amun-code/sources/forcing.F90

!!******************************************************************************
!!
!!  This file is part of the AMUN source code, a program to perform
!!  Newtonian or relativistic magnetohydrodynamical simulations on uniform or
!!  adaptive mesh.
!!
!!  Copyright (C) 2017-2020 Grzegorz Kowal <grzegorz@amuncode.org>
!!
!!  This program 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 3 of the License, or
!!  (at your option) any later version.
!!
!!  This program 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 this program.  If not, see <http://www.gnu.org/licenses/>.
!!
!!*****************************************************************************
!!
!! module: FORCING
!!
!!  This module provides energy injection, e.g. turbulence driving, supernova
!!  explosions, etc.
!!
!!*****************************************************************************
!
module forcing

#ifdef PROFILE
! import external subroutines
!
  use timers, only : set_timer, start_timer, stop_timer
#endif /* PROFILE */

! module variables are not implicit by default
!
  implicit none

#ifdef PROFILE
! timer indices
!
  integer, save :: ifi, ifu
#endif /* PROFILE */

! interfaces for procedure pointers
!
  abstract interface
    subroutine update_forcing_iface(t, dt)
      real(kind=8), intent(in)  :: t, dt
    end subroutine
  end interface

! pointers to the forcing methods
!
  procedure(update_forcing_iface), pointer, save :: update_forcing => null()

! flag indicating if the energy injection is enabled and if it is done in
! the Fourier space
!
  logical, save :: forcing_enabled = .false.
  logical, save :: forcing_fourier = .false.
  logical, save :: energy_profile  = .false.

! implemented injection methods
!
  integer, parameter :: injection_none     = 0
  integer, parameter :: injection_eddy     = 1
  integer, parameter :: injection_oh       = 2
  integer, parameter :: injection_alvelius = 3

! implemented driving force spectral profiles
!
  integer, parameter :: profile_powerlaw  = 0
  integer, parameter :: profile_normal    = 1
  integer, parameter :: profile_parabolic = 2

! driving force method and spectral profile
!
  integer, save :: injection_method  = injection_none
  integer, save :: injection_profile = profile_powerlaw

! forcing parameters
!
  real(kind=8), save :: power = 1.0d+00
  real(kind=8), save :: rate  = 1.0d+03
  real(kind=8), save :: amp   = 1.0d-03
  real(kind=8), save :: del   = 1.0d-01

! Fourier profile parameters
!
  integer     , save :: nmodes = 0
  real(kind=8), save :: fpower = 1.0d+00
  real(kind=8), save :: lscale = 5.0d-01
  real(kind=8), save :: tscale = 1.0d+00
  real(kind=8), save :: vscale = 1.0d+00
  real(kind=8), save :: ascale = 1.0d+00
  real(kind=8), save :: kf     = 2.0d+00
  real(kind=8), save :: kl     = 0.0d+00
  real(kind=8), save :: ku     = 4.0d+00
  real(kind=8), save :: kc     = 1.0d+00
  real(kind=8), save :: pidx   = - 5.0d+00 / 3.0d+00
  real(kind=8), save :: fmin   = 1.0d-08
  real(kind=8), save :: sole   = 1.0d+00
  real(kind=8), save :: norm   = 3.0d+00 / sqrt(2.0d+00)

! module parameters
!
  real(kind=8), save :: dinj = 0.0d+00
  real(kind=8), save :: einj = 0.0d+00
  real(kind=8), save :: rinj = 0.0d+00
  real(kind=8), save :: arms = 0.0d+00

! bound for the driving mode wave number
!
  integer, save :: kmax = 4

! arrays for driving mode positions and amplitudes
!
  integer     , dimension(:,:  ), allocatable :: kmodes
  real(kind=8), dimension(:)    , allocatable :: fmodes

! vector of driving directions in Alfvelius method
!
  real(kind=8), dimension(:,:)  , allocatable :: e1vecs, e2vecs

! solenoidal projection tensor
!
  real(kind=8), dimension(:,:,:), allocatable :: kprjct

! array for driving mode coefficients
!
  complex(kind=8), dimension(:,:), allocatable :: fcoefs

! velocity Fourier coefficients in Alfvelius method
!
  complex(kind=8), dimension(:,:), allocatable :: vcoefs

! by default everything is private
!
  private

! declare public subroutines
!
  public :: initialize_forcing, finalize_forcing, print_forcing
  public :: update_forcing
  public :: forcing_enabled, einj, rinj, arms

!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
!
  contains
!
!===============================================================================
!!
!!***  PUBLIC SUBROUTINES  *****************************************************
!!
!===============================================================================
!
!===============================================================================
!
! subroutine INITIALIZE_FORCING:
! -----------------------------
!
!   Subroutine initializes module FORCING.
!
!   Arguments:
!
!     verbose - flag determining if the subroutine should be verbose;
!     status  - return flag of the procedure execution status;
!
!===============================================================================
!
  subroutine initialize_forcing(verbose, status)

! import external procedures and variables
!
    use parameters, only : get_parameter
    use random    , only : randomn, gaussian_complex

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    logical, intent(in)  :: verbose
    integer, intent(out) :: status

! local variables
!
    character(len=32) :: wfmt = "(1x,'WARNING:',1x,a)"
    character(len=64) :: injection      = "none"
    character(len=64) :: profile_type   = "gauss"
    character(len=64) :: profile_energy = "off"
    logical           :: test
    integer           :: i, j, k = 0, l, k2
    real(kind=8)      :: kl2, ku2, kv2, kv
    real(kind=8)      :: fa, fi, uu

! local vectors
!
    real(kind=8), dimension(NDIMS) :: u, v

! allocatable arrays
!
  integer, dimension(:), allocatable :: kcount
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! set timer descriptions
!
    call set_timer('forcing:: initialization', ifi)
    call set_timer('forcing:: update'        , ifu)

! start accounting time for initialization/finalization
!
    call start_timer(ifi)
#endif /* PROFILE */

! reset the status flag
!
    status = 0

! obtain the chosen injection method
!
    call get_parameter("injection_method", injection)

! select the energy injection method
!
    select case(trim(injection))
    case("eddies", "eddy")
      forcing_enabled  = .true.
      injection_method = injection_eddy
      update_forcing   => update_forcing_eddies

      call get_parameter("injection_power", power)
      call get_parameter("injection_rate" , rate )
      call get_parameter("eddy_amplitude" , amp  )
      call get_parameter("eddy_width"     , del  )

    case("Ornstein–Uhlenbeck", "ornstein–uhlenbeck", "oh")
      forcing_enabled  = .true.
      forcing_fourier  = .true.
      injection_method = injection_oh
      update_forcing   => update_forcing_oh

