source: palm/trunk/SOURCE/init_1d_model.f90 @ 817

 SUBROUTINE init_1d_model

!------------------------------------------------------------------------------!
! Current revisions:
! -----------------
! 
!
! Former revisions:
! -----------------
! $Id: init_1d_model.f90 668 2010-12-23 13:22:58Z raasch $
!
! 667 2010-12-23 12:06:00Z suehring/gryschka
! replaced mirror boundary conditions for u and v  at the ground
! by dirichlet boundary conditions
!
! 254 2009-03-05 15:33:42Z heinze
! Output of messages replaced by message handling routine.
!
! 184 2008-08-04 15:53:39Z letzel
! provisional solution for run_control_1d output: add 'CALL check_open( 15 )'
!
! 135 2007-11-22 12:24:23Z raasch
! Bugfix: absolute value of f must be used when calculating the Blackadar
! mixing length
!
! 82 2007-04-16 15:40:52Z raasch
! Preprocessor strings for different linux clusters changed to "lc",
! routine local_flush is used for buffer flushing
!
! 75 2007-03-22 09:54:05Z raasch
! Bugfix: preset of tendencies te_em, te_um, te_vm,
! moisture renamed humidity
!
! RCS Log replace by Id keyword, revision history cleaned up
!
! Revision 1.21  2006/06/02 15:19:57  raasch
! cpp-directives extended for lctit
!
! Revision 1.1  1998/03/09 16:22:10  raasch
! Initial revision
!
!
! Description:
! ------------
! 1D-model to initialize the 3D-arrays.
! The temperature profile is set as steady and a corresponding steady solution 
! of the wind profile is being computed.
! All subroutines required can be found within this file.
!------------------------------------------------------------------------------!

    USE arrays_3d
    USE indices
    USE model_1d
    USE control_parameters

    IMPLICIT NONE

    CHARACTER (LEN=9) ::  time_to_string
    INTEGER ::  k
    REAL    ::  lambda

!
!-- Allocate required 1D-arrays
    ALLOCATE( e1d(nzb:nzt+1),    e1d_m(nzb:nzt+1),   e1d_p(nzb:nzt+1), &
              kh1d(nzb:nzt+1),   kh1d_m(nzb:nzt+1),  km1d(nzb:nzt+1),  &
              km1d_m(nzb:nzt+1), l_black(nzb:nzt+1), l1d(nzb:nzt+1),   &
              l1d_m(nzb:nzt+1),  rif1d(nzb:nzt+1),   te_e(nzb:nzt+1),  &
              te_em(nzb:nzt+1),  te_u(nzb:nzt+1),    te_um(nzb:nzt+1), &
              te_v(nzb:nzt+1),   te_vm(nzb:nzt+1),    u1d(nzb:nzt+1),   &
              u1d_m(nzb:nzt+1),  u1d_p(nzb:nzt+1),   v1d(nzb:nzt+1),   &
              v1d_m(nzb:nzt+1),  v1d_p(nzb:nzt+1) )

!
!-- Initialize arrays
    IF ( constant_diffusion )  THEN
       km1d   = km_constant
       km1d_m = km_constant
       kh1d   = km_constant / prandtl_number
       kh1d_m = km_constant / prandtl_number
    ELSE
       e1d = 0.0; e1d_m = 0.0; e1d_p = 0.0
       kh1d = 0.0; kh1d_m = 0.0; km1d = 0.0; km1d_m = 0.0
       rif1d = 0.0
!
!--    Compute the mixing length
       l_black(nzb) = 0.0

       IF ( TRIM( mixing_length_1d ) == 'blackadar' )  THEN
!
!--       Blackadar mixing length
          IF ( f /= 0.0 )  THEN
             lambda = 2.7E-4 * SQRT( ug(nzt+1)**2 + vg(nzt+1)**2 ) / &
                               ABS( f ) + 1E-10
          ELSE
             lambda = 30.0
          ENDIF

          DO  k = nzb+1, nzt+1
             l_black(k) = kappa * zu(k) / ( 1.0 + kappa * zu(k) / lambda )
          ENDDO

       ELSEIF ( TRIM( mixing_length_1d ) == 'as_in_3d_model' )  THEN
!
!--       Use the same mixing length as in 3D model
          l_black(1:nzt) = l_grid
          l_black(nzt+1) = l_black(nzt)

       ENDIF

!
!--    Adjust mixing length to the prandtl mixing length (within the prandtl
!--    layer)
       IF ( adjust_mixing_length  .AND.  prandtl_layer )  THEN
          k = nzb+1
          l_black(k) = MIN( l_black(k), kappa * zu(k) )
       ENDIF
    ENDIF
    l1d   = l_black
    l1d_m = l_black
    u1d   = u_init
    u1d_m = u_init
    u1d_p = u_init
    v1d   = v_init
    v1d_m = v_init
    v1d_p = v_init

