mod_cak_force.t

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!> Module to include CAK radiation line force in (magneto)hydrodynamic models
!> Computes both the force from free electrons and the force from an ensemble of
!> lines (various possibilities for the latter).
!> There is an option to only simulate the pure radial CAK force (with various
!> corrections applied) as well as the full vector CAK force. Depending on the
!> chosen option additional output are the CAK line force component(s) and,
!> when doing a 1-D radial force, the finite disc factor.
!>
!> USAGE:
!>
!>  1. Include a cak_list in the .par file and activate (m)hd_cak_force in the
!>     (m)hd_list
!>  2. Create a mod_usr.t file for the problem with appropriate initial and
!>     boundary conditions
!>  3. In the mod_usr.t header call the mod_cak_force module to have access to
!>     global variables from mod_cak_force, which may be handy for printing or
!>     the computation of other variables inside mod_usr.t
!>  4. In usr_init of mod_usr.t call the set_cak_force_norm routine and pass
!>     along the stellar radius and wind temperature---this is needed to
!>     correctly compute the (initial) force normalisation inside mod_cak_force
!>  5. Ensure that the order of calls in usr_init is similar as for test problem
!>     CAKwind_spherical_1D: first reading usr.par list; then set unit scales;
!>     then call (M)HD_activate; then call set_cak_force_norm. This order avoids
!>     an incorrect force normalisation and code crash
!>
!> Developed by Florian Driessen (2022)
module mod_cak_force
  use mod_physics, only: phys_get_pthermal, physics_type
  implicit none
 
  !> Line-ensemble parameters in the Gayley (1995) formalism
  real(8), public :: cak_alpha, gayley_qbar, gayley_q0
 
  !> Ray positions + weights for impact parameter and azimuthal radiation angle
  real(8), allocatable, private :: ay(:), wy(:), aphi(:), wphi(:)
 
  !> The adiabatic index
  real(8), private :: cak_gamma
 
  !> Variables needed to compute force normalisation fnorm in initialisation
  real(8), private :: lum, dlum, drstar, dke, dclight
 
  !> To enforce a floor temperature when doing adiabatic (M)HD
  real(8), private :: tfloor
 
  !> Switch to choose between the 1-D CAK line force options
  integer :: cak_1d_opt
 
  ! Avoid magic numbers in code for 1-D CAK line force option
  integer, parameter, private :: radstream=0, fdisc=1, fdisc_cutoff=2
 
  !> Amount of rays in radiation polar and radiation azimuthal direction
  integer :: nthetaray, nphiray
 
  !> Extra slots to store quantities in w-array
  integer :: gcak1_, gcak2_, gcak3_, fdf_
 
  !> To treat source term in split or unsplit (default) fashion
  logical :: cak_split=.false.
 
  !> To activate the original CAK 1-D line force computation
  logical :: cak_1d_force=.false.
 
  !> To activate the vector CAK line force computation
  logical :: cak_vector_force=.false.
 
  !> To activate the pure radial vector CAK line force computation
  logical :: fix_vector_force_1d=.false.
 
  !> Public method
  public :: set_cak_force_norm
  
contains
 
  !> Read this module's parameters from a file
  subroutine cak_params_read(files)
    use mod_global_parameters, only: unitpar
 
    character(len=*), intent(in) :: files(:)
 
    ! Local variable
    integer :: n
 
    namelist /cak_list/ cak_alpha, gayley_qbar, gayley_q0, cak_1d_opt, &
                        cak_split, cak_1d_force, cak_vector_force, &
                        nphiray, nthetaray, fix_vector_force_1d
 
    do n = 1,size(files)
       open(unitpar, file=trim(files(n)), status="old")
       read(unitpar, cak_list, end=111)
       111 close(unitpar)
    enddo
 
  end subroutine cak_params_read
 
  !> Initialize the module
  subroutine cak_init(phys_gamma)
    use mod_global_parameters
    use mod_comm_lib, only: mpistop
 
    real(8), intent(in) :: phys_gamma
 
    cak_gamma = phys_gamma
 
    ! Set some defaults when user does not
    cak_alpha   = 0.65d0
    gayley_qbar = 2000.0d0
    gayley_q0   = 2000.0d0
    cak_1d_opt  = fdisc
    nthetaray   = 6
    nphiray     = 6
 
