source: palm/trunk/SOURCE/microphysics.f90 @ 1102

 MODULE microphysics_mod

!--------------------------------------------------------------------------------!
! 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: microphysics.f90 1093 2013-02-02 12:58:49Z raasch $
!
! 1092 2013-02-02 11:24:22Z raasch
! unused variables removed
! file put under GPL
!
! 1065 2012-11-22 17:42:36Z hoffmann
! Sedimentation process implemented according to Stevens and Seifert (2008).
! Turbulence effects on autoconversion and accretion added (Seifert, Nuijens 
! and Stevens, 2010).
!
! 1053 2012-11-13 17:11:03Z hoffmann
! initial revision
! 
! Description:
! ------------
! Calculate cloud microphysics according to the two moment bulk 
! scheme by Seifert and Beheng (2006).
!------------------------------------------------------------------------------!

    PRIVATE
    PUBLIC dsd_properties, autoconversion, accretion, selfcollection_breakup, &
           evaporation_rain, sedimentation_cloud, sedimentation_rain

    INTERFACE dsd_properties
       MODULE PROCEDURE dsd_properties
       MODULE PROCEDURE dsd_properties_ij
    END INTERFACE dsd_properties

    INTERFACE autoconversion
       MODULE PROCEDURE autoconversion
       MODULE PROCEDURE autoconversion_ij
    END INTERFACE autoconversion

    INTERFACE accretion
       MODULE PROCEDURE accretion
       MODULE PROCEDURE accretion_ij
    END INTERFACE accretion

    INTERFACE selfcollection_breakup
       MODULE PROCEDURE selfcollection_breakup
       MODULE PROCEDURE selfcollection_breakup_ij
    END INTERFACE selfcollection_breakup

    INTERFACE evaporation_rain
       MODULE PROCEDURE evaporation_rain
       MODULE PROCEDURE evaporation_rain_ij
    END INTERFACE evaporation_rain

    INTERFACE sedimentation_cloud
       MODULE PROCEDURE sedimentation_cloud
       MODULE PROCEDURE sedimentation_cloud_ij
    END INTERFACE sedimentation_cloud
 
    INTERFACE sedimentation_rain
       MODULE PROCEDURE sedimentation_rain
       MODULE PROCEDURE sedimentation_rain_ij
    END INTERFACE sedimentation_rain

 CONTAINS


!------------------------------------------------------------------------------!
! Call for all grid points
!------------------------------------------------------------------------------!
    SUBROUTINE dsd_properties

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices

       IMPLICIT NONE

       INTEGER ::  i, j, k

 
       DO  i = nxl, nxr
          DO  j = nys, nyn
             DO  k = nzb_2d(j,i)+1, nzt

             ENDDO
          ENDDO
       ENDDO

    END SUBROUTINE dsd_properties

    SUBROUTINE autoconversion

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices

       IMPLICIT NONE

       INTEGER ::  i, j, k

 
       DO  i = nxl, nxr
          DO  j = nys, nyn
             DO  k = nzb_2d(j,i)+1, nzt

             ENDDO
          ENDDO
       ENDDO

    END SUBROUTINE autoconversion

    SUBROUTINE accretion

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices

       IMPLICIT NONE

       INTEGER ::  i, j, k

 
       DO  i = nxl, nxr
          DO  j = nys, nyn
             DO  k = nzb_2d(j,i)+1, nzt

             ENDDO
          ENDDO
       ENDDO

    END SUBROUTINE accretion

    SUBROUTINE selfcollection_breakup

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices

       IMPLICIT NONE

       INTEGER ::  i, j, k

 
       DO  i = nxl, nxr
          DO  j = nys, nyn
             DO  k = nzb_2d(j,i)+1, nzt

             ENDDO
          ENDDO
       ENDDO

    END SUBROUTINE selfcollection_breakup

    SUBROUTINE evaporation_rain

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices

       IMPLICIT NONE

       INTEGER ::  i, j, k

 
       DO  i = nxl, nxr
          DO  j = nys, nyn
             DO  k = nzb_2d(j,i)+1, nzt