!
!-- Set initial horizontal velocities at the lowest grid levels to a very small
!-- value in order to avoid too small time steps caused by the diffusion limit
!-- in the initial phase of a run (at k=1, dz/2 occurs in the limiting formula!)
    u1d(0:1)   = 0.1
    u1d_m(0:1) = 0.1
    u1d_p(0:1) = 0.1
    v1d(0:1)   = 0.1
    v1d_m(0:1) = 0.1
    v1d_p(0:1) = 0.1

!
!-- For u*, theta* and the momentum fluxes plausible values are set
    IF ( prandtl_layer )  THEN
       us1d = 0.1   ! without initial friction the flow would not change
    ELSE
       e1d(nzb+1)  = 1.0
       km1d(nzb+1) = 1.0
       us1d = 0.0
    ENDIF
    ts1d = 0.0
    usws1d = 0.0; usws1d_m = 0.0
    vsws1d = 0.0; vsws1d_m = 0.0
    z01d = roughness_length
    IF ( humidity .OR. passive_scalar )  qs1d = 0.0

!
!-- Tendencies must be preset in order to avoid runtime errors within the
!-- first Runge-Kutta step
    te_em = 0.0
    te_um = 0.0
    te_vm = 0.0

!
!-- Set start time in hh:mm:ss - format
    simulated_time_chr = time_to_string( simulated_time_1d )

!
!-- Integrate the 1D-model equations using the leap-frog scheme
    CALL time_integration_1d


 END SUBROUTINE init_1d_model



 SUBROUTINE time_integration_1d

!------------------------------------------------------------------------------!
! Description:
! ------------
! Leap-frog time differencing scheme for the 1D-model.
!------------------------------------------------------------------------------!

    USE arrays_3d
    USE control_parameters
    USE indices
    USE model_1d
    USE pegrid

    IMPLICIT NONE

    CHARACTER (LEN=9) ::  time_to_string
    INTEGER ::  k
    REAL    ::  a, b, dissipation, dpt_dz, flux, kmzm, kmzp, l_stable, pt_0, &
                uv_total

!
!-- Determine the time step at the start of a 1D-simulation and
!-- determine and printout quantities used for run control
    CALL timestep_1d
    CALL run_control_1d

!
!-- Start of time loop
    DO  WHILE ( simulated_time_1d < end_time_1d  .AND.  .NOT. stop_dt_1d )

!
!--    Depending on the timestep scheme, carry out one or more intermediate
!--    timesteps

       intermediate_timestep_count = 0
       DO  WHILE ( intermediate_timestep_count < &
                   intermediate_timestep_count_max )

          intermediate_timestep_count = intermediate_timestep_count + 1

          CALL timestep_scheme_steering

!
!--       Compute all tendency terms. If a Prandtl-layer is simulated, k starts 
!--       at nzb+2.
          DO  k = nzb_diff, nzt

             kmzm = 0.5 * ( km1d_m(k-1) + km1d_m(k) )
             kmzp = 0.5 * ( km1d_m(k) + km1d_m(k+1) )
!
!--          u-component
             te_u(k) =  f * ( v1d(k) - vg(k) ) + ( &
                              kmzp * ( u1d_m(k+1) - u1d_m(k) ) * ddzu(k+1) &
                            - kmzm * ( u1d_m(k) - u1d_m(k-1) ) * ddzu(k)   &
                                              ) * ddzw(k)
!
!--          v-component
             te_v(k) = -f * ( u1d(k) - ug(k) ) + ( &
                              kmzp * ( v1d_m(k+1) - v1d_m(k) ) * ddzu(k+1) &
                            - kmzm * ( v1d_m(k) - v1d_m(k-1) ) * ddzu(k)   &
                                              ) * ddzw(k)
          ENDDO
          IF ( .NOT. constant_diffusion )  THEN
             DO  k = nzb_diff, nzt
!
!--             TKE
                kmzm = 0.5 * ( km1d_m(k-1) + km1d_m(k) )
                kmzp = 0.5 * ( km1d_m(k) + km1d_m(k+1) )
                IF ( .NOT. humidity )  THEN
                   pt_0 = pt_init(k)
                   flux =  ( pt_init(k+1)-pt_init(k-1) ) * dd2zu(k)
                ELSE
                   pt_0 = pt_init(k) * ( 1.0 + 0.61 * q_init(k) )
                   flux = ( ( pt_init(k+1) - pt_init(k-1) ) +                 &
                            0.61 * pt_init(k) * ( q_init(k+1) - q_init(k-1) ) &
                          ) * dd2zu(k)
                ENDIF