    call cak_params_read(par_files)
 
    if (cak_1d_force) then
      gcak1_ = var_set_extravar("gcak1", "gcak1")
      fdf_   = var_set_extravar("fdfac", "fdfac")
    endif
 
    if (cak_vector_force) then
      gcak1_ = var_set_extravar("gcak1", "gcak1")
      gcak2_ = var_set_extravar("gcak2", "gcak2")
      gcak3_ = var_set_extravar("gcak3", "gcak3")
      call rays_init(nthetaray,nphiray)
    endif
 
    ! Some sanity checks
    if ((cak_alpha <= 0.0d0) .or. (cak_alpha > 1.0d0)) then
      call mpistop('CAK error: choose alpha in [0,1[')
    endif
 
    if ((gayley_qbar < 0.0d0) .or. (gayley_q0 < 0.0d0)) then
      call mpistop('CAK error: chosen Qbar or Q0 is < 0')
    endif
 
    if (cak_1d_force .and. cak_vector_force) then
      call mpistop('CAK error: choose either 1-D or vector force')
    endif
 
  end subroutine cak_init
 
  !> Compute some (unitless) variables for CAK force normalisation
  subroutine set_cak_force_norm(rstar,twind)
    use mod_global_parameters
    use mod_constants
 
    real(8), intent(in) :: rstar, twind
    double precision :: const_kappae_local
    const_kappae_local=0.34d0
 
    lum     = 4.0d0*dpi * rstar**2.0d0 * const_sigma * twind**4.0d0
    dke     = const_kappae_local * unit_density * unit_length
    dclight = const_c/unit_velocity
    dlum    = lum/(unit_density * unit_length**5.0d0 / unit_time**3.0d0)
    drstar  = rstar/unit_length
    tfloor  = twind/unit_temperature
 
  end subroutine set_cak_force_norm
  
  !> w[iw]=w[iw]+qdt*S[wCT,qtC,x] where S is the source based on wCT within ixO
  subroutine cak_add_source(qdt,ixI^L,ixO^L,wCT,w,x,energy,qsourcesplit,active)
    use mod_global_parameters
    use mod_comm_lib, only: mpistop
 
    integer, intent(in)    :: ixI^L, ixO^L
    real(8), intent(in)    :: qdt, x(ixI^S,1:ndim), wCT(ixI^S,1:nw)
    real(8), intent(inout) :: w(ixI^S,1:nw)
    logical, intent(in)    :: energy, qsourcesplit
    logical, intent(inout) :: active
 
    ! Local variables
    real(8) :: gl(ixO^S,1:3), ge(ixO^S), ptherm(ixI^S), pmin(ixI^S)
    integer :: idir
 
    ! By default add source in unsplit fashion together with the fluxes
    if (qsourcesplit .eqv. cak_split) then
 
      active = .true.
 
      ! Thomson force
      call get_gelectron(ixi^l,ixo^l,wct,x,ge)
 
      ! CAK line force
      gl(ixo^s,1:3) = 0.0d0
 
      if (cak_1d_force) then
        call get_cak_force_radial(ixi^l,ixo^l,wct,w,x,gl)
      elseif (cak_vector_force) then
        call get_cak_force_vector(ixi^l,ixo^l,wct,w,x,gl)
      else
        call mpistop("No valid force option")
      endif
 
      ! Update conservative vars: w = w + qdt*gsource
      do idir = 1,ndir
        if (idir == 1) gl(ixo^s,idir) = gl(ixo^s,idir) + ge(ixo^s)
        
        w(ixo^s,iw_mom(idir)) = w(ixo^s,iw_mom(idir)) &
                                + qdt * gl(ixo^s,idir) * wct(ixo^s,iw_rho)
                                
        if (energy) then
          w(ixo^s,iw_e) = w(ixo^s,iw_e) + qdt * gl(ixo^s,idir) * wct(ixo^s,iw_mom(idir))
        endif
      enddo
 
