SUBROUTINE prandtl_fluxes

!--------------------------------------------------------------------------------!
! This file is part of PALM.
!
! PALM 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.
!
! PALM 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
! PALM. If not, see <http://www.gnu.org/licenses/>.
!
! Copyright 1997-2012  Leibniz University Hannover
!--------------------------------------------------------------------------------!
!
! Current revisions:
! -----------------
!
!
! Former revisions:
! -----------------
! $Id: prandtl_fluxes.f90 1037 2012-10-22 14:10:22Z witha $
!
! 1036 2012-10-22 13:43:42Z raasch
! code put under GPL (PALM 3.9)
!
! 1015 2012-09-27 09:23:24Z raasch
! OpenACC statements added
!
! 978 2012-08-09 08:28:32Z fricke
! roughness length for scalar quantities z0h added
!
! 759 2011-09-15 13:58:31Z raasch
! Bugfix for ts limitation
!
! 709 2011-03-30 09:31:40Z raasch
! formatting adjustments
!
! 667 2010-12-23 12:06:00Z suehring/gryschka
! Changed surface boundary conditions for u and v from mirror to Dirichlet.
! Therefore u(uzb,:,:) and v(nzb,:,:) are now representative for height z0.
! nxl-1, nxr+1, nys-1, nyn+1 replaced by nxlg, nxrg, nysg, nyng
!
! 315 2009-05-13 10:57:59Z raasch
! Saturation condition at (sea) surface is not used in precursor runs (only
! in the following coupled runs)
! Bugfix: qsws was calculated in case of constant heatflux = .FALSE.
!
! 187 2008-08-06 16:25:09Z letzel
! Bugfix: modification of the calculation of the vertical turbulent momentum
! fluxes u'w' and v'w'
! Bugfix: change definition of us_wall from 1D to 2D
! Change: modification of the integrated version of the profile function for 
! momentum for unstable stratification (does not effect results)
!
! 108 2007-08-24 15:10:38Z letzel
! assume saturation at k=nzb_s_inner(j,i) for atmosphere coupled to ocean
!
! 75 2007-03-22 09:54:05Z raasch
! moisture renamed humidity
!
! RCS Log replace by Id keyword, revision history cleaned up
!
! Revision 1.19  2006/04/26 12:24:35  raasch
! +OpenMP directives and optimization (array assignments replaced by DO loops)
!
! Revision 1.1  1998/01/23 10:06:06  raasch
! Initial revision
!
!
! Description:
! ------------
! Diagnostic computation of vertical fluxes in the Prandtl layer from the
! values of the variables at grid point k=1
!------------------------------------------------------------------------------!

    USE arrays_3d
    USE control_parameters
    USE grid_variables
    USE indices

    IMPLICIT NONE

    INTEGER ::  i, j, k
    LOGICAL ::  coupled_run
    REAL    ::  a, b, e_q, rifm, uv_total, z_p

!
!-- Data information for accelerators
    !$acc data present( e, nzb_u_inner, nzb_v_inner, nzb_s_inner, pt, q, qs ) &
    !$acc      present( qsws, rif, shf, ts, u, us, usws, v, vpt, vsws, zu, zw, z0, z0h )
! 
!-- Compute theta*
    IF ( constant_heatflux )  THEN
!
!--    For a given heat flux in the Prandtl layer:
!--    for u* use the value from the previous time step
       !$OMP PARALLEL DO
       !$acc kernels do
       DO  i = nxlg, nxrg
          DO  j = nysg, nyng
             ts(j,i) = -shf(j,i) / ( us(j,i) + 1E-30 )
!
!--          ts must be limited, because otherwise overflow may occur in case of
!--          us=0 when computing rif further below
             IF ( ts(j,i) < -1.05E5 )  ts(j,i) = -1.0E5
             IF ( ts(j,i) >   1.0E5 )  ts(j,i) =  1.0E5
          ENDDO
       ENDDO

    ELSE
!
!--    For a given surface temperature:
!--    (the Richardson number is still the one from the previous time step)
       !$OMP PARALLEL DO PRIVATE( a, b, k, z_p )
       !$acc kernels do
       DO  i = nxlg, nxrg
          DO  j = nysg, nyng

             k   = nzb_s_inner(j,i)
             z_p = zu(k+1) - zw(k)