! get the solenoidal factor
!
      call get_parameter("solenoidal_factor", sole)
      if (sole < 0.0d+00 .or. sole > 1.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,wfmt) "'solenoidal_factor' must be between 0.0 and 1.0!"
          write(*,wfmt) "Resetting it to the default value: 1.0!"
        end if
        sole = 1.0d+00
      end if

    case("Alvelius", "alvelius", "al")
      forcing_enabled  = .true.
      forcing_fourier  = .true.
      injection_method = injection_alvelius
      update_forcing   => update_forcing_alvelius

    case default
      injection_method = injection_none
      if (verbose .and. injection /= "none") then
        write(*,*)
        write(*,wfmt) "Unknown injection method! Forcing disabled!"
      end if
    end select

! initialization of the Fourier space driving
!
    if (forcing_fourier) then

! get the driving power
!
      call get_parameter("driving_power", fpower)
      if (fpower <= 0.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,wfmt) "'driving_power' must be larger than zero!"
          write(*,wfmt) "Resetting it to the default value: 1.0!"
        end if
        fpower = 1.0d+00
      end if

! get the autocorrelation/turnover time
!
      call get_parameter("driving_timescale", tscale)
      if (tscale <= 0.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,wfmt) "'driving_timescale' must be larger than zero!"
          write(*,wfmt) "Resetting it to the default value: 1.0!"
        end if
        tscale = 1.0d+00
      end if

      call get_parameter("injection_scale"    , kf)
      call get_parameter("injection_width"    , kc)
      call get_parameter("lower_scale"        , kl)
      call get_parameter("upper_scale"        , ku)
      call get_parameter("amplitude_threshold", fmin)

      if (kf < 1.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,wfmt) "'injection_scale' must be larger than 1.0!"
          write(*,wfmt) "Resetting it to the default value: 2.0!"
        end if
        kf = 2.0d+00
      end if
      if (kc <= 0.0d+00) then
        if (verbose) then
          write(*,*)
          write(*,wfmt) "'injection_width' must be larger than 0.0!"
          write(*,wfmt) "Resetting it to the default value: 1.0!"
        end if
        kc = 1.0d+00
      end if
      if (kc > kf) then
        if (verbose) then
          write(*,*)
          write(*,wfmt) "'injection_width' must be smaller than 'injection_scale'!"
          write(*,wfmt) "Resetting it to half of 'injection_scale'!"
        end if
        kc = 5.0d-01 * kf
      end if
      if (kl >= kf) then
        if (verbose) then
          write(*,*)
          write(*,wfmt) "'lower_scale' must be smaller than 'injection_scale'!"
          write(*,wfmt) "Setting it to 'injection_scale' - 'injection_width'!"
        end if
        kl = kf - kc
      end if
      if (ku <= kf) then
        if (verbose) then
          write(*,*)
          write(*,wfmt) "'upper_scale' must be larger than 'injection_scale'!"
          write(*,wfmt) "Setting it to 'injection_scale' + 'injection_width'!"
        end if
        ku = kf + kc
      end if
      fmin = max(1.0d-08, fmin)
      kl2  = kl**2
      ku2  = ku**2

! get the upper bound for the wave number
!
      kmax = ceiling(ku)

! allocate arrays to count the modes with the same wave number
!
      allocate(kcount(NDIMS * kmax**2), stat = status)

      if (status == 0) then

! initialize mode counts
!
        kcount(:) =  0

! count the modes with the same k, get the minimum sufficient number of
! driving modes
!
#if NDIMS == 3
        do i = 1, kmax
          do j = -kmax, kmax
            do k = -kmax, kmax
              k2  = i**2 + j**2 + k**2
              kv2 = 1.0d+00 * k2
              if (kl2 <= kv2 .and. kv2 <= ku2) then
                nmodes     = nmodes + 1
                kcount(k2) = kcount(k2) + 2
              end if
            end do
          end do
        end do
        do j = 1, kmax
          do k = -kmax, kmax
            k2  = j**2 + k**2
            kv2 = 1.0d+00 * k2
            if (kl2 <= kv2 .and. kv2 <= ku2) then
              nmodes     = nmodes + 1
              kcount(k2) = kcount(k2) + 2
            end if
          end do
        end do
        do k = 1, kmax
          k2  = k**2
          kv2 = 1.0d+00 * k2
          if (kl2 <= kv2 .and. kv2 <= ku2) then
            nmodes     = nmodes + 1
            kcount(k2) = kcount(k2) + 2
          end if
        end do
#else /* NDIMS == 3 */
        do i = 1, kmax
          do j = -kmax, kmax
            k2  = i**2 + j**2
            kv2 = 1.0d+00 * k2
            if (kl2 <= kv2 .and. kv2 <= ku2) then
              nmodes     = nmodes + 1
              kcount(k2) = kcount(k2) + 2
            end if
          end do
        end do
        do j = 1, kmax
          k2  = j**2
          kv2 = 1.0d+00 * k2
          if (kl2 <= kv2 .and. kv2 <= ku2) then
            nmodes     = nmodes + 1
            kcount(k2) = kcount(k2) + 2
          end if
        end do
#endif /* NDIMS == 3 */

! allocate arrays for driving modes (positions, amplitudes and coefficients)
!
        allocate(kmodes(nmodes,NDIMS), fmodes(nmodes),                         &
                 fcoefs(nmodes,NDIMS), stat = status)

