| 79 |
|
|
| 80 |
\end{enumerate} |
\end{enumerate} |
| 81 |
|
|
| 82 |
\subsubsection{Checkout from CVS} |
\subsection{Method 1 - Checkout from CVS} |
| 83 |
\label{sect:cvs_checkout} |
\label{sect:cvs_checkout} |
| 84 |
|
|
| 85 |
If CVS is available on your system, we strongly encourage you to use it. CVS |
If CVS is available on your system, we strongly encourage you to use it. CVS |
| 169 |
\end{verbatim} |
\end{verbatim} |
| 170 |
|
|
| 171 |
|
|
| 172 |
\subsubsection{Conventional download method} |
\subsection{Method 2 - Tar file download} |
| 173 |
\label{sect:conventionalDownload} |
\label{sect:conventionalDownload} |
| 174 |
|
|
| 175 |
If you do not have CVS on your system, you can download the model as a |
If you do not have CVS on your system, you can download the model as a |
| 1059 |
>> for n=1:11; imagesc(eta(:,:,n)');axis ij;colorbar;pause(.5);end |
>> for n=1:11; imagesc(eta(:,:,n)');axis ij;colorbar;pause(.5);end |
| 1060 |
\end{verbatim} |
\end{verbatim} |
| 1061 |
|
|
|
\section[Customizing MITgcm]{Doing it yourself: customizing the code} |
|
|
|
|
|
When you are ready to run the model in the configuration you want, the |
|
|
easiest thing is to use and adapt the setup of the case studies |
|
|
experiment (described previously) that is the closest to your |
|
|
configuration. Then, the amount of setup will be minimized. In this |
|
|
section, we focus on the setup relative to the ``numerical model'' |
|
|
part of the code (the setup relative to the ``execution environment'' |
|
|
part is covered in the parallel implementation section) and on the |
|
|
variables and parameters that you are likely to change. |
|
|
|
|
|
\subsection{Configuration and setup} |
|
|
|
|
|
The CPP keys relative to the ``numerical model'' part of the code are |
|
|
all defined and set in the file \textit{CPP\_OPTIONS.h }in the |
|
|
directory \textit{ model/inc }or in one of the \textit{code |
|
|
}directories of the case study experiments under |
|
|
\textit{verification.} The model parameters are defined and declared |
|
|
in the file \textit{model/inc/PARAMS.h }and their default values are |
|
|
set in the routine \textit{model/src/set\_defaults.F. }The default |
|
|
values can be modified in the namelist file \textit{data }which needs |
|
|
to be located in the directory where you will run the model. The |
|
|
parameters are initialized in the routine |
|
|
\textit{model/src/ini\_parms.F}. Look at this routine to see in what |
|
|
part of the namelist the parameters are located. |
|
|
|
|
|
In what follows the parameters are grouped into categories related to |
|
|
the computational domain, the equations solved in the model, and the |
|
|
simulation controls. |
|
|
|
|
|
\subsection{Computational domain, geometry and time-discretization} |
|
|
|
|
|
\begin{description} |
|
|
\item[dimensions] \ |
|
|
|
|
|
The number of points in the x, y, and r directions are represented |
|
|
by the variables \textbf{sNx}, \textbf{sNy} and \textbf{Nr} |
|
|
respectively which are declared and set in the file |
|
|
\textit{model/inc/SIZE.h}. (Again, this assumes a mono-processor |
|
|
calculation. For multiprocessor calculations see the section on |
|
|
parallel implementation.) |
|
|
|
|
|
\item[grid] \ |
|
|
|
|
|
Three different grids are available: cartesian, spherical polar, and |
|
|
curvilinear (which includes the cubed sphere). The grid is set |
|
|
through the logical variables \textbf{usingCartesianGrid}, |
|
|
\textbf{usingSphericalPolarGrid}, and \textbf{usingCurvilinearGrid}. |
|
|
In the case of spherical and curvilinear grids, the southern |
|
|
boundary is defined through the variable \textbf{phiMin} which |
|
|
corresponds to the latitude of the southern most cell face (in |
|
|
degrees). The resolution along the x and y directions is controlled |
|
|
by the 1D arrays \textbf{delx} and \textbf{dely} (in meters in the |
|
|
case of a cartesian grid, in degrees otherwise). The vertical grid |
|
|
spacing is set through the 1D array \textbf{delz} for the ocean (in |
|
|
meters) or \textbf{delp} for the atmosphere (in Pa). The variable |
|
|
\textbf{Ro\_SeaLevel} represents the standard position of Sea-Level |
|
|
in ``R'' coordinate. This is typically set to 0m for the ocean |
|
|
(default value) and 10$^{5}$Pa for the atmosphere. For the |
|
|
atmosphere, also set the logical variable \textbf{groundAtK1} to |
|
|