             IF ( rif(j,i) >= 0.0 )  THEN
!
!--             Stable stratification
                ts(j,i) = kappa * ( pt(k+1,j,i) - pt(k,j,i) ) / (           &
                                  LOG( z_p / z0h(j,i) ) +                   &
                                  5.0 * rif(j,i) * ( z_p - z0h(j,i) ) / z_p &
                                                                )
             ELSE
!
!--             Unstable stratification
                a = SQRT( 1.0 - 16.0 * rif(j,i) )
                b = SQRT( 1.0 - 16.0 * rif(j,i) * z0h(j,i) / z_p )

                ts(j,i) = kappa * ( pt(k+1,j,i) - pt(k,j,i) ) /  (          &
                          LOG( z_p / z0h(j,i) ) -                           &
                          2.0 * LOG( ( 1.0 + a ) / ( 1.0 + b ) ) )
             ENDIF

          ENDDO
       ENDDO
    ENDIF

!
!-- Compute z_p/L (corresponds to the Richardson-flux number)
    IF ( .NOT. humidity )  THEN
       !$OMP PARALLEL DO PRIVATE( k, z_p )
       !$acc kernels do
       DO  i = nxlg, nxrg
          DO  j = nysg, nyng
             k   = nzb_s_inner(j,i)
             z_p = zu(k+1) - zw(k)
             rif(j,i) = z_p * kappa * g * ts(j,i) / &
                        ( pt(k+1,j,i) * ( us(j,i)**2 + 1E-30 ) )
!
!--          Limit the value range of the Richardson numbers.
!--          This is necessary for very small velocities (u,v --> 0), because
!--          the absolute value of rif can then become very large, which in
!--          consequence would result in very large shear stresses and very
!--          small momentum fluxes (both are generally unrealistic).
             IF ( rif(j,i) < rif_min )  rif(j,i) = rif_min
             IF ( rif(j,i) > rif_max )  rif(j,i) = rif_max
          ENDDO
       ENDDO
    ELSE
       !$OMP PARALLEL DO PRIVATE( k, z_p )
       !$acc kernels do
       DO  i = nxlg, nxrg
          DO  j = nysg, nyng
             k   = nzb_s_inner(j,i)
             z_p = zu(k+1) - zw(k)
             rif(j,i) = z_p * kappa * g *                            &
                        ( ts(j,i) + 0.61 * pt(k+1,j,i) * qs(j,i) ) / &
                        ( vpt(k+1,j,i) * ( us(j,i)**2 + 1E-30 ) )
!
!--          Limit the value range of the Richardson numbers.
!--          This is necessary for very small velocities (u,v --> 0), because
!--          the absolute value of rif can then become very large, which in
!--          consequence would result in very large shear stresses and very
!--          small momentum fluxes (both are generally unrealistic).
             IF ( rif(j,i) < rif_min )  rif(j,i) = rif_min
             IF ( rif(j,i) > rif_max )  rif(j,i) = rif_max
          ENDDO
       ENDDO       
    ENDIF

!
!-- Compute u* at the scalars' grid points
    !$OMP PARALLEL DO PRIVATE( a, b, k, uv_total, z_p )
    !$acc kernels do
    DO  i = nxl, nxr
       DO  j = nys, nyn

          k   = nzb_s_inner(j,i)
          z_p = zu(k+1) - zw(k)

!
!--       Compute the absolute value of the horizontal velocity 
!--       (relative to the surface)
          uv_total = SQRT( ( 0.5 * ( u(k+1,j,i) + u(k+1,j,i+1)        &
                                   - u(k,j,i)   - u(k,j,i+1) ) )**2 + &
                           ( 0.5 * ( v(k+1,j,i) + v(k+1,j+1,i)        &
                                   - v(k,j,i)   - v(k,j+1,i) ) )**2 )    