             ENDDO
          ENDDO
       ENDDO

    END SUBROUTINE evaporation_rain

    SUBROUTINE sedimentation_cloud

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices

       IMPLICIT NONE

       INTEGER ::  i, j, k

 
       DO  i = nxl, nxr
          DO  j = nys, nyn
             DO  k = nzb_2d(j,i)+1, nzt

             ENDDO
          ENDDO
       ENDDO

    END SUBROUTINE sedimentation_cloud

    SUBROUTINE sedimentation_rain

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices

       IMPLICIT NONE

       INTEGER ::  i, j, k

 
       DO  i = nxl, nxr
          DO  j = nys, nyn
             DO  k = nzb_2d(j,i)+1, nzt

             ENDDO
          ENDDO
       ENDDO


    END SUBROUTINE sedimentation_rain


!------------------------------------------------------------------------------!
! Call for grid point i,j
!------------------------------------------------------------------------------!
    SUBROUTINE dsd_properties_ij( i, j )

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices
       USE control_parameters
       USE user
   
       IMPLICIT NONE

       INTEGER ::  i, j, k

       DO  k = nzb_2d(j,i)+1, nzt

          IF ( qr(k,j,i) <= eps_sb )  THEN
             qr(k,j,i) = 0.0
          ELSE
!
!--          Adjust number of raindrops to avoid nonlinear effects in 
!--          sedimentation and evaporation of rain drops due to too small or
!--          too big weights of rain drops (Stevens and Seifert, 2008).
             IF ( nr(k,j,i) * xrmin > qr(k,j,i) * hyrho(k) )  THEN
                nr(k,j,i) = qr(k,j,i) * hyrho(k) / xrmin
             ELSEIF ( nr(k,j,i) * xrmax < qr(k,j,i) * hyrho(k) )  THEN
                nr(k,j,i) = qr(k,j,i) * hyrho(k) / xrmax
             ENDIF
             xr(k) = hyrho(k) * qr(k,j,i) / nr(k,j,i)
!
!--          Weight averaged diameter of rain drops:
             dr(k) = ( hyrho(k) * qr(k,j,i) / nr(k,j,i) *                     &
                       dpirho_l )**( 1.0 / 3.0 )
!
!--          Shape parameter of gamma distribution (Milbrandt and Yau, 2005;
!--          Stevens and Seifert, 2008):
             mu_r(k) = 10.0 * ( 1.0 + TANH( 1.2E3 * ( dr(k) - 1.4E-3 ) ) )
!
!--          Slope parameter of gamma distribution (Seifert, 2008):
             lambda_r(k) = ( ( mu_r(k) + 3.0 ) * ( mu_r(k) + 2.0 ) *          &
                             ( mu_r(k) + 1.0 ) )**( 1.0 / 3.0 ) / dr(k)
          ENDIF
       ENDDO

    END SUBROUTINE dsd_properties_ij

    SUBROUTINE autoconversion_ij( i, j )

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices
       USE control_parameters
       USE statistics
       USE grid_variables
   
       IMPLICIT NONE

       INTEGER ::  i, j, k
       REAL    ::  k_au, autocon, phi_au, tau_cloud, xc, nu_c, rc,   &
                   l_mix, re_lambda, alpha_cc, r_cc, sigma_cc, epsilon

       k_au = k_cc / ( 20.0 * x0 )

       DO  k = nzb_2d(j,i)+1, nzt

          IF ( ql(k,j,i) > 0.0 )  THEN
!
!--          Intern time scale of coagulation (Seifert and Beheng, 2006):
!--          (1.0 - ql(k,j,i) / ( ql(k,j,i) + qr(k,j,i) ))
             tau_cloud = 1.0 - ql(k,j,i) / ( ql(k,j,i) + qr(k,j,i) + 1.0E-20 )
!
!--          Universal function for autoconversion process 
!--          (Seifert and Beheng, 2006):
             phi_au    = 600.0 * tau_cloud**0.68 * ( 1.0 - tau_cloud**0.68 )**3
!
!--          Shape parameter of gamma distribution (Geoffroy et al., 2010):
!--          (Use constant nu_c = 1.0 instead?)
             nu_c      = 1.0 !MAX( 0.0, 1580.0 * hyrho(k) * ql(k,j,i) - 0.28 )
!
!--          Mean weight of cloud droplets:
             xc        = hyrho(k) * ql(k,j,i) / nc
!
!--          Parameterized turbulence effects on autoconversion (Seifert, 
!--          Nuijens and Stevens, 2010)
             IF ( turbulence )  THEN
!
!--             Weight averaged radius of cloud droplets:
                rc = 0.5 * ( xc * dpirho_l )**( 1.0 / 3.0 )