      ! Impose fixed floor temperature to mimic stellar heating
      if (energy) then
        call phys_get_pthermal(w,x,ixi^l,ixo^l,ptherm)
        pmin(ixo^s) = w(ixo^s,iw_rho) * tfloor
 
        where (ptherm(ixo^s) < pmin(ixo^s))
          w(ixo^s,iw_e) = w(ixo^s,iw_e) + (pmin(ixo^s) - ptherm(ixo^s))/(cak_gamma - 1.0d0)
        endwhere
      endif
    endif
 
  end subroutine cak_add_source
 
  !> 1-D CAK line force in the Gayley line-ensemble distribution parametrisation
  subroutine get_cak_force_radial(ixI^L,ixO^L,wCT,w,x,gcak)
    use mod_global_parameters
    use mod_comm_lib, only: mpistop
 
    integer, intent(in)    :: ixI^L, ixO^L
    real(8), intent(in)    :: wCT(ixI^S,1:nw), x(ixI^S,1:ndim)
    real(8), intent(inout) :: w(ixI^S,1:nw)
    real(8), intent(inout) :: gcak(ixO^S,1:3)
  
    ! Local variables
    real(8) :: vr(ixI^S), dvrdr(ixO^S)
    real(8) :: beta_fd(ixO^S), fdfac(ixO^S), taus(ixO^S), ge(ixO^S)
  
    vr(ixi^s) = wct(ixi^s,iw_mom(1)) / wct(ixi^s,iw_rho)
    call get_velocity_gradient(ixi^l,ixo^l,vr,x,1,dvrdr)
 
    if (physics_type == 'hd') then
      ! Monotonic flow to avoid multiple resonances and radiative coupling
      dvrdr(ixo^s) = abs(dvrdr(ixo^s))
    else
      ! Allow material to fallback to the star in a magnetosphere model
      dvrdr(ixo^s) = max(dvrdr(ixo^s), smalldouble)
    endif
  
    ! Thomson force
    call get_gelectron(ixi^l,ixo^l,wct,x,ge)
 
    ! Sobolev optical depth for line ensemble (tau = Qbar * t_r) and the force
    select case (cak_1d_opt)
    case(radstream, fdisc)
      taus(ixo^s)   = gayley_qbar * dke * dclight * wct(ixo^s,iw_rho)/dvrdr(ixo^s)
      gcak(ixo^s,1) = gayley_qbar/(1.0d0 - cak_alpha) &
                      * ge(ixo^s)/taus(ixo^s)**cak_alpha
 
    case(fdisc_cutoff)
      taus(ixo^s)   = gayley_q0 * dke * dclight * wct(ixo^s,iw_rho)/dvrdr(ixo^s)
      gcak(ixo^s,1) = gayley_qbar * ge(ixo^s)                                  &
                      * ( (1.0d0 + taus(ixo^s))**(1.0d0 - cak_alpha) - 1.0d0 ) &
                      / ( (1.0d0 - cak_alpha) * taus(ixo^s) )
    case default
      call mpistop("Error in force computation.")
    end select
 
    ! Finite disk factor parameterisation (Owocki & Puls 1996)
    beta_fd(ixo^s) = ( 1.0d0 - vr(ixo^s)/(x(ixo^s,1) * dvrdr(ixo^s)) ) &
                      * (drstar/x(ixo^s,1))**2.0d0
 
    select case (cak_1d_opt)
    case(radstream)
      fdfac(ixo^s) = 1.0d0
    case(fdisc, fdisc_cutoff)
      where (beta_fd(ixo^s) >= 1.0d0)
        fdfac(ixo^s) = 1.0d0/(1.0d0 + cak_alpha)
      elsewhere (beta_fd(ixo^s) < -1.0d10)
        fdfac(ixo^s) = abs(beta_fd(ixo^s))**cak_alpha / (1.0d0 + cak_alpha)
      elsewhere (abs(beta_fd(ixo^s)) > 1.0d-3)
        fdfac(ixo^s) = (1.0d0 - (1.0d0 - beta_fd(ixo^s))**(1.0d0 + cak_alpha)) &
                       / (beta_fd(ixo^s)*(1.0d0 + cak_alpha))
      elsewhere
        fdfac(ixo^s) = 1.0d0 - 0.5d0*cak_alpha*beta_fd(ixo^s) &
                       * (1.0d0 + 1.0d0/3.0d0 * (1.0d0 - cak_alpha)*beta_fd(ixo^s))
      endwhere
    end select
 
