source: palm/trunk/SOURCE/wall_fluxes.f90 @ 188

 MODULE wall_fluxes_mod
!------------------------------------------------------------------------------!
! Actual revisions:
! -----------------
! Bugfix: change definition of us_wall from 1D to 2D:
!         Modification of the evaluation of the vertical turbulent momentum 
!         fluxes u'w' and v'w'; the first usws that is computed corresponds 
!         to -u'w'/u* and not as priorily assumed to (-u'w')**0.5, the first 
!         vsws that is computed corresponds to -v'w'/u* and not as priorily
!         assumed to (-v'w')**0.5. Therefore, the intermediate result for 
!         usws has to be multiplied by -u* instead by itself in order to 
!         get u'w'. Accordingly, the intermediate result for vsws has to be 
!         multiplied by -u* instead by itself in order to get v'w'.
!         This requires the calculation of us_wall (and vel_total, u_i, v_i, ws)
!         also in wall_fluxes_e.
! Bugfix: storage of rifs to rifs_wall in wall_fluxes_e removed
! Change: add 'minus' sign to fluxes produced by subroutine wall_fluxes_e for 
!         consistency with subroutine wall_fluxes
! Change: Modification of the integrated version of the profile function for 
!         momentum for unstable stratification
!
! Former revisions:
! -----------------
! $Id: wall_fluxes.f90 187 2008-08-06 16:25:09Z letzel $
! Initial version (2007/03/07)
!
! Description:
! ------------
! Calculates momentum fluxes at vertical walls assuming Monin-Obukhov
! similarity.
! Indices: usvs a=1, vsus b=1, wsvs c1=1, wsus c2=1 (other=0).
! The all-gridpoint version of wall_fluxes_e is not used so far, because
! it gives slightly different results from the ij-version for some unknown
! reason.
!------------------------------------------------------------------------------!
    PRIVATE
    PUBLIC wall_fluxes, wall_fluxes_e
    
    INTERFACE wall_fluxes
       MODULE PROCEDURE wall_fluxes
       MODULE PROCEDURE wall_fluxes_ij
    END INTERFACE wall_fluxes
    
    INTERFACE wall_fluxes_e
       MODULE PROCEDURE wall_fluxes_e
       MODULE PROCEDURE wall_fluxes_e_ij
    END INTERFACE wall_fluxes_e
 
 CONTAINS

!------------------------------------------------------------------------------!
! Call for all grid points
!------------------------------------------------------------------------------!
    SUBROUTINE wall_fluxes( wall_flux, a, b, c1, c2, nzb_uvw_inner, &
                            nzb_uvw_outer, wall )

       USE arrays_3d
       USE control_parameters
       USE grid_variables
       USE indices
       USE statistics

       IMPLICIT NONE

       INTEGER ::  i, j, k, wall_index

       INTEGER, DIMENSION(nys-1:nyn+1,nxl-1:nxr+1) ::  nzb_uvw_inner, &
                                                       nzb_uvw_outer
       REAL ::  a, b, c1, c2, h1, h2, zp
       REAL ::  pts, pt_i, rifs, u_i, v_i, us_wall, vel_total, ws, wspts

       REAL, DIMENSION(nys-1:nyn+1,nxl-1:nxr+1)   ::  wall
       REAL, DIMENSION(nzb:nzt+1,nys:nyn,nxl:nxr) ::  wall_flux


       zp         = 0.5 * ( (a+c1) * dy + (b+c2) * dx )
       wall_flux  = 0.0
       wall_index = NINT( a+ 2*b + 3*c1 + 4*c2 )

       DO  i = nxl, nxr
          DO  j = nys, nyn

             IF ( wall(j,i) /= 0.0 )  THEN
!
!--             All subsequent variables are computed for the respective
!--             location where the respective flux is defined.
                DO  k = nzb_uvw_inner(j,i)+1, nzb_uvw_outer(j,i)

