!> @file lpm_advec.f90 !------------------------------------------------------------------------------! ! 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 . ! ! Copyright 1997-2017 Leibniz Universitaet Hannover !------------------------------------------------------------------------------! ! ! Current revisions: ! ------------------ ! ! ! Former revisions: ! ----------------- ! $Id: lpm_advec.f90 2417 2017-09-06 15:22:27Z schwenkel $ ! Particle loops adapted for sub-box structure, i.e. for each sub-box the ! particle loop runs from start_index up to end_index instead from 1 to ! number_of_particles. This way, it is possible to skip unnecessary ! computations for particles that already completed the LES timestep. ! ! 2318 2017-07-20 17:27:44Z suehring ! Get topography top index via Function call ! ! 2317 2017-07-20 17:27:19Z suehring ! ! 2232 2017-05-30 17:47:52Z suehring ! Adjustments to new topography and surface concept ! ! 2100 2017-01-05 16:40:16Z suehring ! Prevent extremely large SGS-velocities in regions where TKE is zero, e.g. ! at the begin of simulations and/or in non-turbulent regions. ! ! 2000 2016-08-20 18:09:15Z knoop ! Forced header and separation lines into 80 columns ! ! 1936 2016-06-13 13:37:44Z suehring ! Formatting adjustments ! ! 1929 2016-06-09 16:25:25Z suehring ! Put stochastic equation in an extra subroutine. ! Set flag for stochastic equation to communicate whether a particle is near ! topography. This case, memory and drift term are disabled in the Weil equation. ! ! Enable vertical logarithmic interpolation also above topography. This case, ! set a lower limit for the friction velocity, as it can become very small ! in narrow street canyons, leading to too large particle speeds. ! ! 1888 2016-04-21 12:20:49Z suehring ! Bugfix concerning logarithmic interpolation of particle speed ! ! 1822 2016-04-07 07:49:42Z hoffmann ! Random velocity fluctuations for particles added. Terminal fall velocity ! for droplets is calculated from a parameterization (which is better than ! the previous, physically correct calculation, which demands a very short ! time step that is not used in the model). ! ! Unused variables deleted. ! ! 1691 2015-10-26 16:17:44Z maronga ! Renamed prandtl_layer to constant_flux_layer. ! ! 1685 2015-10-08 07:32:13Z raasch ! TKE check for negative values (so far, only zero value was checked) ! offset_ocean_nzt_m1 removed ! ! 1682 2015-10-07 23:56:08Z knoop ! Code annotations made doxygen readable ! ! 1583 2015-04-15 12:16:27Z suehring ! Bugfix: particle advection within Prandtl-layer in case of Galilei ! transformation. ! ! 1369 2014-04-24 05:57:38Z raasch ! usage of module interfaces removed ! ! 1359 2014-04-11 17:15:14Z hoffmann ! New particle structure integrated. ! Kind definition added to all floating point numbers. ! ! 1322 2014-03-20 16:38:49Z raasch ! REAL constants defined as wp_kind ! ! 1320 2014-03-20 08:40:49Z raasch ! ONLY-attribute added to USE-statements, ! kind-parameters added to all INTEGER and REAL declaration statements, ! kinds are defined in new module kinds, ! revision history before 2012 removed, ! comment fields (!:) to be used for variable explanations added to ! all variable declaration statements ! ! 1314 2014-03-14 18:25:17Z suehring ! Vertical logarithmic interpolation of horizontal particle speed for particles ! between roughness height and first vertical grid level. ! ! 1036 2012-10-22 13:43:42Z raasch ! code put under GPL (PALM 3.9) ! ! 849 2012-03-15 10:35:09Z raasch ! initial revision (former part of advec_particles) ! ! ! Description: ! ------------ !> Calculation of new particle positions due to advection using a simple Euler !> scheme. Particles may feel inertia effects. SGS transport can be included !> using the stochastic model of Weil et al. (2004, JAS, 61, 2877-2887). !------------------------------------------------------------------------------! SUBROUTINE lpm_advec (ip,jp,kp) USE arrays_3d, & ONLY: de_dx, de_dy, de_dz, diss, e, km, u, v, w, zu, zw USE cpulog USE pegrid USE control_parameters, & ONLY: atmos_ocean_sign, cloud_droplets, constant_flux_layer, dt_3d, & dt_3d_reached_l, dz, g, kappa, topography, u_gtrans, v_gtrans USE grid_variables, & ONLY: ddx, dx, ddy, dy USE indices, & ONLY: nzb, nzt USE kinds USE particle_attributes, & ONLY: block_offset, c_0, dt_min_part, grid_particles, & iran_part, log_z_z0, number_of_particles, number_of_sublayers, & particles, particle_groups, offset_ocean_nzt, sgs_wf_part, & use_sgs_for_particles, vertical_particle_advection, z0_av_global USE statistics, & ONLY: hom USE surface_mod, & ONLY: get_topography_top_index, surf_def_h, surf_lsm_h, surf_usm_h IMPLICIT NONE INTEGER(iwp) :: agp !