                IF ( dissipation_1d == 'detering' )  THEN
!
!--                According to Detering, c_e=0.064
                   dissipation = 0.064 * e1d_m(k) * SQRT( e1d_m(k) ) / l1d_m(k)
                ELSEIF ( dissipation_1d == 'as_in_3d_model' )  THEN
                   dissipation = ( 0.19 + 0.74 * l1d_m(k) / l_grid(k) ) &
                                 * e1d_m(k) * SQRT( e1d_m(k) ) / l1d_m(k)
                ENDIF

                te_e(k) = km1d(k) * ( ( ( u1d(k+1) - u1d(k-1) ) * dd2zu(k) )**2&
                                    + ( ( v1d(k+1) - v1d(k-1) ) * dd2zu(k) )**2&
                                    )                                          &
                                    - g / pt_0 * kh1d(k) * flux                &
                                    +            (                             &
                                 kmzp * ( e1d_m(k+1) - e1d_m(k) ) * ddzu(k+1)  &
                               - kmzm * ( e1d_m(k) - e1d_m(k-1) ) * ddzu(k)    &
                                                 ) * ddzw(k)                   &
                               - dissipation
             ENDDO
          ENDIF

!
!--       Tendency terms at the top of the Prandtl-layer.
!--       Finite differences of the momentum fluxes are computed using half the 
!--       normal grid length (2.0*ddzw(k)) for the sake of enhanced accuracy 
          IF ( prandtl_layer )  THEN

             k = nzb+1
             kmzm = 0.5 * ( km1d_m(k-1) + km1d_m(k) )
             kmzp = 0.5 * ( km1d_m(k) + km1d_m(k+1) )
             IF ( .NOT. humidity )  THEN
                pt_0 = pt_init(k)
                flux =  ( pt_init(k+1)-pt_init(k-1) ) * dd2zu(k)
             ELSE
                pt_0 = pt_init(k) * ( 1.0 + 0.61 * q_init(k) )
                flux = ( ( pt_init(k+1) - pt_init(k-1) ) +                 &
                         0.61 * pt_init(k) * ( q_init(k+1) - q_init(k-1) ) &
                       ) * dd2zu(k)
             ENDIF

             IF ( dissipation_1d == 'detering' )  THEN
!
!--             According to Detering, c_e=0.064
                dissipation = 0.064 * e1d_m(k) * SQRT( e1d_m(k) ) / l1d_m(k)
             ELSEIF ( dissipation_1d == 'as_in_3d_model' )  THEN
                dissipation = ( 0.19 + 0.74 * l1d_m(k) / l_grid(k) ) &
                              * e1d_m(k) * SQRT( e1d_m(k) ) / l1d_m(k)
             ENDIF

!
!--          u-component
             te_u(k) = f * ( v1d(k) - vg(k) ) + ( &
                       kmzp * ( u1d_m(k+1) - u1d_m(k) ) * ddzu(k+1) + usws1d_m &
                                               ) * 2.0 * ddzw(k)
!
!--          v-component
             te_v(k) = -f * ( u1d(k) - ug(k) ) + ( &
                       kmzp * ( v1d_m(k+1) - v1d_m(k) ) * ddzu(k+1) + vsws1d_m &
                                           ) * 2.0 * ddzw(k)
!
!--          TKE
             te_e(k) = km1d(k) * ( ( ( u1d(k+1) - u1d(k-1) ) * dd2zu(k) )**2   &
                                 + ( ( v1d(k+1) - v1d(k-1) ) * dd2zu(k) )**2   &
                                 )                                             &
                                 - g / pt_0 * kh1d(k) * flux                   &
                                 +           (                                 &
                              kmzp * ( e1d_m(k+1) - e1d_m(k) ) * ddzu(k+1)     &
                            - kmzm * ( e1d_m(k) - e1d_m(k-1) ) * ddzu(k)       &
                                              ) * ddzw(k)                      &
                            - dissipation
          ENDIF

!
!--       Prognostic equations for all 1D variables
          DO  k = nzb+1, nzt

             u1d_p(k) = ( 1. - tsc(1) ) * u1d_m(k) + &
                        tsc(1) * u1d(k) + dt_1d * ( tsc(2) * te_u(k) + &
                                                    tsc(3) * te_um(k) ) 
             v1d_p(k) = ( 1. - tsc(1) ) * v1d_m(k) + &
                        tsc(1) * v1d(k) + dt_1d * ( tsc(2) * te_v(k) + &
                                                    tsc(3) * te_vm(k) ) 

          ENDDO
          IF ( .NOT. constant_diffusion )  THEN
             DO  k = nzb+1, nzt

                e1d_p(k) = ( 1. - tsc(1) ) * e1d_m(k) + &
                           tsc(1) * e1d(k) + dt_1d * ( tsc(2) * te_e(k) + &
                                                       tsc(3) * te_em(k) ) 

             ENDDO
!
!--          Eliminate negative TKE values, which can result from the
!--          integration due to numerical inaccuracies. In such cases the TKE
!--          value is reduced to 10 percent of its old value.
             WHERE ( e1d_p < 0.0 )  e1d_p = 0.1 * e1d
          ENDIF

!
!--       Calculate tendencies for the next Runge-Kutta step
          IF ( timestep_scheme(1:5) == 'runge' ) THEN
             IF ( intermediate_timestep_count == 1 )  THEN