!
!--                (1) Compute rifs, u_i, v_i, ws, pt' and w'pt'
                   rifs  = rif_wall(k,j,i,wall_index)

                   u_i   = a * u(k,j,i) + c1 * 0.25 * &
                           ( u(k+1,j,i+1) + u(k+1,j,i) + u(k,j,i+1) + u(k,j,i) )

                   v_i   = b * v(k,j,i) + c2 * 0.25 * &
                           ( v(k+1,j+1,i) + v(k+1,j,i) + v(k,j+1,i) + v(k,j,i) )

                   ws    = ( c1 + c2 ) * w(k,j,i) + 0.25 * (                   &
                     a * ( w(k-1,j,i-1) + w(k-1,j,i) + w(k,j,i-1) + w(k,j,i) ) &
                   + b * ( w(k-1,j-1,i) + w(k-1,j,i) + w(k,j-1,i) + w(k,j,i) ) &
                                                           )
                   pt_i  = 0.5 * ( pt(k,j,i) + a *  pt(k,j,i-1) + &
                                   b * pt(k,j-1,i) + ( c1 + c2 ) * pt(k+1,j,i) )

                   pts   = pt_i - hom(k,1,4,0)
                   wspts = ws * pts

!
!--                (2) Compute wall-parallel absolute velocity vel_total
                   vel_total = SQRT( ws**2 + (a+c1) * u_i**2 + (b+c2) * v_i**2 )

!
!--                (3) Compute wall friction velocity us_wall
                   IF ( rifs >= 0.0 )  THEN

!
!--                   Stable stratification (and neutral)
                      us_wall = kappa * vel_total / ( LOG( zp / z0(j,i) ) +    &
                                            5.0 * rifs * ( zp - z0(j,i) ) / zp &
                                                    )
                   ELSE

!
!--                   Unstable stratification
                      h1 = SQRT( SQRT( 1.0 - 16.0 * rifs ) )
                      h2 = SQRT( SQRT( 1.0 - 16.0 * rifs * z0(j,i) / zp ) )

                      us_wall = kappa * vel_total / (                          &
                           LOG( zp / z0(j,i) ) -                               &
                           LOG( ( 1.0 + h1 )**2 * ( 1.0 + h1**2 ) / (          &
                                ( 1.0 + h2 )**2 * ( 1.0 + h2**2 )   ) ) +      &
                                2.0 * ( ATAN( h1 ) - ATAN( h2 ) )              &
                                                    )
                   ENDIF

!
!--                (4) Compute zp/L (corresponds to neutral Richardson flux
!--                    number rifs)
                   rifs = -1.0 * zp * kappa * g * wspts / ( pt_i * &
                                                        ( us_wall**3 + 1E-30 ) )

!
!--                Limit the value range of the Richardson numbers.
!--                This is necessary for very small velocities (u,w --> 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 ( rifs < rif_min )  rifs = rif_min
                   IF ( rifs > rif_max )  rifs = rif_max

!
!--                (5) Compute wall_flux (u'v', v'u', w'v', or w'u')
                   IF ( rifs >= 0.0 )  THEN

!
!--                   Stable stratification (and neutral)
                      wall_flux(k,j,i) = kappa *                               &
                              ( a*u(k,j,i) + b*v(k,j,i) + (c1+c2)*w(k,j,i) ) / &
                              (  LOG( zp / z0(j,i) ) +                         &
                                 5.0 * rifs * ( zp - z0(j,i) ) / zp            &
                              )
                   ELSE

!
!--                   Unstable stratification
                      h1 = SQRT( SQRT( 1.0 - 16.0 * rifs ) )
                      h2 = SQRT( SQRT( 1.0 - 16.0 * rifs * z0(j,i) / zp ) )

                      wall_flux(k,j,i) = kappa *                               &
                           ( a*u(k,j,i) + b*v(k,j,i) + (c1+c2)*w(k,j,i) ) / (  &
                           LOG( zp / z0(j,i) ) -                               &
                           LOG( ( 1.0 + h1 )**2 * ( 1.0 + h1**2 ) / (          &
                                ( 1.0 + h2 )**2 * ( 1.0 + h2**2 )   ) ) +      &
                                2.0 * ( ATAN( h1 ) - ATAN( h2 ) )              &
                                                                            )
                   ENDIF
                   wall_flux(k,j,i) = -wall_flux(k,j,i) * us_wall