< loop variable INTEGER(iwp) :: gp_outside_of_building(1:8) !< number of grid points used for particle interpolation in case of topography INTEGER(iwp) :: i !< index variable along x INTEGER(iwp) :: ip !< index variable along x INTEGER(iwp) :: ilog !< index variable along x INTEGER(iwp) :: j !< index variable along y INTEGER(iwp) :: jp !< index variable along y INTEGER(iwp) :: jlog !< index variable along y INTEGER(iwp) :: k !< index variable along z INTEGER(iwp) :: k_wall !< vertical index of topography top INTEGER(iwp) :: kp !< index variable along z INTEGER(iwp) :: kw !< index variable along z INTEGER(iwp) :: n !< loop variable over all particles in a grid box INTEGER(iwp) :: nb !< block number particles are sorted in INTEGER(iwp) :: num_gp !< number of adjacent grid points inside topography INTEGER(iwp) :: surf_start !< Index on surface data-type for current grid box INTEGER(iwp), DIMENSION(0:7) :: start_index !< start particle index for current block INTEGER(iwp), DIMENSION(0:7) :: end_index !< start particle index for current block REAL(wp) :: aa !< dummy argument for horizontal particle interpolation REAL(wp) :: bb !< dummy argument for horizontal particle interpolation REAL(wp) :: cc !< dummy argument for horizontal particle interpolation REAL(wp) :: d_sum !< dummy argument for horizontal particle interpolation in case of topography REAL(wp) :: d_z_p_z0 !< inverse of interpolation length for logarithmic interpolation REAL(wp) :: dd !< dummy argument for horizontal particle interpolation REAL(wp) :: de_dx_int_l !< x/y-interpolated TKE gradient (x) at particle position at lower vertical level REAL(wp) :: de_dx_int_u !< x/y-interpolated TKE gradient (x) at particle position at upper vertical level REAL(wp) :: de_dy_int_l !< x/y-interpolated TKE gradient (y) at particle position at lower vertical level REAL(wp) :: de_dy_int_u !< x/y-interpolated TKE gradient (y) at particle position at upper vertical level REAL(wp) :: de_dt !< temporal derivative of TKE experienced by the particle REAL(wp) :: de_dt_min !< lower level for temporal TKE derivative REAL(wp) :: de_dz_int_l !< x/y-interpolated TKE gradient (z) at particle position at lower vertical level REAL(wp) :: de_dz_int_u !< x/y-interpolated TKE gradient (z) at particle position at upper vertical level REAL(wp) :: diameter !< diamter of droplet REAL(wp) :: diss_int_l !< x/y-interpolated dissipation at particle position at lower vertical level REAL(wp) :: diss_int_u !< x/y-interpolated dissipation at particle position at upper vertical level REAL(wp) :: dt_gap !< remaining time until particle time integration reaches LES time REAL(wp) :: dt_particle_m !< previous particle time step REAL(wp) :: e_int_l !< x/y-interpolated TKE at particle position at lower vertical level REAL(wp) :: e_int_u !< x/y-interpolated TKE at particle position at upper vertical level REAL(wp) :: e_mean_int !< horizontal mean TKE at particle height REAL(wp) :: exp_arg !< REAL(wp) :: exp_term !< REAL(wp) :: gg !< dummy argument for horizontal particle interpolation REAL(wp) :: height_p !< dummy argument for logarithmic interpolation REAL(wp) :: lagr_timescale !< Lagrangian timescale REAL(wp) :: location(1:30,1:3) !< wall locations REAL(wp) :: log_z_z0_int !< logarithmus used for surface_layer interpolation REAL(wp) :: random_gauss !< REAL(wp) :: RL !< Lagrangian autocorrelation coefficient REAL(wp) :: rg1 !< Gaussian distributed random number REAL(wp) :: rg2 !< Gaussian distributed random number REAL(wp) :: rg3 !< Gaussian distributed random number REAL(wp) :: sigma !< velocity standard deviation REAL(wp) :: u_int_l !< x/y-interpolated u-component at particle position at lower vertical level REAL(wp) :: u_int_u !< x/y-interpolated u-component at particle position at upper vertical level REAL(wp) :: us_int !< friction velocity at particle grid box REAL(wp) :: usws_int !< surface momentum flux (u component) at particle grid box REAL(wp) :: v_int_l !< x/y-interpolated v-component at particle position at lower vertical level REAL(wp) :: v_int_u !< x/y-interpolated v-component at particle position at upper vertical level REAL(wp) :: vsws_int !< surface momentum flux (u component) at particle grid box REAL(wp) :: vv_int !< REAL(wp) :: w_int_l !< x/y-interpolated w-component at particle position at lower vertical level REAL(wp) :: w_int_u !