# Title: Coastally-trapped waves
#
# Description:
#
# A simulation is initialised with an exact solution found by Lamb
# \cite{lamb} for inertia-gravity waves in a circular, flat-bottomed
# basin on an $f$-plane.
#
# The waves should rotate unchanged around the basin. This is a good
# test of the shallow-water solver with Coriolis terms and non-trivial
# coastlines.
#
# Figure \ref{solution} illustrates the solution obtained after three
# wave periods.
#
# Following \cite{rutgers-waves}, Table \ref{correlation} presents the
# phase error and the maximum correlation between the computed
# solution and the exact solution. Note that these results are not
# directly comparable to \cite{rutgers-waves}, because the timestep
# used is much larger. The time-explicit codes tested in
# \cite{rutgers-waves} have stronger stability requirements than
# Gerris.
#
# \begin{figure}[htbp]
# \caption{\label{solution}Surface-height after a rotation of 3 wave periods for a
# resolution of 9.375 km.}
# \begin{center}
# \includegraphics[width=0.6\hsize]{solution.eps}
# \end{center}
# \end{figure}
#
# \begin{table}[htbp]
# \caption{\label{correlation}Maximum correlation and phase error as functions of resolution.}
# \begin{center}
# \input{correlation.tex}
# \end{center}
# \end{table}
#
# Author: St\'ephane Popinet
# Command: sh waves.sh waves.gfs
# Version: 1.0.0
# Required files: waves.sh solution.gfv correlation.ref
# Generated files: correlation correlation.tex solution.eps
#
1 0 GfsOcean GfsBox GfsGEdge {} {
  Time { end = 37.80501984 dtmax = 0.1 }
  PhysicalParams { g = 5.87060327757e-3 }
  Global {
    #include <gsl/gsl_sf_bessel.h>
    @link -lgsl -lgslcblas
    
    #define Ik(k,r,D) (gsl_sf_bessel_Inu ((k) - 1., (r)/(D))/(D)\
                          - (k)/(r)*gsl_sf_bessel_Inu ((k), (r)/(D)))

    static double D = 8.83906519983e-2;
    static double k = 3.;
    static double sigma = 0.4986;
    static double a = 1./2555510.;
    
    static double pwave (double x, double y, double angle) {
      double theta = atan2 (y, x) + angle*M_PI/180.;
      double r = sqrt (x*x + y*y);
      return a*cos (k*theta)*gsl_sf_bessel_Inu (k, r/D);
    }

    static double ur (double theta, double r) {
      return -a*D*D/5.87060327757e-3*sin (k*theta)*(sigma*Ik (k, r, D) - 
          k/r*gsl_sf_bessel_Inu (k, r/D));
    }

    static double vt (double theta, double r) {
      return a*D*D/5.87060327757e-3*cos (k*theta)*(Ik (k, r, D) - 
       	  k*sigma/r*gsl_sf_bessel_Inu (k, r/D));
    }

    static double uwave (double x, double y) {
      double theta = atan2 (y, x);
      double r = sqrt (x*x + y*y);
      return ur (theta, r)*cos (theta) - vt (theta, r)*sin (theta);
    }

    static double vwave (double x, double y) {
      double theta = atan2 (y, x);
      double r = sqrt (x*x + y*y);
      return ur (theta, r)*sin (theta) + vt (theta, r)*cos (theta);
    }
  }
  Init {} {
    P = pwave(cx, cy, 0)
    U = uwave(x, y)
    V = vwave(x, y)
    H = 1
  }
  Refine LEVEL
  Solid (- ellipse (0, 0, 0.25, 0.25)) { scale = 1.999 }
  AdvectionParams { scheme = none }
  ApproxProjectionParams { tolerance = 1e-9 weighted = 0 }
  SourceCoriolis {} 1.
  EventFilter { istep = 1 } U 4
  EventFilter { istep = 1 } V 4
  EventScript { start = end } { echo -n "-30 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -30)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-29 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -29)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-28 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -28)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-27 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -27)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-26 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -26)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-25 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -25)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-24 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -24)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-23 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -23)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-22 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -22)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-21 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -21)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-20 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -20)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-19 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -19)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-18 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -18)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-17 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -17)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-16 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -16)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-15 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -15)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-14 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -14)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-13 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -13)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-12 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -12)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-11 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -11)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-10 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -10)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-9 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -9)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-8 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -8)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-7 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -7)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-6 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -6)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-5 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -5)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-4 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -4)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-3 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -3)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-2 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -2)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "-1 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, -1)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.1 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.1)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.2 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.2)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.3 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.3)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.4 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.4)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.5 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.5)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.6 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.6)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.7 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.7)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.8 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.8)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "0.9 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 0.9)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.1 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.1)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.2 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.2)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.3 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.3)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.4 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.4)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.5 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.5)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.6 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.6)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.7 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.7)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.8 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.8)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "1.9 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 1.9)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.1 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.1)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.2 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.2)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.3 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.3)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.4 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.4)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.5 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.5)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.6 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.6)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.7 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.7)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.8 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.8)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "2.9 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 2.9)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.1 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.1)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.2 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.2)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.3 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.3)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.4 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.4)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.5 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.5)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.6 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.6)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.7 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.7)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.8 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.8)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "3.9 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 3.9)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.1 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.1)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.2 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.2)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.3 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.3)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.4 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.4)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.5 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.5)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.6 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.6)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.7 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.7)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.8 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.8)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "4.9 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 4.9)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "5 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 5)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "6 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 6)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "7 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 7)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "8 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 8)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "9 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 9)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "10 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 10)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "11 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 11)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "12 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 12)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "13 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 13)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "14 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 14)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "15 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 15)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "16 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 16)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "17 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 17)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "18 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 18)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "19 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 19)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "20 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 20)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "21 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 21)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "22 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 22)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "23 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 23)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "24 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 24)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "25 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 25)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "26 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 26)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "27 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 27)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "28 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 28)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "29 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 29)
    unbiased = 1
  }
  EventScript { start = end } { echo -n "30 " }
  OutputCorrelation { start = end } stdout { v = P } {
    s = pwave(cx, cy, 30)
    unbiased = 1
  }
  OutputSimulation { start = end } sim-LEVEL
}
GfsBox {
  front = Boundary
}