source: palm/trunk/SOURCE/lpm_collision_kernels.f90 @ 1018

 MODULE lpm_collision_kernels_mod

!------------------------------------------------------------------------------!
! Current revisions:
! -----------------
!
!
! Former revisions:
! -----------------
! $Id: lpm_collision_kernels.f90 1008 2012-09-19 14:49:14Z raasch $
!
! 1007 2012-09-19 14:30:36Z franke
! converted all units to SI units and replaced some parameters by corresponding
! PALM parameters
! Bugfix: factor in calculation of enhancement factor for collision efficencies
! changed from 10. to 1.0
!
! 849 2012-03-15 10:35:09Z raasch
! routine collision_efficiency_rogers added (moved from former advec_particles
! to here)
!
! 835 2012-02-22 11:21:19Z raasch $
! Bugfix: array diss can be used only in case of Wang kernel
!
! 828 2012-02-21 12:00:36Z raasch
! code has been completely reformatted, routine colker renamed
! recalculate_kernel,
! routine init_kernels added, radius is now communicated to the collision
! routines by array radclass
!
! Bugfix: transformation factor for dissipation changed from 1E5 to 1E4
!
! 825 2012-02-19 03:03:44Z raasch
! routine renamed from wang_kernel to lpm_collision_kernels,
! turbulence_effects on collision replaced by wang_kernel
!
! 799 2011-12-21 17:48:03Z franke
! speed optimizations and formatting
! Bugfix: iq=1 is not allowed (routine effic) 
! Bugfix: replaced stop by ec=0.0 in case of very small ec (routine effic) 
!
! 790 2011-11-29 03:11:20Z raasch
! initial revision
!
! Description:
! ------------
! This module calculates collision efficiencies either due to pure gravitational
! effects (Hall kernel, see Hall, 1980: J. Atmos. Sci., 2486-2507) or
! including the effects of (SGS) turbulence (Wang kernel, see Wang and
! Grabowski, 2009: Atmos. Sci. Lett., 10, 1-8). The original code has been
! provided by L.-P. Wang but is substantially reformatted and speed optimized
! here.
!
! ATTENTION:
! Physical quantities (like g, densities, etc.) used in this module still
! have to be adjusted to those values used in the main PALM code.
! Also, quantities in CGS-units should be converted to SI-units eventually.
!------------------------------------------------------------------------------!

    USE arrays_3d
    USE cloud_parameters
    USE constants
    USE particle_attributes
    USE pegrid


    IMPLICIT NONE

    PRIVATE

    PUBLIC  ckernel, collision_efficiency_rogers, init_kernels, &
            rclass_lbound, rclass_ubound, recalculate_kernel

    REAL ::  epsilon, eps2, rclass_lbound, rclass_ubound, urms, urms2

    REAL, DIMENSION(:),   ALLOCATABLE ::  epsclass, radclass, winf
    REAL, DIMENSION(:,:), ALLOCATABLE ::  ec, ecf, gck, hkernel, hwratio
    REAL, DIMENSION(:,:,:), ALLOCATABLE ::  ckernel

    SAVE

!
!-- Public interfaces
    INTERFACE collision_efficiency_rogers
       MODULE PROCEDURE collision_efficiency_rogers
    END INTERFACE collision_efficiency_rogers

    INTERFACE init_kernels
       MODULE PROCEDURE init_kernels
    END INTERFACE init_kernels

    INTERFACE recalculate_kernel
       MODULE PROCEDURE recalculate_kernel
    END INTERFACE recalculate_kernel


    CONTAINS


    SUBROUTINE init_kernels
!------------------------------------------------------------------------------!
! Initialization of the collision efficiency matrix with fixed radius and
! dissipation classes, calculated at simulation start only.
!------------------------------------------------------------------------------!

       IMPLICIT NONE

       INTEGER ::  i, j, k


!
!--    Calculate collision efficiencies for fixed radius- and dissipation
!--    classes
       IF ( collision_kernel(6:9) == 'fast' )  THEN

          ALLOCATE( ckernel(1:radius_classes,1:radius_classes,               &
                    0:dissipation_classes), epsclass(1:dissipation_classes), &
                    radclass(1:radius_classes) )

!
!--       Calculate the radius class bounds with logarithmic distances
!--       in the interval [1.0E-6, 2.0E-4] m
          rclass_lbound = LOG( 1.0E-6 )
          rclass_ubound = LOG( 2.0E-4 )
          radclass(1)   = 1.0E-6
          DO  i = 2, radius_classes
             radclass(i) = EXP( rclass_lbound +                                &
                                ( rclass_ubound - rclass_lbound ) * ( i-1.0 ) /&
                                ( radius_classes - 1.0 ) )
!             IF ( myid == 0 )  THEN
!                PRINT*, 'i=', i, ' r = ', radclass(i)*1.0E6
!             ENDIF
          ENDDO

!
!--       Set the class bounds for dissipation in interval [0.0, 0.1] m**2/s**3
          DO  i = 1, dissipation_classes
             epsclass(i) = 0.1 * REAL( i ) / dissipation_classes
!             IF ( myid == 0 )  THEN
!                PRINT*, 'i=', i, ' eps = ', epsclass(i)
!             ENDIF
          ENDDO
!
!--       Calculate collision efficiencies of the Wang/ayala kernel
          ALLOCATE( ec(1:radius_classes,1:radius_classes),  &
                    ecf(1:radius_classes,1:radius_classes), &
                    gck(1:radius_classes,1:radius_classes), &
                    winf(1:radius_classes) )