!
!--                store rifs for next time step
                   rif_wall(k,j,i,wall_index) = rifs

                ENDDO

             ENDIF

          ENDDO
       ENDDO

    END SUBROUTINE wall_fluxes



!------------------------------------------------------------------------------!
! Call for all grid point i,j
!------------------------------------------------------------------------------!
    SUBROUTINE wall_fluxes_ij( i, j, nzb_w, nzt_w, wall_flux, a, b, c1, c2 )

       USE arrays_3d
       USE control_parameters
       USE grid_variables
       USE indices
       USE statistics

       IMPLICIT NONE

       INTEGER ::  i, j, k, nzb_w, nzt_w, wall_index
       REAL    ::  a, b, c1, c2, h1, h2, zp

       REAL ::  pts, pt_i, rifs, u_i, v_i, us_wall, vel_total, ws, wspts

       REAL, DIMENSION(nzb:nzt+1) ::  wall_flux


       zp         = 0.5 * ( (a+c1) * dy + (b+c2) * dx )
       wall_flux  = 0.0
       wall_index = NINT( a+ 2*b + 3*c1 + 4*c2 )

!
!--    All subsequent variables are computed for the respective location where 
!--    the respective flux is defined.
       DO  k = nzb_w, nzt_w

!
!--       (1) Compute rifs, u_i, v_i, ws, pt' and w'pt'
          rifs  = rif_wall(k,j,i,wall_index)

          u_i   = a * u(k,j,i) + c1 * 0.25 * &
                  ( u(k+1,j,i+1) + u(k+1,j,i) + u(k,j,i+1) + u(k,j,i) )

          v_i   = b * v(k,j,i) + c2 * 0.25 * &
                  ( v(k+1,j+1,i) + v(k+1,j,i) + v(k,j+1,i) + v(k,j,i) )

          ws    = ( c1 + c2 ) * w(k,j,i) + 0.25 * (                            &
                     a * ( w(k-1,j,i-1) + w(k-1,j,i) + w(k,j,i-1) + w(k,j,i) ) &
                   + b * ( w(k-1,j-1,i) + w(k-1,j,i) + w(k,j-1,i) + w(k,j,i) ) &
                                                  )
          pt_i  = 0.5 * ( pt(k,j,i) + a *  pt(k,j,i-1) + b * pt(k,j-1,i)  &
                          + ( c1 + c2 ) * pt(k+1,j,i) )

          pts   = pt_i - hom(k,1,4,0)
          wspts = ws * pts

!
!--       (2) Compute wall-parallel absolute velocity vel_total
          vel_total = SQRT( ws**2 + ( a+c1 ) * u_i**2 + ( b+c2 ) * v_i**2 )

!
!--       (3) Compute wall friction velocity us_wall
          IF ( rifs >= 0.0 )  THEN

!
!--          Stable stratification (and neutral)
             us_wall = kappa * vel_total / ( LOG( zp / z0(j,i) ) +             &
                                            5.0 * rifs * ( zp - z0(j,i) ) / zp &
                                           )
          ELSE

!
!--          Unstable stratification
             h1 = SQRT( SQRT( 1.0 - 16.0 * rifs ) )
             h2 = SQRT( SQRT( 1.0 - 16.0 * rifs * z0(j,i) / zp ) )

             us_wall = kappa * vel_total / (                          &
                  LOG( zp / z0(j,i) ) -                               &
                  LOG( ( 1.0 + h1 )**2 * ( 1.0 + h1**2 ) / (          &
                       ( 1.0 + h2 )**2 * ( 1.0 + h2**2 )   ) ) +      &
                       2.0 * ( ATAN( h1 ) - ATAN( h2 ) )              &
                                           )
          ENDIF

!
!--       (4) Compute zp/L (corresponds to neutral Richardson flux number 
!--           rifs)
          rifs = -1.0 * zp * kappa * g * wspts / ( pt_i * (us_wall**3 + 1E-30) )