    ! Correct radial line force for finite disc (if applicable)
    gcak(ixo^s,1) = gcak(ixo^s,1) * fdfac(ixo^s)
    gcak(ixo^s,2) = 0.0d0
    gcak(ixo^s,3) = 0.0d0
      
    ! Fill the nwextra slots for output
    w(ixo^s,gcak1_) = gcak(ixo^s,1)
    w(ixo^s,fdf_)   = fdfac(ixo^s)
    
  end subroutine get_cak_force_radial
 
  !> Vector CAK line force in the Gayley line-ensemble distribution parametrisation
  subroutine get_cak_force_vector(ixI^L,ixO^L,wCT,w,x,gcak)
    use mod_global_parameters
    use mod_usr_methods
 
    ! Subroutine arguments
    integer, intent(in)    :: ixI^L, ixO^L
    real(8), intent(in)    :: wCT(ixI^S,1:nw), x(ixI^S,1:ndim)
    real(8), intent(inout) :: w(ixI^S,1:nw)
    real(8), intent(inout) :: gcak(ixO^S,1:3)
 
    ! Local variables
    real(8) :: a1, a2, a3, wyray, y, wpray, phiray, wtot, mustar, dvndn
    real(8) :: costp, costp2, sintp, cospp, sinpp, cott0
    real(8) :: vr(ixI^S), vt(ixI^S), vp(ixI^S)
    real(8) :: vrr(ixI^S), vtr(ixI^S), vpr(ixI^S)
    real(8) :: dvrdr(ixO^S), dvtdr(ixO^S), dvpdr(ixO^S)
    real(8) :: dvrdt(ixO^S), dvtdt(ixO^S), dvpdt(ixO^S)
    real(8) :: dvrdp(ixO^S), dvtdp(ixO^S), dvpdp(ixO^S)
    integer :: ix^D, itray, ipray
    
    ! Initialisation to have full velocity strain tensor expression at all times
    vt(ixo^s) = 0.0d0; vtr(ixo^s) = 0.0d0
    vp(ixo^s) = 0.0d0; vpr(ixo^s) = 0.0d0
    cott0 = 0.0d0
    dvrdr(ixo^s) = 0.0d0; dvtdr(ixo^s) = 0.0d0; dvpdr(ixo^s) = 0.0d0
    dvrdt(ixo^s) = 0.0d0; dvtdt(ixo^s) = 0.0d0; dvpdt(ixo^s) = 0.0d0
    dvrdp(ixo^s) = 0.0d0; dvtdp(ixo^s) = 0.0d0; dvpdp(ixo^s) = 0.0d0
 
    ! Populate velocity field(s) depending on dimensions and directions
    vr(ixi^s)  = wct(ixi^s,iw_mom(1)) / wct(ixi^s,iw_rho)
    vrr(ixi^s) = vr(ixi^s) / x(ixi^s,1)
 
    {^nooned
    vt(ixi^s)  = wct(ixi^s,iw_mom(2)) / wct(ixi^s,iw_rho)
    vtr(ixi^s) = vt(ixi^s) / x(ixi^s,1)
    
    if (ndir > 2) then
      vp(ixi^s)  = wct(ixi^s,iw_mom(3)) / wct(ixi^s,iw_rho)
      vpr(ixi^s) = vp(ixi^s) / x(ixi^s,1)
    endif
    }
    
    ! Derivatives of velocity field in each coordinate direction (r=1,t=2,p=3)
    call get_velocity_gradient(ixi^l,ixo^l,vr,x,1,dvrdr)
    
