richardson_cpu.f90

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module richardson_cpu
  use num_types, only : rp
  use utils, only : neko_error, neko_warning
  use logger, only : log_size, neko_log
  use math, only : glsum, glmin, glmax
  implicit none
  private
 
  public :: richardson_compute_cpu
 
  abstract interface
     function tau_interface(magu, Ri_b, h, z0, l, kappa) result(tau)
       import rp
       real(kind=rp), intent(in) :: magu, ri_b, h, z0, l, kappa
       real(kind=rp) :: tau
     end function tau_interface
 
     function heat_flux_interface(ti, ts, Ri_b, h, magu, z0h, Pr,&
          l, utau, kappa) result(heat_flux)
       import rp
       real(kind=rp), intent(in) :: ts, ti, ri_b, h, magu
       real(kind=rp), intent(in) :: z0h, pr, l, utau, kappa
       real(kind=rp) :: heat_flux
     end function heat_flux_interface
 
  end interface
 
  ! These will point to the correct functions
  ! depending on stability regime and bc_type.
  procedure(tau_interface), pointer :: tau_ptr => null()
  procedure(heat_flux_interface), pointer :: heat_flux_ptr => null()
 
contains
 
  subroutine compute_ri_b(bc_type, g_dot_n, hi, ti, ts, magu, kappa, q, Ri_b)
    character(len=*), intent(in) :: bc_type
    real(kind=rp), intent(in) :: hi, ti, ts
    real(kind=rp), intent(in) :: magu, kappa, g_dot_n
    real(kind=rp), intent(inout) :: q, ri_b
 
    select case (bc_type)
    case ("neumann")
       ri_b = - g_dot_n*hi / ti*q / (magu**3*kappa**2)
    case ("dirichlet")
       ri_b = g_dot_n*hi/ti*(ti - ts)/magu**2
    case default
       call neko_error("Invalid specified temperature b.c. type ('neumann' or 'dirichlet'?)")
    end select
  end subroutine compute_ri_b
 
  subroutine assign_bc_value(bc_type,bc_value,q,ts,ti,kappa,utau,z0h,hi)
    character(len=*), intent(in) :: bc_type
    real(kind=rp), intent(in) :: hi, ti, kappa, utau, z0h,bc_value
    real(kind=rp), intent(inout) :: q,ts
 
    select case (bc_type)
    case ("neumann")
       ! ts not used
       q = bc_value
    case ("dirichlet")
       ts = bc_value
       q = kappa*utau*(ts - ti)/log(hi/z0h)
    case default
       call neko_error("Invalid specified temperature b.c. type ('neumann' or 'dirichlet'?)")
    end select
  end subroutine assign_bc_value
 
  subroutine set_stability_regime(Ri_b,Ri_threshold)
    real(kind=rp), intent(in) :: ri_b, ri_threshold
 
    if (ri_b > ri_threshold) then
       tau_ptr => tau_stable
       heat_flux_ptr => heat_flux_stable
    elseif (ri_b < -ri_threshold) then
       tau_ptr => tau_convective
       heat_flux_ptr => heat_flux_convective
    else
       tau_ptr => tau_neutral
       heat_flux_ptr => heat_flux_neutral
    end if
  end subroutine set_stability_regime
 
  subroutine richardson_compute_cpu(u, v, w, temp, ind_r, ind_s, ind_t, ind_e, &
       n_x, n_y, n_z, h, tau_x, tau_y, tau_z, n_nodes, lx, nelv, &
       kappa, mu_w, rho_w, g_vec, Pr, z0, z0h_in, bc_type, bc_value, tstep, &
       Ri_b_diagn, L_ob_diagn, utau_diagn, magu_diagn, ti_diagn, ts_diagn,&
       q_diagn, h_x_idx, h_y_idx, h_z_idx)
    integer, intent(in) :: n_nodes, lx, nelv, tstep
    real(kind=rp), dimension(lx, lx, lx, nelv), intent(in) :: u, v, w, temp
    integer, intent(in), dimension(n_nodes) :: ind_r, ind_s, ind_t, ind_e
    real(kind=rp), dimension(n_nodes), intent(in) :: n_x, n_y, n_z, h
    real(kind=rp), intent(in) :: kappa, z0, z0h_in, bc_value, pr
    real(kind=rp), dimension(3), intent(in) :: g_vec
    real(kind=rp), dimension(n_nodes), intent(in) :: mu_w, rho_w
    real(kind=rp) :: g_dot_n
    character(len=*), intent(in) :: bc_type
    real(kind=rp), dimension(n_nodes), intent(inout) :: tau_x, tau_y, tau_z
    integer :: i
    real(kind=rp) :: ui, vi, wi, hi, rho, mu
    real(kind=rp) :: normu, z0h
    real(kind=rp) :: l
    real(kind=rp), parameter :: tol = 0.001_rp
    real(kind=rp), parameter :: nr_step = 0.001_rp
    real(kind=rp), parameter :: ri_threshold = 0.0001_rp
    character(len=LOG_SIZE) :: log_buf
    real(kind=rp) :: utau, ri_b, l_ob, magu, q, ti, ts
    real(kind=rp), dimension(n_nodes), intent(inout) :: ri_b_diagn, l_ob_diagn
    real(kind=rp), dimension(n_nodes), intent(inout) :: utau_diagn, magu_diagn
    real(kind=rp), dimension(n_nodes), intent(inout) :: ti_diagn, ts_diagn
    real(kind=rp), dimension(n_nodes), intent(inout) :: q_diagn
    integer, dimension(n_nodes), intent(in) :: h_x_idx
    integer, dimension(n_nodes), intent(in) :: h_y_idx
    integer, dimension(n_nodes), intent(in) :: h_z_idx
 