< x/y-interpolated w-component at particle position at upper vertical level REAL(wp) :: w_s !< terminal velocity of droplets REAL(wp) :: x !< dummy argument for horizontal particle interpolation REAL(wp) :: y !< dummy argument for horizontal particle interpolation REAL(wp) :: z_p !< surface layer height (0.5 dz) REAL(wp), PARAMETER :: a_rog = 9.65_wp !< parameter for fall velocity REAL(wp), PARAMETER :: b_rog = 10.43_wp !< parameter for fall velocity REAL(wp), PARAMETER :: c_rog = 0.6_wp !< parameter for fall velocity REAL(wp), PARAMETER :: k_cap_rog = 4.0_wp !< parameter for fall velocity REAL(wp), PARAMETER :: k_low_rog = 12.0_wp !< parameter for fall velocity REAL(wp), PARAMETER :: d0_rog = 0.745_wp !< separation diameter REAL(wp), DIMENSION(1:30) :: d_gp_pl !< dummy argument for particle interpolation scheme in case of topography REAL(wp), DIMENSION(1:30) :: de_dxi !< horizontal TKE gradient along x at adjacent wall REAL(wp), DIMENSION(1:30) :: de_dyi !< horizontal TKE gradient along y at adjacent wall REAL(wp), DIMENSION(1:30) :: de_dzi !< horizontal TKE gradient along z at adjacent wall REAL(wp), DIMENSION(1:30) :: dissi !< dissipation at adjacent wall REAL(wp), DIMENSION(1:30) :: ei !< TKE at adjacent wall REAL(wp), DIMENSION(number_of_particles) :: term_1_2 !< flag to communicate whether a particle is near topography or not REAL(wp), DIMENSION(number_of_particles) :: dens_ratio !< REAL(wp), DIMENSION(number_of_particles) :: de_dx_int !< horizontal TKE gradient along x at particle position REAL(wp), DIMENSION(number_of_particles) :: de_dy_int !< horizontal TKE gradient along y at particle position REAL(wp), DIMENSION(number_of_particles) :: de_dz_int !< horizontal TKE gradient along z at particle position REAL(wp), DIMENSION(number_of_particles) :: diss_int !< dissipation at particle position REAL(wp), DIMENSION(number_of_particles) :: dt_particle !< particle time step REAL(wp), DIMENSION(number_of_particles) :: e_int !< TKE at particle position REAL(wp), DIMENSION(number_of_particles) :: fs_int !< weighting factor for subgrid-scale particle speed REAL(wp), DIMENSION(number_of_particles) :: u_int !< u-component of particle speed REAL(wp), DIMENSION(number_of_particles) :: v_int !< v-component of particle speed REAL(wp), DIMENSION(number_of_particles) :: w_int !< w-component of particle speed REAL(wp), DIMENSION(number_of_particles) :: xv !< x-position REAL(wp), DIMENSION(number_of_particles) :: yv !< y-position REAL(wp), DIMENSION(number_of_particles) :: zv !< z-position REAL(wp), DIMENSION(number_of_particles, 3) :: rg !< vector of Gaussian distributed random numbers CALL cpu_log( log_point_s(44), 'lpm_advec', 'continue' ) ! !-- Determine height of Prandtl layer and distance between Prandtl-layer !-- height and horizontal mean roughness height, which are required for !-- vertical logarithmic interpolation of horizontal particle speeds !-- (for particles below first vertical grid level). z_p = zu(nzb+1) - zw(nzb) d_z_p_z0 = 1.0_wp / ( z_p - z0_av_global ) start_index = grid_particles(kp,jp,ip)%start_index end_index = grid_particles(kp,jp,ip)%end_index xv = particles(1:number_of_particles)%x yv = particles(1:number_of_particles)%y zv = particles(1:number_of_particles)%z DO nb = 0, 7 i = ip j = jp + block_offset(nb)%j_off k = kp + block_offset(nb)%k_off ! !-- Interpolate u velocity-component DO n = start_index(nb), end_index(nb) ! !-- Interpolation of the u velocity component onto particle position. !-- Particles are interpolation bi-linearly in the horizontal and a !-- linearly in the vertical. An exception is made for particles below !-- the first vertical grid level in case of a prandtl layer. In this !-- case the horizontal particle velocity components are determined using !-- Monin-Obukhov relations (if branch). !-- First, check if particle is located below first vertical grid level !-- above topography (Prandtl-layer height) ilog = ( particles(n)%x + 0.5_wp * dx ) * ddx jlog = ( particles(n)%y + 0.5_wp * dy ) * ddy ! !-- Determine vertical index of topography top k_wall = get_topography_top_index( jlog, ilog, 's' ) IF ( constant_flux_layer .AND. zv(n) - zw(k_wall) < z_p ) THEN ! !-- Resolved-scale horizontal particle velocity is zero below z0. IF ( zv(n) - zw(k_wall) < z0_av_global ) THEN u_int(n) = 0.0_wp ELSE ! !-- Determine the sublayer. Further used as index. height_p = ( zv(n) - zw(k_wall) - z0_av_global ) & * REAL( number_of_sublayers, KIND=wp ) & * d_z_p_z0 ! !