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# Title: "Garden sprinkler effect" in wave model
#
# Description:
#
# The wave model is used to reproduce the classical "Garden Sprinkler
# Effect" (GSE), a numerical artifact of the discrete directions of
# wave propagation (see Tolman, 2002).
#
# A spatially-Gaussian wave spectrum is initialised in a 5000
# km-squared domain. The other parameters are those of Tolman, 2002.
#
# The final (t = 5 days) significant wave height for different model
# runs is illustrated in Figure \ref{end}. The interval between the
# isolines is 0.1 metres as in Figure 1 of Tolman, 2002. For a small
# number of discrete directions (24), the GSE is evident and the
# results closely match those of Tolman both for the constant
# resolution and the adaptive version of the code. For larger number
# of directions (60 and 120), the results do not show any obvious GSE
# and match the corresponding results of Tolman (Figure 1.b of Tolman,
# 2002 but note that the spatial resolution of Tolman is finer, 25 km
# rather than 78 km here).
#
# \begin{figure}[htbp]
# \caption{\label{end}Final (t = 5 days) significant wave height for
# different model runs.}
# \begin{center}
# \includegraphics[width=\hsize]{end.eps}
# \end{center}
# \end{figure}
#
# The evolution in time of the significant wave height together with
# the corresponding adaptive discretisation is illustrated in Figure
# \ref{mesh} for 120 directions. The mesh is adapted according to the
# spatial gradient in the significant wave height. This results in
# substantial savings in computational cost as illustrated by the
# timings given in Table \ref{cpu}. The computational cost with 120
# directions is comparable to the cost with 24 directions on a regular
# (i.e. non-adaptive) mesh. This demonstrates that the GSE can be
# alleviated -- at comparable computational cost -- by combining
# adaptive refinement with a refined discretisation in direction
# space.
#
# \begin{figure}[htbp]
# \caption{\label{mesh}Evolution of the significant wave height and
# adaptive mesh. 120 directions.}
# \begin{center}
# \includegraphics[width=\hsize]{mesh.eps}
# \end{center}
# \end{figure}
#
# \begin{table}[htbp]
# \caption{\label{cpu}CPU time for the four models of Figure \ref{end}.}
# \begin{center}
# \input{cpu.tex}
# \end{center}
# \end{table}
#
# Author: St\'ephane Popinet
# Command: sh garden.sh
# Version: 1.2.1
# Required files: garden.sh end.gfv mesh.gfv
# Running time: 41 minutes
# Generated files: end.eps mesh.eps cpu.tex
#
1 0 GfsWave GfsBox GfsGEdge {} {
Refine 6
# Default time units for wave model is hours
# 120 hours = 5 days
Time { end = 120 }
# Default length units for wave model is km
PhysicalParams { L = 5000 }
# Define some useful functions
Global {
/* gaussian distribution */
static double gaussian (double f, double fmean, double fsigma) {
return exp (-((f - fmean)*(f - fmean))/(2.*fsigma*fsigma));
}
/* cos(theta)^n distribution */
static double costheta (double theta, double thetam, double thetapower) {
double a = cos (theta - thetam);
return a > 0. ? pow (a, thetapower) : 0.;
}
}
# Initialise the wave spectrum
InitWave {} {
/* This function defines the spectral distribution:
* a gaussian in frequency space and
* a cos(theta)^2 distribution in direction space
*/
return gaussian (Frequency, 0.1, 0.02)*
costheta (Direction, 30.*M_PI/180., 2.);
} {
/* This function defines the significant wave height:
* the energy is a gaussian bump in (x,y) space,
* the maximum significant wave height is 2.5
*/
x -= -2000.;
y -= -2000.;
double Hsmax = 2.5;
double E = (Hsmax*Hsmax/16.)*gaussian (sqrt (x*x + y*y), 0., 150.);
return 4.*sqrt (E);
}
AdaptGradient { istep = 1 } { cmax = 0.04 minlevel = MINLEVEL maxlevel = 6 } Hs
OutputTime { istep = 1 } log-MINLEVEL-NTHETA
OutputScalarStats { step = 12 } hs-MINLEVEL-NTHETA { v = Hs }
OutputSimulation { step = 12 } sim-MINLEVEL-NTHETA-%g.gfs
EventScript { step = 12 } { gzip -f sim-*-*-*.gfs }
OutputSimulation { start = end } end-MINLEVEL-NTHETA.gfs
EventScript { start = end } { gzip -f end-*-*.gfs }
OutputPPM { step = 12 } { ppm2mpeg > hs-MINLEVEL-NTHETA.mpg } { v = Hs maxlevel = 7 }
} {
# Number of discretised directions (default is 24)
ntheta = NTHETA
}
GfsBox {}
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