          DO  k = 1, dissipation_classes

             epsilon = epsclass(k)
             urms    = 2.02 * ( epsilon / 0.04 )**( 1.0 / 3.0 )

             CALL turbsd
             CALL turb_enhance_eff
             CALL effic

             DO  j = 1, radius_classes
                DO  i = 1, radius_classes
                   ckernel(i,j,k) = ec(i,j) * gck(i,j) * ecf(i,j)
                ENDDO
             ENDDO

          ENDDO

!
!--       Calculate collision efficiencies of the Hall kernel
          ALLOCATE( hkernel(1:radius_classes,1:radius_classes), &
                    hwratio(1:radius_classes,1:radius_classes) )

          CALL fallg
          CALL effic

          DO  j = 1, radius_classes
             DO  i =  1, radius_classes
                hkernel(i,j) = pi * ( radclass(j) + radclass(i) )**2 &
                                  * ec(i,j) * ABS( winf(j) - winf(i) )
                ckernel(i,j,0) = hkernel(i,j)  ! hall kernel stored on index 0
              ENDDO
          ENDDO

!
!--       Test output of efficiencies
          IF ( j == -1 )  THEN

             PRINT*, '*** Hall kernel'
             WRITE ( *,'(5X,20(F4.0,1X))' ) ( radclass(i)*1.0E6, &
                                              i = 1,radius_classes )
             DO  j = 1, radius_classes
                WRITE ( *,'(F4.0,1X,20(F8.4,1X))' ) radclass(j),  &
                                          ( hkernel(i,j), i = 1,radius_classes )
             ENDDO

             DO  k = 1, dissipation_classes
                DO  i = 1, radius_classes
                   DO  j = 1, radius_classes
                      IF ( hkernel(i,j) == 0.0 )  THEN
                         hwratio(i,j) = 9999999.9
                      ELSE
                         hwratio(i,j) = ckernel(i,j,k) / hkernel(i,j)
                      ENDIF
                   ENDDO
                ENDDO

                PRINT*, '*** epsilon = ', epsclass(k)
                WRITE ( *,'(5X,20(F4.0,1X))' ) ( radclass(i)*1.0E6, &
                                                 i = 1,radius_classes )
                DO  j = 1, radius_classes
!                   WRITE ( *,'(F4.0,1X,20(F4.2,1X))' ) radclass(j)*1.0E6, &
!                                       ( ckernel(i,j,k), i = 1,radius_classes )
                   WRITE ( *,'(F4.0,1X,20(F8.4,1X))' ) radclass(j)*1.0E6, &
                                          ( hwratio(i,j), i = 1,radius_classes )
                ENDDO
             ENDDO

          ENDIF

          DEALLOCATE( ec, ecf, epsclass, gck, hkernel, winf )

       ELSEIF( collision_kernel == 'hall'  .OR.  collision_kernel == 'wang' ) &
       THEN
!
!--       Initial settings for Hall- and Wang-Kernel
!--       To be done: move here parts from turbsd, fallg, ecoll, etc.
       ENDIF

    END SUBROUTINE init_kernels


!------------------------------------------------------------------------------!
! Calculation of collision kernels during each timestep and for each grid box
!------------------------------------------------------------------------------!
    SUBROUTINE recalculate_kernel( i1, j1, k1 )

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE cpulog
       USE indices
       USE interfaces
       USE particle_attributes

       IMPLICIT NONE

       INTEGER ::  i, i1, j, j1, k1, pend, pstart


       pstart = prt_start_index(k1,j1,i1)
       pend   = prt_start_index(k1,j1,i1) + prt_count(k1,j1,i1) - 1
       radius_classes = prt_count(k1,j1,i1)

       ALLOCATE( ec(1:radius_classes,1:radius_classes), &
                 radclass(1:radius_classes), winf(1:radius_classes) )

!
!--    Store particle radii on the radclass array
       radclass(1:radius_classes) = particles(pstart:pend)%radius

       IF ( wang_kernel )  THEN
          epsilon = diss(k1,j1,i1)   ! dissipation rate in m**2/s**3
       ELSE
          epsilon = 0.0
       ENDIF
       urms    = 2.02 * ( epsilon / 0.04 )**( 0.33333333333 )

       IF ( wang_kernel  .AND.  epsilon > 1.0E-7 )  THEN
!
!--       Call routines to calculate efficiencies for the Wang kernel
          ALLOCATE( gck(1:radius_classes,1:radius_classes), &
                    ecf(1:radius_classes,1:radius_classes) )

          CALL turbsd
          CALL turb_enhance_eff
          CALL effic

          DO  j = 1, radius_classes
             DO  i =  1, radius_classes
                ckernel(pstart+i-1,pstart+j-1,1) = ec(i,j) * gck(i,j) * ecf(i,j)
             ENDDO
          ENDDO

          DEALLOCATE( gck, ecf )

       ELSE
!
!--       Call routines to calculate efficiencies for the Hall kernel
          CALL fallg
          CALL effic

          DO  j = 1, radius_classes
             DO  i =  1, radius_classes
                ckernel(pstart+i-1,pstart+j-1,1) = pi *                       &
                                          ( radclass(j) + radclass(i) )**2    &
                                          * ec(i,j) * ABS( winf(j) - winf(i) )
             ENDDO
          ENDDO

       ENDIF

       DEALLOCATE( ec, radclass, winf )

    END SUBROUTINE recalculate_kernel


!------------------------------------------------------------------------------!
! Calculation of gck
! This is from Aayala 2008b, page 37ff.
! Necessary input parameters: water density, radii of droplets, air density,
! air viscosity, turbulent dissipation rate, taylor microscale reynolds number,
! gravitational acceleration  --> to be replaced by PALM parameters
!------------------------------------------------------------------------------!
    SUBROUTINE turbsd

       USE constants
       USE cloud_parameters
       USE particle_attributes
       USE arrays_3d
       USE control_parameters

       IMPLICIT NONE

       INTEGER ::  i, j

       LOGICAL, SAVE ::  first = .TRUE.