                DO  k = nzb+1, nzt
                   te_um(k) = te_u(k)
                   te_vm(k) = te_v(k)
                ENDDO

                IF ( .NOT. constant_diffusion )  THEN
                   DO k = nzb+1, nzt
                      te_em(k) = te_e(k)
                   ENDDO
                ENDIF

             ELSEIF ( intermediate_timestep_count < &
                         intermediate_timestep_count_max )  THEN

                DO  k = nzb+1, nzt
                   te_um(k) = -9.5625 * te_u(k) + 5.3125 * te_um(k)
                   te_vm(k) = -9.5625 * te_v(k) + 5.3125 * te_vm(k)
                ENDDO

                IF ( .NOT. constant_diffusion )  THEN
                   DO k = nzb+1, nzt
                      te_em(k) = -9.5625 * te_e(k) + 5.3125 * te_em(k)
                   ENDDO
                ENDIF

             ENDIF
          ENDIF


!
!--       Boundary conditions for the prognostic variables.
!--       At the top boundary (nzt+1) u,v and e keep their initial values 
!--       (ug(nzt+1), vg(nzt+1), 0), at the bottom boundary the mirror 
!--       boundary condition applies to u and v.
!--       The boundary condition for e is set further below ( (u*/cm)**2 ).
         ! u1d_p(nzb) = -u1d_p(nzb+1)
         ! v1d_p(nzb) = -v1d_p(nzb+1)

          u1d_p(nzb) = 0.0
          v1d_p(nzb) = 0.0

!
!--       If necessary, apply the time filter
          IF ( asselin_filter_factor /= 0.0  .AND. &
               timestep_scheme(1:5) /= 'runge' )  THEN

             u1d = u1d + asselin_filter_factor * ( u1d_p - 2.0 * u1d + u1d_m )
             v1d = v1d + asselin_filter_factor * ( v1d_p - 2.0 * v1d + v1d_m )

             IF ( .NOT. constant_diffusion )  THEN
                e1d = e1d + asselin_filter_factor * &
                            ( e1d_p - 2.0 * e1d + e1d_m )
             ENDIF

          ENDIF

!
!--       Swap the time levels in preparation for the next time step.
          IF ( timestep_scheme(1:4) == 'leap' )  THEN
             u1d_m  = u1d
             v1d_m  = v1d
             IF ( .NOT. constant_diffusion )  THEN
                e1d_m  = e1d
                kh1d_m = kh1d  ! The old diffusion quantities are required for
                km1d_m = km1d  ! explicit diffusion in the leap-frog scheme.
                l1d_m  = l1d
                IF ( prandtl_layer )  THEN
                   usws1d_m = usws1d
                   vsws1d_m = vsws1d
                ENDIF
             ENDIF
          ENDIF
          u1d  = u1d_p
          v1d  = v1d_p
          IF ( .NOT. constant_diffusion )  THEN
             e1d  = e1d_p
          ENDIF

!
!--       Compute diffusion quantities
          IF ( .NOT. constant_diffusion )  THEN

!
!--          First compute the vertical fluxes in the Prandtl-layer
             IF ( prandtl_layer )  THEN
!
!--             Compute theta* using Rif numbers of the previous time step
                IF ( rif1d(1) >= 0.0 )  THEN
!
!--                Stable stratification
                   ts1d = kappa * ( pt_init(nzb+1) - pt_init(nzb) ) /      &
                          ( LOG( zu(nzb+1) / z01d ) + 5.0 * rif1d(nzb+1) * &
                                          ( zu(nzb+1) - z01d ) / zu(nzb+1) &
                          )
                ELSE
!
!--                Unstable stratification
                   a = SQRT( 1.0 - 16.0 * rif1d(nzb+1) )
                   b = SQRT( 1.0 - 16.0 * rif1d(nzb+1) / zu(nzb+1) * z01d )
!
!--                In the borderline case the formula for stable stratification 
!--                must be applied, because otherwise a zero division would
!--                occur in the argument of the logarithm.
                   IF ( a == 0.0  .OR.  b == 0.0 )  THEN
                      ts1d = kappa * ( pt_init(nzb+1) - pt_init(nzb) ) /      &
                             ( LOG( zu(nzb+1) / z01d ) + 5.0 * rif1d(nzb+1) * &
                                             ( zu(nzb+1) - z01d ) / zu(nzb+1) &
                             )
                   ELSE
                      ts1d = kappa * ( pt_init(nzb+1) - pt_init(nzb) ) / &
                             LOG( (a-1.0) / (a+1.0) * (b+1.0) / (b-1.0) )
                   ENDIF
                ENDIF

             ENDIF    ! prandtl_layer

!
!--          Compute the Richardson-flux numbers,
!--          first at the top of the Prandtl-layer using u* of the previous
!--          time step (+1E-30, if u* = 0), then in the remaining area. There
!--          the rif-numbers of the previous time step are used.