        if (status == 0) then

! prepare wave vectors
!
          l = 0
#if NDIMS == 3
          do i = 1, kmax
            do j = -kmax, kmax
              do k = -kmax, kmax
                k2  = i**2 + j**2 + k**2
                kv2 = 1.0d+00 * k2
                if (kl2 <= kv2 .and. kv2 <= ku2) then
                  l = l + 1
                  kmodes(l,1)   = i
                  kmodes(l,2)   = j
                  kmodes(l,3)   = k
                end if
              end do
            end do
          end do
          do j = 1, kmax
            do k = -kmax, kmax
              k2  = j**2 + k**2
              kv2 = 1.0d+00 * k2
              if (kl2 <= kv2 .and. kv2 <= ku2) then
                l = l + 1
                kmodes(l,1)   = 0
                kmodes(l,2)   = j
                kmodes(l,3)   = k
              end if
            end do
          end do
          do k = 1, kmax
            k2  = k**2
            kv2 = 1.0d+00 * k2
            if (kl2 <= kv2 .and. kv2 <= ku2) then
              l = l + 1
              kmodes(l,1)   = 0
              kmodes(l,2)   = 0
              kmodes(l,3)   = k
            end if
          end do
#else /* NDIMS == 3 */
          do i = 1, kmax
            do j = -kmax, kmax
              k2  = i**2 + j**2
              kv2 = 1.0d+00 * k2
              if (kl2 <= kv2 .and. kv2 <= ku2) then
                l = l + 1
                kmodes(l,1) = i
                kmodes(l,2) = j
              end if
            end do
          end do
          do j = 1, kmax
            k2  = j**2
            kv2 = 1.0d+00 * k2
            if (kl2 <= kv2 .and. kv2 <= ku2) then
              l = l + 1
              kmodes(l,1) = 0
              kmodes(l,2) = j
            end if
          end do
#endif /* NDIMS == 3 */

! determine if the profile is set by energy or amplitude
!
          call get_parameter("energy_profile", profile_energy)
          select case(trim(profile_energy))
          case ("on", "ON", "t", "T", "y", "Y", "true", "TRUE", "yes", "YES")
            energy_profile = .true.
          case default
            energy_profile = .false.
            kcount(:)      = 1
          end select

! initialize driving mode amplitudes depending on the chosen profile
!
          call get_parameter("injection_profile", profile_type)
          select case(trim(profile_type))

! --- power-law profile ---
!
          case("power-law", "powerlaw", "band")

            injection_profile = profile_powerlaw

! get the power index of the power-law distribution
!
            call get_parameter("power_index", pidx)

! calculate amplitudes of driving modes
!
            fi = 0.0d+00
            do l = 1, nmodes
              k2        = sum(kmodes(l,:)**2)
              kv        = sqrt(1.0d+00 * k2)
              fa        = kv**pidx / kcount(k2)
              fi        = fi + fa
              fmodes(l) = sqrt(fa)
            end do
            do l = 1, nmodes
              k2        = sum(kmodes(l,:)**2)
              fmodes(l) = fmodes(l) / sqrt(fi)
            end do

! --- normal distribution profile ---
!
          case("gauss", "gaussian", "normal")

            injection_profile = profile_normal

! calculate amplitudes of driving modes
!
            fi = 0.0d+00
            do l = 1, nmodes
              k2        = sum(kmodes(l,:)**2)
              kv        = sqrt(1.0d+00 * k2)
              fa        = exp(- 0.5d+00 * ((kv - kf) / kc)**2) / kcount(k2)
              fi        = fi + fa
              fmodes(l) = sqrt(fa)
            end do
            do l = 1, nmodes
              k2        = sum(kmodes(l,:)**2)
              fmodes(l) = fmodes(l) / sqrt(fi)
            end do

! --- parabolic profile ---
!
          case("parabolic")

            injection_profile = profile_parabolic

! correct the injection scale and width
!
            kf = 0.5d+00 * (ku + kl)
            kc = 0.5d+00 * (ku - kl)

! calculate amplitudes of driving modes
!
            fi = 0.0d+00
            do l = 1, nmodes
              k2        = sum(kmodes(l,:)**2)
              kv        = sqrt(1.0d+00 * k2)
              fa        = 4.0d+00 * (kv - kl) * (ku - kv) / (ku - kl)**2
              fa        = max(0.0d+00, fa)**2 / kcount(k2)
              fi        = fi + fa
              fmodes(l) = sqrt(fa)
            end do
            do l = 1, nmodes
              k2        = sum(kmodes(l,:)**2)
              fmodes(l) = fmodes(l) / sqrt(fi)
            end do

! --- normal distribution profile ---
!
          case default

! warn if the specified profile is not implemented and the normal one is used
!
            if (verbose .and. .not. (profile_type == "normal" .or.             &
                                     profile_type == "gauss"  .or.             &
                                     profile_type == "gaussian")) then
              write(*,*)
              write(*,wfmt) "Unknown spectral profile of driving!"
              write(*,wfmt) "Available profiles: 'power-law', 'normal', 'parabolic'."
              write(*,wfmt) "Using default normal distribution profile 'gaussian'."

            end if

            injection_profile = profile_normal

! calculate amplitudes of driving modes
!
            fi = 0.0d+00
            do l = 1, nmodes
              k2        = sum(kmodes(l,:)**2)
              kv        = sqrt(1.0d+00 * k2)
              fa        = exp(- 0.5d+00 * ((kv - kf) / kc)**2) / kcount(k2)
              fi        = fi + fa
              fmodes(l) = sqrt(fa)
            end do
            do l = 1, nmodes
              k2        = sum(kmodes(l,:)**2)
              fmodes(l) = fmodes(l) / sqrt(fi)
            end do

          end select

! === Ornstein–Uhlenbeck driving ===
!
          if (injection_method == injection_oh) then

! get the characteristic driving length, velocity, time, and acceleration scales
!
            lscale = 1.0d+00 / kf
            vscale = lscale / tscale
            ascale = vscale / tscale * sqrt(2.0d+00 / tscale)
            fpower = lscale**2 / tscale**3

! normalize the driving amplitude profile to the driving power
!
            fmodes(:) = ascale * fmodes(:)

! allocate arrays for projection tensors
!
            allocate(kprjct(nmodes,NDIMS,NDIMS), stat = status)

            if (status == 0) then

! generate the driving mode coefficients with random phases and directions
!
              do l = 1, nmodes

! prepare the projection tensors
!
                uu = sum(kmodes(l,:)**2)
                fa = (1.0d+00 - 2.0d+00 * sole) / uu
                do j = 1, NDIMS
                  kprjct(l,j,j) = sole
                  do i = 1, NDIMS
                    kprjct(l,i,j) = kprjct(l,i,j) + fa * kmodes(l,i) * kmodes(l,j)
                  end do
                end do

! generate a random vector, project it and normalize
!
                test = .true.
                do while (test)
                  do i = 1, NDIMS
                    v(i) = randomn()
                  end do
                  do i = 1, NDIMS
                    u(i) = dot_product(v(:), kprjct(l,:,i))
                  end do
                  uu = dot_product(u(:), u(:))
                  if (uu > 0.0d+00) then
                    u(:) = u(:) / sqrt(uu)
                    test = .false.
                  end if
                end do ! while