       REAL ::  ao, ao_gr, bbb, be, b1, b2, ccc, c1, c1_gr, c2, d1, d2, eta, &
                e1, e2, fao_gr, fr, grfin, lambda, lambda_re, lf, rc, rrp,   &
                sst, tauk, tl, t2, tt, t1, vk, vrms1xy, vrms2xy, v1, v1v2xy, &
                v1xysq, v2, v2xysq, wrfin, wrgrav2, wrtur2xy, xx, yy, z

       REAL, DIMENSION(1:radius_classes) ::  st, tau


!
!--    Initial assignment of constants
       IF ( first )  THEN

          first = .FALSE.

       ENDIF

       lambda    = urms * SQRT( 15.0 * molecular_viscosity / epsilon )    ! in m
       lambda_re = urms**2 * SQRT( 15.0 / epsilon / molecular_viscosity )
       tl        = urms**2 / epsilon                       ! in s
       lf        = 0.5 * urms**3 / epsilon                 ! in m
       tauk      = SQRT( molecular_viscosity / epsilon )                  ! in s
       eta       = ( molecular_viscosity**3 / epsilon )**0.25             ! in m
       vk        = eta / tauk

       ao = ( 11.0 + 7.0 * lambda_re ) / ( 205.0 + lambda_re )
       tt = SQRT( 2.0 * lambda_re / ( SQRT( 15.0 ) * ao ) ) * tauk   ! in s

       CALL fallg    ! gives winf in m/s

       DO  i = 1, radius_classes
          tau(i) = winf(i) / g    ! in s
          st(i)  = tau(i) / tauk
       ENDDO

!
!--    Calculate wr (from Aayala 2008b, page 38f)
       z   = tt / tl
       be  = SQRT( 2.0 ) * lambda / lf
       bbb = SQRT( 1.0 - 2.0 * be**2 )
       d1  = ( 1.0 + bbb ) / ( 2.0 * bbb )
       e1  = lf * ( 1.0 + bbb ) * 0.5   ! in m
       d2  = ( 1.0 - bbb ) * 0.5 / bbb
       e2  = lf * ( 1.0 - bbb ) * 0.5   ! in m
       ccc = SQRT( 1.0 - 2.0 * z**2 )
       b1  = ( 1.0 + ccc ) * 0.5 / ccc
       c1  = tl * ( 1.0 + ccc ) * 0.5   ! in s
       b2  = ( 1.0 - ccc ) * 0.5 / ccc
       c2  = tl * ( 1.0 - ccc ) * 0.5   ! in s

       DO  i = 1, radius_classes

          v1 = winf(i)        ! in m/s
          t1 = tau(i)         ! in s

          DO  j = 1, i
             rrp = radclass(i) + radclass(j)
             v2  = winf(j)                                 ! in m/s
             t2  = tau(j)                                  ! in s

             v1xysq  = b1 * d1 * phi_w(c1,e1,v1,t1) - b1 * d2 * phi_w(c1,e2,v1,t1) &
                     - b2 * d1 * phi_w(c2,e1,v1,t1) + b2 * d2 * phi_w(c2,e2,v1,t1)
             v1xysq  = v1xysq * urms**2 / t1                ! in m**2/s**2
             vrms1xy = SQRT( v1xysq )                       ! in m/s

             v2xysq  = b1 * d1 * phi_w(c1,e1,v2,t2) - b1 * d2 * phi_w(c1,e2,v2,t2) &
                     - b2 * d1 * phi_w(c2,e1,v2,t2) + b2 * d2 * phi_w(c2,e2,v2,t2)
             v2xysq  = v2xysq * urms**2 / t2                ! in m**2/s**2
             vrms2xy = SQRT( v2xysq )                       ! in m/s

             IF ( winf(i) >= winf(j) )  THEN
                v1 = winf(i)
                t1 = tau(i)
                v2 = winf(j)
                t2 = tau(j)
             ELSE
                v1 = winf(j)
                t1 = tau(j)
                v2 = winf(i)
                t2 = tau(i)
             ENDIF

             v1v2xy   =  b1 * d1 * zhi(c1,e1,v1,t1,v2,t2) - &
                         b1 * d2 * zhi(c1,e2,v1,t1,v2,t2) - &
                         b2 * d1 * zhi(c2,e1,v1,t1,v2,t2) + &
                         b2 * d2* zhi(c2,e2,v1,t1,v2,t2)
             fr       = d1 * EXP( -rrp / e1 ) - d2 * EXP( -rrp / e2 )
             v1v2xy   = v1v2xy * fr * urms**2 / tau(i) / tau(j)  ! in m**2/s**2
             wrtur2xy = vrms1xy**2 + vrms2xy**2 - 2.0 * v1v2xy   ! in m**2/s**2
             IF ( wrtur2xy < 0.0 )  wrtur2xy = 0.0
             wrgrav2  = pi / 8.0 * ( winf(j) - winf(i) )**2
             wrfin    = SQRT( ( 2.0 / pi ) * ( wrtur2xy + wrgrav2) )   ! in m/s