                alpha_cc = ( a_1 + a_2 * nu_c ) / ( 1.0 + a_3 * nu_c )
                r_cc     = ( b_1 + b_2 * nu_c ) / ( 1.0 + b_3 * nu_c )
                sigma_cc = ( c_1 + c_2 * nu_c ) / ( 1.0 + c_3 * nu_c )
!
!--             Mixing length (neglecting distance to ground and stratification)
                l_mix = ( dx * dy * dzu(k) )**( 1.0 / 3.0 )
!
!--             Limit dissipation rate according to Seifert, Nuijens and 
!--             Stevens (2010)
                epsilon = MIN( 0.06, diss(k,j,i) )
!
!--             Compute Taylor-microscale Reynolds number:
                re_lambda = 6.0 / 11.0 * ( l_mix / c_const )**( 2.0 / 3.0 ) *  &
                            SQRT( 15.0 / kin_vis_air ) * epsilon**( 1.0 / 6.0 )
!
!--             The factor of 1.0E4 is needed to convert the dissipation rate
!--             from m2 s-3 to cm2 s-3.
                k_au = k_au * ( 1.0 +                                          &
                       epsilon * 1.0E4 * ( re_lambda * 1.0E-3 )**0.25 *        &
                       ( alpha_cc * EXP( -1.0 * ( ( rc - r_cc ) /              &
                       sigma_cc )**2 ) + beta_cc ) )
             ENDIF
!
!--          Autoconversion rate (Seifert and Beheng, 2006):
             autocon   = k_au * ( nu_c + 2.0 ) * ( nu_c + 4.0 ) /              &
                         ( nu_c + 1.0 )**2 * ql(k,j,i)**2 * xc**2 *            &
                         ( 1.0 + phi_au / ( 1.0 - tau_cloud )**2 ) *           &
                         rho_surface
             autocon   = MIN( autocon, ql(k,j,i) / ( dt_3d *                   &
                              weight_substep(intermediate_timestep_count) ) )
!
!--          Tendencies for q, qr, nr, pt:
             tend_qr(k,j,i) = tend_qr(k,j,i) + autocon
             tend_q(k,j,i)  = tend_q(k,j,i) - autocon
             tend_nr(k,j,i) = tend_nr(k,j,i) + autocon / x0 * hyrho(k)
             tend_pt(k,j,i) = tend_pt(k,j,i) + autocon * l_d_cp * pt_d_t(k) 
          ENDIF

       ENDDO

    END SUBROUTINE autoconversion_ij

    SUBROUTINE accretion_ij( i, j )

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices
       USE control_parameters
       USE statistics
       
       IMPLICIT NONE

       INTEGER ::  i, j, k
       REAL    ::  accr, phi_ac, tau_cloud, k_cr

       DO  k = nzb_2d(j,i)+1, nzt
          IF ( ( ql(k,j,i) > 0.0 )  .AND.  ( qr(k,j,i) > eps_sb ) )  THEN
!
!--          Intern time scale of coagulation (Seifert and Beheng, 2006):
             tau_cloud = 1.0 - ql(k,j,i) / ( ql(k,j,i) + qr(k,j,i) + 1.0E-20) 
!
!--          Universal function for accretion process 
!--          (Seifert and Beheng, 2001):
             phi_ac = tau_cloud / ( tau_cloud + 5.0E-5 ) 
             phi_ac = ( phi_ac**2 )**2
!
!--          Parameterized turbulence effects on autoconversion (Seifert, 
!--          Nuijens and Stevens, 2010). The factor of 1.0E4 is needed to 
!--          convert the dissipation (diss) from m2 s-3 to cm2 s-3.
             IF ( turbulence )  THEN
                k_cr = k_cr0 * ( 1.0 + 0.05 *                            &
                                 MIN( 600.0, diss(k,j,i) * 1.0E4 )**0.25 )
             ELSE
                k_cr = k_cr0                       
             ENDIF
!
!--          Accretion rate (Seifert and Beheng, 2006):
             accr = k_cr * ql(k,j,i) * qr(k,j,i) * phi_ac *              &
                    SQRT( rho_surface * hyrho(k) )
             accr = MIN( accr, ql(k,j,i) / ( dt_3d *                     &
                    weight_substep(intermediate_timestep_count) ) )
!
!--          Tendencies for q, qr, pt:
             tend_qr(k,j,i) = tend_qr(k,j,i) + accr
             tend_q(k,j,i)  = tend_q(k,j,i) - accr
             tend_pt(k,j,i) = tend_pt(k,j,i) + accr * l_d_cp * pt_d_t(k) 
          ENDIF
       ENDDO