             IF ( prandtl_layer )  THEN
                IF ( .NOT. humidity )  THEN
                   pt_0 = pt_init(nzb+1)
                   flux = ts1d
                ELSE
                   pt_0 = pt_init(nzb+1) * ( 1.0 + 0.61 * q_init(nzb+1) )
                   flux = ts1d + 0.61 * pt_init(k) * qs1d
                ENDIF
                rif1d(nzb+1) = zu(nzb+1) * kappa * g * flux / &
                               ( pt_0 * ( us1d**2 + 1E-30 ) )
             ENDIF

             DO  k = nzb_diff, nzt
                IF ( .NOT. humidity )  THEN
                   pt_0 = pt_init(k)
                   flux = ( pt_init(k+1) - pt_init(k-1) ) * dd2zu(k)
                ELSE
                   pt_0 = pt_init(k) * ( 1.0 + 0.61 * q_init(k) )
                   flux = ( ( pt_init(k+1) - pt_init(k-1) )                    &
                            + 0.61 * pt_init(k) * ( q_init(k+1) - q_init(k-1) )&
                          ) * dd2zu(k)
                ENDIF
                IF ( rif1d(k) >= 0.0 )  THEN
                   rif1d(k) = g / pt_0 * flux /                              &
                              (  ( ( u1d(k+1) - u1d(k-1) ) * dd2zu(k) )**2   &
                               + ( ( v1d(k+1) - v1d(k-1) ) * dd2zu(k) )**2   &
                               + 1E-30                                       &
                              )
                ELSE
                   rif1d(k) = g / pt_0 * flux /                              &
                              (  ( ( u1d(k+1) - u1d(k-1) ) * dd2zu(k) )**2   &
                               + ( ( v1d(k+1) - v1d(k-1) ) * dd2zu(k) )**2   &
                               + 1E-30                                       &
                              ) * ( 1.0 - 16.0 * rif1d(k) )**0.25
                ENDIF
             ENDDO
!
!--          Richardson-numbers must remain restricted to a realistic value
!--          range. It is exceeded excessively for very small velocities
!--          (u,v --> 0).
             WHERE ( rif1d < rif_min )  rif1d = rif_min
             WHERE ( rif1d > rif_max )  rif1d = rif_max

!
!--          Compute u* from the absolute velocity value
             IF ( prandtl_layer )  THEN
                uv_total = SQRT( u1d(nzb+1)**2 + v1d(nzb+1)**2 )

                IF ( rif1d(nzb+1) >= 0.0 )  THEN
!
!--                Stable stratification
                   us1d = kappa * uv_total / (                                 &
                             LOG( zu(nzb+1) / z01d ) + 5.0 * rif1d(nzb+1) *    &
                                              ( zu(nzb+1) - z01d ) / zu(nzb+1) &
                                             )
                ELSE
!
!--                Unstable stratification
                   a = 1.0 / SQRT( SQRT( 1.0 - 16.0 * rif1d(nzb+1) ) )
                   b = 1.0 / SQRT( SQRT( 1.0 - 16.0 * rif1d(nzb+1) / zu(nzb+1) &
                                                    * z01d ) )
!
!--                In the borderline case the formula for stable stratification 
!--                must be applied, because otherwise a zero division would
!--                occur in the argument of the logarithm.
                   IF ( a == 1.0  .OR.  b == 1.0 )  THEN
                      us1d = kappa * uv_total / (                            &
                             LOG( zu(nzb+1) / z01d ) +                       &
                             5.0 * rif1d(nzb+1) * ( zu(nzb+1) - z01d ) /     &
                                                  zu(nzb+1) )
                   ELSE
                      us1d = kappa * uv_total / (                              &
                                 LOG( (1.0+b) / (1.0-b) * (1.0-a) / (1.0+a) ) +&
                                 2.0 * ( ATAN( b ) - ATAN( a ) )               &
                                                )
                   ENDIF
                ENDIF

!
!--             Compute the momentum fluxes for the diffusion terms
                usws1d  = - u1d(nzb+1) / uv_total * us1d**2
                vsws1d  = - v1d(nzb+1) / uv_total * us1d**2

!
!--             Boundary condition for the turbulent kinetic energy at the top
!--             of the Prandtl-layer. c_m = 0.4 according to Detering.
!--             Additional Neumann condition de/dz = 0 at nzb is set to ensure
!--             compatibility with the 3D model.
                IF ( ibc_e_b == 2 )  THEN
                   e1d(nzb+1) = ( us1d / 0.1 )**2
!                  e1d(nzb+1) = ( us1d / 0.4 )**2  !not used so far, see also
                                                   !prandtl_fluxes
                ENDIF
                e1d(nzb) = e1d(nzb+1)