\texttt{'.TRUE.'} which puts the first level (k=1) at the lower |
|
|
boundary (ground). |
|
|
|
|
|
For the cartesian grid case, the Coriolis parameter $f$ is set |
|
|
through the variables \textbf{f0} and \textbf{beta} which correspond |
|
|
to the reference Coriolis parameter (in s$^{-1}$) and |
|
|
$\frac{\partial f}{ \partial y}$(in m$^{-1}$s$^{-1}$) respectively. |
|
|
If \textbf{beta } is set to a nonzero value, \textbf{f0} is the |
|
|
value of $f$ at the southern edge of the domain. |
|
|
|
|
|
\item[topography - full and partial cells] \ |
|
|
|
|
|
The domain bathymetry is read from a file that contains a 2D (x,y) |
|
|
map of depths (in m) for the ocean or pressures (in Pa) for the |
|
|
atmosphere. The file name is represented by the variable |
|
|
\textbf{bathyFile}. The file is assumed to contain binary numbers |
|
|
giving the depth (pressure) of the model at each grid cell, ordered |
|
|
with the x coordinate varying fastest. The points are ordered from |
|
|
low coordinate to high coordinate for both axes. The model code |
|
|
applies without modification to enclosed, periodic, and double |
|
|
periodic domains. Periodicity is assumed by default and is |
|
|
suppressed by setting the depths to 0m for the cells at the limits |
|
|
of the computational domain (note: not sure this is the case for the |
|
|
atmosphere). The precision with which to read the binary data is |
|
|
controlled by the integer variable \textbf{readBinaryPrec} which can |
|
|
take the value \texttt{32} (single precision) or \texttt{64} (double |
|
|
precision). See the matlab program \textit{gendata.m} in the |
|
|
\textit{input} directories under \textit{verification} to see how |
|
|
the bathymetry files are generated for the case study experiments. |
|
|
|
|
|
To use the partial cell capability, the variable \textbf{hFacMin} |
|
|
needs to be set to a value between 0 and 1 (it is set to 1 by |
|
|
default) corresponding to the minimum fractional size of the cell. |
|
|
For example if the bottom cell is 500m thick and \textbf{hFacMin} is |
|
|
set to 0.1, the actual thickness of the cell (i.e. used in the code) |
|
|
can cover a range of discrete values 50m apart from 50m to 500m |
|
|
depending on the value of the bottom depth (in \textbf{bathyFile}) |
|
|
at this point. |
|
|
|
|
|
Note that the bottom depths (or pressures) need not coincide with |
|
|
the models levels as deduced from \textbf{delz} or \textbf{delp}. |
|
|
The model will interpolate the numbers in \textbf{bathyFile} so that |
|
|
they match the levels obtained from \textbf{delz} or \textbf{delp} |
|
|
and \textbf{hFacMin}. |
|
|
|
|
|
(Note: the atmospheric case is a bit more complicated than what is |
|
|
written here I think. To come soon...) |
|
|
|
|
|
\item[time-discretization] \ |
|
|
|
|
|
The time steps are set through the real variables \textbf{deltaTMom} |
|
|
and \textbf{deltaTtracer} (in s) which represent the time step for |
|
|
the momentum and tracer equations, respectively. For synchronous |
|
|
integrations, simply set the two variables to the same value (or you |
|
|
can prescribe one time step only through the variable |
|
|
\textbf{deltaT}). The Adams-Bashforth stabilizing parameter is set |
|
|
through the variable \textbf{abEps} (dimensionless). The stagger |
|
|
baroclinic time stepping can be activated by setting the logical |
|
|
variable \textbf{staggerTimeStep} to \texttt{'.TRUE.'}. |
|
|
|
|
|
\end{description} |
|
|
|
|
|
|
|
|
\subsection{Equation of state} |
|
|
|
|
|
First, because the model equations are written in terms of |
|
|
perturbations, a reference thermodynamic state needs to be specified. |
|
|
This is done through the 1D arrays \textbf{tRef} and \textbf{sRef}. |
|
|
\textbf{tRef} specifies the reference potential temperature profile |
|
|
(in $^{o}$C for the ocean and $^{o}$K for the atmosphere) starting |
|
|
from the level k=1. Similarly, \textbf{sRef} specifies the reference |
|
|
salinity profile (in ppt) for the ocean or the reference specific |
|
|
humidity profile (in g/kg) for the atmosphere. |
|
|
|
|
|
The form of the equation of state is controlled by the character |
|
|
variables \textbf{buoyancyRelation} and \textbf{eosType}. |
|
|
\textbf{buoyancyRelation} is set to \texttt{'OCEANIC'} by default and |
|
|
needs to be set to \texttt{'ATMOSPHERIC'} for atmosphere simulations. |
|
|
In this case, \textbf{eosType} must be set to \texttt{'IDEALGAS'}. |