! calculate the initial driving mode coefficient
!
                fcoefs(l,:) = fmodes(l) * u(:) * gaussian_complex()

              end do ! l = 1, nmodes

! calculate the rms of acceleration
!
              arms = sqrt(abs(sum(fcoefs(:,:) * conjg(fcoefs(:,:)))) / NDIMS)

            end if ! allocation of kprjct

          end if ! injection = 'OH'

! === Alvelius driving ===
!
          if (injection_method == injection_alvelius) then

! get the characteristic driving length, velocity, time, and acceleration scales
!
            lscale = 1.0d+00 / kf
            tscale = sqrt(5.0d-01 * lscale / sqrt(fpower))
            vscale = lscale / tscale
            ascale = vscale / tscale

! normalize the driving amplitude profile to the driving power
!
            fmodes(:) = ascale * fmodes(:)

! allocate perpendicular vectors
!
            allocate(e1vecs(nmodes,NDIMS), e2vecs(nmodes,NDIMS), vcoefs(nmodes,NDIMS), stat = status)

            if (status == 0) then

! generate random vectors, perpendicular to the wave vector
!
              do l = 1, nmodes

                k2 = sum(kmodes(l,:)**2)
                kv = sqrt(1.0d+00 * k2)

                u(:) = kmodes(l,:) / kv

                test = .true.
                do while (test)
                  do i = 1, NDIMS
                    v(i) = randomn()
                  end do
                  uu = dot_product(v(:), v(:))
                  if (uu > 0.0d+00) then
                    v(:) = v(:) / sqrt(uu)
                    uu = dot_product(u(:), v(:))
                    if (abs(uu) <= 9.0d-01) then
                      v(:) = v(:) - uu * u(:)
                      uu = dot_product(v(:), v(:))
                      if (uu > 0.0d+00) then
                        v(:) = v(:) / sqrt(uu)

                        e1vecs(l,:) = v(:)
#if NDIMS == 3
                        e2vecs(l,1) = u(2) * v(3) - u(3) * v(2)
                        e2vecs(l,2) = u(3) * v(1) - u(1) * v(3)
                        e2vecs(l,3) = u(1) * v(2) - u(2) * v(1)
#else /* NDIMS == 3 */
                        e2vecs(l,:) = 0.0d+00
#endif /* NDIMS == 3 */

                        test = .false.
                      end if
                    end if
                  end if
                end do ! while

                e1vecs(l,:) = e1vecs(l,:) / sqrt(sum(e1vecs(l,:)**2))
                e2vecs(l,:) = e2vecs(l,:) / sqrt(sum(e2vecs(l,:)**2))

              end do

              fcoefs(:,:) = cmplx(0.0d+00, 0.0d+00)

! calculate the rms of acceleration
!
              arms = sqrt(abs(sum(fcoefs(:,:) * conjg(fcoefs(:,:)))) / NDIMS)

            end if ! allocation of e1 and e2 vectors

          end if ! injection = 'Alvelius'

        end if ! allocation of driving mode arrays
      end if ! allocation of kcount

! deallocate mode counter
!
      if (allocated(kcount)) deallocate(kcount)

    end if ! Fourier forcing

#ifdef PROFILE
! stop accounting time
!
    call stop_timer(ifi)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine initialize_forcing
!
!===============================================================================
!
! subroutine FINALIZE_FORCING:
! ---------------------------
!
!   Subroutine releases memory used by the module variables.
!
!   Arguments:
!
!     status  - return flag of the procedure execution status;
!
!===============================================================================
!
  subroutine finalize_forcing(status)

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    integer, intent(out) :: status
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for initialization/finalization
!
    call start_timer(ifi)
#endif /* PROFILE */

    if (allocated(kmodes)) deallocate(kmodes)
    if (allocated(fmodes)) deallocate(fmodes)
    if (allocated(fcoefs)) deallocate(fcoefs)
    if (allocated(kprjct)) deallocate(kprjct)

    if (allocated(e1vecs)) deallocate(e1vecs)
    if (allocated(e2vecs)) deallocate(e2vecs)
    if (allocated(vcoefs)) deallocate(vcoefs)

#ifdef PROFILE
! stop accounting time
!
    call stop_timer(ifi)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine finalize_forcing
!
!===============================================================================
!
! subroutine PRINT_FORCING:
! ------------------------
!
!   Subroutine prints module parameters.
!
!   Arguments:
!
!     verbose - a logical flag turning the information printing;
!
!===============================================================================
!
  subroutine print_forcing(verbose)

! import external procedures and variables
!
    use helpers  , only : print_section, print_parameter

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    logical, intent(in) :: verbose

! local variables
!
    character(len=64) :: injection    = "none"
    character(len=64) :: profile_type = "gaussian"
!
!-------------------------------------------------------------------------------
!
    if (verbose) then

      select case(injection_method)
      case(injection_eddy)
        injection = "random eddies"
      case(injection_oh)
        injection = "Ornstein–Uhlenbeck"
      case(injection_alvelius)
        injection = "Alvelius"
      case default
        injection = "none"
      end select

      call print_section(verbose, "Forcing")
      call print_parameter(verbose, "injection method", injection)

      select case(injection_method)
      case(injection_eddy)
        call print_parameter(verbose, "injection power", power)
        call print_parameter(verbose, "injection rate" , rate)
        call print_parameter(verbose, "eddy amplitude" , amp)
        call print_parameter(verbose, "eddy width"     , del)
      end select

      if (forcing_fourier) then
        call print_parameter(verbose, "driving power"         , fpower)
        call print_parameter(verbose, "driving length scale"  , lscale)
        call print_parameter(verbose, "driving time scale"    , tscale)
        call print_parameter(verbose, "driving velocity scale", vscale)
        call print_parameter(verbose, "driving acceleration scale", ascale)
        if (injection_method == injection_oh) then
          call print_parameter(verbose, "solenoidal factor"   , sole)
        end if
        select case(injection_profile)
        case(profile_powerlaw)
          profile_type = "power-law"
        case(profile_normal)
          profile_type = "gaussian"
        case(profile_parabolic)
          profile_type = "parabolic"
        case default
          profile_type = "unknown"
        end select