                IF ( humidity .OR. passive_scalar ) THEN
!
!--                Compute q* 
                   IF ( rif1d(1) >= 0.0 )  THEN
!
!--                Stable stratification
                   qs1d = kappa * ( q_init(nzb+1) - q_init(nzb) ) /        &
                          ( LOG( zu(nzb+1) / z01d ) + 5.0 * rif1d(nzb+1) * &
                                          ( zu(nzb+1) - z01d ) / zu(nzb+1) &
                          )
                ELSE
!
!--                Unstable stratification
                   a = SQRT( 1.0 - 16.0 * rif1d(nzb+1) )
                   b = SQRT( 1.0 - 16.0 * rif1d(nzb+1) / zu(nzb+1) * z01d )
!
!--                In the borderline case the formula for stable stratification 
!--                must be applied, because otherwise a zero division would
!--                occur in the argument of the logarithm.
                   IF ( a == 1.0  .OR.  b == 1.0 )  THEN
                      qs1d = kappa * ( q_init(nzb+1) - q_init(nzb) ) /        &
                             ( LOG( zu(nzb+1) / z01d ) + 5.0 * rif1d(nzb+1) * &
                                             ( zu(nzb+1) - z01d ) / zu(nzb+1) &
                             )
                   ELSE
                      qs1d = kappa * ( q_init(nzb+1) - q_init(nzb) ) / &
                             LOG( (a-1.0) / (a+1.0) * (b+1.0) / (b-1.0) )
                   ENDIF
                ENDIF                
                ELSE
                   qs1d = 0.0
                ENDIF             

             ENDIF   !  prandtl_layer

!
!--          Compute the diabatic mixing length
             IF ( mixing_length_1d == 'blackadar' )  THEN
                DO  k = nzb+1, nzt
                   IF ( rif1d(k) >= 0.0 )  THEN
                      l1d(k) = l_black(k) / ( 1.0 + 5.0 * rif1d(k) )
                   ELSE
                      l1d(k) = l_black(k) * ( 1.0 - 16.0 * rif1d(k) )**0.25
                   ENDIF
                   l1d(k) = l_black(k)
                ENDDO

             ELSEIF ( mixing_length_1d == 'as_in_3d_model' )  THEN
                DO  k = nzb+1, nzt
                   dpt_dz = ( pt_init(k+1) - pt_init(k-1) ) * dd2zu(k)
                   IF ( dpt_dz > 0.0 )  THEN
                      l_stable = 0.76 * SQRT( e1d(k) ) / &
                                     SQRT( g / pt_init(k) * dpt_dz ) + 1E-5
                   ELSE
                      l_stable = l_grid(k)
                   ENDIF
                   l1d(k) = MIN( l_grid(k), l_stable )
                ENDDO
             ENDIF

!
!--          Adjust mixing length to the prandtl mixing length
             IF ( adjust_mixing_length  .AND.  prandtl_layer )  THEN
                k = nzb+1
                IF ( rif1d(k) >= 0.0 )  THEN
                   l1d(k) = MIN( l1d(k), kappa * zu(k) / ( 1.0 + 5.0 * &
                                                           rif1d(k) ) )
                ELSE
                   l1d(k) = MIN( l1d(k), kappa * zu(k) *          &
                                  SQRT( SQRT( 1.0 - 16.0 * rif1d(k) ) ) )
                ENDIF
             ENDIF

!
!--          Compute the diffusion coefficients for momentum via the
!--          corresponding Prandtl-layer relationship and according to
!--          Prandtl-Kolmogorov, respectively. The unstable stratification is
!--          computed via the adiabatic mixing length, for the unstability has
!--          already been taken account of via the TKE (cf. also Diss.).
             IF ( prandtl_layer )  THEN
                IF ( rif1d(nzb+1) >= 0.0 )  THEN
                   km1d(nzb+1) = us1d * kappa * zu(nzb+1) / &
                                 ( 1.0 + 5.0 * rif1d(nzb+1) )
                ELSE
                   km1d(nzb+1) = us1d * kappa * zu(nzb+1) * &
                                 ( 1.0 - 16.0 * rif1d(nzb+1) )**0.25
                ENDIF
             ENDIF
             DO  k = nzb_diff, nzt
!                km1d(k) = 0.4 * SQRT( e1d(k) ) !changed: adjustment to 3D-model
                km1d(k) = 0.1 * SQRT( e1d(k) )
                IF ( rif1d(k) >= 0.0 )  THEN
                   km1d(k) = km1d(k) * l1d(k)
                ELSE
                   km1d(k) = km1d(k) * l_black(k)
                ENDIF
             ENDDO

!
!--          Add damping layer
             DO  k = damp_level_ind_1d+1, nzt+1
                km1d(k) = 1.1 * km1d(k-1)
                km1d(k) = MIN( km1d(k), 10.0 )
             ENDDO

!
!--          Compute the diffusion coefficient for heat via the relationship
!--          kh = phim / phih * km
             DO  k = nzb+1, nzt
                IF ( rif1d(k) >= 0.0 )  THEN
                   kh1d(k) = km1d(k)
                ELSE
                   kh1d(k) = km1d(k) * ( 1.0 - 16.0 * rif1d(k) )**0.25
                ENDIF
             ENDDO