    do i=1, n_nodes
       ! Sample the variables
       ui = u(ind_r(i), ind_s(i), ind_t(i), ind_e(i))
       vi = v(ind_r(i), ind_s(i), ind_t(i), ind_e(i))
       wi = w(ind_r(i), ind_s(i), ind_t(i), ind_e(i))
       ti = temp(ind_r(i), ind_s(i), ind_t(i), ind_e(i))
       hi = h(i)
       rho = rho_w(i)
       mu = mu_w(i)
 
       ! Project on horizontal directions
       normu = ui * n_x(i) + vi * n_y(i) + wi * n_z(i)
       ui = ui - normu * n_x(i)
       vi = vi - normu * n_y(i)
       wi = wi - normu * n_z(i)
 
       ! Compute velocity magnitude
       magu = sqrt(ui**2 + vi**2 + wi**2)
       magu = max(magu, 1.0e-6_rp)
       utau = magu*kappa / log(hi/z0)
 
       ! Compute thermal roughness length from Zilitinkevich, 1995
       if (z0h_in < 0) then
          ! z0h_in is interpreted as -C_Zil (Zilitinkevich constant) for z0h
          z0h = z0 * exp(z0h_in*sqrt((utau*z0)/(mu/rho)))
       else
          z0h = z0h_in
       end if
 
       ! Get q, ts based on bc_type
       ! Maybe redundant, but needed to initialise Rib
       call assign_bc_value(bc_type,bc_value,q,ts,ti,kappa,utau,z0h,hi)
 
       ! Compute g along the normal (generalisation for hills and similar)
       g_dot_n = abs(g_vec(1)*n_x(i) + g_vec(2)*n_y(i) + g_vec(3)*n_z(i))
 
       ! Compute Richardson and set stability accordingly
       call compute_ri_b(bc_type, g_dot_n, hi, ti, ts, magu, kappa, q, ri_b)
       call set_stability_regime(ri_b, ri_threshold)
 
       ! Set length scale
       l = kappa * hi
       ! Compute u*
       utau = sqrt(tau_ptr(magu, ri_b, hi, z0, l, kappa))
       select case (bc_type)
       case ("neumann")
          !!! TEMPORARY: neutral log-law approximation
          ts = ti - (q * pr * log(hi/z0h)) / (max(utau, 1e-6_rp) * kappa)
          q = q
       case ("dirichlet")
          ! Compute q
          q = heat_flux_ptr(ti, ts, ri_b, hi, magu, z0h, pr, l, utau, kappa)
       case default
          call neko_error("Invalid specified temperature b.c. type ('neumann' or 'dirichlet'?)")
       end select
 