    {^nooned
    call get_velocity_gradient(ixi^l,ixo^l,vr,x,2,dvrdt)
    call get_velocity_gradient(ixi^l,ixo^l,vt,x,1,dvtdr)
    call get_velocity_gradient(ixi^l,ixo^l,vt,x,2,dvtdt)
 
    if (ndir > 2) then
      call get_velocity_gradient(ixi^l,ixo^l,vp,x,1,dvpdr)
      call get_velocity_gradient(ixi^l,ixo^l,vp,x,2,dvpdt)
    endif
    }
    {^ifthreed
    call get_velocity_gradient(ixi^l,ixo^l,vr,x,3,dvrdp)
    call get_velocity_gradient(ixi^l,ixo^l,vt,x,3,dvtdp)
    call get_velocity_gradient(ixi^l,ixo^l,vp,x,3,dvpdp)
    }
 
    ! Get total acceleration from all rays at a certain grid point
    {do ix^db=ixomin^db,ixomax^db\}
      ! Loop over the rays; first theta then phi radiation angle
      ! Get weights from current ray and their position
      do itray = 1,nthetaray
        wyray  = wy(itray)
        y      = ay(itray)
 
        do ipray = 1,nphiray
          wpray = wphi(ipray)
          phiray  = aphi(ipray)
 
          ! Redistribute the phi rays by a small offset
          ! if (mod(itp,3) == 1) then
          !   phip = phip + dphi/3.0d0
          ! elseif (mod(itp,3) == 2) then
          !   phip = phip - dphi/3.0d0
          ! endif
          
          ! === Geometrical factors ===
          ! Make y quadrature linear in mu, not mu**2; better for gtheta,gphi
          ! y -> mu quadrature is preserved; y=0 <=> mu=1; y=1 <=> mu=mustar
          mustar = sqrt(max(1.0d0 - (drstar/x(ix^d,1))**2.0d0, 0.0d0))
          costp  = 1.0d0 - y*(1.0d0 - mustar)
          costp2 = costp*costp
          sintp  = sqrt(max(1.0d0 - costp2, 0.0d0))
          sinpp  = sin(phiray)
          cospp  = cos(phiray)
          {^nooned cott0  = cos(x(ix^d,2))/sin(x(ix^d,2))}
 
          ! More weight close to star, less farther away
          wtot  = wyray * wpray * (1.0d0 - mustar)
 
          ! Convenients a la Cranmer & Owocki (1995)
          a1 = costp
          a2 = sintp * cospp
          a3 = sintp * sinpp
 
          ! Get total velocity gradient for one ray with given (theta', phi')
          dvndn = a1*a1 * dvrdr(ix^d) + a2*a2 * (dvtdt(ix^d) + vrr(ix^d))  &
                 + a3*a3 * (dvpdp(ix^d) + cott0 * vtr(ix^d) + vrr(ix^d))   &
                 + a1*a2 * (dvtdr(ix^d) + dvrdt(ix^d) - vtr(ix^d))         &
                 + a1*a3 * (dvpdr(ix^d) + dvrdp(ix^d) - vpr(ix^d))         &
                 + a2*a3 * (dvpdt(ix^d) + dvtdp(ix^d) - cott0 * vpr(ix^d))
 
          ! No multiple resonances in CAK
          dvndn = abs(dvndn)
 
          ! Convert gradient back from wind coordinates (r',theta',phi') to
          ! stellar coordinates (r,theta,phi)
          gcak(ix^d,1) = gcak(ix^d,1) + (dvndn/wct(ix^d,iw_rho))**cak_alpha * a1 * wtot
          gcak(ix^d,2) = gcak(ix^d,2) + (dvndn/wct(ix^d,iw_rho))**cak_alpha * a2 * wtot
          gcak(ix^d,3) = gcak(ix^d,3) + (dvndn/wct(ix^d,iw_rho))**cak_alpha * a3 * wtot
        enddo
      enddo
    {enddo\}
 
    ! Normalisation for line force
    ! NOTE: extra 1/pi factor comes from integration in radiation Phi angle
    gcak(ixo^s,:) = (dke*gayley_qbar)**(1.0d0 - cak_alpha)/(1.0d0 - cak_alpha)    &
                    * dlum/(4.0d0*dpi*drstar**2.0d0 * dclight**(1.0d0+cak_alpha)) &
                    * gcak(ixo^s,:)/dpi
 
    if (fix_vector_force_1d) then
      gcak(ixo^s,2) = 0.0d0
      gcak(ixo^s,3) = 0.0d0
    endif
          