          ENDIF   ! .NOT. constant_diffusion

!
!--       The Runge-Kutta scheme needs the recent diffusion quantities
          IF ( timestep_scheme(1:5) == 'runge' )  THEN
             u1d_m  = u1d
             v1d_m  = v1d
             IF ( .NOT. constant_diffusion )  THEN
                e1d_m  = e1d
                kh1d_m = kh1d
                km1d_m = km1d
                l1d_m  = l1d
                IF ( prandtl_layer )  THEN
                   usws1d_m = usws1d
                   vsws1d_m = vsws1d
                ENDIF
             ENDIF
          ENDIF


       ENDDO   ! intermediate step loop

!
!--    Increment simulated time and output times
       current_timestep_number_1d = current_timestep_number_1d + 1
       simulated_time_1d          = simulated_time_1d + dt_1d
       simulated_time_chr         = time_to_string( simulated_time_1d )
       time_pr_1d                 = time_pr_1d          + dt_1d
       time_run_control_1d        = time_run_control_1d + dt_1d

!
!--    Determine and print out quantities for run control
       IF ( time_run_control_1d >= dt_run_control_1d )  THEN
          CALL run_control_1d
          time_run_control_1d = time_run_control_1d - dt_run_control_1d
       ENDIF

!
!--    Profile output on file
       IF ( time_pr_1d >= dt_pr_1d )  THEN
          CALL print_1d_model
          time_pr_1d = time_pr_1d - dt_pr_1d
       ENDIF

!
!--    Determine size of next time step
       CALL timestep_1d

    ENDDO   ! time loop


 END SUBROUTINE time_integration_1d


 SUBROUTINE run_control_1d

!------------------------------------------------------------------------------!
! Description:
! ------------
! Compute and print out quantities for run control of the 1D model.
!------------------------------------------------------------------------------!

    USE constants
    USE indices
    USE model_1d
    USE pegrid
    USE control_parameters

    IMPLICIT NONE

    INTEGER ::  k
    REAL    ::  alpha, energy, umax, uv_total, vmax

!
!-- Output
    IF ( myid == 0 )  THEN
!
!--    If necessary, write header
       IF ( .NOT. run_control_header_1d )  THEN
          CALL check_open( 15 )
          WRITE ( 15, 100 )
          run_control_header_1d = .TRUE.
       ENDIF

!
!--    Compute control quantities
!--    grid level nzp is excluded due to mirror boundary condition
       umax = 0.0; vmax = 0.0; energy = 0.0
       DO  k = nzb+1, nzt+1
          umax = MAX( ABS( umax ), ABS( u1d(k) ) )
          vmax = MAX( ABS( vmax ), ABS( v1d(k) ) )
          energy = energy + 0.5 * ( u1d(k)**2 + v1d(k)**2 )
       ENDDO
       energy = energy / REAL( nzt - nzb + 1 )

       uv_total = SQRT( u1d(nzb+1)**2 + v1d(nzb+1)**2 )
       IF ( ABS( v1d(nzb+1) ) .LT. 1.0E-5 )  THEN
          alpha = ACOS( SIGN( 1.0 , u1d(nzb+1) ) )
       ELSE
          alpha = ACOS( u1d(nzb+1) / uv_total )
          IF ( v1d(nzb+1) <= 0.0 )  alpha = 2.0 * pi - alpha
       ENDIF
       alpha = alpha / ( 2.0 * pi ) * 360.0

       WRITE ( 15, 101 )  current_timestep_number_1d, simulated_time_chr, &
                          dt_1d, umax, vmax, us1d, alpha, energy
!
!--    Write buffer contents to disc immediately
       CALL local_flush( 15 )

    ENDIF

!
!-- formats
100 FORMAT (///'1D-Zeitschrittkontrollausgaben:'/ &
              &'------------------------------'// &
           &'ITER.  HH:MM:SS    DT      UMAX   VMAX    U*   ALPHA   ENERG.'/ &
           &'-------------------------------------------------------------')
101 FORMAT (I5,2X,A9,1X,F6.2,2X,F6.2,1X,F6.2,2X,F5.3,2X,F5.1,2X,F7.2)


 END SUBROUTINE run_control_1d



 SUBROUTINE timestep_1d

!------------------------------------------------------------------------------!
! Description:
! ------------
! Compute the time step w.r.t. the diffusion criterion
!------------------------------------------------------------------------------!

    USE arrays_3d
    USE indices
    USE model_1d
    USE pegrid
    USE control_parameters

    IMPLICIT NONE

    INTEGER ::  k
    REAL    ::  div, dt_diff, fac, percent_change, value


!
!-- Compute the currently feasible time step according to the diffusion
!-- criterion. At nzb+1 the half grid length is used.
    IF ( timestep_scheme(1:4) == 'leap' )  THEN
       fac = 0.25
    ELSE
       fac = 0.35
    ENDIF
    dt_diff = dt_max_1d
    DO  k = nzb+2, nzt
       value   = fac * dzu(k) * dzu(k) / ( km1d(k) + 1E-20 )
       dt_diff = MIN( value, dt_diff )
    ENDDO
    value   = fac * zu(nzb+1) * zu(nzb+1) / ( km1d(nzb+1) + 1E-20 )
    dt_1d = MIN( value, dt_diff )

!
!-- Set flag when the time step becomes too small
    IF ( dt_1d < ( 0.00001 * dt_max_1d ) )  THEN
       stop_dt_1d = .TRUE.