|
|
For the ocean, two forms of the equation of state are available: |
|
|
linear (set \textbf{eosType} to \texttt{'LINEAR'}) and a polynomial |
|
|
approximation to the full nonlinear equation ( set \textbf{eosType} to |
|
|
\texttt{'POLYNOMIAL'}). In the linear case, you need to specify the |
|
|
thermal and haline expansion coefficients represented by the variables |
|
|
\textbf{tAlpha} (in K$^{-1}$) and \textbf{sBeta} (in ppt$^{-1}$). For |
|
|
the nonlinear case, you need to generate a file of polynomial |
|
|
coefficients called \textit{POLY3.COEFFS}. To do this, use the program |
|
|
\textit{utils/knudsen2/knudsen2.f} under the model tree (a Makefile is |
|
|
available in the same directory and you will need to edit the number |
|
|
and the values of the vertical levels in \textit{knudsen2.f} so that |
|
|
they match those of your configuration). |
|
|
|
|
|
There there are also higher polynomials for the equation of state: |
|
|
\begin{description} |
|
|
\item[\texttt{'UNESCO'}:] The UNESCO equation of state formula of |
|
|
Fofonoff and Millard \cite{fofonoff83}. This equation of state |
|
|
assumes in-situ temperature, which is not a model variable; {\em its |
|
|
use is therefore discouraged, and it is only listed for |
|
|
completeness}. |
|
|
\item[\texttt{'JMD95Z'}:] A modified UNESCO formula by Jackett and |
|
|
McDougall \cite{jackett95}, which uses the model variable potential |
|
|
temperature as input. The \texttt{'Z'} indicates that this equation |
|
|
of state uses a horizontally and temporally constant pressure |
|
|
$p_{0}=-g\rho_{0}z$. |
|
|
\item[\texttt{'JMD95P'}:] A modified UNESCO formula by Jackett and |
|
|
McDougall \cite{jackett95}, which uses the model variable potential |
|
|
temperature as input. The \texttt{'P'} indicates that this equation |
|
|
of state uses the actual hydrostatic pressure of the last time |
|
|
step. Lagging the pressure in this way requires an additional pickup |
|
|
file for restarts. |
|
|
\item[\texttt{'MDJWF'}:] The new, more accurate and less expensive |
|
|
equation of state by McDougall et~al. \cite{mcdougall03}. It also |
|
|
requires lagging the pressure and therefore an additional pickup |
|
|
file for restarts. |
|
|
\end{description} |
|
|
For none of these options an reference profile of temperature or |
|
|
salinity is required. |
|
|
|
|
|
\subsection{Momentum equations} |
|
|
|
|
|
In this section, we only focus for now on the parameters that you are |
|
|
likely to change, i.e. the ones relative to forcing and dissipation |
|
|
for example. The details relevant to the vector-invariant form of the |
|
|
equations and the various advection schemes are not covered for the |
|
|
moment. We assume that you use the standard form of the momentum |
|
|
equations (i.e. the flux-form) with the default advection scheme. |
|
|
Also, there are a few logical variables that allow you to turn on/off |
|
|
various terms in the momentum equation. These variables are called |
|
|
\textbf{momViscosity, momAdvection, momForcing, useCoriolis, |
|
|
momPressureForcing, momStepping} and \textbf{metricTerms }and are |
|
|
assumed to be set to \texttt{'.TRUE.'} here. Look at the file |
|
|
\textit{model/inc/PARAMS.h }for a precise definition of these |
|
|
variables. |
|
|
|
|
|
\begin{description} |
|
|
\item[initialization] \ |
|
|
|
|
|
The velocity components are initialized to 0 unless the simulation |
|
|
is starting from a pickup file (see section on simulation control |
|
|
parameters). |
|
|
|
|
|
\item[forcing] \ |
|
|
|
|
|
This section only applies to the ocean. You need to generate |
|
|
wind-stress data into two files \textbf{zonalWindFile} and |
|
|
\textbf{meridWindFile} corresponding to the zonal and meridional |
|
|
components of the wind stress, respectively (if you want the stress |
|
|
to be along the direction of only one of the model horizontal axes, |
|
|
you only need to generate one file). The format of the files is |
|
|
similar to the bathymetry file. The zonal (meridional) stress data |
|
|
are assumed to be in Pa and located at U-points (V-points). As for |
|
|
the bathymetry, the precision with which to read the binary data is |
|
|
controlled by the variable \textbf{readBinaryPrec}. See the matlab |
|
|
program \textit{gendata.m} in the \textit{input} directories under |
|
|
\textit{verification} to see how simple analytical wind forcing data |