       ! Distribute according to the velocity vector and bound magu to avoid 0 division
       magu = max(magu, 1.0e-6_rp)
       tau_x(i) = -rho*utau**2 * ui / magu
       tau_y(i) = -rho*utau**2 * vi / magu
       tau_z(i) = -rho*utau**2 * wi / magu
       if (abs(ri_b) <= ri_threshold) then
          ! Neutral (L_ob undefined)
          l_ob = 1e10_rp
       else
          l_ob = -(ts*utau**3)/(kappa*g_dot_n*q)
       end if
 
       ri_b_diagn(i) = ri_b
       l_ob_diagn(i) = l_ob
       utau_diagn(i) = utau
       magu_diagn(i) = magu
       ti_diagn(i) = ti
       ts_diagn(i) = temp(ind_r(i)-h_x_idx(i), ind_s(i)-h_y_idx(i), ind_t(i)-h_z_idx(i), ind_e(i))
       q_diagn(i) = q
    end do
 
  end subroutine richardson_compute_cpu
 
  function tau_stable(magu, Ri_b, h, z0, l, kappa) result(tau)
    real(kind=rp), intent(in) :: magu, ri_b, h, z0, l, kappa
    real(kind=rp) :: tau
 
    tau = magu**2/(log(h/z0)**2) * f_tau_stable(ri_b)/ &
         f_tau_stable(0.0_rp) * (l/h)**2
  end function tau_stable
 
  function heat_flux_stable(ti, ts, Ri_b, h, magu, z0h, Pr,&
       l, utau, kappa) result(heat_flux)
    real(kind=rp), intent(in) :: ts, ti, ri_b, h, magu
    real(kind=rp), intent(in) :: z0h, pr, l, utau, kappa
    real(kind=rp) :: heat_flux
 
    heat_flux = (ti - ts)/(log(h/z0h)) * &
         f_theta_stable(ri_b)/abs(f_theta_stable(0.0_rp)) * &
         (l/h) * utau/pr
  end function heat_flux_stable
 
  function f_tau_stable(Ri_b) result(f_tau)
    real(kind=rp), intent(in) :: ri_b
    real(kind=rp) :: f_tau
 
    f_tau = 0.17 * (0.25 + 0.75 / (1.0 + 4.0*ri_b))
  end function f_tau_stable
 
  function f_theta_stable(Ri_b) result(f_theta)
    real(kind=rp), intent(in) :: ri_b
    real(kind=rp) :: f_theta
 
    f_theta = -0.145 / (1.0 + 4.0 * ri_b)
  end function f_theta_stable
 
  function tau_convective(magu, Ri_b, h, z0, l, kappa) result(tau)
    real(kind=rp), intent(in) :: magu, ri_b, h, z0, l, kappa
    real(kind=rp) :: tau
    real(kind=rp) :: a, b, c
 
    a = kappa / log(h/z0)
    b = 2.0
    c = 7.4 * a**2 * b * (h/z0)**0.5
 
    tau = a**2 * magu**2 * f_tau_convective(ri_b, c)
  end function tau_convective
 
  function heat_flux_convective(ti, ts, Ri_b, h, magu, z0h, Pr,&
       l, utau, kappa) result(heat_flux)
    real(kind=rp), intent(in) :: ts, ti, ri_b, h, magu
    real(kind=rp), intent(in) :: z0h, pr, l, utau, kappa
    real(kind=rp) :: heat_flux
    real(kind=rp) :: a, b, c
 
    a = kappa / log(h/z0h)
    b = 2.0
    c = 5.3 * a**2 * b * (h/z0h)**0.5
 
    heat_flux = - a**2 / 0.74 * magu * &
         (ti - ts) * f_theta_convective(ri_b, c)
 
  end function heat_flux_convective
 
  function f_tau_convective(Ri_b, c) result(f_tau)
    real(kind=rp), intent(in) :: ri_b, c
    real(kind=rp) :: f_tau
 
    f_tau = 1.0 - 2*ri_b / (1.0 + c * abs(ri_b)**0.5)
  end function f_tau_convective
 
  function f_theta_convective(Ri_b, c) result(f_theta)
    real(kind=rp), intent(in) :: ri_b, c
    real(kind=rp) :: f_theta
 
    f_theta = 1.0 - 2*ri_b / (1.0 + c * abs(ri_b)**0.5)
  end function f_theta_convective
 
  function tau_neutral(magu, Ri_b, h, z0, l, kappa) result(tau)
    real(kind=rp), intent(in) :: magu, ri_b, h, z0, l, kappa
    real(kind=rp) :: tau
 
    tau = (kappa*magu/log(h/z0))**2
  end function tau_neutral
 
  function heat_flux_neutral(ti, ts, Ri_b, h, magu, z0h, Pr,&
       l, utau, kappa) result(heat_flux)
    real(kind=rp), intent(in) :: ts, ti, ri_b, h, magu
    real(kind=rp), intent(in) :: z0h, pr, l, utau, kappa
    real(kind=rp) :: heat_flux
 
    heat_flux = kappa*utau * (ti - ts)/log(h/z0h)
  end function heat_flux_neutral
 
end module richardson_cpu