!
!--       Limit the value range of the Richardson numbers.
!--       This is necessary for very small velocities (u,w --> 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 ( rifs < rif_min )  rifs = rif_min
          IF ( rifs > rif_max )  rifs = rif_max

!
!--       (5) Compute wall_flux (u'v', v'u', w'v', or w'u')
          IF ( rifs >= 0.0 )  THEN

!
!--          Stable stratification (and neutral)
             wall_flux(k) = kappa *                                          &
                            ( a*u(k,j,i) + b*v(k,j,i) + (c1+c2)*w(k,j,i) ) / &
                            (  LOG( zp / z0(j,i) ) +                         &
                               5.0 * rifs * ( zp - z0(j,i) ) / zp            &
                            )
          ELSE

!
!--          Unstable stratification
             h1 = SQRT( SQRT( 1.0 - 16.0 * rifs ) )
             h2 = SQRT( SQRT( 1.0 - 16.0 * rifs * z0(j,i) / zp ) )

             wall_flux(k) = kappa *                               &
                  ( a*u(k,j,i) + b*v(k,j,i) + (c1+c2)*w(k,j,i) ) / (  &
                  LOG( zp / z0(j,i) ) -                               &
                  LOG( ( 1.0 + h1 )**2 * ( 1.0 + h1**2 ) / (          &
                       ( 1.0 + h2 )**2 * ( 1.0 + h2**2 )   ) ) +      &
                       2.0 * ( ATAN( h1 ) - ATAN( h2 ) )              &
                                                                   )
          ENDIF
          wall_flux(k) = -wall_flux(k) * us_wall

!
!--       store rifs for next time step
          rif_wall(k,j,i,wall_index) = rifs

       ENDDO

    END SUBROUTINE wall_fluxes_ij



!------------------------------------------------------------------------------!
! Call for all grid points
!------------------------------------------------------------------------------!
    SUBROUTINE wall_fluxes_e( wall_flux, a, b, c1, c2, wall )

!------------------------------------------------------------------------------!
! Description:
! ------------
! Calculates momentum fluxes at vertical walls for routine production_e
! assuming Monin-Obukhov similarity.
! Indices: usvs a=1, vsus b=1, wsvs c1=1, wsus c2=1 (other=0).
!------------------------------------------------------------------------------!

       USE arrays_3d
       USE control_parameters
       USE grid_variables
       USE indices
       USE statistics

       IMPLICIT NONE

       INTEGER ::  i, j, k, kk, wall_index
       REAL    ::  a, b, c1, c2, h1, h2, u_i, v_i, us_wall, vel_total, vel_zp, &
                   ws, zp

       REAL ::  rifs

       REAL, DIMENSION(nys-1:nyn+1,nxl-1:nxr+1)   ::  wall
       REAL, DIMENSION(nzb:nzt+1,nys:nyn,nxl:nxr) ::  wall_flux


       zp         = 0.5 * ( (a+c1) * dy + (b+c2) * dx )
       wall_flux  = 0.0
       wall_index = NINT( a+ 2*b + 3*c1 + 4*c2 )

       DO  i = nxl, nxr
          DO  j = nys, nyn

             IF ( wall(j,i) /= 0.0 )  THEN
!
!--             All subsequent variables are computed for scalar locations.
                DO  k = nzb_diff_s_inner(j,i)-1, nzb_diff_s_outer(j,i)-2
!
!--                (1) Compute rifs, u_i, v_i, and ws
                   IF ( k == nzb_diff_s_inner(j,i)-1 )  THEN
                      kk = nzb_diff_s_inner(j,i)-1
                   ELSE
                      kk = k-1
                   ENDIF
                   rifs  = 0.5 * ( rif_wall(k,j,i,wall_index) +                &
                          a * rif_wall(k,j,i+1,1) +  b * rif_wall(k,j+1,i,2) + &
                          c1 * rif_wall(kk,j,i,3) + c2 * rif_wall(kk,j,i,4)    &
                                 )