    END SUBROUTINE accretion_ij


    SUBROUTINE selfcollection_breakup_ij( i, j )

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices
       USE control_parameters
       USE statistics
   
       IMPLICIT NONE

       INTEGER ::  i, j, k
       REAL    ::  selfcoll, breakup, phi_br, phi_sc

       DO  k = nzb_2d(j,i)+1, nzt
          IF ( qr(k,j,i) > eps_sb )  THEN
!
!--          Selfcollection rate (Seifert and Beheng, 2006):
!--          pirho_l**( 1.0 / 3.0 ) is necessary to convert [lambda_r] = m-1 to 
!--          kg**( 1.0 / 3.0 ).
             phi_sc = 1.0 !( 1.0 + kappa_rr / lambda_r(k) *                        &
                      !pirho_l**( 1.0 / 3.0 ) )**( -9 )

             selfcoll = k_rr * nr(k,j,i) * qr(k,j,i) * phi_sc *               &
                        SQRT( hyrho(k) * rho_surface )
!
!--          Collisional breakup rate (Seifert, 2008):
             IF ( dr(k) >= 0.3E-3 )  THEN
                phi_br  = k_br * ( dr(k) - 1.1E-3 )
                breakup = selfcoll * ( phi_br + 1.0 )
             ELSE
                breakup = 0.0
             ENDIF

             selfcoll =  MAX( breakup - selfcoll, -nr(k,j,i) / ( dt_3d *      &
                              weight_substep(intermediate_timestep_count) ) )
!
!--          Tendency for nr:
             tend_nr(k,j,i) = tend_nr(k,j,i) + selfcoll
          ENDIF          
       ENDDO

    END SUBROUTINE selfcollection_breakup_ij

    SUBROUTINE evaporation_rain_ij( i, j )
!
!--    Evaporation of precipitable water. Condensation is neglected for 
!--    precipitable water.

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices
       USE control_parameters
       USE statistics

       IMPLICIT NONE

       INTEGER ::  i, j, k
       REAL    ::  evap, alpha, e_s, q_s, t_l, sat, temp, g_evap, f_vent, &
                   mu_r_2, mu_r_5d2, nr_0