-- Calculate LOG(z/z0) for exact particle height. Therefore, !-- interpolate linearly between precalculated logarithm. log_z_z0_int = log_z_z0(INT(height_p)) & + ( height_p - INT(height_p) ) & * ( log_z_z0(INT(height_p)+1) & - log_z_z0(INT(height_p)) & ) ! !-- Get friction velocity and momentum flux from new surface data !-- types. IF ( surf_def_h(0)%start_index(jlog,ilog) <= & surf_def_h(0)%end_index(jlog,ilog) ) THEN surf_start = surf_def_h(0)%start_index(jlog,ilog) !-- Limit friction velocity. In narrow canyons or holes the !-- friction velocity can become very small, resulting in a too !-- large particle speed. us_int = MAX( surf_def_h(0)%us(surf_start), 0.01_wp ) usws_int = surf_def_h(0)%usws(surf_start) ELSEIF ( surf_lsm_h%start_index(jlog,ilog) <= & surf_lsm_h%end_index(jlog,ilog) ) THEN surf_start = surf_lsm_h%start_index(jlog,ilog) us_int = MAX( surf_lsm_h%us(surf_start), 0.01_wp ) usws_int = surf_lsm_h%usws(surf_start) ELSEIF ( surf_usm_h%start_index(jlog,ilog) <= & surf_usm_h%end_index(jlog,ilog) ) THEN surf_start = surf_usm_h%start_index(jlog,ilog) us_int = MAX( surf_usm_h%us(surf_start), 0.01_wp ) usws_int = surf_usm_h%usws(surf_start) ENDIF ! !-- Neutral solution is applied for all situations, e.g. also for !-- unstable and stable situations. Even though this is not exact !-- this saves a lot of CPU time since several calls of intrinsic !-- FORTRAN procedures (LOG, ATAN) are avoided, This is justified !-- as sensitivity studies revealed no significant effect of !-- using the neutral solution also for un/stable situations. u_int(n) = -usws_int / ( us_int * kappa + 1E-10_wp ) & * log_z_z0_int - u_gtrans ENDIF ! !-- Particle above the first grid level. Bi-linear interpolation in the !-- horizontal and linear interpolation in the vertical direction. ELSE x = xv(n) + ( 0.5_wp - i ) * dx y = yv(n) - j * dy 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 u_int_l = ( ( gg - aa ) * u(k,j,i) + ( gg - bb ) * u(k,j,i+1) & + ( gg - cc ) * u(k,j+1,i) + ( gg - dd ) * & u(k,j+1,i+1) ) / ( 3.0_wp * gg ) - u_gtrans IF ( k == nzt ) THEN u_int(n) = u_int_l ELSE u_int_u = ( ( gg-aa ) * u(k+1,j,i) + ( gg-bb ) * u(k+1,j,i+1) & + ( gg-cc ) * u(k+1,j+1,i) + ( gg-dd ) * & u(k+1,j+1,i+1) ) / ( 3.0_wp * gg ) - u_gtrans u_int(n) = u_int_l + ( zv(n) - zu(k) ) / dz * & ( u_int_u - u_int_l ) ENDIF ENDIF ENDDO i = ip + block_offset(nb)%i_off j = jp k = kp + block_offset(nb)%k_off ! !-- Same procedure for interpolation of the v velocity-component DO n = start_index(nb), end_index(nb) ilog = ( particles(n)%x + 0.5_wp * dx ) * ddx jlog = ( particles(n)%y + 0.5_wp * dy ) * ddy ! !-- Determine vertical index of topography top k_wall = get_topography_top_index( jlog,ilog, 's' ) IF ( constant_flux_layer .AND. zv(n) - zw(k_wall) < z_p ) THEN IF ( zv(n) - zw(k_wall) < z0_av_global ) THEN ! !-- Resolved-scale horizontal particle velocity is zero below z0. v_int(n) = 0.0_wp ELSE ! !-- Determine the sublayer. Further used as index. Please note, !-- logarithmus can not be reused from above, as in in case of !-- topography particle on u-grid can be above surface-layer height, !-- whereas it can be below on v-grid. height_p = ( zv(n) - zw(k_wall) - z0_av_global ) & * REAL( number_of_sublayers, KIND=wp ) & * d_z_p_z0 ! !-- Calculate LOG(z/z0) for exact particle height. Therefore, !-- interpolate linearly between precalculated logarithm. log_z_z0_int = log_z_z0(INT(height_p)) & + ( height_p - INT(height_p) ) & * ( log_z_z0(INT(height_p)+1) & - log_z_z0(INT(height_p)) & ) ! !-- Get friction velocity and momentum flux from new surface data !-- types. IF ( surf_def_h(0)%start_index(jlog,ilog) <= & surf_def_h(0)%end_index(jlog,ilog) ) THEN surf_start = surf_def_h(0)%start_index(jlog,ilog) !-- Limit friction velocity. In narrow canyons or holes the !-- friction velocity can become very small, resulting in a too !-- large particle speed. us_int = MAX( surf_def_h(0)%us(surf_start), 0.01_wp ) vsws_int = surf_def_h(0)%usws(surf_start) ELSEIF ( surf_lsm_h%start_index(jlog,ilog) <= & surf_lsm_h%end_index(jlog,ilog) ) THEN surf_start = surf_lsm_h%start_index(jlog,ilog) us_int = MAX( surf_lsm_h%us(surf_start), 0.01_wp ) vsws_int = surf_lsm_h%usws(surf_start) ELSEIF ( surf_usm_h%start_index(jlog,ilog) <= & surf_usm_h%end_index(jlog,ilog) ) THEN surf_start = surf_usm_h%start_index(jlog,ilog) us_int = MAX( surf_usm_h%us(surf_start), 0.01_wp ) vsws_int = surf_usm_h%usws(surf_start) ENDIF ! !-- Neutral solution is applied for all situations, e.g. also for !