        call print_parameter(verbose, "injection profile", profile_type)
        if (energy_profile) then
          call print_parameter(verbose, "profile of", "energy")
        else
          call print_parameter(verbose, "profile of", "amplitude")
        end if
        call print_parameter(verbose, "injection scale"  , kf)
        call print_parameter(verbose, "lower scale limit", kl)
        call print_parameter(verbose, "upper scale limit", ku)
        if (injection_profile == profile_powerlaw) then
          call print_parameter(verbose, "power-law index", pidx)
        end if
        if (injection_profile == profile_normal) then
          call print_parameter(verbose, "gaussian profile width", kc)
        end if
        call print_parameter(verbose, "amplitude threshold", fmin)
        call print_parameter(verbose, "number of modes"    , nmodes)
      end if

    end if

!-------------------------------------------------------------------------------
!
  end subroutine print_forcing
!
!===============================================================================
!
! subroutine UPDATE_FORCING_EDDIES:
! --------------------------------
!
!   Subroutine adds the energy injection terms.
!
!   Arguments:
!
!     t, dt - time and its increment;
!
!===============================================================================
!
  subroutine update_forcing_eddies(t, dt)

! import external procedures and variables
!
    use coordinates, only : xmin, ymin, zmin, xlen, ylen, zlen
    use random     , only : randomu, randomn

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    real(kind=8), intent(in) :: t, dt

! local variables
!
    integer                    :: ni, n
    real(kind=8)               :: tmp
    real(kind=8), dimension(3) :: xp, ap
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for forcing term update
!
    call start_timer(ifu)
#endif /* PROFILE */

! proceed only if forcing is enabled
!
    if (forcing_enabled) then

! calculate the number of eddies to be injected during this interval
!
      dinj = dinj + rate * dt
      ni   = int(floor(dinj))
      dinj = dinj - 1.0d+00 * ni

! inject the required number of eddies
!
      if (ni > 0) then

        do n = 1, ni

! get random position
!
          xp(1) = xmin + xlen * randomu()
          xp(2) = ymin + ylen * randomu()
          xp(3) = zmin + zlen * randomu()

! get random orientation
!
#if NDIMS == 3
          tmp = 0.0d+00
          do while(tmp == 0.0d+00)
            ap(1) = randomn()
            ap(2) = randomn()
            ap(3) = randomn()
            tmp   = sqrt(ap(1)**2 + ap(2)**2 + ap(3)**2)
          end do
          ap(:) = amp * (ap(:) / tmp) / del
#else /* NDIMS == 3 */
          ap(:) = sign(1.0d+00, randomn()) * (/ 0.0d+00, 0.0d+00, amp / del /)
#endif /* NDIMS == 3 */
          ap(:) = ap(:) * dt

! iterate over data blocks and add forcing components
!
          call inject_eddy(xp(:), ap(:))

        end do

      end if

    end if ! forcing enabled

#ifdef PROFILE
! stop accounting time
!
    call stop_timer(ifu)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine update_forcing_eddies
!
!===============================================================================
!
! subroutine UPDATE_FORCING_OH:
! ----------------------------
!
!   Subroutine adds the energy injection terms for Ornstein–Uhlenbeck driving.
!
!   Arguments:
!
!     t, dt - time and its increment;
!
!===============================================================================
!
  subroutine update_forcing_oh(t, dt)

! import external procedures and variables
!
    use random, only : randomn, gaussian_complex

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    real(kind=8), intent(in) :: t, dt

! local variables
!
    logical      :: test
    integer      :: i, j, k, l
    real(kind=8) :: acoeff, dcoeff
    real(kind=8) :: dinj, uu

! local vectors
!
    complex(kind=8), dimension(NDIMS) :: finj
    real(kind=8)   , dimension(NDIMS) :: u, v
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for forcing term update
!
    call start_timer(ifu)
#endif /* PROFILE */

! proceed only if forcing is enabled
!
    if (forcing_enabled) then

! calculate drift and diffusion coefficients
!
      acoeff = exp(- dt / tscale)
      dcoeff = sqrt(1.0d+00 - acoeff**2)

! iterate over driving modes
!
      do l = 1, nmodes

! generate a random vector, project it and normalize
!
        test = .true.
        do while (test)
          do i = 1, NDIMS
            v(i) = randomn()
          end do
          do i = 1, NDIMS
            u(i) = dot_product(v(:), kprjct(l,:,i))
          end do
          uu = dot_product(u(:), u(:))
          if (uu > 0.0d+00) then
            u(:) = u(:) / sqrt(uu)
            test = .false.
          end if
        end do ! while

! generate gaussian random vector along the driving modes
!
        finj(:)     = fmodes(l) * u(:) * gaussian_complex()

! advance the driving force modes
!
        fcoefs(l,:) = acoeff * fcoefs(l,:) + dcoeff * finj(:)

      end do

! calculate the rms of acceleration
!
      arms = sqrt(abs(sum(fcoefs(:,:) * conjg(fcoefs(:,:)))) / NDIMS)

! store previously injected energy
!
      dinj = einj

! inject driving modes
!
      call inject_fmodes(dt)

! calculate the energy injected during this step
!
      dinj = einj - dinj

! calculate the injection rate
!
      rinj = dinj / dt

    end if ! forcing enabled
#ifdef PROFILE
! stop accounting time
!
    call stop_timer(ifu)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine update_forcing_oh
!
!===============================================================================
!
! subroutine UPDATE_FORCING_ALVELIUS:
! ----------------------------------
!
!   Subroutine adds the energy injection terms for Alvelius driving.
!
!   Arguments:
!
!     t, dt - time and its increment;
!
!===============================================================================
!
  subroutine update_forcing_alvelius(t, dt)

! import external procedures and variables
!
    use constants, only : pi2
    use random   , only : randomu

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    real(kind=8), intent(in) :: t, dt

! local variables
!
    logical         :: test
    integer         :: i, j, k, l
    real(kind=8)    :: th1, th2, phi, psi, ga, gb, dinj, sqdt
    complex(kind=8) :: aran, bran, xi1, xi2
!
!-------------------------------------------------------------------------------
!
#ifdef PROFILE
! start accounting time for forcing term update
!
    call start_timer(ifu)
#endif /* PROFILE */

! proceed only if forcing is enabled
!
    if (forcing_enabled) then

! determine velocify coefficients in Fourier space
!
      call get_vcoefs()

! square root of dt
!
      sqdt = sqrt(dt)

! iterate over driving modes
!
      do l = 1, nmodes

! get xi1 and xi2
!
        xi1 = dot_product(vcoefs(l,:), e1vecs(l,:))
        xi2 = dot_product(vcoefs(l,:), e2vecs(l,:))