                   u_i   = 0.5 * ( u(k,j,i) + u(k,j,i+1) )
                   v_i   = 0.5 * ( v(k,j,i) + v(k,j+1,i) )
                   ws    = 0.5 * ( w(k,j,i) + w(k-1,j,i) )
!
!--                (2) Compute wall-parallel absolute velocity vel_total and
!--                interpolate appropriate velocity component vel_zp.
                   vel_total = SQRT( ws**2 + (a+c1) * u_i**2 + (b+c2) * v_i**2 )
                   vel_zp = 0.5 * ( a * u_i + b * v_i + (c1+c2) * ws )
!
!--                (3) Compute wall friction velocity us_wall
                   IF ( rifs >= 0.0 )  THEN

!
!--                   Stable stratification (and neutral)
                      us_wall = kappa * vel_total / ( LOG( zp / z0(j,i) ) +    &
                                            5.0 * rifs * ( zp - z0(j,i) ) / zp &
                                                    )
                   ELSE

!
!--                   Unstable stratification
                      h1 = SQRT( SQRT( 1.0 - 16.0 * rifs ) )
                      h2 = SQRT( SQRT( 1.0 - 16.0 * rifs * z0(j,i) / zp ) )

                      us_wall = kappa * vel_total / (                          &
                           LOG( zp / z0(j,i) ) -                               &
                           LOG( ( 1.0 + h1 )**2 * ( 1.0 + h1**2 ) / (          &
                                ( 1.0 + h2 )**2 * ( 1.0 + h2**2 )   ) ) +      &
                                2.0 * ( ATAN( h1 ) - ATAN( h2 ) )              &
                                                    )
                   ENDIF

!
!--                Skip step (4) of wall_fluxes, because here rifs is already
!--                available from (1)
!
!--                (5) Compute wall_flux (u'v', v'u', w'v', or w'u')

                   IF ( rifs >= 0.0 )  THEN

!
!--                   Stable stratification (and neutral)
                      wall_flux(k,j,i) = kappa *  vel_zp / &
                          ( LOG( zp/z0(j,i) ) + 5.0*rifs * ( zp-z0(j,i) ) / zp )
                   ELSE

!
!--                   Unstable stratification
                      h1 = SQRT( SQRT( 1.0 - 16.0 * rifs ) )
                      h2 = SQRT( SQRT( 1.0 - 16.0 * rifs * z0(j,i) / zp ) )

                      wall_flux(k,j,i) = kappa * vel_zp / (                    &
                           LOG( zp / z0(j,i) ) -                               &
                           LOG( ( 1.0 + h1 )**2 * ( 1.0 + h1**2 ) / (          &
                                ( 1.0 + h2 )**2 * ( 1.0 + h2**2 )   ) ) +      &
                                2.0 * ( ATAN( h1 ) - ATAN( h2 ) )              &
                                                          )
                   ENDIF
                   wall_flux(k,j,i) = - wall_flux(k,j,i) * us_wall

                ENDDO

             ENDIF

          ENDDO
       ENDDO

    END SUBROUTINE wall_fluxes_e



!------------------------------------------------------------------------------!
! Call for grid point i,j
!------------------------------------------------------------------------------!
    SUBROUTINE wall_fluxes_e_ij( i, j, nzb_w, nzt_w, wall_flux, a, b, c1, c2 )

       USE arrays_3d
       USE control_parameters
       USE grid_variables
       USE indices
       USE statistics

       IMPLICIT NONE

       INTEGER ::  i, j, k, kk, nzb_w, nzt_w, wall_index
       REAL    ::  a, b, c1, c2, h1, h2, u_i, v_i, us_wall, vel_total, vel_zp, &
                   ws, zp

       REAL ::  rifs

       REAL, DIMENSION(nzb:nzt+1) ::  wall_flux


       zp         = 0.5 * ( (a+c1) * dy + (b+c2) * dx )
       wall_flux  = 0.0
       wall_index = NINT( a+ 2*b + 3*c1 + 4*c2 )