          IF ( rif(j,i) >= 0.0 )  THEN
!
!--          Stable stratification
             us(j,i) = kappa * uv_total / (                                &
                                  LOG( z_p / z0(j,i) ) +                   &
                                  5.0 * rif(j,i) * ( z_p - z0(j,i) ) / z_p &
                                          )
          ELSE
!
!--          Unstable stratification
             a = SQRT( SQRT( 1.0 - 16.0 * rif(j,i) ) )
             b = SQRT( SQRT( 1.0 - 16.0 * rif(j,i) / z_p * z0(j,i) ) )

             us(j,i) = kappa * uv_total / (                                  &
                       LOG( z_p / z0(j,i) ) -                                &
                       LOG( ( 1.0 + a )**2 * ( 1.0 + a**2 ) / (              &
                            ( 1.0 + b )**2 * ( 1.0 + b**2 )   ) ) +          &
                            2.0 * ( ATAN( a ) - ATAN( b ) )                  &
                                           )
          ENDIF
       ENDDO
    ENDDO

!
!-- Values of us at ghost point locations are needed for the evaluation of usws 
!-- and vsws.
    !$acc update host( us )
    CALL exchange_horiz_2d( us )
    !$acc update device( us )

!
!-- Compute u'w' for the total model domain.
!-- First compute the corresponding component of u* and square it.
    !$OMP PARALLEL DO PRIVATE( a, b, k, rifm, z_p )
    !$acc kernels do
    DO  i = nxl, nxr
       DO  j = nys, nyn

          k   = nzb_u_inner(j,i)
          z_p = zu(k+1) - zw(k)

!
!--       Compute Richardson-flux number for this point
          rifm = 0.5 * ( rif(j,i-1) + rif(j,i) )
          IF ( rifm >= 0.0 )  THEN
!
!--          Stable stratification
             usws(j,i) = kappa * ( u(k+1,j,i) - u(k,j,i) )/ (              &
                                     LOG( z_p / z0(j,i) ) +               &
                                     5.0 * rifm * ( z_p - z0(j,i) ) / z_p &
                                              )
          ELSE
!
!--          Unstable stratification
             a = SQRT( SQRT( 1.0 - 16.0 * rifm ) )
             b = SQRT( SQRT( 1.0 - 16.0 * rifm / z_p * z0(j,i) ) )

             usws(j,i) = kappa * ( u(k+1,j,i) - u(k,j,i) ) / (            &
                         LOG( z_p / z0(j,i) ) -                           &
                         LOG( (1.0 + a )**2 * ( 1.0 + a**2 ) / (          &
                              (1.0 + b )**2 * ( 1.0 + b**2 )   ) ) +      &
                              2.0 * ( ATAN( a ) - ATAN( b ) )             &
                                                 )
          ENDIF
          usws(j,i) = -usws(j,i) * 0.5 * ( us(j,i-1) + us(j,i) )
       ENDDO
    ENDDO

!
!-- Compute v'w' for the total model domain.
!-- First compute the corresponding component of u* and square it.
    !$OMP PARALLEL DO PRIVATE( a, b, k, rifm, z_p )
    !$acc kernels do
    DO  i = nxl, nxr
       DO  j = nys, nyn

          k   = nzb_v_inner(j,i)
          z_p = zu(k+1) - zw(k)

!
!--       Compute Richardson-flux number for this point
          rifm = 0.5 * ( rif(j-1,i) + rif(j,i) )
          IF ( rifm >= 0.0 )  THEN
!
!--          Stable stratification
             vsws(j,i) = kappa * ( v(k+1,j,i) -  v(k,j,i) ) / (           &
                                     LOG( z_p / z0(j,i) ) +               &
                                     5.0 * rifm * ( z_p - z0(j,i) ) / z_p &
                                              )
          ELSE
!
!--          Unstable stratification
             a = SQRT( SQRT( 1.0 - 16.0 * rifm ) )
             b = SQRT( SQRT( 1.0 - 16.0 * rifm / z_p * z0(j,i) ) )

             vsws(j,i) = kappa * ( v(k+1,j,i) - v(k,j,i) ) / (            &
                         LOG( z_p / z0(j,i) ) -                           &
                         LOG( (1.0 + a )**2 * ( 1.0 + a**2 ) / (          &
                              (1.0 + b )**2 * ( 1.0 + b**2 )   ) ) +      &
                              2.0 * ( ATAN( a ) - ATAN( b ) )             &
                                                 )
          ENDIF
          vsws(j,i) = -vsws(j,i) * 0.5 * ( us(j-1,i) + us(j,i) )
       ENDDO
    ENDDO