! get random angles theta1, theta2, and phi
!
        phi = pi2 * randomu()
        ga  = sin(phi)
        gb  = cos(phi)
        psi = pi2 * randomu()
        th1 = - ga * dimag(xi1) + gb * (sin(psi) * dreal(xi2) - cos(psi) * dimag(xi2))
        if (th1 /= 0.0d+00) then
          th1 = atan((ga * dreal(xi1) + gb * (sin(psi) * dimag(xi2) + cos(psi) * dreal(xi2))) / th1)
        end if
        th2 = psi + th1

! calculate amplitude
!
        aran = cmplx(cos(th1), sin(th1)) * ga
        bran = cmplx(cos(th2), sin(th2)) * gb

! advance the driving force modes
!
        fcoefs(l,:) = fmodes(l) * (aran * e1vecs(l,:)                          &
                                 + bran * e2vecs(l,:)) / sqdt

      end do

! calculate the rms of acceleration
!
      arms = sqrt(abs(sum(fcoefs(:,:) * conjg(fcoefs(:,:)))) / NDIMS)

! store previously injected energy
!
      dinj = einj

! inject driving modes
!
      call inject_fmodes(dt)

! calculate the energy injected during this step
!
      dinj = einj - dinj

! calculate the injection rate
!
      rinj = dinj / dt

    end if ! forcing enabled

#ifdef PROFILE
! stop accounting time
!
    call stop_timer(ifu)
#endif /* PROFILE */

!-------------------------------------------------------------------------------
!
  end subroutine update_forcing_alvelius
!
!===============================================================================
!!
!!***  PRIVATE SUBROUTINES  ****************************************************
!!
!===============================================================================
!
!===============================================================================
!
! subroutine INJECT_EDDY:
! ----------------------
!
!   Subroutine injects one eddy to the domain at given position.
!
!   Arguments:
!
!     xp, ap - eddy position and amplitude vectors;
!
!===============================================================================
!
  subroutine inject_eddy(xp, ap)

! include external variables
!
    use blocks, only : block_data, list_data

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    real(kind=8), dimension(3), intent(in) :: xp, ap

! local pointers
!
    type(block_data), pointer :: pdata
!
!-------------------------------------------------------------------------------
!
! assign pdata with the first block on the data block list
!
    pdata => list_data

! iterate over all data blocks
!
    do while (associated(pdata))

! inject eddy into the current block
!
      call inject_eddy_block(pdata, xp(:), ap(:))

! assign pdata to the next block
!
      pdata => pdata%next

    end do ! over data blocks

!-------------------------------------------------------------------------------
!
  end subroutine inject_eddy
!
!===============================================================================
!
! subroutine INJECT_EDDY_BLOCK:
! ----------------------------
!
!   Subroutine injects one eddy to the local block.
!
!   Arguments:
!
!     xp, ap - eddy position and amplitude vectors;
!
!===============================================================================
!
  subroutine inject_eddy_block(pdata, xp, ap)

! include external variables
!
    use blocks     , only : block_data
    use coordinates, only : nm => bcells
    use coordinates, only : ax, ay, az
    use coordinates, only : xlen, ylen, zlen
    use coordinates, only : periodic
    use equations  , only : nv
    use equations  , only : idn, imx, imy, imz, ien

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    type(block_data), pointer , intent(inout) :: pdata
    real(kind=8), dimension(3), intent(in)    :: xp, ap

! local variables
!
    integer      :: i, j, k = 1
    real(kind=8) :: x2, y2, z2, r2
    real(kind=8) :: fx, fy, fz, fp, e1, e2

! local arrays
!
    real(kind=8), dimension(nm)    :: x, y, z
!
!-------------------------------------------------------------------------------
!
! prepare block coordinates
!
    if (periodic(1)) then
      fx   = 0.5d+00 * xlen
      x(:) = pdata%meta%xmin + ax(pdata%meta%level,:) - xp(1)
      x(:) = abs(x(:) / fx)
      x(:) = (abs(1.0d+00 - abs(1.0d+00 - abs(x(:))))) * fx / del
    else
      x(:) = (pdata%meta%xmin + ax(pdata%meta%level,:) - xp(1)) / del
    end if
    if (periodic(2)) then
      fy   = 0.5d+00 * ylen
      y(:) = pdata%meta%ymin + ay(pdata%meta%level,:) - xp(2)
      y(:) = abs(y(:) / fy)
      y(:) = (abs(1.0d+00 - abs(1.0d+00 - abs(y(:))))) * fy / del
    else
      y(:) = (pdata%meta%ymin + ay(pdata%meta%level,:) - xp(2)) / del
    end if
#if NDIMS == 3
    if (periodic(3)) then
      fz   = 0.5d+00 * zlen
      z(:) = pdata%meta%zmin + az(pdata%meta%level,:) - xp(3)
      z(:) = abs(z(:) / fz)
      z(:) = (abs(1.0d+00 - abs(1.0d+00 - abs(z(:))))) * fz / del
    else
      z(:) = (pdata%meta%zmin + az(pdata%meta%level,:) - xp(3)) / del
    end if
#else /* NDIMS == 3 */
    z(:) = 0.0d+00
#endif /* NDIMS == 3 */

! iterate over the block coordinates
!
    if (ien > 0) then
#if NDIMS == 3
      do k = 1, nm
        z2 = z(k)**2
#endif /* NDIMS == 3 */
        do j = 1, nm
          y2 = y(j)**2
          do i = 1, nm
            x2 = x(i)**2
#if NDIMS == 3
            r2 = x2 + y2 + z2
#else /* NDIMS == 3 */
            r2 = x2 + y2
#endif /* NDIMS == 3 */

            fp = pdata%u(idn,i,j,k) * exp(- 0.5d+00 * r2)