!
!--          Calculate gr
             IF ( st(j) > st(i) )  THEN
                sst = st(j)
             ELSE
                sst = st(i)
             ENDIF

             xx = -0.1988 * sst**4 + 1.5275 * sst**3 - 4.2942 * sst**2 + &
                   5.3406 * sst
             IF ( xx < 0.0 )  xx = 0.0
             yy = 0.1886 * EXP( 20.306 / lambda_re )

             c1_gr  =  xx / ( g / vk * tauk )**yy

             ao_gr  = ao + ( pi / 8.0) * ( g / vk * tauk )**2
             fao_gr = 20.115 * SQRT( ao_gr / lambda_re )
             rc     = SQRT( fao_gr * ABS( st(j) - st(i) ) ) * eta   ! in cm

             grfin  = ( ( eta**2 + rc**2 ) / ( rrp**2 + rc**2) )**( c1_gr*0.5 )
             IF ( grfin < 1.0 )  grfin = 1.0

             gck(i,j) = 2.0 * pi * rrp**2 * wrfin * grfin           ! in cm**3/s
             gck(j,i) = gck(i,j)

          ENDDO
       ENDDO

    END SUBROUTINE turbsd


!------------------------------------------------------------------------------!
! phi_w as a function
!------------------------------------------------------------------------------!
    REAL FUNCTION phi_w( a, b, vsett, tau0 )

       IMPLICIT NONE

       REAL ::  a, aa1, b, tau0, vsett

       aa1 = 1.0 / tau0 + 1.0 / a + vsett / b
       phi_w = 1.0 / aa1  - 0.5 * vsett / b / aa1**2  ! in s

    END FUNCTION phi_w


!------------------------------------------------------------------------------!
! zhi as a function
!------------------------------------------------------------------------------!
    REAL FUNCTION zhi( a, b, vsett1, tau1, vsett2, tau2 )

       IMPLICIT NONE

       REAL ::  a, aa1, aa2, aa3, aa4, aa5, aa6, b, tau1, tau2, vsett1, vsett2

       aa1 = vsett2 / b - 1.0 / tau2 - 1.0 / a
       aa2 = vsett1 / b + 1.0 / tau1 + 1.0 / a
       aa3 = ( vsett1 - vsett2 ) / b + 1.0 / tau1 + 1.0 / tau2
       aa4 = ( vsett2 / b )**2 - ( 1.0 / tau2 + 1.0 / a )**2
       aa5 = vsett2 / b + 1.0 / tau2 + 1.0 / a
       aa6 = 1.0 / tau1 - 1.0 / a + ( 1.0 / tau2 + 1.0 / a) * vsett1 / vsett2
       zhi = (1.0 / aa1 - 1.0 / aa2 ) * ( vsett1 - vsett2 ) * 0.5 / b / aa3**2 &
           + (4.0 / aa4 - 1.0 / aa5**2 - 1.0 / aa1**2 ) * vsett2 * 0.5 / b /aa6&
           + (2.0 * ( b / aa2 - b / aa1 ) - vsett1 / aa2**2 + vsett2 / aa1**2 )&
           * 0.5 / b / aa3      ! in s**2

    END FUNCTION zhi


!------------------------------------------------------------------------------!
! Calculation of terminal velocity winf following Equations 10-138 to 10-145
! from (Pruppacher and Klett, 1997)
!------------------------------------------------------------------------------!
    SUBROUTINE fallg

       USE constants
       USE cloud_parameters
       USE particle_attributes
       USE arrays_3d
       USE control_parameters

       IMPLICIT NONE

       INTEGER ::  i, j

       LOGICAL, SAVE ::  first = .TRUE.

       REAL, SAVE ::  cunh, eta, phy, py, rho_a, sigma, stb, stok, &
                      t0, xlamb

       REAL ::  bond, x, xrey, y

       REAL, DIMENSION(1:7), SAVE  ::  b
       REAL, DIMENSION(1:6), SAVE  ::  c

!
!--    Initial assignment of constants
       IF ( first )  THEN

          first = .FALSE.
          b = (/  -0.318657E1,  0.992696E0, -0.153193E-2, -0.987059E-3, &
                 -0.578878E-3, 0.855176E-4, -0.327815E-5 /)
          c = (/  -0.500015E1,  0.523778E1,  -0.204914E1,   0.475294E0, &
                 -0.542819E-1, 0.238449E-2 /)

!
!--       Parameter values for p = 1013,25 hPa and T = 293,15 K
          eta   = 1.818E-5         ! in kg/(m s)
          xlamb = 6.6E-8           ! in m
          rho_a = 1.204            ! in kg/m**3
          cunh  = 1.26 * xlamb     ! in m
          sigma = 0.07363          ! in kg/s**2
          stok  = 2.0  * g * ( rho_l - rho_a ) / ( 9.0 * eta ) ! in 1/(m s)
          stb   = 32.0 * rho_a * ( rho_l - rho_a) * g / (3.0 * eta * eta)
          phy   = sigma**3 * rho_a**2 / ( eta**4 * g * ( rho_l - rho_a ) )
          py    = phy**( 1.0 / 6.0 )

       ENDIF

       DO  j = 1, radius_classes

          IF ( radclass(j) <= 1.0E-5 ) THEN

             winf(j) = stok * ( radclass(j)**2 + cunh * radclass(j) )