-- unstable and stable situations. Even though this is not exact !-- this saves a lot of CPU time since several calls of intrinsic !-- FORTRAN procedures (LOG, ATAN) are avoided, This is justified !-- as sensitivity studies revealed no significant effect of !-- using the neutral solution also for un/stable situations. v_int(n) = -vsws_int / ( us_int * kappa + 1E-10_wp ) & * log_z_z0_int - v_gtrans ENDIF ELSE x = xv(n) - i * dx y = yv(n) + ( 0.5_wp - j ) * dy 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 v_int_l = ( ( gg - aa ) * v(k,j,i) + ( gg - bb ) * v(k,j,i+1) & + ( gg - cc ) * v(k,j+1,i) + ( gg - dd ) * v(k,j+1,i+1) & ) / ( 3.0_wp * gg ) - v_gtrans IF ( k == nzt ) THEN v_int(n) = v_int_l ELSE v_int_u = ( ( gg-aa ) * v(k+1,j,i) + ( gg-bb ) * v(k+1,j,i+1) & + ( gg-cc ) * v(k+1,j+1,i) + ( gg-dd ) * v(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) - v_gtrans v_int(n) = v_int_l + ( zv(n) - zu(k) ) / dz * & ( v_int_u - v_int_l ) ENDIF ENDIF ENDDO i = ip + block_offset(nb)%i_off j = jp + block_offset(nb)%j_off k = kp - 1 ! !-- Same procedure for interpolation of the w velocity-component DO n = start_index(nb), end_index(nb) IF ( vertical_particle_advection(particles(n)%group) ) THEN x = xv(n) - i * dx y = yv(n) - j * dy 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 w_int_l = ( ( gg - aa ) * w(k,j,i) + ( gg - bb ) * w(k,j,i+1) & + ( gg - cc ) * w(k,j+1,i) + ( gg - dd ) * w(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( k == nzt ) THEN w_int(n) = w_int_l ELSE w_int_u = ( ( gg-aa ) * w(k+1,j,i) + & ( gg-bb ) * w(k+1,j,i+1) + & ( gg-cc ) * w(k+1,j+1,i) + & ( gg-dd ) * w(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) w_int(n) = w_int_l + ( zv(n) - zw(k) ) / dz * & ( w_int_u - w_int_l ) ENDIF ELSE w_int(n) = 0.0_wp ENDIF ENDDO ENDDO !-- Interpolate and calculate quantities needed for calculating the SGS !-- velocities IF ( use_sgs_for_particles .AND. .NOT. cloud_droplets ) THEN IF ( topography == 'flat' ) THEN DO nb = 0,7 i = ip + block_offset(nb)%i_off j = jp + block_offset(nb)%j_off k = kp + block_offset(nb)%k_off DO n = start_index(nb), end_index(nb) ! !-- Interpolate TKE x = xv(n) - i * dx y = yv(n) - j * dy 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_int_l = ( ( gg-aa ) * e(k,j,i) + ( gg-bb ) * e(k,j,i+1) & + ( gg-cc ) * e(k,j+1,i) + ( gg-dd ) * e(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( k+1 == nzt+1 ) THEN e_int(n) = e_int_l ELSE e_int_u = ( ( gg - aa ) * e(k+1,j,i) + & ( gg - bb ) * e(k+1,j,i+1) + & ( gg - cc ) * e(k+1,j+1,i) + & ( gg - dd ) * e(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) e_int(n) = e_int_l + ( zv(n) - zu(k) ) / dz * & ( e_int_u - e_int_l ) ENDIF ! !-- Needed to avoid NaN particle velocities (this might not be !-- required any more) IF ( e_int(n) <= 0.0_wp ) THEN e_int(n) = 1.0E-20_wp ENDIF ! !-- Interpolate the TKE gradient along x (adopt incides i,j,k and !-- all position variables from above (TKE)) de_dx_int_l = ( ( gg - aa ) * de_dx(k,j,i) + & ( gg - bb ) * de_dx(k,j,i+1) + & ( gg - cc ) * de_dx(k,j+1,i) + & ( gg - dd ) * de_dx(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( ( k+1 == nzt+1 ) .OR. ( k == nzb ) ) THEN de_dx_int(n) = de_dx_int_l ELSE de_dx_int_u = ( ( gg - aa ) * de_dx(k+1,j,i) + & ( gg - bb ) * de_dx(k+1,j,i+1) + & ( gg - cc ) * de_dx(k+1,j+1,i) + & ( gg - dd ) * de_dx(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) de_dx_int(n) = de_dx_int_l + ( zv(n) - zu(k) ) / dz * & ( de_dx_int_u - de_dx_int_l ) ENDIF ! !-- Interpolate the TKE gradient along y de_dy_int_l = ( ( gg - aa ) * de_dy(k,j,i) + & ( gg - bb ) * de_dy(k,j,i+1) + & ( gg - cc ) * de_dy(k,j+1,i) + & ( gg - dd ) * de_dy(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( ( k+1 == nzt+1 ) .OR. ( k == nzb ) ) THEN de_dy_int(n) = de_dy_int_l ELSE de_dy_int_u = ( ( gg - aa ) * de_dy(k+1,j,i) + & ( gg - bb ) * de_dy(k+1,j,i+1) + & ( gg - cc ) * de_dy(k+1,j+1,i) + & ( gg - dd ) * de_dy(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) de_dy_int(n) = de_dy_int_l + ( zv(n) - zu(k) ) / dz * & ( de_dy_int_u - de_dy_int_l ) ENDIF ! !-- Interpolate the TKE gradient along z IF ( zv(n) < 0.5_wp * dz ) THEN de_dz_int(n) = 0.0_wp ELSE de_dz_int_l = ( ( gg - aa ) * de_dz(k,j,i) + & ( gg - bb ) * de_dz(k,j,i+1) + & ( gg - cc ) * de_dz(k,j+1,i) + & ( gg - dd ) * de_dz(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( ( k+1 == nzt+1 ) .OR. ( k == nzb ) ) THEN de_dz_int(n) = de_dz_int_l ELSE de_dz_int_u = ( ( gg - aa ) * de_dz(k+1,j,i) + & ( gg - bb ) * de_dz(k+1,j,i+1) + & ( gg - cc ) * de_dz(k+1,j+1,i) + & ( gg - dd ) * de_dz(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) de_dz_int(n) = de_dz_int_l + ( zv(n) - zu(k) ) / dz * & ( de_dz_int_u - de_dz_int_l ) ENDIF ENDIF ! !