!
!--    All subsequent variables are computed for scalar locations.
       DO  k = nzb_w, nzt_w

!
!--       (1) Compute rifs, u_i, v_i, and ws
          IF ( k == nzb_w )  THEN
             kk = nzb_w
          ELSE
             kk = k-1
          ENDIF
          rifs  = 0.5 * ( rif_wall(k,j,i,wall_index) +                         &
                          a * rif_wall(k,j,i+1,1) +  b * rif_wall(k,j+1,i,2) + &
                          c1 * rif_wall(kk,j,i,3) + c2 * rif_wall(kk,j,i,4)    &
                        )

          u_i   = 0.5 * ( u(k,j,i) + u(k,j,i+1) )
          v_i   = 0.5 * ( v(k,j,i) + v(k,j+1,i) )
          ws    = 0.5 * ( w(k,j,i) + w(k-1,j,i) )
!
!--       (2) Compute wall-parallel absolute velocity vel_total and
!--       interpolate appropriate velocity component vel_zp.
          vel_total = SQRT( ws**2 + (a+c1) * u_i**2 + (b+c2) * v_i**2 )
          vel_zp = 0.5 * ( a * u_i + b * v_i + (c1+c2) * ws )
!
!--       (3) Compute wall friction velocity us_wall
          IF ( rifs >= 0.0 )  THEN

!
!--          Stable stratification (and neutral)
             us_wall = kappa * vel_total / ( LOG( zp / z0(j,i) ) +             &
                                            5.0 * rifs * ( zp - z0(j,i) ) / zp &
                                           )
          ELSE

!
!--          Unstable stratification
             h1 = SQRT( SQRT( 1.0 - 16.0 * rifs ) )
             h2 = SQRT( SQRT( 1.0 - 16.0 * rifs * z0(j,i) / zp ) )

             us_wall = kappa * vel_total / (                          &
                  LOG( zp / z0(j,i) ) -                               &
                  LOG( ( 1.0 + h1 )**2 * ( 1.0 + h1**2 ) / (          &
                       ( 1.0 + h2 )**2 * ( 1.0 + h2**2 )   ) ) +      &
                       2.0 * ( ATAN( h1 ) - ATAN( h2 ) )              &
                                           )
          ENDIF

!
!--       Skip step (4) of wall_fluxes, because here rifs is already
!--       available from (1)
!
!--       (5) Compute wall_flux (u'v', v'u', w'v', or w'u')
!--       First interpolate the velocity (this is different from 
!--       subroutine wall_fluxes because fluxes in subroutine 
!--       wall_fluxes_e are defined at scalar locations).
          vel_zp = 0.5 * (       a * ( u(k,j,i) + u(k,j,i+1) ) +  &
                                 b * ( v(k,j,i) + v(k,j+1,i) ) +  &
                           (c1+c2) * ( w(k,j,i) + w(k-1,j,i) )    &
                         )

          IF ( rifs >= 0.0 )  THEN

!
!--          Stable stratification (and neutral)
             wall_flux(k) = kappa *  vel_zp / &
                          ( LOG( zp/z0(j,i) ) + 5.0*rifs * ( zp-z0(j,i) ) / zp )
          ELSE

!
!--          Unstable stratification
             h1 = SQRT( SQRT( 1.0 - 16.0 * rifs ) )
             h2 = SQRT( SQRT( 1.0 - 16.0 * rifs * z0(j,i) / zp ) )

             wall_flux(k) = kappa * vel_zp / (                        &
                  LOG( zp / z0(j,i) ) -                               &
                  LOG( ( 1.0 + h1 )**2 * ( 1.0 + h1**2 ) / (          &
                       ( 1.0 + h2 )**2 * ( 1.0 + h2**2 )   ) ) +      &
                       2.0 * ( ATAN( h1 ) - ATAN( h2 ) )              &
                                                 )
          ENDIF
          wall_flux(k) = - wall_flux(k) * us_wall

       ENDDO

    END SUBROUTINE wall_fluxes_e_ij

 END MODULE wall_fluxes_mod