          ELSEIF ( radclass(j) > 1.0E-5  .AND.  radclass(j) <= 5.35E-4 )  THEN

             x = LOG( stb * radclass(j)**3 )
             y = 0.0

             DO  i = 1, 7
                y = y + b(i) * x**(i-1)
             ENDDO
!
!--          Note: this Eq. is wrong in (Pruppacher and Klett, 1997, p. 418)
!--          for correct version see (Beard, 1976)
             xrey = ( 1.0 + cunh / radclass(j) ) * EXP( y ) 

             winf(j) = xrey * eta / ( 2.0 * rho_a * radclass(j) )

          ELSEIF ( radclass(j) > 5.35E-4 )  THEN

             IF ( radclass(j) > 0.0035 )  THEN
                bond = g * ( rho_l - rho_a ) * 0.0035**2 / sigma
             ELSE
               bond = g * ( rho_l - rho_a ) * radclass(j)**2 / sigma
             ENDIF

             x = LOG( 16.0 * bond * py / 3.0 )
             y = 0.0

             DO  i = 1, 6
                y = y + c(i) * x**(i-1)
             ENDDO

             xrey = py * EXP( y )

             IF ( radclass(j) > 0.0035 )  THEN
                winf(j) = xrey * eta / ( 2.0 * rho_a * 0.0035 )
             ELSE
                winf(j) = xrey * eta / ( 2.0 * rho_a * radclass(j) )
             ENDIF

          ENDIF

       ENDDO

    END SUBROUTINE fallg


!------------------------------------------------------------------------------!
! Calculation of collision efficencies for the Hall kernel
!------------------------------------------------------------------------------!
    SUBROUTINE effic

       USE arrays_3d
       USE cloud_parameters
       USE constants
       USE particle_attributes

       IMPLICIT NONE

       INTEGER ::  i, iq, ir, j, k, kk

       INTEGER, DIMENSION(:), ALLOCATABLE ::  ira

       LOGICAL, SAVE ::  first = .TRUE.

       REAL ::  ek, particle_radius, pp, qq, rq

       REAL, DIMENSION(1:21), SAVE ::  rat
       REAL, DIMENSION(1:15), SAVE ::  r0
       REAL, DIMENSION(1:15,1:21), SAVE ::  ecoll

!
!--    Initial assignment of constants
       IF ( first )  THEN

         first = .FALSE.
         r0  = (/ 6.0, 8.0, 10.0, 15.0, 20.0, 25.0, 30.0, 40.0, 50.0, 60., &
                  70.0, 100.0, 150.0, 200.0, 300.0 /)
         rat = (/ 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, &
                  0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, &
                  1.00 /)

         ecoll(:,1) = (/0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001, &
                        0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001/)
         ecoll(:,2) = (/0.003, 0.003, 0.003, 0.004, 0.005, 0.005, 0.005, &
                        0.010, 0.100, 0.050, 0.200, 0.500, 0.770, 0.870, 0.970/)
         ecoll(:,3) = (/0.007, 0.007, 0.007, 0.008, 0.009, 0.010, 0.010, &
                        0.070, 0.400, 0.430, 0.580, 0.790, 0.930, 0.960, 1.000/)
         ecoll(:,4) = (/0.009, 0.009, 0.009, 0.012, 0.015, 0.010, 0.020, &
                        0.280, 0.600, 0.640, 0.750, 0.910, 0.970, 0.980, 1.000/)
         ecoll(:,5) = (/0.014, 0.014, 0.014, 0.015, 0.016, 0.030, 0.060, &
                        0.500, 0.700, 0.770, 0.840, 0.950, 0.970, 1.000, 1.000/)
         ecoll(:,6) = (/0.017, 0.017, 0.017, 0.020, 0.022, 0.060, 0.100, &
                        0.620, 0.780, 0.840, 0.880, 0.950, 1.000, 1.000, 1.000/)
         ecoll(:,7) = (/0.030, 0.030, 0.024, 0.022, 0.032, 0.062, 0.200, &
                        0.680, 0.830, 0.870, 0.900, 0.950, 1.000, 1.000, 1.000/)
         ecoll(:,8) = (/0.025, 0.025, 0.025, 0.036, 0.043, 0.130, 0.270, &
                        0.740, 0.860, 0.890, 0.920, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,9) = (/0.027, 0.027, 0.027, 0.040, 0.052, 0.200, 0.400, &
                        0.780, 0.880, 0.900, 0.940, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,10)= (/0.030, 0.030, 0.030, 0.047, 0.064, 0.250, 0.500, &
                        0.800, 0.900, 0.910, 0.950, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,11)= (/0.040, 0.040, 0.033, 0.037, 0.068, 0.240, 0.550, &
                        0.800, 0.900, 0.910, 0.950, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,12)= (/0.035, 0.035, 0.035, 0.055, 0.079, 0.290, 0.580, &
                        0.800, 0.900, 0.910, 0.950, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,13)= (/0.037, 0.037, 0.037, 0.062, 0.082, 0.290, 0.590, &
                        0.780, 0.900, 0.910, 0.950, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,14)= (/0.037, 0.037, 0.037, 0.060, 0.080, 0.290, 0.580, &
                        0.770, 0.890, 0.910, 0.950, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,15)= (/0.037, 0.037, 0.037, 0.041, 0.075, 0.250, 0.540, &
                        0.760, 0.880, 0.920, 0.950, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,16)= (/0.037, 0.037, 0.037, 0.052, 0.067, 0.250, 0.510, &
                        0.770, 0.880, 0.930, 0.970, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,17)= (/0.037, 0.037, 0.037, 0.047, 0.057, 0.250, 0.490, &
                        0.770, 0.890, 0.950, 1.000, 1.000, 1.000, 1.000, 1.000/)
         ecoll(:,18)= (/0.036, 0.036, 0.036, 0.042, 0.048, 0.230, 0.470, &
                        0.780, 0.920, 1.000, 1.020, 1.020, 1.020, 1.020, 1.020/)
         ecoll(:,19)= (/0.040, 0.040, 0.035, 0.033, 0.040, 0.112, 0.450, &
                        0.790, 1.010, 1.030, 1.040, 1.040, 1.040, 1.040, 1.040/)
         ecoll(:,20)= (/0.033, 0.033, 0.033, 0.033, 0.033, 0.119, 0.470, &
                        0.950, 1.300, 1.700, 2.300, 2.300, 2.300, 2.300, 2.300/)
         ecoll(:,21)= (/0.027, 0.027, 0.027, 0.027, 0.027, 0.125, 0.520, &
                        1.400, 2.300, 3.000, 4.000, 4.000, 4.000, 4.000, 4.000/)
       ENDIF