!
!-- If required compute q*
    IF ( humidity  .OR.  passive_scalar )  THEN
       IF ( constant_waterflux )  THEN
!
!--       For a given water flux in the Prandtl layer:
          !$OMP PARALLEL DO
          !$acc kernels do
          DO  i = nxlg, nxrg
             DO  j = nysg, nyng
                qs(j,i) = -qsws(j,i) / ( us(j,i) + 1E-30 )
             ENDDO
          ENDDO
          
       ELSE
          coupled_run = ( coupling_mode == 'atmosphere_to_ocean' .AND. run_coupled )
          !$OMP PARALLEL DO PRIVATE( a, b, k, z_p )
          !$acc kernels do
          DO  i = nxlg, nxrg
             DO  j = nysg, nyng

                k   = nzb_s_inner(j,i)
                z_p = zu(k+1) - zw(k)

!
!--             Assume saturation for atmosphere coupled to ocean (but not
!--             in case of precursor runs)
                IF ( coupled_run )  THEN
                   e_q = 6.1 * &
                        EXP( 0.07 * ( MIN(pt(0,j,i),pt(1,j,i)) - 273.15 ) )
                   q(k,j,i) = 0.622 * e_q / ( surface_pressure - e_q )
                ENDIF
                IF ( rif(j,i) >= 0.0 )  THEN
!
!--                Stable stratification
                   qs(j,i) = kappa * ( q(k+1,j,i) - q(k,j,i) ) / (          &
                                  LOG( z_p / z0h(j,i) ) +                   &
                                  5.0 * rif(j,i) * ( z_p - z0h(j,i) ) / z_p &
                                                                 )
                ELSE
!
!--                Unstable stratification
                   a = SQRT( 1.0 - 16.0 * rif(j,i) ) 
                   b = SQRT( 1.0 - 16.0 * rif(j,i) * z0h(j,i) / z_p ) 
 
                   qs(j,i) = kappa * ( q(k+1,j,i) - q(k,j,i) ) /   (        &
                             LOG( z_p / z0h(j,i) ) -                        &
                              2.0 * LOG( (1.0 + a ) / ( 1.0 + b ) ) )
                ENDIF

             ENDDO
          ENDDO
       ENDIF
    ENDIF

!
!-- Exchange the boundaries for the momentum fluxes (only for sake of 
!-- completeness)
    !$acc update host( usws, vsws )
    CALL exchange_horiz_2d( usws )
    CALL exchange_horiz_2d( vsws )
    !$acc update device( usws, vsws )
    IF ( humidity  .OR.  passive_scalar )  THEN
       !$acc update host( qsws )
       CALL exchange_horiz_2d( qsws )
       !$acc update device( qsws )
    ENDIF

!
!-- Compute the vertical kinematic heat flux
    IF ( .NOT. constant_heatflux )  THEN
       !$OMP PARALLEL DO
       !$acc kernels do
       DO  i = nxlg, nxrg
          DO  j = nysg, nyng
             shf(j,i) = -ts(j,i) * us(j,i)
          ENDDO
       ENDDO
    ENDIF

!
!-- Compute the vertical water/scalar flux
    IF ( .NOT. constant_waterflux .AND. ( humidity .OR. passive_scalar ) ) THEN
       !$OMP PARALLEL DO
       !$acc kernels do
       DO  i = nxlg, nxrg
          DO  j = nysg, nyng
             qsws(j,i) = -qs(j,i) * us(j,i)
          ENDDO
       ENDDO
    ENDIF

!
!-- Bottom boundary condition for the TKE
    IF ( ibc_e_b == 2 )  THEN
       !$OMP PARALLEL DO
       !$acc kernels do
       DO  i = nxlg, nxrg
          DO  j = nysg, nyng
             e(nzb_s_inner(j,i)+1,j,i) = ( us(j,i) / 0.1 )**2
!
!--          As a test: cm = 0.4
!            e(nzb_s_inner(j,i)+1,j,i) = ( us(j,i) / 0.4 )**2
             e(nzb_s_inner(j,i),j,i)   = e(nzb_s_inner(j,i)+1,j,i)
          ENDDO
       ENDDO
    ENDIF

    !$acc end data

 END SUBROUTINE prandtl_fluxes