       DO  k = nzb_2d(j,i)+1, nzt
          IF ( qr(k,j,i) > eps_sb )  THEN
!
!--          Actual liquid water temperature:
             t_l = t_d_pt(k) * pt(k,j,i)
!
!--          Saturation vapor pressure at t_l:
             e_s = 610.78 * EXP( 17.269 * ( t_l - 273.16 ) / ( t_l - 35.86 ) )
!
!--          Computation of saturation humidity:
             q_s = 0.622 * e_s / ( hyp(k) - 0.378 * e_s )
             alpha = 0.622 * l_d_r * l_d_cp / ( t_l * t_l )
             q_s = q_s * ( 1.0 + alpha * q(k,j,i) ) / ( 1.0 + alpha * q_s )
!
!--          Oversaturation:
             sat = MIN( 0.0, ( q(k,j,i) - ql(k,j,i) ) / q_s - 1.0 )
!
!--          Actual temperature:
             temp = t_l + l_d_cp * ql(k,j,i)
!
!--          
             g_evap = ( l_v / ( r_v * temp ) - 1.0 ) * l_v /                  &
                      ( thermal_conductivity_l * temp ) + rho_l * r_v * temp /&
                      ( diff_coeff_l * e_s )
             g_evap = 1.0 / g_evap
!
!--          Compute ventilation factor and intercept parameter 
!--          (Seifert and Beheng, 2006; Seifert, 2008):
             IF ( ventilation_effect )  THEN
                mu_r_2   = mu_r(k) + 2.0
                mu_r_5d2 = mu_r(k) + 2.5 
                f_vent = a_vent * gamm( mu_r_2 ) *                            &
                         lambda_r(k)**( -mu_r_2 ) +                           &
                         b_vent * schmidt_p_1d3 *                             &
                         SQRT( a_term / kin_vis_air ) * gamm( mu_r_5d2 ) *    &
                         lambda_r(k)**( -mu_r_5d2 ) *                         &
                         ( 1.0 - 0.5 * ( b_term / a_term ) *                  &
                         ( lambda_r(k) /                                      &
                         (       c_term + lambda_r(k) ) )**mu_r_5d2 -         &
                                 0.125 * ( b_term / a_term )**2 *             &
                         ( lambda_r(k) /                                      &
                         ( 2.0 * c_term + lambda_r(k) ) )**mu_r_5d2 -         &
                                 0.0625 * ( b_term / a_term )**3 *            &
                         ( lambda_r(k) /                                      &
                         ( 3.0 * c_term + lambda_r(k) ) )**mu_r_5d2 -         &
                                 0.0390625 * ( b_term / a_term )**4 *         &
                         ( lambda_r(k) /                                      &
                         ( 4.0 * c_term + lambda_r(k) ) )**mu_r_5d2 )
                nr_0   = nr(k,j,i) * lambda_r(k)**( mu_r(k) + 1.0 ) /         &
                         gamm( mu_r(k) + 1.0 ) 
             ELSE
                f_vent = 1.0
                nr_0   = nr(k,j,i) * dr(k)
             ENDIF
!
!--          Evaporation rate of rain water content (Seifert and Beheng, 2006):
             evap = 2.0 * pi * nr_0 * g_evap * f_vent * sat /    &
                    hyrho(k)
             evap = MAX( evap, -qr(k,j,i) / ( dt_3d *                         &
                         weight_substep(intermediate_timestep_count) ) )
!
!--          Tendencies for q, qr, nr, pt:
             tend_qr(k,j,i) = tend_qr(k,j,i) + evap
             tend_q(k,j,i)  = tend_q(k,j,i) - evap
             tend_nr(k,j,i) = tend_nr(k,j,i) + c_evap * evap / xr(k) * hyrho(k)
             tend_pt(k,j,i) = tend_pt(k,j,i) + evap * l_d_cp * pt_d_t(k) 
          ENDIF          
       ENDDO

    END SUBROUTINE evaporation_rain_ij

    SUBROUTINE sedimentation_cloud_ij( i, j )

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices
       USE control_parameters
       
       IMPLICIT NONE

       INTEGER ::  i, j, k
       REAL    ::  sed_q_const, sigma_gc = 1.3, k_st = 1.2E8
!
!--    Sedimentation of cloud droplets (Heus et al., 2010):
       sed_q_const = k_st * ( 3.0 / ( 4.0 * pi * rho_l ))**( 2.0 / 3.0 ) *    &
                     EXP( 5.0 * LOG( sigma_gc )**2 )

       sed_q = 0.0

       DO  k = nzb_2d(j,i)+1, nzt
          IF ( ql(k,j,i) > 0.0 )  THEN
             sed_q(k) = sed_q_const * nc**( -2.0 / 3.0 ) *                    &
                        ( ql(k,j,i) * hyrho(k) )**( 5.0 / 3.0 )
          ENDIF
       ENDDO
!
!--   Tendency for q, pt:
       DO  k = nzb_2d(j,i)+1, nzt
          tend_q(k,j,i)  = tend_q(k,j,i) + ( sed_q(k+1) - sed_q(k) ) *        &
                           ddzu(k+1) / hyrho(k) 
          tend_pt(k,j,i) = tend_pt(k,j,i) - ( sed_q(k+1) - sed_q(k) ) *       &
                           ddzu(k+1) / hyrho(k) * l_d_cp * pt_d_t(k) 
       ENDDO

    END SUBROUTINE sedimentation_cloud_ij

    SUBROUTINE sedimentation_rain_ij( i, j )

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE indices
       USE control_parameters
       USE statistics
       
       IMPLICIT NONE

       INTEGER ::  i, j, k, k_run
       REAL    ::  c_run, d_max, d_mean, d_min, dt_sedi, flux, z_run

       REAL, DIMENSION(nzb:nzt) :: c_nr, c_qr, nr_slope, qr_slope, w_nr, w_qr

!
!--    Computation of sedimentation flux. Implementation according to Stevens
!--    and Seifert (2008).