|
|
are generated for the case study experiments. |
|
|
|
|
|
There is also the possibility of prescribing time-dependent periodic |
|
|
forcing. To do this, concatenate the successive time records into a |
|
|
single file (for each stress component) ordered in a (x,y,t) fashion |
|
|
and set the following variables: \textbf{periodicExternalForcing }to |
|
|
\texttt{'.TRUE.'}, \textbf{externForcingPeriod }to the period (in s) |
|
|
of which the forcing varies (typically 1 month), and |
|
|
\textbf{externForcingCycle} to the repeat time (in s) of the forcing |
|
|
(typically 1 year -- note: \textbf{ externForcingCycle} must be a |
|
|
multiple of \textbf{externForcingPeriod}). With these variables set |
|
|
up, the model will interpolate the forcing linearly at each |
|
|
iteration. |
|
|
|
|
|
\item[dissipation] \ |
|
|
|
|
|
The lateral eddy viscosity coefficient is specified through the |
|
|
variable \textbf{viscAh} (in m$^{2}$s$^{-1}$). The vertical eddy |
|
|
viscosity coefficient is specified through the variable |
|
|
\textbf{viscAz} (in m$^{2}$s$^{-1}$) for the ocean and |
|
|
\textbf{viscAp} (in Pa$^{2}$s$^{-1}$) for the atmosphere. The |
|
|
vertical diffusive fluxes can be computed implicitly by setting the |
|
|
logical variable \textbf{implicitViscosity }to \texttt{'.TRUE.'}. |
|
|
In addition, biharmonic mixing can be added as well through the |
|
|
variable \textbf{viscA4} (in m$^{4}$s$^{-1}$). On a spherical polar |
|
|
grid, you might also need to set the variable \textbf{cosPower} |
|
|
which is set to 0 by default and which represents the power of |
|
|
cosine of latitude to multiply viscosity. Slip or no-slip conditions |
|
|
at lateral and bottom boundaries are specified through the logical |
|
|
variables \textbf{no\_slip\_sides} and \textbf{no\_slip\_bottom}. If |
|
|
set to \texttt{'.FALSE.'}, free-slip boundary conditions are |
|
|
applied. If no-slip boundary conditions are applied at the bottom, a |
|
|
bottom drag can be applied as well. Two forms are available: linear |
|
|
(set the variable \textbf{bottomDragLinear} in s$ ^{-1}$) and |
|
|
quadratic (set the variable \textbf{bottomDragQuadratic} in |
|
|
m$^{-1}$). |
|
|
|
|
|
The Fourier and Shapiro filters are described elsewhere. |
|
|
|
|
|
\item[C-D scheme] \ |
|
|
|
|
|
If you run at a sufficiently coarse resolution, you will need the |
|
|
C-D scheme for the computation of the Coriolis terms. The |
|
|
variable\textbf{\ tauCD}, which represents the C-D scheme coupling |
|
|
timescale (in s) needs to be set. |
|
|
|
|
|
\item[calculation of pressure/geopotential] \ |
|
|
|
|
|
First, to run a non-hydrostatic ocean simulation, set the logical |
|
|
variable \textbf{nonHydrostatic} to \texttt{'.TRUE.'}. The pressure |
|
|
field is then inverted through a 3D elliptic equation. (Note: this |
|
|
capability is not available for the atmosphere yet.) By default, a |
|
|
hydrostatic simulation is assumed and a 2D elliptic equation is used |
|
|
to invert the pressure field. The parameters controlling the |
|
|
behaviour of the elliptic solvers are the variables |
|
|
\textbf{cg2dMaxIters} and \textbf{cg2dTargetResidual } for |
|
|
the 2D case and \textbf{cg3dMaxIters} and |
|
|
\textbf{cg3dTargetResidual} for the 3D case. You probably won't need to |
|
|
alter the default values (are we sure of this?). |
|
|
|
|
|
For the calculation of the surface pressure (for the ocean) or |
|
|
surface geopotential (for the atmosphere) you need to set the |
|
|
logical variables \textbf{rigidLid} and \textbf{implicitFreeSurface} |
|
|
(set one to \texttt{'.TRUE.'} and the other to \texttt{'.FALSE.'} |
|
|
depending on how you want to deal with the ocean upper or atmosphere |
|
|
lower boundary). |
|
|
|
|
|
\end{description} |
|
|
|
|
|
\subsection{Tracer equations} |
|
|
|
|
|
This section covers the tracer equations i.e. the potential |
|
|
temperature equation and the salinity (for the ocean) or specific |
|
|
humidity (for the atmosphere) equation. As for the momentum equations, |
|
|
we only describe for now the parameters that you are likely to change. |
|
|
The logical variables \textbf{tempDiffusion} \textbf{tempAdvection} |
|
|
\textbf{tempForcing}, and \textbf{tempStepping} allow you to turn |
|
|
on/off terms in the temperature equation (same thing for salinity or |
|
|
specific humidity with variables \textbf{saltDiffusion}, |
|
|