#if NDIMS == 3
            fx = (- ap(3) * y(j) + ap(2) * z(k)) * fp
            fy = (- ap(1) * z(k) + ap(3) * x(i)) * fp
            fz = (- ap(2) * x(i) + ap(1) * y(j)) * fp
#else /* NDIMS == 3 */
            fx = (- ap(3) * y(j)) * fp
            fy = (+ ap(3) * x(i)) * fp
            fz = 0.0d+00
#endif /* NDIMS == 3 */

            e1 = 0.5d+00 * sum(pdata%u(imx:imx,i,j,k)**2) / pdata%u(idn,i,j,k)

            pdata%u(imx,i,j,k) = pdata%u(imx,i,j,k) + fx
            pdata%u(imy,i,j,k) = pdata%u(imy,i,j,k) + fy
            pdata%u(imz,i,j,k) = pdata%u(imz,i,j,k) + fz

            e2 = 0.5d+00 * sum(pdata%u(imx:imz,i,j,k)**2) / pdata%u(idn,i,j,k)

            pdata%u(ien,i,j,k) = pdata%u(ien,i,j,k) + (e2 - e1)

          end do
        end do
#if NDIMS == 3
      end do
#endif /* NDIMS == 3 */
    else
#if NDIMS == 3
      do k = 1, nm
        z2 = z(k)**2
#endif /* NDIMS == 3 */
        do j = 1, nm
          y2 = y(j)**2
          do i = 1, nm
            x2 = x(i)**2
#if NDIMS == 3
            r2 = x2 + y2 + z2
#else /* NDIMS == 3 */
            r2 = x2 + y2
#endif /* NDIMS == 3 */

            fp = pdata%u(idn,i,j,k) * exp(- 0.5d+00 * r2)

#if NDIMS == 3
            fx = (- ap(3) * y(j) + ap(2) * z(k)) * fp
            fy = (- ap(1) * z(k) + ap(3) * x(i)) * fp
            fz = (- ap(2) * x(i) + ap(1) * y(j)) * fp
#else /* NDIMS == 3 */
            fx = (- ap(3) * y(j)) * fp
            fy = (+ ap(3) * x(i)) * fp
            fz = 0.0d+00
#endif /* NDIMS == 3 */

            pdata%u(imx,i,j,k) = pdata%u(imx,i,j,k) + fx
            pdata%u(imy,i,j,k) = pdata%u(imy,i,j,k) + fy
            pdata%u(imz,i,j,k) = pdata%u(imz,i,j,k) + fz

          end do
        end do
#if NDIMS == 3
      end do
#endif /* NDIMS == 3 */
    end if

!-------------------------------------------------------------------------------
!
  end subroutine inject_eddy_block
!
!===============================================================================
!
! subroutine INJECT_FMODES:
! ------------------------
!
!   Subroutine injects Fourier driving modes.
!
!   Arguments:
!
!     dt - the time increment;
!
!===============================================================================
!
  subroutine inject_fmodes(dt)

! include external variables
!
    use blocks, only : block_data, list_data

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    real(kind=8), intent(in) :: dt

! local pointers
!
    type(block_data), pointer :: pdata
!
!-------------------------------------------------------------------------------
!
! assign pdata with the first block on the data block list
!
    pdata => list_data

! iterate over all data blocks
!
    do while (associated(pdata))

! inject eddy into the current block
!
      call inject_fmodes_block(pdata, dt)

! assign pdata to the next block
!
      pdata => pdata%next

    end do ! over data blocks

!-------------------------------------------------------------------------------
!
  end subroutine inject_fmodes
!
!===============================================================================
!
! subroutine INJECT_FMODES_BLOCK:
! ------------------------------
!
!   Subroutine injects Fourier driving modes to the local block.
!
!   Arguments:
!
!     pdata - a pointer to the data block;
!     dt    - the time increment;
!
!===============================================================================
!
  subroutine inject_fmodes_block(pdata, dt)

! include external variables
!
    use blocks     , only : block_data
    use constants  , only : pi2
    use coordinates, only : nm => bcells, nb, ne
    use coordinates, only : ax, ay, az, advol
    use equations  , only : idn, imx, imy, imz, ien, ivx, ivy, ivz

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    type(block_data), pointer, intent(inout) :: pdata
    real(kind=8)             , intent(in)    :: dt

! local variables
!
    integer      :: i, j, k = 1, l, n
    real(kind=8) :: cs, sn, tt
    real(kind=8) :: dvol

! local arrays
!
    real(kind=8), dimension(nm):: x, y, z, kx, ky, kz
    real(kind=8), dimension(nm):: snkx, snky, snkz
    real(kind=8), dimension(nm):: cskx, csky, cskz
#if NDIMS == 3
    real(kind=8), dimension(3,nm,nm,nm) :: acc
    real(kind=8), dimension(  nm,nm,nm) :: den
#else /* NDIMS == 3 */
    real(kind=8), dimension(2,nm,nm, 1) :: acc
    real(kind=8), dimension(  nm,nm, 1) :: den
#endif /* NDIMS == 3 */
!
!-------------------------------------------------------------------------------
!
! prepare block coordinates
!
    x(:) = - pi2 * (pdata%meta%xmin + ax(pdata%meta%level,:))
    y(:) = - pi2 * (pdata%meta%ymin + ay(pdata%meta%level,:))
#if NDIMS == 3
    z(:) = - pi2 * (pdata%meta%zmin + az(pdata%meta%level,:))
#else /* NDIMS == 3 */
    z(:) = 0.0d+00
#endif /* NDIMS == 3 */
    dvol = advol(pdata%meta%level)

! reset the acceleration
!
    acc(:,:,:,:) = 0.0d+00

! iterate over driving modes and compute the acceleration in the real space
!
    do l = 1, nmodes

      if (fmodes(l) > fmin) then

        kx(:) = kmodes(l,1) * x(:)
        ky(:) = kmodes(l,2) * y(:)
#if NDIMS == 3
        kz(:) = kmodes(l,3) * z(:)
#endif /* NDIMS == 3 */
        cskx(:) = cos(kx(:))
        snkx(:) = sin(kx(:))
        csky(:) = cos(ky(:))
        snky(:) = sin(ky(:))
#if NDIMS == 3
        cskz(:) = cos(kz(:))
        snkz(:) = sin(kz(:))
#endif /* NDIMS == 3 */

#if NDIMS == 3
        do k = 1, nm
#endif /* NDIMS == 3 */
          do j = 1, nm
            do i = 1, nm

              cs = cskx(i) * csky(j) - snkx(i) * snky(j)
              sn = cskx(i) * snky(j) + snkx(i) * csky(j)
#if NDIMS == 3
              tt = cs
              cs = tt * cskz(k) - sn * snkz(k)
              sn = tt * snkz(k) + sn * cskz(k)
#endif /* NDIMS == 3 */

              acc(:,i,j,k) = acc(:,i,j,k) + (dreal(fcoefs(l,:)) * cs           &
                                           + dimag(fcoefs(l,:)) * sn)
            end do
          end do
#if NDIMS == 3
        end do
#endif /* NDIMS == 3 */

      end if

    end do ! l = 1, nmodes

! multiply the acceleration by density and time step
!
    do n = 1, NDIMS
      acc(n,:,:,:) = pdata%u(idn,:,:,:) * acc(n,:,:,:) * dt
    end do