-- Interpolate the dissipation of TKE diss_int_l = ( ( gg - aa ) * diss(k,j,i) + & ( gg - bb ) * diss(k,j,i+1) + & ( gg - cc ) * diss(k,j+1,i) + & ( gg - dd ) * diss(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( k == nzt ) THEN diss_int(n) = diss_int_l ELSE diss_int_u = ( ( gg - aa ) * diss(k+1,j,i) + & ( gg - bb ) * diss(k+1,j,i+1) + & ( gg - cc ) * diss(k+1,j+1,i) + & ( gg - dd ) * diss(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) diss_int(n) = diss_int_l + ( zv(n) - zu(k) ) / dz * & ( diss_int_u - diss_int_l ) ENDIF ! !-- Set flag for stochastic equation. term_1_2(n) = 1.0_wp ENDDO ENDDO ELSE ! non-flat topography, e.g., buildings DO nb = 0, 7 DO n = start_index(nb), end_index(nb) i = particles(n)%x * ddx j = particles(n)%y * ddy k = ( zv(n) + 0.5_wp * dz * atmos_ocean_sign ) / dz & + offset_ocean_nzt ! only exact if eq.dist ! !-- In case that there are buildings it has to be determined !-- how many of the gridpoints defining the particle box are !-- situated within a building !-- gp_outside_of_building(1): i,j,k !-- gp_outside_of_building(2): i,j+1,k !-- gp_outside_of_building(3): i,j,k+1 !-- gp_outside_of_building(4): i,j+1,k+1 !-- gp_outside_of_building(5): i+1,j,k !-- gp_outside_of_building(6): i+1,j+1,k !-- gp_outside_of_building(7): i+1,j,k+1 !-- gp_outside_of_building(8): i+1,j+1,k+1 gp_outside_of_building = 0 location = 0.0_wp num_gp = 0 ! !-- Determine vertical index of topography top at (j,i) k_wall = get_topography_top_index( j, i, 's' ) ! !-- To do: Reconsider order of computations in order to avoid !-- unnecessary index calculations. IF ( k > k_wall .OR. k_wall == 0 ) THEN num_gp = num_gp + 1 gp_outside_of_building(1) = 1 location(num_gp,1) = i * dx location(num_gp,2) = j * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j,i) dissi(num_gp) = diss(k,j,i) de_dxi(num_gp) = de_dx(k,j,i) de_dyi(num_gp) = de_dy(k,j,i) de_dzi(num_gp) = de_dz(k,j,i) ENDIF ! !-- Determine vertical index of topography top at (j+1,i) k_wall = get_topography_top_index( j+1, i, 's' ) IF ( k > k_wall .OR. k_wall == 0 ) THEN num_gp = num_gp + 1 gp_outside_of_building(2) = 1 location(num_gp,1) = i * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j+1,i) dissi(num_gp) = diss(k,j+1,i) de_dxi(num_gp) = de_dx(k,j+1,i) de_dyi(num_gp) = de_dy(k,j+1,i) de_dzi(num_gp) = de_dz(k,j+1,i) ENDIF ! !-- Determine vertical index of topography top at (j,i) k_wall = get_topography_top_index( j, i, 's' ) IF ( k+1 > k_wall .OR. k_wall == 0 ) THEN num_gp = num_gp + 1 gp_outside_of_building(3) = 1 location(num_gp,1) = i * dx location(num_gp,2) = j * dy location(num_gp,3) = (k+1) * dz - 0.5_wp * dz ei(num_gp) = e(k+1,j,i) dissi(num_gp) = diss(k+1,j,i) de_dxi(num_gp) = de_dx(k+1,j,i) de_dyi(num_gp) = de_dy(k+1,j,i) de_dzi(num_gp) = de_dz(k+1,j,i) ENDIF ! !-- Determine vertical index of topography top at (j+1,i) k_wall = get_topography_top_index( j+1, i, 's' ) IF ( k+1 > k_wall .OR. k_wall == 0 ) THEN num_gp = num_gp + 1 gp_outside_of_building(4) = 1 location(num_gp,1) = i * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = (k+1) * dz - 0.5_wp * dz ei(num_gp) = e(k+1,j+1,i) dissi(num_gp) = diss(k+1,j+1,i) de_dxi(num_gp) = de_dx(k+1,j+1,i) de_dyi(num_gp) = de_dy(k+1,j+1,i) de_dzi(num_gp) = de_dz(k+1,j+1,i) ENDIF ! !-- Determine vertical index of topography top at (j,i+1) k_wall = get_topography_top_index( j, i+1, 's' ) IF ( k > k_wall .OR. k_wall == 0 ) THEN num_gp = num_gp + 1 gp_outside_of_building(5) = 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = j * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j,i+1) dissi(num_gp) = diss(k,j,i+1) de_dxi(num_gp) = de_dx(k,j,i+1) de_dyi(num_gp) = de_dy(k,j,i+1) de_dzi(num_gp) = de_dz(k,j,i+1) ENDIF ! !-- Determine vertical index of topography top at (j+1,i+1) k_wall = get_topography_top_index( j+1, i+1, 's' ) IF ( k > k_wall .OR. k_wall == 0 ) THEN num_gp = num_gp + 1 gp_outside_of_building(6) = 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j+1,i+1) dissi(num_gp) = diss(k,j+1,i+1) de_dxi(num_gp) = de_dx(k,j+1,i+1) de_dyi(num_gp) = de_dy(k,j+1,i+1) de_dzi(num_gp) = de_dz(k,j+1,i+1) ENDIF ! !