verbatim}
294  this line sets  this line sets
295  the initial and reference values of potential temperature at each model  the initial and reference values of potential temperature at each model
296  level in units of $^{\circ}$C.  level in units of $^{\circ}\mathrm{C}$.
297  The entries are ordered from surface to depth. For each  The entries are ordered from surface to depth. For each
298  depth level the inital and reference profiles will be uniform in  depth level the initial and reference profiles will be uniform in
299  $x$ and $y$.  $x$ and $y$.
300    
301  \fbox{  \fbox{
# Line 296  $x$ and $y$. Line 307  $x$ and $y$.
307    
308  \item Line 6,  \item Line 6,
309  \begin{verbatim} viscAz=1.E-2, \end{verbatim}  \begin{verbatim} viscAz=1.E-2, \end{verbatim}
310  this line sets the vertical laplacian dissipation coefficient to  this line sets the vertical Laplacian dissipation coefficient to
311  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions
312  for this operator are specified later. This variable is copied into  for this operator are specified later. This variable is copied into
313  model general vertical coordinate variable {\bf viscAr}.  model general vertical coordinate variable {\bf viscAr}.
# Line 311  model general vertical coordinate variab Line 322  model general vertical coordinate variab
322  \begin{verbatim}  \begin{verbatim}
323  viscAh=4.E2,  viscAh=4.E2,
324  \end{verbatim}  \end{verbatim}
325  this line sets the horizontal laplacian frictional dissipation coefficient to  this line sets the horizontal Laplacian frictional dissipation coefficient to
326  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions  $1 \times 10^{-2} {\rm m^{2}s^{-1}}$. Boundary conditions
327  for this operator are specified later.  for this operator are specified later.
328    
# Line 320  for this operator are specified later. Line 331  for this operator are specified later.
331  no_slip_sides=.FALSE.  no_slip_sides=.FALSE.
332  \end{verbatim}  \end{verbatim}
333  this line selects a free-slip lateral boundary condition for  this line selects a free-slip lateral boundary condition for
334  the horizontal laplacian friction operator  the horizontal Laplacian friction operator
335  e.g. $\frac{\partial u}{\partial y}$=0 along boundaries in $y$ and  e.g. $\frac{\partial u}{\partial y}$=0 along boundaries in $y$ and
336  $\frac{\partial v}{\partial x}$=0 along boundaries in $x$.  $\frac{\partial v}{\partial x}$=0 along boundaries in $x$.
337    
# Line 329  $\frac{\partial v}{\partial x}$=0 along Line 340  $\frac{\partial v}{\partial x}$=0 along
340  no_slip_bottom=.TRUE.  no_slip_bottom=.TRUE.
341  \end{verbatim}  \end{verbatim}
342  this line selects a no-slip boundary condition for bottom  this line selects a no-slip boundary condition for bottom
343  boundary condition in the vertical laplacian friction operator  boundary condition in the vertical Laplacian friction operator
344  e.g. $u=v=0$ at $z=-H$, where $H$ is the local depth of the domain.  e.g. $u=v=0$ at $z=-H$, where $H$ is the local depth of the domain.
345    
346  \item Line 10,  \item Line 10,
# Line 382  usingSphericalPolarGrid=.TRUE., Line 393  usingSphericalPolarGrid=.TRUE.,
393  \end{verbatim}  \end{verbatim}
394  This line requests that the simulation be performed in a  This line requests that the simulation be performed in a
395  spherical polar coordinate system. It affects the interpretation of  spherical polar coordinate system. It affects the interpretation of
396  grid inoput parameters, for exampl {\bf delX} and {\bf delY} and  grid input parameters, for example {\bf delX} and {\bf delY} and
397  causes the grid generation routines to initialise an internal grid based  causes the grid generation routines to initialize an internal grid based
398  on spherical polar geometry.  on spherical polar geometry.