       IF ( intermediate_timestep_count == 1 )  prr(:,j,i) = 0.0

       dt_sedi = dt_3d * weight_substep(intermediate_timestep_count)

       w_nr = 0.0
       w_qr = 0.0
!
!--    Compute velocities 
       DO  k = nzb_s_inner(j,i)+1, nzt
          IF ( qr(k,j,i) > eps_sb )  THEN
             w_nr(k) = MAX( 0.1, MIN( 20.0, a_term - b_term * ( 1.0 +          &
                       c_term / lambda_r(k) )**( -1.0 * ( mu_r(k) + 1.0 ) ) ) )
             w_qr(k) = MAX( 0.1, MIN( 20.0, a_term - b_term * ( 1.0 +          &
                       c_term / lambda_r(k) )**( -1.0 * ( mu_r(k) + 4.0 ) ) ) )
          ELSE
             w_nr(k) = 0.0
             w_qr(k) = 0.0
          ENDIF
       ENDDO
!
!--    Adjust boundary values
       w_nr(nzb_2d(j,i)) = w_nr(nzb_2d(j,i)+1)
       w_qr(nzb_2d(j,i)) = w_qr(nzb_2d(j,i)+1)
       w_nr(nzt) = w_nr(nzt-1)
       w_qr(nzt) = w_qr(nzt-1)
!
!--    Compute Courant number
       DO  k = nzb_s_inner(j,i)+1, nzt-1
          c_nr(k) = 0.25 * ( w_nr(k-1) + 2.0 * w_nr(k) + w_nr(k+1) ) * &
                    dt_sedi * ddzu(k)
          c_qr(k) = 0.25 * ( w_qr(k-1) + 2.0 * w_qr(k) + w_qr(k+1) ) * &
                    dt_sedi * ddzu(k)
       ENDDO
!
!--    Limit slopes with monotonized centered (MC) limiter (van Leer, 1977):
       IF ( limiter_sedimentation )  THEN

          qr(nzb_s_inner(j,i),j,i) = 0.0
          nr(nzb_s_inner(j,i),j,i) = 0.0
          qr(nzt,j,i) = 0.0
          nr(nzt,j,i) = 0.0

          DO k = nzb_s_inner(j,i)+1, nzt-1
             d_mean = 0.5 * ( qr(k+1,j,i) + qr(k-1,j,i) )
             d_min  = qr(k,j,i) - MIN( qr(k+1,j,i), qr(k,j,i), qr(k-1,j,i) )
             d_max  = MAX( qr(k+1,j,i), qr(k,j,i), qr(k-1,j,i) ) - qr(k,j,i)

             qr_slope(k) = SIGN(1.0, d_mean) * MIN ( 2.0 * d_min, 2.0 * d_max, &
                                                     ABS( d_mean ) )

             d_mean = 0.5 * ( nr(k+1,j,i) + nr(k-1,j,i) )
             d_min  = nr(k,j,i) - MIN( nr(k+1,j,i), nr(k,j,i), nr(k-1,j,i) )
             d_max  = MAX( nr(k+1,j,i), nr(k,j,i), nr(k-1,j,i) ) - nr(k,j,i)

             nr_slope(k) = SIGN(1.0, d_mean) * MIN ( 2.0 * d_min, 2.0 * d_max, &
                                                     ABS( d_mean ) )
          ENDDO