! calculate the kinetic energy before the injection
!
    den(:,:,:) = sum(pdata%u(imx:imz,:,:,:)**2, 1)

! add the momentum increment
!
    pdata%u(imx,:,:,:) = pdata%u(imx,:,:,:) + acc(1,:,:,:)
    pdata%u(imy,:,:,:) = pdata%u(imy,:,:,:) + acc(2,:,:,:)
#if NDIMS == 3
    pdata%u(imz,:,:,:) = pdata%u(imz,:,:,:) + acc(3,:,:,:)
#endif /* NDIMS == 3 */

! calculate the injected energy
!
    den(:,:,:) = 5.0d-01 * (sum(pdata%u(imx:imz,:,:,:)**2, 1) - den(:,:,:))    &
               / pdata%u(idn,:,:,:)

! calculate the injected energy
!
#if NDIMS == 3
    einj = einj + sum(den(nb:ne,nb:ne,nb:ne)) * dvol
#else /* NDIMS == 3 */
    einj = einj + sum(den(nb:ne,nb:ne, 1   )) * dvol
#endif /* NDIMS == 3 */

! add the energy increment
!
    if (ien > 0) then
      pdata%u(ien,:,:,:) = pdata%u(ien,:,:,:) + den(:,:,:)
    end if

!-------------------------------------------------------------------------------
!
  end subroutine inject_fmodes_block
!
!===============================================================================
!
! subroutine GET_VCOEFS:
! ---------------------
!
!   Subroutine determines velocity Fourier coefficients.
!
!
!===============================================================================
!
  subroutine get_vcoefs()

! include external variables
!
    use blocks  , only : block_data, list_data
#ifdef MPI
    use mpitools, only : reduce_sum_complex_array
#endif /* MPI */

! local variables are not implicit by default
!
    implicit none

! local variables
!
    integer :: status

! local pointers
!
    type(block_data), pointer :: pdata
!
!-------------------------------------------------------------------------------
!
! reset vcoefs
!
    vcoefs(:,:) = cmplx(0.0d+00, 0.0d+00)

! assign pdata with the first block on the data block list
!
    pdata => list_data

! iterate over all data blocks
!
    do while (associated(pdata))

! get contribution of velocity coefficients from the current block
!
      call get_vcoefs_block(pdata)

! assign pdata to the next block
!
      pdata => pdata%next

    end do ! over data blocks

#ifdef MPI
! reduce velocity coefficients over all processes
!
    call reduce_sum_complex_array(size(vcoefs), vcoefs, status)
#endif /* MPI */

!-------------------------------------------------------------------------------
!
  end subroutine get_vcoefs
!
!===============================================================================
!
! subroutine GET_VCOEFS_BLOCK:
! ---------------------------
!
!   Subroutine determines Fourier coefficients of velocity.
!
!   Arguments:
!
!     pdata - a pointer to the data block;
!
!===============================================================================
!
  subroutine get_vcoefs_block(pdata)

! include external variables
!
    use blocks     , only : block_data
    use constants  , only : pi2
    use coordinates, only : nm => bcells, nb, ne
    use coordinates, only : ax, ay, az, advol
    use equations  , only : ivx, ivy, ivz

! local variables are not implicit by default
!
    implicit none

! subroutine arguments
!
    type(block_data), pointer, intent(inout) :: pdata

! local variables
!
    integer         :: i, j, k = 1, l
    real(kind=8)    :: cs, sn, tt, dvol
    complex(kind=8) :: cf

! local arrays
!
    real(kind=8), dimension(nm):: x, y, z, kx, ky, kz
    real(kind=8), dimension(nm):: snkx, snky, snkz
    real(kind=8), dimension(nm):: cskx, csky, cskz
!
!-------------------------------------------------------------------------------
!
! prepare block coordinates
!
    x(:) = pi2 * (pdata%meta%xmin + ax(pdata%meta%level,:))
    y(:) = pi2 * (pdata%meta%ymin + ay(pdata%meta%level,:))
#if NDIMS == 3
    z(:) = pi2 * (pdata%meta%zmin + az(pdata%meta%level,:))
#else /* NDIMS == 3 */
    z(:) = 0.0d+00
#endif /* NDIMS == 3 */
    dvol = advol(pdata%meta%level)

! iterate over driving modes and compute the velocity Fourier coefficients
!
    do l = 1, nmodes

      kx(:) = kmodes(l,1) * x(:)
      ky(:) = kmodes(l,2) * y(:)
#if NDIMS == 3
      kz(:) = kmodes(l,3) * z(:)
#endif /* NDIMS == 3 */
      cskx(:) = cos(kx(:))
      snkx(:) = sin(kx(:))
      csky(:) = cos(ky(:))
      snky(:) = sin(ky(:))
#if NDIMS == 3
      cskz(:) = cos(kz(:))
      snkz(:) = sin(kz(:))
#endif /* NDIMS == 3 */

#if NDIMS == 3
      do k = nb, ne
#endif /* NDIMS == 3 */
        do j = nb, ne
          do i = nb, ne

            cs = cskx(i) * csky(j) - snkx(i) * snky(j)
            sn = cskx(i) * snky(j) + snkx(i) * csky(j)
#if NDIMS == 3
            tt = cs
            cs = tt * cskz(k) - sn * snkz(k)
            sn = tt * snkz(k) + sn * cskz(k)
#endif /* NDIMS == 3 */

            cf = cmplx(cs, sn) * dvol

            vcoefs(l,1) = vcoefs(l,1) + pdata%q(ivx,i,j,k) * cf
            vcoefs(l,2) = vcoefs(l,2) + pdata%q(ivy,i,j,k) * cf
#if NDIMS == 3
            vcoefs(l,3) = vcoefs(l,3) + pdata%q(ivz,i,j,k) * cf
#endif /* NDIMS == 3 */

          end do
        end do
#if NDIMS == 3
      end do
#endif /* NDIMS == 3 */

    end do ! l = 1, nmodes

!-------------------------------------------------------------------------------
!
  end subroutine get_vcoefs_block

!===============================================================================
!
end module forcing