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# Title: Cyclone-generated wave field
#
# Description:
#
# The wave model is used to simulate the evolution of the wave field
# generated by a model cyclone. See Popinet et al, 2009, Ocean
# Modelling for a detailed description.
#
# The simulation domain is $3328\times 3328$ km and the evolution is
# simulated for 48 hours, during which the maximum wind velocity
# increases from 0 to 50 m/s.
#
# The evolution of the significant wave height and corresponding
# adaptive mesh is illustrated in Figure \ref{hs}. The maximum
# significant wave height and wind speed evolutions are illustrated in
# Figure \ref{hsmax}.
#
# Interestingly the extrema of significant wave height are shifted
# compared to the wind speed extrema (Figure \ref{shifted}). This
# shift has also been observed in real cyclone-generated wave fields.
#
# \begin{figure}
# \caption{\label{hs}Evolution of the significant wave height (left
# column) and adaptive mesh (right column).}
# \begin{center}
# \begin{tabular}{cc}
# \includegraphics*[width=0.45\hsize]{hs-12.eps} &
# \includegraphics*[width=0.45\hsize]{mesh-12.eps} \\
# \multicolumn{2}{c}{$t=12$ hours} \\
# \\
# \includegraphics*[width=0.45\hsize]{hs-24.eps} &
# \includegraphics*[width=0.45\hsize]{mesh-24.eps} \\
# \multicolumn{2}{c}{$t=24$ hours} \\
# \\
# \includegraphics*[width=0.45\hsize]{hs-36.eps} &
# \includegraphics*[width=0.45\hsize]{mesh-36.eps} \\
# \multicolumn{2}{c}{$t=36$ hours} \\
# \\
# \includegraphics*[width=0.45\hsize]{hs-48.eps} &
# \includegraphics*[width=0.45\hsize]{mesh-48.eps} \\
# \multicolumn{2}{c}{$t=48$ hours}
# \end{tabular}
# \end{center}
# \end{figure}
#
# \begin{figure}
# \caption{\label{hsmax}Evolution of the maximum significant wave
# height and maximum wind speed.}
# \begin{center}
# \includegraphics*[width=\hsize]{hsmax.eps}
# \end{center}
# \end{figure}
#
# \begin{figure}
# \caption{\label{shifted}Significant wave height (colours) and wind field.}
# \begin{center}
# \includegraphics*[width=0.8\hsize]{shifted.eps}
# \end{center}
# \end{figure}
#
# Author: St\'ephane Popinet
# Command: gerris2D -DLEVEL=7 -DALPHA=3. cyclone.gfs | gfsview2D cyclone.gfv
# Version: 20091105
# Required files: cyclone.gfv hs.gfv mesh.gfv shifted.gfv
# Running time: 15 minutes
# Generated files: hs-12.eps hs-36.eps mesh-12.eps mesh-36.eps hs-24.eps hs-48.eps mesh-24.eps mesh-48.eps hsmax.eps shifted.eps
#
# Use the spectral wave model 'GfsWave'
1 0 GfsWave GfsBox GfsGEdge {} {
Refine 4
# Run for 48 hours
Time { end = 48 }
# Domain size is 3328 km
PhysicalParams { L = 3328 }
# Define some useful functions
Global {
/* gaussian distribution */
static double gaussian (double f, double fmean, double fsigma) {
return exp (-((f - fmean)*(f - fmean))/(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.;
}
/* Holland cyclone model */
static double holland(double r, double Rmax, double Vmax) {
if (r < Rmax/1e3) return 0.;
return Vmax*pow(Rmax/r, 2)*exp(2.*(1. - Rmax/r));
}
/* Position of the center of the cyclone */
double ut = 555./24.; /* km/h */
static double xc (double t) {
return 0.;
}
static double yc (double t) {
return 1110. - ut*t;
}
/* Intensity of the cyclone as a function of time */
static double vmax (double t) {
return 50.*(t < 25. ? t/25. : 1.);
}
/* velocity components */
static double ur (double x, double y, double t) {
x -= xc (t);
y -= yc (t);
double r = sqrt (x*x + y*y);
return holland (r, 100., vmax (t))*y/r;
}
static double vr (double x, double y, double t) {
x -= xc (t);
y -= yc (t);
double r = sqrt (x*x + y*y);
return - holland (r, 100., vmax (t))*x/r;
}
}
# Use source terms from WaveWatch III
GModule wavewatch
Init { istep = 1 } {
# Wind at 10 metres
U10 = ur(x, y, t)
V10 = vr(x, y, t)
}
# Adapt the mesh according to the error in significant wave height
AdaptError { istep = 1 } { cmax = 0.1 minlevel = 4 maxlevel = LEVEL c = Hse } Hs
# Adapt the mesh according to the error in the norm of the forcing wind field
AdaptError { istep = 1 } {
cmax = 0.2 minlevel = 4 maxlevel = LEVEL c = Ve
} sqrt(U10*U10 + V10*V10)
# Output time at every timestep
OutputTime { istep = 1 } stderr
# Output simulation size
OutputBalance { istep = 1 } stderr
# Output timing statistics every 100 timesteps
OutputTiming { istep = 100 } stderr
# Output significant wave height to file hs, every quarter of an hour
OutputScalarStats { step = 0.25 } hs { v = Hs }
# Output norm of wind velocity to file vr, every quarter of an hour
OutputScalarStats { step = 0.25 } vr { v = sqrt(U10*U10 + V10*V10) }
# Output simulation to standard output for visualisation with GfsView
OutputSimulation { istep = 10 } stdout
# Output simulation results every 4 hours
OutputSimulation { step = 4 } sim-%g.gfs
# Compress the files to save disk space
EventScript { step = 4 } { gzip -f sim-*.gfs }
# Create movies of significant wave height and level of refinement
OutputPPM { istep = 1 } { ppm2mpeg > hs.mpg } { v = Hs maxlevel = 9 }
OutputPPM { istep = 1 } { ppm2mpeg > level.mpg } { v = Level min = 4 max = LEVEL maxlevel = 9 }
# Create figures at the end of the simulation
EventScript { start = end } {
for i in 12 24 36 48; do
echo "Save hs-$i.eps { format = EPS }" | gfsview-batch2D sim-$i.gfs.gz hs.gfv
echo "Save mesh-$i.eps { format = EPS }" | gfsview-batch2D sim-$i.gfs.gz mesh.gfv
done
echo "Save shifted.eps { format = EPS }" | gfsview-batch2D sim-48.gfs.gz shifted.gfv
cat <<EOF | gnuplot
set term postscript eps color lw 2 18
set output 'hsmax.eps'
set xlabel 'Time (hours)'
set ylabel 'Amplitude (m or m/s)'
set xtics 0,12,48
set grid
plot 'hs' u 3:11 w l t 'max(Hs)', 'vr' u 3:11 w l t 'max(|U10|)'
EOF
}
} {
# Garden Sprinkler Effect alleviation parameter
alpha_s = ALPHA
}
GfsBox {}
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