399    
400  \fbox{  \fbox{
# Line 399  phiMin=0., Line 410  phiMin=0.,
410  This line sets the southern boundary of the modeled  This line sets the southern boundary of the modeled
411  domain to $0^{\circ}$ latitude. This value affects both the  domain to $0^{\circ}$ latitude. This value affects both the
412  generation of the locally orthogonal grid that the model  generation of the locally orthogonal grid that the model
413  uses internally and affects the initialisation of the coriolis force.  uses internally and affects the initialization of the coriolis force.
414  Note - it is not required to set  Note - it is not required to set
415  a longitude boundary, since the absolute longitude does  a longitude boundary, since the absolute longitude does
416  not alter the kernel equation discretisation.  not alter the kernel equation discretisation.
# Line 473  notes. Line 484  notes.
484  \end{small}  \end{small}
485    
486  \subsubsection{File {\it input/data.pkg}}  \subsubsection{File {\it input/data.pkg}}
487    \label{www:tutorials}
488    
489  This file uses standard default values and does not contain  This file uses standard default values and does not contain
490  customisations for this experiment.  customizations for this experiment.
491    
492  \subsubsection{File {\it input/eedata}}  \subsubsection{File {\it input/eedata}}
493    \label{www:tutorials}
494    
495  This file uses standard default values and does not contain  This file uses standard default values and does not contain
496  customisations for this experiment.  customizations for this experiment.
497    
498  \subsubsection{File {\it input/windx.sin\_y}}  \subsubsection{File {\it input/windx.sin\_y}}
499    \label{www:tutorials}
500    
501  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)  The {\it input/windx.sin\_y} file specifies a two-dimensional ($x,y$)
502  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.  map of wind stress ,$\tau_{x}$, values. The units used are $Nm^{-2}$.
# Line 493  in MITgcm. The included matlab program { Line 507  in MITgcm. The included matlab program {
507  code for creating the {\it input/windx.sin\_y} file.  code for creating the {\it input/windx.sin\_y} file.
508    
509  \subsubsection{File {\it input/topog.box}}  \subsubsection{File {\it input/topog.box}}
510    \label{www:tutorials}
511    
512    
513  The {\it input/topog.box} file specifies a two-dimensional ($x,y$)  The {\it input/topog.box} file specifies a two-dimensional ($x,y$)
# Line 504  The included matlab program {\it input/g Line 519  The included matlab program {\it input/g
519  code for creating the {\it input/topog.box} file.  code for creating the {\it input/topog.box} file.
520    
521  \subsubsection{File {\it code/SIZE.h}}  \subsubsection{File {\it code/SIZE.h}}
522    \label{www:tutorials}
523    
524  Two lines are customized in this file for the current experiment  Two lines are customized in this file for the current experiment
525    
# Line 530  the vertical domain extent in grid point Line 546  the vertical domain extent in grid point
546  \end{small}  \end{small}
547    
548  \subsubsection{File {\it code/CPP\_OPTIONS.h}}  \subsubsection{File {\it code/CPP\_OPTIONS.h}}
549    \label{www:tutorials}
550    
551  This file uses standard default values and does not contain  This file uses standard default values and does not contain
552  customisations for this experiment.  customizations for this experiment.
553    
554    
555  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}  \subsubsection{File {\it code/CPP\_EEOPTIONS.h}}
556    \label{www:tutorials}
557    
558  This file uses standard default values and does not contain  This file uses standard default values and does not contain
559  customisations for this experiment.  customizations for this experiment.
560    
561  \subsubsection{Other Files }  \subsubsection{Other Files }
562    \label{www:tutorials}
563    
564  Other files relevant to this experiment are  Other files relevant to this experiment are
565  \begin{itemize}  \begin{itemize}
# Line 550  coriolis variables {\bf fCorU}. Line 569  coriolis variables {\bf fCorU}.
569  \item {\it model/src/ini\_parms.F},  \item {\it model/src/ini\_parms.F},
570  \item {\it input/windx.sin\_y},  \item {\it input/windx.sin\_y},
571  \end{itemize}  \end{itemize}
572  contain the code customisations and parameter settings for this  contain the code customizations and parameter settings for this
573  experiements. Below we describe the customisations  experiments. Below we describe the customizations
574  to these files associated with this experiment.  to these files associated with this experiment.
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