       WRITE( message_string, * ) 'timestep has exceeded the lower limit &', &
                                  'dt_1d = ',dt_1d,' s   simulation stopped!'
       CALL message( 'timestep_1d', 'PA0192', 1, 2, 0, 6, 0 )
       
    ENDIF

    IF ( timestep_scheme(1:4) == 'leap' )  THEN

!
!--    The current time step will only be changed if the new time step exceeds
!--    its previous value by 5 % or falls 2 % below. After a time step
!--    reduction at least 30 iterations must be done with this value before a
!--    new reduction will be allowed again.
!--    The control parameters for application of Euler- or leap-frog schemes are
!--    set accordingly.
       percent_change = dt_1d / old_dt_1d - 1.0
       IF ( percent_change > 0.05  .OR.  percent_change < -0.02 )  THEN

!
!--       Each time step increase is by at most 2 %
          IF ( percent_change > 0.0  .AND.  simulated_time_1d /= 0.0 )  THEN
             dt_1d = 1.02 * old_dt_1d
          ENDIF

!
!--       A more or less simple new time step value is obtained taking only the
!--       first two significant digits
          div = 1000.0
          DO  WHILE ( dt_1d < div )
             div = div / 10.0
          ENDDO
          dt_1d = NINT( dt_1d * 100.0 / div ) * div / 100.0

!
!--       Now the time step can be changed.
          IF ( percent_change < 0.0 )  THEN
!
!--          Time step reduction
             old_dt_1d = dt_1d
             last_dt_change_1d = current_timestep_number_1d
          ELSE
!
!--          Time step will only be increased if at least 30 iterations have
!--          been done since the previous time step change, and of course at
!--          simulation start, respectively.
             IF ( current_timestep_number_1d >= last_dt_change_1d + 30  .OR. &
                     simulated_time_1d == 0.0 )  THEN
                old_dt_1d = dt_1d
                last_dt_change_1d = current_timestep_number_1d
             ELSE
                dt_1d = old_dt_1d
             ENDIF
          ENDIF
       ELSE
!
!--       No time step change since the difference is too small
          dt_1d = old_dt_1d
       ENDIF

    ELSE    ! Runge-Kutta

!--    A more or less simple new time step value is obtained taking only the
!--    first two significant digits
       div = 1000.0
       DO  WHILE ( dt_1d < div )
          div = div / 10.0
       ENDDO
       dt_1d = NINT( dt_1d * 100.0 / div ) * div / 100.0

       old_dt_1d = dt_1d
       last_dt_change_1d = current_timestep_number_1d

    ENDIF

 END SUBROUTINE timestep_1d



 SUBROUTINE print_1d_model

!------------------------------------------------------------------------------!
! Description:
! ------------
! List output of profiles from the 1D-model
!------------------------------------------------------------------------------!

    USE arrays_3d
    USE indices
    USE model_1d
    USE pegrid
    USE control_parameters

    IMPLICIT NONE


    INTEGER ::  k


    IF ( myid == 0 )  THEN
!
!--    Open list output file for profiles from the 1D-model
       CALL check_open( 17 )

!
!--    Write Header
       WRITE ( 17, 100 )  TRIM( run_description_header ), &
                          TRIM( simulated_time_chr )
       WRITE ( 17, 101 )

!
!--    Write the values
       WRITE ( 17, 102 )
       WRITE ( 17, 101 )
       DO  k = nzt+1, nzb, -1
          WRITE ( 17, 103)  k, zu(k), u1d(k), v1d(k), pt_init(k), e1d(k), &
                            rif1d(k), km1d(k), kh1d(k), l1d(k), zu(k), k
       ENDDO
       WRITE ( 17, 101 )
       WRITE ( 17, 102 )
       WRITE ( 17, 101 )

!
!--    Write buffer contents to disc immediately
       CALL local_flush( 17 )

    ENDIF

!
!-- Formats
100 FORMAT (//1X,A/1X,10('-')/' 1d-model profiles'/ &
            ' Time: ',A)
101 FORMAT (1X,79('-'))
102 FORMAT ('   k     zu      u      v     pt      e    rif    Km    Kh     ', &
            'l      zu      k')
103 FORMAT (1X,I4,1X,F7.1,1X,F6.2,1X,F6.2,1X,F6.2,1X,F6.2,1X,F5.2,1X,F5.2, &
            1X,F5.2,1X,F6.2,1X,F7.1,2X,I4)


 END SUBROUTINE print_1d_model