-- Determine vertical index of topography top at (j,i+1) k_wall = get_topography_top_index( j, i+1, 's' ) IF ( k+1 > k_wall .OR. k_wall == 0 ) THEN num_gp = num_gp + 1 gp_outside_of_building(7) = 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = j * dy location(num_gp,3) = (k+1) * dz - 0.5_wp * dz ei(num_gp) = e(k+1,j,i+1) dissi(num_gp) = diss(k+1,j,i+1) de_dxi(num_gp) = de_dx(k+1,j,i+1) de_dyi(num_gp) = de_dy(k+1,j,i+1) de_dzi(num_gp) = de_dz(k+1,j,i+1) ENDIF ! !-- Determine vertical index of topography top at (j+1,i+1) k_wall = get_topography_top_index( j+1, i+1, 's' ) IF ( k+1 > k_wall .OR. k_wall == 0) THEN num_gp = num_gp + 1 gp_outside_of_building(8) = 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = (k+1) * dz - 0.5_wp * dz ei(num_gp) = e(k+1,j+1,i+1) dissi(num_gp) = diss(k+1,j+1,i+1) de_dxi(num_gp) = de_dx(k+1,j+1,i+1) de_dyi(num_gp) = de_dy(k+1,j+1,i+1) de_dzi(num_gp) = de_dz(k+1,j+1,i+1) ENDIF ! !-- If all gridpoints are situated outside of a building, then the !-- ordinary interpolation scheme can be used. IF ( num_gp == 8 ) THEN x = particles(n)%x - i * dx y = particles(n)%y - j * dy 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_int_l = ( ( gg - aa ) * e(k,j,i) + ( gg - bb ) * e(k,j,i+1) & + ( gg - cc ) * e(k,j+1,i) + ( gg - dd ) * e(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( k == nzt ) THEN e_int(n) = e_int_l ELSE e_int_u = ( ( gg - aa ) * e(k+1,j,i) + & ( gg - bb ) * e(k+1,j,i+1) + & ( gg - cc ) * e(k+1,j+1,i) + & ( gg - dd ) * e(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) e_int(n) = e_int_l + ( zv(n) - zu(k) ) / dz * & ( e_int_u - e_int_l ) ENDIF ! !-- Needed to avoid NaN particle velocities (this might not be !-- required any more) IF ( e_int(n) <= 0.0_wp ) THEN e_int(n) = 1.0E-20_wp ENDIF ! !-- Interpolate the TKE gradient along x (adopt incides i,j,k !-- and all position variables from above (TKE)) de_dx_int_l = ( ( gg - aa ) * de_dx(k,j,i) + & ( gg - bb ) * de_dx(k,j,i+1) + & ( gg - cc ) * de_dx(k,j+1,i) + & ( gg - dd ) * de_dx(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( ( k == nzt ) .OR. ( k == nzb ) ) THEN de_dx_int(n) = de_dx_int_l ELSE de_dx_int_u = ( ( gg - aa ) * de_dx(k+1,j,i) + & ( gg - bb ) * de_dx(k+1,j,i+1) + & ( gg - cc ) * de_dx(k+1,j+1,i) + & ( gg - dd ) * de_dx(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) de_dx_int(n) = de_dx_int_l + ( zv(n) - zu(k) ) / & dz * ( de_dx_int_u - de_dx_int_l ) ENDIF ! !-- Interpolate the TKE gradient along y de_dy_int_l = ( ( gg - aa ) * de_dy(k,j,i) + & ( gg - bb ) * de_dy(k,j,i+1) + & ( gg - cc ) * de_dy(k,j+1,i) + & ( gg - dd ) * de_dy(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( ( k+1 == nzt+1 ) .OR. ( k == nzb ) ) THEN de_dy_int(n) = de_dy_int_l ELSE de_dy_int_u = ( ( gg - aa ) * de_dy(k+1,j,i) + & ( gg - bb ) * de_dy(k+1,j,i+1) + & ( gg - cc ) * de_dy(k+1,j+1,i) + & ( gg - dd ) * de_dy(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) de_dy_int(n) = de_dy_int_l + ( zv(n) - zu(k) ) / & dz * ( de_dy_int_u - de_dy_int_l ) ENDIF ! !-- Interpolate the TKE gradient along z IF ( zv(n) < 0.5_wp * dz ) THEN de_dz_int(n) = 0.0_wp ELSE de_dz_int_l = ( ( gg - aa ) * de_dz(k,j,i) + & ( gg - bb ) * de_dz(k,j,i+1) + & ( gg - cc ) * de_dz(k,j+1,i) + & ( gg - dd ) * de_dz(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( ( k+1 == nzt+1 ) .OR. ( k == nzb ) ) THEN de_dz_int(n) = de_dz_int_l ELSE de_dz_int_u = ( ( gg - aa ) * de_dz(k+1,j,i) + & ( gg - bb ) * de_dz(k+1,j,i+1) + & ( gg - cc ) * de_dz(k+1,j+1,i) + & ( gg - dd ) * de_dz(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) de_dz_int(n) = de_dz_int_l + ( zv(n) - zu(k) ) /& dz * ( de_dz_int_u - de_dz_int_l ) ENDIF ENDIF ! !-- Interpolate the dissipation of TKE diss_int_l = ( ( gg - aa ) * diss(k,j,i) + & ( gg - bb ) * diss(k,j,i+1) + & ( gg - cc ) * diss(k,j+1,i) + & ( gg - dd ) * diss(k,j+1,i+1) & ) / ( 3.0_wp * gg ) IF ( k == nzt ) THEN diss_int(n) = diss_int_l ELSE diss_int_u = ( ( gg - aa ) * diss(k+1,j,i) + & ( gg - bb ) * diss(k+1,j,i+1) + & ( gg - cc ) * diss(k+1,j+1,i) + & ( gg - dd ) * diss(k+1,j+1,i+1) & ) / ( 3.0_wp * gg ) diss_int(n) = diss_int_l + ( zv(n) - zu(k) ) / dz *& ( diss_int_u - diss_int_l ) ENDIF ! !-- Set flag for stochastic equation. term_1_2(n) = 1.0_wp ELSE ! !-- If wall between gridpoint 1 and gridpoint 5, then !