       ELSE
          nr_slope = 0.0
          qr_slope = 0.0
       ENDIF
!
!--    Compute sedimentation flux
       DO  k = nzt-2, nzb_s_inner(j,i)+1, -1
!
!--       Sum up all rain drop number densities which contribute to the flux 
!--       through k-1/2
          flux  = 0.0
          z_run = 0.0 ! height above z(k)
          k_run = k
          c_run = MIN( 1.0, c_nr(k) )
          DO WHILE ( c_run > 0.0  .AND.  k_run <= nzt-1 )
             flux  = flux + hyrho(k_run) *                                    &
                     ( nr(k_run,j,i) + nr_slope(k_run) * ( 1.0 - c_run ) *    &
                     0.5 ) * c_run * dzu(k_run)
             z_run = z_run + dzu(k_run)
             k_run = k_run + 1
             c_run = MIN( 1.0, c_nr(k_run) - z_run * ddzu(k_run) )
          ENDDO
!
!--       It is not allowed to sediment more rain drop number density than 
!--       available
          flux = MIN( flux,                                                   &
                      hyrho(k) * dzu(k) * nr(k,j,i) + sed_nr(k+1) * dt_sedi )

          sed_nr(k)      = flux / dt_sedi
          tend_nr(k,j,i) = tend_nr(k,j,i) + ( sed_nr(k+1) - sed_nr(k) ) *     &
                                            ddzu(k+1) / hyrho(k)
!
!--       Sum up all rain water content which contributes to the flux 
!--       through k-1/2
          flux  = 0.0
          z_run = 0.0 ! height above z(k)
          k_run = k
          c_run = MIN( 1.0, c_qr(k) )
          DO WHILE ( c_run > 0.0  .AND.  k_run <= nzt-1 )
             flux  = flux + hyrho(k_run) *                                    &
                     ( qr(k_run,j,i) + qr_slope(k_run) * ( 1.0 - c_run ) *    &
                     0.5 ) * c_run * dzu(k_run)
             z_run = z_run + dzu(k_run)
             k_run = k_run + 1
             c_run = MIN( 1.0, c_qr(k_run) - z_run * ddzu(k_run) )
          ENDDO
!
!--       It is not allowed to sediment more rain water content than available
          flux = MIN( flux,                                                   &
                      hyrho(k) * dzu(k) * qr(k,j,i) + sed_qr(k+1) * dt_sedi )

          sed_qr(k)      = flux / dt_sedi
          tend_qr(k,j,i) = tend_qr(k,j,i) + ( sed_qr(k+1) - sed_qr(k) ) *     &
                                            ddzu(k+1) / hyrho(k)
!
!--       Compute the rain rate
          prr(k,j,i) = prr(k,j,i) + sed_qr(k) / hyrho(k) *                    &
                       weight_substep(intermediate_timestep_count) 
       ENDDO
!
!--    Precipitation amount
       IF ( intermediate_timestep_count == intermediate_timestep_count_max    &
            .AND.  ( dt_do2d_xy - time_do2d_xy ) <                            &
            precipitation_amount_interval )  THEN

          precipitation_amount(j,i) = precipitation_amount(j,i) +   &
                                      prr(nzb_2d(j,i)+1,j,i) *      &
                                      hyrho(nzb_2d(j,i)+1) * dt_3d
       ENDIF

    END SUBROUTINE sedimentation_rain_ij

!
!-- This function computes the gamma function (Press et al., 1992). 
!-- The gamma function is needed for the calculation of the evaporation 
!-- of rain drops.
    FUNCTION gamm( xx ) 
       
       USE cloud_parameters
    
       IMPLICIT NONE 
 
       REAL    ::  gamm, ser, tmp, x_gamm, xx, y_gamm
       INTEGER ::  j 
 
       x_gamm = xx 
       y_gamm = x_gamm 
       tmp = x_gamm + 5.5 
       tmp = ( x_gamm + 0.5 ) * LOG( tmp ) - tmp 
       ser = 1.000000000190015 
       do j = 1, 6 
          y_gamm = y_gamm + 1.0 
          ser    = ser + cof( j ) / y_gamm 
       enddo 
! 
!--    Until this point the algorithm computes the logarithm of the gamma 
!--    function. Hence, the exponential function is used.  
!       gamm = EXP( tmp + LOG( stp * ser / x_gamm ) ) 
       gamm = EXP( tmp ) * stp * ser / x_gamm 
       RETURN 
 
    END FUNCTION gamm 

 END MODULE microphysics_mod