src/earth/greens.cc

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// Copyright (C) 2004-2007 Jed Brown and Ed Bueler
//
// This file is part of PISM.
//
// PISM is free software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the Free Software
// Foundation; either version 2 of the License, or (at your option) any later
// version.
//
// PISM is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more
// details.
//
// You should have received a copy of the GNU General Public License
// along with PISM; if not, write to the Free Software
// Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA

#include <cmath>
#include <gsl/gsl_sf_bessel.h>
#include <gsl/gsl_integration.h>
#include "matlablike.hh"

#include "greens.hh"


double ge_integrand(unsigned ndimMUSTBETWO, const double* xiANDeta, void* paramsIN) {
  // Matlab:  function z=integrand(xi,eta,dx,dy,p,q)

  if (ndimMUSTBETWO != 2) { perror("ge_integrand only defined for 2 variables"); }
  
  // data here is from Lingle & Clark (1985), but claims to give G^E(r) for Farrell (1972)
  double rmkm[42] =
    {0.0, 0.011,  0.111,  1.112,  2.224,  3.336,  4.448,  6.672,  8.896,  11.12, 17.79,
          22.24,  27.80,  33.36,  44.48,  55.60,  66.72,  88.96,  111.2,  133.4, 177.9,
          222.4,  278.0,  333.6,  444.8,  556.0,  667.2,  778.4,  889.6, 1001.0, 1112.0,
         1334.0, 1779.0, 2224.0, 2780.0, 3336.0, 4448.0, 5560.0, 6672.0, 7784.0, 8896.0,
        10008.0};
  // rm = rmkm * 1e3 (remember to convert to meters); GE /(10^12 rm) is
  //    vertical displacement in meters
  // (GE(r=0) has been computed by linear extrapolation:  GE(0) := -33.6488)
  double GE[42] =
    {-33.6488, -33.64, -33.56, -32.75, -31.86, -30.98, -30.12, -28.44, -26.87, -25.41,
               -21.80, -20.02, -18.36, -17.18, -15.71, -14.91, -14.41, -13.69, -13.01,
               -12.31, -10.95, -9.757, -8.519, -7.533, -6.131, -5.237, -4.660, -4.272,
               -3.999, -3.798, -3.640, -3.392, -2.999, -2.619, -2.103, -1.530, -0.292,
                0.848,  1.676,  2.083,  2.057,  1.643};  

  struct ge_params*  params = (struct ge_params*) paramsIN;
  const double  dx = params->dx;
  const double  dy = params->dy;
  const int     p = params->p;
  const int     q = params->q;
  const double  xishift = (double) p * dx - xiANDeta[0];
  const double  etashift = (double) q * dy - xiANDeta[1];
  const double  r = sqrt(xishift * xishift + etashift * etashift);
  
  double  z;
  if (r < 0.01) {
    z = GE[0]/(rmkm[1] * 1.0e3 * 1.0e12);
  } else if (r > rmkm[41] * 1.0e3) {
    z = 0.0;
  } else {
    z = interp1_linear((double*) rmkm,(double*) GE,42,r / 1.0e3) / (r * 1.0e12);
  }
  return z;
}


double vd_integrand (double kap, void * paramsIN) {
// Matlab:  function y=integrand(kap,rg,D,t,eta,R0,rk)
//            beta=rg + D*kap.^4;
//            expdiff=exp(-beta*t./(2*eta*kap))-ones(size(kap));
//            y=expdiff.*besselj(1.0,kap*R0).*besselj(0.0,kap*rk)./beta;

  struct vd_params*  params = (struct vd_params*) paramsIN;
  const double       t = params->t;
  const double       R0 = params->R0;
  const double       rk = params->rk; 
  const double       rho = params->rho; 
  const double       grav = params->grav; 
  const double       D = params->D;
  const double       eta = params->eta;
  const double       beta = rho * grav + D * pow(kap,4.0);
  const double       expdiff = exp(-beta * t / (2.0 * eta * kap)) - 1.0;
  return expdiff * gsl_sf_bessel_J1(kap * R0) * gsl_sf_bessel_J0(kap * rk) / beta;
}


double viscDisc(double t, double H0, double R0, double r, 
                double rho, double grav, double D, double eta) {
  // t in seconds; H0, R0, r in meters

  const double      ABSTOL = 1.0e-10;
  const double      RELTOL = 1.0e-14;
  const int         N_gsl_workspace = 1000;
  gsl_integration_workspace*
                    w = gsl_integration_workspace_alloc(N_gsl_workspace);
  double*           pts;
  const int         lengthpts = 142;
  
  // Matlab:  pts=[10.^(-3:-0.05:-10) 1.0e-14];
  pts = new double[lengthpts];
  for (int j=0; j < lengthpts-1; j++) {
    pts[j] = pow(10.0,-3.0 - 0.05 * (double) j);
  }
  pts[lengthpts-1] = 1.0e-14;

  // result=quadl(@integrand,pts(1),100.0*pts(1),TOL,0,rg,D,t,eta,R0,rk); % kap->infty tail
  gsl_function      F;
  struct vd_params  params = { t, R0, r, rho, grav, D, eta };
  double            result, error;
  F.function = &vd_integrand;
  F.params = &params;
  // regarding tolerance: request is for convergence of all digits and relative tolerance RELTOL
  gsl_integration_qag (&F, pts[1], 100.0*pts[1], ABSTOL, RELTOL, N_gsl_workspace, 
                       GSL_INTEG_GAUSS21, w, &result, &error);

  double  sum = result; 
  // for j=1:length(pts)-1
  //   result=result+quadl(@integrand,pts(j+1),pts(j),TOL,0,rg,D,t,eta,R0,rk);
  // end
  for (int j=0; j < lengthpts-1; j++) {
    gsl_integration_qag (&F, pts[j+1], pts[j], ABSTOL, RELTOL, N_gsl_workspace, 
                         GSL_INTEG_GAUSS21, w, &result, &error);
    sum += result;
  }
  
  delete [] pts;
  gsl_integration_workspace_free(w);
  // u(k)=rhoi*g*H0*R0*result;
  return rho * grav * H0 * R0 * sum;
}