!
!--    Calculate the radius class index of particles with respect to array r
!--    Radius has to be in µm
       ALLOCATE( ira(1:radius_classes) )
       DO  j = 1, radius_classes
          particle_radius = radclass(j) * 1.0E6
          DO  k = 1, 15
             IF ( particle_radius < r0(k) )  THEN
                ira(j) = k
                EXIT
             ENDIF
          ENDDO
          IF ( particle_radius >= r0(15) )  ira(j) = 16
       ENDDO

!
!--    Two-dimensional linear interpolation of the collision efficiency.
!--    Radius has to be in µm
       DO  j = 1, radius_classes
          DO  i = 1, j

             ir = ira(j)
             rq = radclass(i) / radclass(j)
             iq = INT( rq * 20 ) + 1
             iq = MAX( iq , 2)

             IF ( ir < 16 )  THEN
                IF ( ir >= 2 )  THEN
                   pp = ( ( radclass(j) * 1.0E06 ) - r0(ir-1) ) / &
                        ( r0(ir) - r0(ir-1) )
                   qq = ( rq- rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
                   ec(j,i) = ( 1.0-pp ) * ( 1.0-qq ) * ecoll(ir-1,iq-1)  &
                             + pp * ( 1.0-qq ) * ecoll(ir,iq-1)          &
                             + qq * ( 1.0-pp ) * ecoll(ir-1,iq)          &
                             + pp * qq * ecoll(ir,iq)
                ELSE
                   qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
                   ec(j,i) = (1.0-qq) * ecoll(1,iq-1) + qq * ecoll(1,iq)
                ENDIF
             ELSE
                qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
                ek = ( 1.0 - qq ) * ecoll(15,iq-1) + qq * ecoll(15,iq)
                ec(j,i) = MIN( ek, 1.0 )
             ENDIF 

             ec(i,j) = ec(j,i)
             IF ( ec(i,j) < 1.0E-20 )  ec(i,j) = 0.0

          ENDDO
       ENDDO

       DEALLOCATE( ira )

    END SUBROUTINE effic


!------------------------------------------------------------------------------!
! Calculation of enhancement factor for collision efficencies due to turbulence
!------------------------------------------------------------------------------!
    SUBROUTINE turb_enhance_eff

       USE constants
       USE cloud_parameters
       USE particle_attributes
       USE arrays_3d

       IMPLICIT NONE

       INTEGER :: i, ik, iq, ir, j, k, kk

       INTEGER, DIMENSION(:), ALLOCATABLE ::  ira

       REAL ::  particle_radius, pp, qq, rq, x1, x2, x3, y1, y2, y3

       LOGICAL, SAVE ::  first = .TRUE.

       REAL, DIMENSION(1:11), SAVE ::  rat
       REAL, DIMENSION(1:7), SAVE  ::  r0
       REAL, DIMENSION(1:7,1:11), SAVE ::  ecoll_100, ecoll_400

!
!--    Initial assignment of constants
       IF ( first )  THEN

          first = .FALSE.

          r0  = (/ 10.0, 20.0, 30.0, 40.0, 50.0, 60.0, 100.0 /)
          rat = (/ 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 /)
!
!--       for 100 cm**2/s**3
          ecoll_100(:,1) = (/1.74,  1.74,  1.773, 1.49,  1.207,  1.207,  1.0 /)
          ecoll_100(:,2) = (/1.46,  1.46,  1.421, 1.245, 1.069,  1.069,  1.0 /)
          ecoll_100(:,3) = (/1.32,  1.32,  1.245, 1.123, 1.000,  1.000,  1.0 /)
          ecoll_100(:,4) = (/1.250, 1.250, 1.148, 1.087, 1.025,  1.025,  1.0 /)
          ecoll_100(:,5) = (/1.186, 1.186, 1.066, 1.060, 1.056,  1.056,  1.0 /)
          ecoll_100(:,6) = (/1.045, 1.045, 1.000, 1.014, 1.028,  1.028,  1.0 /)
          ecoll_100(:,7) = (/1.070, 1.070, 1.030, 1.038, 1.046,  1.046,  1.0 /)
          ecoll_100(:,8) = (/1.000, 1.000, 1.054, 1.042, 1.029,  1.029,  1.0 /)
          ecoll_100(:,9) = (/1.223, 1.223, 1.117, 1.069, 1.021,  1.021,  1.0 /)
          ecoll_100(:,10)= (/1.570, 1.570, 1.244, 1.166, 1.088,  1.088,  1.0 /)
          ecoll_100(:,11)= (/20.3,  20.3,  14.6 , 8.61,  2.60,   2.60 ,  1.0 /)
!
!--       for 400 cm**2/s**3
          ecoll_400(:,1) = (/4.976, 4.976,  3.593, 2.519, 1.445,  1.445,  1.0 /)
          ecoll_400(:,2) = (/2.984, 2.984,  2.181, 1.691, 1.201,  1.201,  1.0 /)
          ecoll_400(:,3) = (/1.988, 1.988,  1.475, 1.313, 1.150,  1.150,  1.0 /)
          ecoll_400(:,4) = (/1.490, 1.490,  1.187, 1.156, 1.126,  1.126,  1.0 /)
          ecoll_400(:,5) = (/1.249, 1.249,  1.088, 1.090, 1.092,  1.092,  1.0 /)
          ecoll_400(:,6) = (/1.139, 1.139,  1.130, 1.091, 1.051,  1.051,  1.0 /)
          ecoll_400(:,7) = (/1.220, 1.220,  1.190, 1.138, 1.086,  1.086,  1.0 /)
          ecoll_400(:,8) = (/1.325, 1.325,  1.267, 1.165, 1.063,  1.063,  1.0 /)
          ecoll_400(:,9) = (/1.716, 1.716,  1.345, 1.223, 1.100,  1.100,  1.0 /)
          ecoll_400(:,10)= (/3.788, 3.788,  1.501, 1.311, 1.120,  1.120,  1.0 /)
          ecoll_400(:,11)= (/36.52, 36.52,  19.16, 22.80,  26.0,   26.0,  1.0 /)