    ! Fill the nwextra slots for output
    w(ixo^s,gcak1_) = gcak(ixo^s,1)
    w(ixo^s,gcak2_) = gcak(ixo^s,2)
    w(ixo^s,gcak3_) = gcak(ixo^s,3)
 
  end subroutine get_cak_force_vector
  
  !> Compute continuum radiation force from Thomson scattering
  subroutine get_gelectron(ixI^L,ixO^L,w,x,ge)
    use mod_global_parameters
 
    integer, intent(in) :: ixI^L, ixO^L
    real(8), intent(in) :: w(ixI^S,1:nw), x(ixI^S,1:ndim)
    real(8), intent(out):: ge(ixO^S)
 
    ge(ixo^s) = dke * dlum/(4.0d0*dpi * dclight * x(ixo^s,1)**2.0d0)
 
  end subroutine get_gelectron
 
  !> Check time step for total radiation contribution
  subroutine cak_get_dt(wprim,ixI^L,ixO^L,dtnew,dx^D,x)
    use mod_global_parameters
 
    integer, intent(in)    :: ixI^L, ixO^L
    real(8), intent(in)    :: dx^D, x(ixI^S,1:ndim)
    real(8), intent(in)    :: wprim(ixI^S,1:nw)
    real(8), intent(inout) :: dtnew
    
    ! Local variables
    real(8) :: ge(ixO^S), max_gr, dt_cak
 
    call get_gelectron(ixi^l,ixo^l,wprim,x,ge)
 
    dtnew = bigdouble
 
    ! Get dt from line force that is saved in the w-array in nwextra slot
    max_gr = max( maxval(abs(ge(ixo^s) + wprim(ixo^s,gcak1_))), epsilon(1.0d0) )
    dt_cak = minval( sqrt(block%dx(ixo^s,1)/max_gr) )
    dtnew  = min(dtnew, courantpar*dt_cak)
 
    {^nooned
    if (cak_vector_force) then
      max_gr = max( maxval(abs(wprim(ixo^s,gcak2_))), epsilon(1.0d0) )
      dt_cak = minval( sqrt(block%dx(ixo^s,1) * block%dx(ixo^s,2)/max_gr) )
      dtnew  = min(dtnew, courantpar*dt_cak)
 
      {^ifthreed
      max_gr = max( maxval(abs(wprim(ixo^s,gcak3_))), epsilon(1.0d0) )
      dt_cak = minval( sqrt(block%dx(ixo^s,1) * sin(block%dx(ixo^s,3))/max_gr) )
      dtnew  = min(dtnew, courantpar*dt_cak)
      }
    endif
    }
 
  end subroutine cak_get_dt
 
  !> Compute velocity gradient in direction 'idir' on a non-uniform grid
  subroutine get_velocity_gradient(ixI^L,ixO^L,vfield,x,idir,grad_vn)
    use mod_global_parameters
 
    integer, intent(in)  :: ixI^L, ixO^L, idir
    real(8), intent(in)  :: vfield(ixI^S), x(ixI^S,1:ndim)
    real(8), intent(out) :: grad_vn(ixO^S)
 
    ! Local variables
    real(8) :: forw(ixO^S), backw(ixO^S), cent(ixO^S)
    integer :: jrx^L, hrx^L{^NOONED,jtx^L, htx^L}{^IFTHREED,jpx^L, hpx^L}
 
    ! Index +1 (j) and index -1 (h) in radial direction; kr(dir,dim)=1, dir=dim
    jrx^l=ixo^l+kr(1,^d);
    hrx^l=ixo^l-kr(1,^d);
 
    {^nooned
    ! Index +1 (j) and index -1 (h) in polar direction
    jtx^l=ixo^l+kr(2,^d);
    htx^l=ixo^l-kr(2,^d);
    }
 
    {^ifthreed
    ! Index +1 (j) and index -1 (h) in azimuthal direction
    jpx^l=ixo^l+kr(3,^d);
    hpx^l=ixo^l-kr(3,^d);
    }
 