\textbf{saltAdvection} etc.). These variables are all assumed here to |
|
|
be set to \texttt{'.TRUE.'}. Look at file \textit{model/inc/PARAMS.h} |
|
|
for a precise definition. |
|
|
|
|
|
\begin{description} |
|
|
\item[initialization] \ |
|
|
|
|
|
The initial tracer data can be contained in the binary files |
|
|
\textbf{hydrogThetaFile} and \textbf{hydrogSaltFile}. These files |
|
|
should contain 3D data ordered in an (x,y,r) fashion with k=1 as the |
|
|
first vertical level. If no file names are provided, the tracers |
|
|
are then initialized with the values of \textbf{tRef} and |
|
|
\textbf{sRef} mentioned above (in the equation of state section). In |
|
|
this case, the initial tracer data are uniform in x and y for each |
|
|
depth level. |
|
|
|
|
|
\item[forcing] \ |
|
|
|
|
|
This part is more relevant for the ocean, the procedure for the |
|
|
atmosphere not being completely stabilized at the moment. |
|
|
|
|
|
A combination of fluxes data and relaxation terms can be used for |
|
|
driving the tracer equations. For potential temperature, heat flux |
|
|
data (in W/m$ ^{2}$) can be stored in the 2D binary file |
|
|
\textbf{surfQfile}. Alternatively or in addition, the forcing can |
|
|
be specified through a relaxation term. The SST data to which the |
|
|
model surface temperatures are restored to are supposed to be stored |
|
|
in the 2D binary file \textbf{thetaClimFile}. The corresponding |
|
|
relaxation time scale coefficient is set through the variable |
|
|
\textbf{tauThetaClimRelax} (in s). The same procedure applies for |
|
|
salinity with the variable names \textbf{EmPmRfile}, |
|
|
\textbf{saltClimFile}, and \textbf{tauSaltClimRelax} for freshwater |
|
|
flux (in m/s) and surface salinity (in ppt) data files and |
|
|
relaxation time scale coefficient (in s), respectively. Also for |
|
|
salinity, if the CPP key \textbf{USE\_NATURAL\_BCS} is turned on, |
|
|
natural boundary conditions are applied i.e. when computing the |
|
|
surface salinity tendency, the freshwater flux is multiplied by the |
|
|
model surface salinity instead of a constant salinity value. |
|
|
|
|
|
As for the other input files, the precision with which to read the |
|
|
data is controlled by the variable \textbf{readBinaryPrec}. |
|
|
Time-dependent, periodic forcing can be applied as well following |
|
|
the same procedure used for the wind forcing data (see above). |
|
|
|
|
|
\item[dissipation] \ |
|
|
|
|
|
Lateral eddy diffusivities for temperature and salinity/specific |
|
|
humidity are specified through the variables \textbf{diffKhT} and |
|
|
\textbf{diffKhS} (in m$^{2}$/s). Vertical eddy diffusivities are |
|
|
specified through the variables \textbf{diffKzT} and |
|
|
\textbf{diffKzS} (in m$^{2}$/s) for the ocean and \textbf{diffKpT |
|
|
}and \textbf{diffKpS} (in Pa$^{2}$/s) for the atmosphere. The |
|
|
vertical diffusive fluxes can be computed implicitly by setting the |
|
|
logical variable \textbf{implicitDiffusion} to \texttt{'.TRUE.'}. |
|
|
In addition, biharmonic diffusivities can be specified as well |
|
|
through the coefficients \textbf{diffK4T} and \textbf{diffK4S} (in |
|
|
m$^{4}$/s). Note that the cosine power scaling (specified through |
|
|
\textbf{cosPower}---see the momentum equations section) is applied to |
|
|
the tracer diffusivities (Laplacian and biharmonic) as well. The |
|
|
Gent and McWilliams parameterization for oceanic tracers is |
|
|
described in the package section. Finally, note that tracers can be |
|
|
also subject to Fourier and Shapiro filtering (see the corresponding |
|
|
section on these filters). |
|
|
|
|
|
\item[ocean convection] \ |
|
|
|
|
|
Two options are available to parameterize ocean convection: one is |
|
|
to use the convective adjustment scheme. In this case, you need to |
|
|
set the variable \textbf{cadjFreq}, which represents the frequency |
|
|
(in s) with which the adjustment algorithm is called, to a non-zero |
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value (if set to a negative value by the user, the model will set it |
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to the tracer time step). The other option is to parameterize |
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convection with implicit vertical diffusion. To do this, set the |
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logical variable \textbf{implicitDiffusion} to \texttt{'.TRUE.'} |