       ENDIF

!
!--    Calculate the radius class index of particles with respect to array r0
!--    Radius has to be in µm
       ALLOCATE( ira(1:radius_classes) )

       DO  j = 1, radius_classes
          particle_radius = radclass(j) * 1.0E6
          DO  k = 1, 7
             IF ( particle_radius < r0(k) )  THEN
                ira(j) = k
                EXIT
             ENDIF
          ENDDO
          IF ( particle_radius >= r0(7) )  ira(j) = 8
       ENDDO

!
!--    Two-dimensional linear interpolation of the collision efficiencies
!--    Radius has to be in µm
       DO  j =  1, radius_classes
          DO  i = 1, j

             ir = ira(j)
             rq = radclass(i) / radclass(j)

             DO  kk = 2, 11
                IF ( rq <= rat(kk) )  THEN
                   iq = kk
                   EXIT
                ENDIF
             ENDDO

             y1 = 0.0001      ! for 0 m**2/s**3

             IF ( ir < 8 )  THEN
                IF ( ir >= 2 )  THEN
                   pp = ( radclass(j)*1.0E6 - r0(ir-1) ) / ( r0(ir) - r0(ir-1) )
                   qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
                   y2 = ( 1.0-pp ) * ( 1.0-qq ) * ecoll_100(ir-1,iq-1) +  &
                                pp * ( 1.0-qq ) * ecoll_100(ir,iq-1)   +  &
                                qq * ( 1.0-pp ) * ecoll_100(ir-1,iq)   +  &
                                pp * qq         * ecoll_100(ir,iq)
                   y3 = ( 1.0-pp ) * ( 1.0-qq ) * ecoll_400(ir-1,iq-1) +  &
                                pp * ( 1.0-qq ) * ecoll_400(ir,iq-1)   +  &
                                qq * ( 1.0-pp ) * ecoll_400(ir-1,iq)   +  &
                                pp * qq         * ecoll_400(ir,iq)
                ELSE
                   qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
                   y2 = ( 1.0-qq ) * ecoll_100(1,iq-1) + qq * ecoll_100(1,iq)
                   y3 = ( 1.0-qq ) * ecoll_400(1,iq-1) + qq * ecoll_400(1,iq)
                ENDIF
             ELSE
                qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
                y2 = ( 1.0-qq ) * ecoll_100(7,iq-1) + qq * ecoll_100(7,iq)
                y3 = ( 1.0-qq ) * ecoll_400(7,iq-1) + qq * ecoll_400(7,iq)
             ENDIF
!
!--          Linear interpolation of dissipation rate in m**2/s**3
             IF ( epsilon <= 0.01 )  THEN
                ecf(j,i) = ( epsilon - 0.01 ) / (   0.0 - 0.01 ) * y1 &
                         + ( epsilon -   0.0 ) / ( 0.01 -   0.0 ) * y2
             ELSEIF ( epsilon <= 0.06 )  THEN
                ecf(j,i) = ( epsilon - 0.04 ) / ( 0.01 - 0.04 ) * y2 &
                         + ( epsilon - 0.01 ) / ( 0.04 - 0.01 ) * y3
             ELSE
                ecf(j,i) = (   0.06 - 0.04 ) / ( 0.01 - 0.04 ) * y2 &
                         + (   0.06 - 0.01 ) / ( 0.04 - 0.01 ) * y3
             ENDIF

             IF ( ecf(j,i) < 1.0 )  ecf(j,i) = 1.0

             ecf(i,j) = ecf(j,i)

          ENDDO
       ENDDO

    END SUBROUTINE turb_enhance_eff



    SUBROUTINE collision_efficiency_rogers( mean_r, r, e)
!------------------------------------------------------------------------------!
! Collision efficiencies from table 8.2 in Rogers and Yau (1989, 3rd edition).
! Values are calculated from table by bilinear interpolation.
!------------------------------------------------------------------------------!

       IMPLICIT NONE

       INTEGER       ::  i, j, k

       LOGICAL, SAVE ::  first = .TRUE.