    ! grad(v.n) on non-uniform grid according to Sundqvist & Veronis (1970)
    select case (idir)
    case(1) ! Radial forward, backward, and central derivatives
      forw(ixo^s)  = (x(ixo^s,1) - x(hrx^s,1)) * vfield(jrx^s) &
                      / ((x(jrx^s,1) - x(ixo^s,1)) * (x(jrx^s,1) - x(hrx^s,1)))
 
      backw(ixo^s) = -(x(jrx^s,1) - x(ixo^s,1)) * vfield(hrx^s) &
                      / ((x(ixo^s,1) - x(hrx^s,1)) * (x(jrx^s,1) - x(hrx^s,1)))
 
      cent(ixo^s)  = (x(jrx^s,1) + x(hrx^s,1) - 2.0d0*x(ixo^s,1)) * vfield(ixo^s) &
                      / ((x(ixo^s,1) - x(hrx^s,1)) * (x(jrx^s,1) - x(ixo^s,1)))
    {^nooned
    case(2) ! Polar forward, backward, and central derivatives
      forw(ixo^s)  = (x(ixo^s,2) - x(htx^s,2)) * vfield(jtx^s) &
                      / (x(ixo^s,1) * (x(jtx^s,2) - x(ixo^s,2)) * (x(jtx^s,2) - x(htx^s,2)))
 
      backw(ixo^s) = -(x(jtx^s,2) - x(ixo^s,2)) * vfield(htx^s) &
                      / ( x(ixo^s,1) * (x(ixo^s,2) - x(htx^s,2)) * (x(jtx^s,2) - x(htx^s,2)))
 
      cent(ixo^s)  = (x(jtx^s,2) + x(htx^s,2) - 2.0d0*x(ixo^s,2)) * vfield(ixo^s) &
                      / ( x(ixo^s,1) * (x(ixo^s,2) - x(htx^s,2)) * (x(jtx^s,2) - x(ixo^s,2)))
    }
    {^ifthreed
    case(3) ! Azimuthal forward, backward, and central derivatives
      forw(ixo^s)  = (x(ixo^s,3) - x(hpx^s,3)) *  vfield(jpx^s) &
                      / ( x(ixo^s,1)*sin(x(ixo^s,2)) * (x(jpx^s,3) - x(ixo^s,3)) * (x(jpx^s,3) - x(hpx^s,3)))
 
      backw(ixo^s) = -(x(jpx^s,3) - x(ixo^s,3)) *  vfield(hpx^s) &
                      / ( x(ixo^s,1)*sin(x(ixo^s,2)) * (x(ixo^s,3) - x(hpx^s,3)) * (x(jpx^s,3) - x(hpx^s,3)))
 
      cent(ixo^s)  = (x(jpx^s,3) + x(hpx^s,3) - 2.0d0*x(ixo^s,3)) *  vfield(ixo^s) &
                      / ( x(ixo^s,1)*sin(x(ixo^s,2)) * (x(ixo^s,3) - x(hpx^s,3)) * (x(jpx^s,3) - x(ixo^s,3)))
    }
    end select
 
    ! Total gradient for given velocity field
    grad_vn(ixo^s) = backw(ixo^s) + cent(ixo^s) + forw(ixo^s)
 
  end subroutine get_velocity_gradient
 
  !> Initialise (theta',phi') radiation angles coming from stellar disc
  subroutine rays_init(ntheta_point,nphi_point)
    use mod_global_parameters
 
    ! Subroutine arguments
    integer, intent(in) :: ntheta_point, nphi_point
 
    ! Local variables
    real(8) :: ymin, ymax, phipmin, phipmax, adum
    integer :: ii
 
    ! Minimum and maximum range of theta and phi rays
    ! NOTE: theta points are cast into y-space
    ymin    = 0.0d0
    ymax    = 1.0d0
    phipmin = -dpi !0.0d0
    phipmax = dpi !2.0d0*dpi
    ! dphi    = (phipmax - phipmin) / nphi_point
 
    if (mype == 0) then
      allocate(ay(ntheta_point))
      allocate(wy(ntheta_point))
      allocate(aphi(nphi_point))
      allocate(wphi(nphi_point))
 