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and the real variable \textbf{ivdc\_kappa} to a value (in m$^{2}$/s) |
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you wish the tracer vertical diffusivities to have when mixing |
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tracers vertically due to static instabilities. Note that |
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\textbf{cadjFreq} and \textbf{ivdc\_kappa}can not both have non-zero |
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value. |
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\end{description} |
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\subsection{Simulation controls} |
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The model ''clock'' is defined by the variable \textbf{deltaTClock} |
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(in s) which determines the IO frequencies and is used in tagging |
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output. Typically, you will set it to the tracer time step for |
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accelerated runs (otherwise it is simply set to the default time step |
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|
\textbf{deltaT}). Frequency of checkpointing and dumping of the model |
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|
state are referenced to this clock (see below). |
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\begin{description} |
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\item[run duration] \ |
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The beginning of a simulation is set by specifying a start time (in |
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s) through the real variable \textbf{startTime} or by specifying an |
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|
initial iteration number through the integer variable |
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|
\textbf{nIter0}. If these variables are set to nonzero values, the |
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|
model will look for a ''pickup'' file \textit{pickup.0000nIter0} to |
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|
restart the integration. The end of a simulation is set through the |
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real variable \textbf{endTime} (in s). Alternatively, you can |
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specify instead the number of time steps to execute through the |
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integer variable \textbf{nTimeSteps}. |
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\item[frequency of output] \ |
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Real variables defining frequencies (in s) with which output files |
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|
are written on disk need to be set up. \textbf{dumpFreq} controls |
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|
the frequency with which the instantaneous state of the model is |
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|
saved. \textbf{chkPtFreq} and \textbf{pchkPtFreq} control the output |
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frequency of rolling and permanent checkpoint files, respectively. |
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See section 1.5.1 Output files for the definition of model state and |
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checkpoint files. In addition, time-averaged fields can be written |
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out by setting the variable \textbf{taveFreq} (in s). The precision |
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with which to write the binary data is controlled by the integer |
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variable w\textbf{riteBinaryPrec} (set it to \texttt{32} or |
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\texttt{64}). |
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\end{description} |
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%%% Local Variables: |
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%%% mode: latex |
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%%% TeX-master: t |
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%%% End: |
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