       REAL          ::  aa, bb, cc, dd, dx, dy, e, gg, mean_r, mean_rm, r, &
                         rm, x, y

       REAL, DIMENSION(1:9), SAVE      ::  collected_r = 0.0
       REAL, DIMENSION(1:19), SAVE     ::  collector_r = 0.0
       REAL, DIMENSION(1:9,1:19), SAVE ::  ef = 0.0

       mean_rm = mean_r * 1.0E06
       rm      = r      * 1.0E06

       IF ( first )  THEN

          collected_r = (/ 2.0, 3.0, 4.0, 6.0, 8.0, 10.0, 15.0, 20.0, 25.0 /)
          collector_r = (/ 10.0, 20.0, 30.0, 40.0, 50.0, 60.0, 80.0, 100.0,  &
                           150.0, 200.0, 300.0, 400.0, 500.0, 600.0, 1000.0, &
                           1400.0, 1800.0, 2400.0, 3000.0 /)

          ef(:,1) = (/0.017, 0.027, 0.037, 0.052, 0.052, 0.052, 0.052, 0.0, &
                      0.0 /)
          ef(:,2) = (/0.001, 0.016, 0.027, 0.060, 0.12, 0.17, 0.17, 0.17, 0.0 /)
          ef(:,3) = (/0.001, 0.001, 0.02,  0.13,  0.28, 0.37, 0.54, 0.55, 0.47/)
          ef(:,4) = (/0.001, 0.001, 0.02,  0.23,  0.4,  0.55, 0.7,  0.75, 0.75/)
          ef(:,5) = (/0.01,  0.01,  0.03,  0.3,   0.4,  0.58, 0.73, 0.75, 0.79/)
          ef(:,6) = (/0.01,  0.01,  0.13,  0.38,  0.57, 0.68, 0.80, 0.86, 0.91/)
          ef(:,7) = (/0.01,  0.085, 0.23,  0.52,  0.68, 0.76, 0.86, 0.92, 0.95/)
          ef(:,8) = (/0.01,  0.14,  0.32,  0.60,  0.73, 0.81, 0.90, 0.94, 0.96/)
          ef(:,9) = (/0.025, 0.25,  0.43,  0.66,  0.78, 0.83, 0.92, 0.95, 0.96/)
          ef(:,10)= (/0.039, 0.3,   0.46,  0.69,  0.81, 0.87, 0.93, 0.95, 0.96/)
          ef(:,11)= (/0.095, 0.33,  0.51,  0.72,  0.82, 0.87, 0.93, 0.96, 0.97/)
          ef(:,12)= (/0.098, 0.36,  0.51,  0.73,  0.83, 0.88, 0.93, 0.96, 0.97/)
          ef(:,13)= (/0.1,   0.36,  0.52,  0.74,  0.83, 0.88, 0.93, 0.96, 0.97/)
          ef(:,14)= (/0.17,  0.4,   0.54,  0.72,  0.83, 0.88, 0.94, 0.98, 1.0 /)
          ef(:,15)= (/0.15,  0.37,  0.52,  0.74,  0.82, 0.88, 0.94, 0.98, 1.0 /)
          ef(:,16)= (/0.11,  0.34,  0.49,  0.71,  0.83, 0.88, 0.94, 0.95, 1.0 /)
          ef(:,17)= (/0.08,  0.29,  0.45,  0.68,  0.8,  0.86, 0.96, 0.94, 1.0 /)
          ef(:,18)= (/0.04,  0.22,  0.39,  0.62,  0.75, 0.83, 0.92, 0.96, 1.0 /)
          ef(:,19)= (/0.02,  0.16,  0.33,  0.55,  0.71, 0.81, 0.90, 0.94, 1.0 /)

       ENDIF

       DO  k = 1, 8
          IF ( collected_r(k) <= mean_rm )  i = k
       ENDDO

       DO  k = 1, 18
          IF ( collector_r(k) <= rm )  j = k
       ENDDO

       IF ( rm < 10.0 )  THEN
          e = 0.0
       ELSEIF ( mean_rm < 2.0 )  THEN
          e = 0.001
       ELSEIF ( mean_rm >= 25.0 )  THEN
          IF( j <= 2 )  e = 0.0
          IF( j == 3 )  e = 0.47
          IF( j == 4 )  e = 0.8
          IF( j == 5 )  e = 0.9
          IF( j >=6  )  e = 1.0
       ELSEIF ( rm >= 3000.0 )  THEN
          IF( i == 1 )  e = 0.02
          IF( i == 2 )  e = 0.16
          IF( i == 3 )  e = 0.33
          IF( i == 4 )  e = 0.55
          IF( i == 5 )  e = 0.71
          IF( i == 6 )  e = 0.81
          IF( i == 7 )  e = 0.90
          IF( i >= 8 )  e = 0.94
       ELSE
          x  = mean_rm - collected_r(i)
          y  = rm - collector_r(j)
          dx = collected_r(i+1) - collected_r(i)
          dy = collector_r(j+1) - collector_r(j)
          aa = x**2 + y**2
          bb = ( dx - x )**2 + y**2
          cc = x**2 + ( dy - y )**2
          dd = ( dx - x )**2 + ( dy - y )**2
          gg = aa + bb + cc + dd

          e = ( (gg-aa)*ef(i,j) + (gg-bb)*ef(i+1,j) + (gg-cc)*ef(i,j+1) + &
                (gg-dd)*ef(i+1,j+1) ) / (3.0*gg)
       ENDIF

    END SUBROUTINE collision_efficiency_rogers

 END MODULE lpm_collision_kernels_mod