      ! theta and phi ray positions and weights: Gauss-Legendre
      call gauss_legendre_quadrature(ymin,ymax,ntheta_point,ay,wy)
      call gauss_legendre_quadrature(phipmin,phipmax,nphi_point,aphi,wphi)
 
      ! theta rays and weights: uniform
      ! dth = 1.0d0 / nthetap
      ! adum = ymin + 0.5d0*dth
      ! do ip = 1,nthetap
      !   ay(ip) = adum
      !   wy(ip) = 1.0d0/nthetap
      !   adum = adum + dth
      !   !print*,'phipoints'
      !   !print*,ip,aphi(ip),wphi(ip),dphi
      ! enddo
 
      ! phi ray position and weights: uniform
      ! adum = phipmin + 0.5d0*dphi
      ! do ii = 1,nphi_point
      !   aphi(ii) = adum
      !   wphi(ii) = 1.0d0/nphi_point
      !   adum     = adum + dphi
      ! enddo
 
      print*, '==========================='
      print*, '    Radiation ray setup    '
      print*, '==========================='
      print*, 'Theta ray points + weights '
      do ii = 1,ntheta_point
        print*,ii,ay(ii),wy(ii)
      enddo
      print*
      print*, 'Phi ray points + weights   '
      do ii = 1,nphi_point
        print*,ii,aphi(ii),wphi(ii)
      enddo
      print*
    endif
 
    call mpi_barrier(icomm,ierrmpi)
 
    !===========================
    ! Broadcast what mype=0 read
    !===========================
    if (npe > 1) then
      call mpi_bcast(ntheta_point,1,mpi_integer,0,icomm,ierrmpi)
      call mpi_bcast(nphi_point,1,mpi_integer,0,icomm,ierrmpi)
 
      if (mype /= 0) then
        allocate(ay(ntheta_point))
        allocate(wy(ntheta_point))
        allocate(aphi(nphi_point))
        allocate(wphi(nphi_point))
      endif
 
      call mpi_bcast(ay,ntheta_point,mpi_double_precision,0,icomm,ierrmpi)
      call mpi_bcast(wy,ntheta_point,mpi_double_precision,0,icomm,ierrmpi)
      call mpi_bcast(aphi,nphi_point,mpi_double_precision,0,icomm,ierrmpi)
      call mpi_bcast(wphi,nphi_point,mpi_double_precision,0,icomm,ierrmpi)
    endif
 
  end subroutine rays_init
  
  !> Fast Gauss-Legendre N-point quadrature algorithm by G. Rybicki
  subroutine gauss_legendre_quadrature(xlow,xhi,n,x,w)
    ! Given the lower and upper limits of integration xlow and xhi, and given n,
    ! this routine returns arrays x and w of length n, containing the abscissas
    ! and weights of the Gauss-Legendre N-point quadrature
    use mod_global_parameters
 
    ! Subroutine arguments
    real(8), intent(in)  :: xlow, xhi
    integer, intent(in)  :: n
    real(8), intent(out) :: x(n), w(n)
 
    ! Local variables
    real(8) :: p1, p2, p3, pp, xl, xm, z, z1
    real(8), parameter :: error=3.0d-14
    integer :: i, j, m
 
    m = (n + 1)/2
    xm = 0.5d0*(xhi + xlow)
    xl = 0.5d0*(xhi - xlow)
 
    do i = 1,m
      z = cos( dpi * (i - 0.25d0)/(n + 0.5d0) )
      z1 = 2.0d0 * z
 
      do while (abs(z1 - z) > error)
        p1 = 1.0d0
        p2 = 0.0d0
 
        do j = 1,n
          p3 = p2
          p2 = p1
          p1 = ( (2.0d0*j - 1.0d0)*z*p2 - (j - 1.0d0)*p3 )/j
        enddo
 
        pp = n*(z*p1 - p2) / (z*z - 1.0d0)
        z1 = z
        z = z1 - p1/pp
      enddo
 
      x(i)     = xm - xl*z
      x(n+1-i) = xm + xl*z
      w(i)     = 2.0d0*xl / ((1.0d0 - z*z) * pp*pp)
      w(n+1-i) = w(i)
    enddo
 
  end subroutine gauss_legendre_quadrature
 
end module mod_cak_force