-- Neumann boundary condition has to be applied IF ( gp_outside_of_building(1) == 1 .AND. & gp_outside_of_building(5) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx + 0.5_wp * dx location(num_gp,2) = j * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j,i) dissi(num_gp) = diss(k,j,i) de_dxi(num_gp) = 0.0_wp de_dyi(num_gp) = de_dy(k,j,i) de_dzi(num_gp) = de_dz(k,j,i) ENDIF IF ( gp_outside_of_building(5) == 1 .AND. & gp_outside_of_building(1) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx + 0.5_wp * dx location(num_gp,2) = j * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j,i+1) dissi(num_gp) = diss(k,j,i+1) de_dxi(num_gp) = 0.0_wp de_dyi(num_gp) = de_dy(k,j,i+1) de_dzi(num_gp) = de_dz(k,j,i+1) ENDIF ! !-- If wall between gridpoint 5 and gridpoint 6, then !-- then Neumann boundary condition has to be applied IF ( gp_outside_of_building(5) == 1 .AND. & gp_outside_of_building(6) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = j * dy + 0.5_wp * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j,i+1) dissi(num_gp) = diss(k,j,i+1) de_dxi(num_gp) = de_dx(k,j,i+1) de_dyi(num_gp) = 0.0_wp de_dzi(num_gp) = de_dz(k,j,i+1) ENDIF IF ( gp_outside_of_building(6) == 1 .AND. & gp_outside_of_building(5) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = j * dy + 0.5_wp * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j+1,i+1) dissi(num_gp) = diss(k,j+1,i+1) de_dxi(num_gp) = de_dx(k,j+1,i+1) de_dyi(num_gp) = 0.0_wp de_dzi(num_gp) = de_dz(k,j+1,i+1) ENDIF ! !-- If wall between gridpoint 2 and gridpoint 6, then !-- Neumann boundary condition has to be applied IF ( gp_outside_of_building(2) == 1 .AND. & gp_outside_of_building(6) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx + 0.5_wp * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j+1,i) dissi(num_gp) = diss(k,j+1,i) de_dxi(num_gp) = 0.0_wp de_dyi(num_gp) = de_dy(k,j+1,i) de_dzi(num_gp) = de_dz(k,j+1,i) ENDIF IF ( gp_outside_of_building(6) == 1 .AND. & gp_outside_of_building(2) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx + 0.5_wp * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j+1,i+1) dissi(num_gp) = diss(k,j+1,i+1) de_dxi(num_gp) = 0.0_wp de_dyi(num_gp) = de_dy(k,j+1,i+1) de_dzi(num_gp) = de_dz(k,j+1,i+1) ENDIF ! !-- If wall between gridpoint 1 and gridpoint 2, then !-- Neumann boundary condition has to be applied IF ( gp_outside_of_building(1) == 1 .AND. & gp_outside_of_building(2) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx location(num_gp,2) = j * dy + 0.5_wp * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j,i) dissi(num_gp) = diss(k,j,i) de_dxi(num_gp) = de_dx(k,j,i) de_dyi(num_gp) = 0.0_wp de_dzi(num_gp) = de_dz(k,j,i) ENDIF IF ( gp_outside_of_building(2) == 1 .AND. & gp_outside_of_building(1) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx location(num_gp,2) = j * dy + 0.5_wp * dy location(num_gp,3) = k * dz - 0.5_wp * dz ei(num_gp) = e(k,j+1,i) dissi(num_gp) = diss(k,j+1,i) de_dxi(num_gp) = de_dx(k,j+1,i) de_dyi(num_gp) = 0.0_wp de_dzi(num_gp) = de_dz(k,j+1,i) ENDIF ! !-- If wall between gridpoint 3 and gridpoint 7, then !-- Neumann boundary condition has to be applied IF ( gp_outside_of_building(3) == 1 .AND. & gp_outside_of_building(7) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx + 0.5_wp * dx location(num_gp,2) = j * dy location(num_gp,3) = k * dz + 0.5_wp * dz ei(num_gp) = e(k+1,j,i) dissi(num_gp) = diss(k+1,j,i) de_dxi(num_gp) = 0.0_wp de_dyi(num_gp) = de_dy(k+1,j,i) de_dzi(num_gp) = de_dz(k+1,j,i) ENDIF IF ( gp_outside_of_building(7) == 1 .AND. & gp_outside_of_building(3) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx + 0.5_wp * dx location(num_gp,2) = j * dy location(num_gp,3) = k * dz + 0.5_wp * dz ei(num_gp) = e(k+1,j,i+1) dissi(num_gp) = diss(k+1,j,i+1) de_dxi(num_gp) = 0.0_wp de_dyi(num_gp) = de_dy(k+1,j,i+1) de_dzi(num_gp) = de_dz(k+1,j,i+1) ENDIF ! !-- If wall between gridpoint 7 and gridpoint 8, then !-- Neumann boundary condition has to be applied IF ( gp_outside_of_building(7) == 1 .AND. & gp_outside_of_building(8) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = j * dy + 0.5_wp * dy location(num_gp,3) = k * dz + 0.5_wp * dz ei(num_gp) = e(k+1,j,i+1) dissi(num_gp) = diss(k+1,j,i+1) de_dxi(num_gp) = de_dx(k+1,j,i+1) de_dyi(num_gp) = 0.0_wp de_dzi(num_gp) = de_dz(k+1,j,i+1) ENDIF IF ( gp_outside_of_building(8) == 1 .AND. & gp_outside_of_building(7) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = j * dy + 0.5_wp * dy location(num_gp,3) = k * dz + 0.5_wp * dz ei(num_gp) = e(k+1,j+1,i+1) dissi(num_gp) = diss(k+1,j+1,i+1) de_dxi(num_gp) = de_dx(k+1,j+1,i+1) de_dyi(num_gp) = 0.0_wp de_dzi(num_gp) = de_dz(k+1,j+1,i+1) ENDIF ! !-- If wall between gridpoint 4 and gridpoint 8, then !-- Neumann boundary condition has to be applied IF ( gp_outside_of_building(4) == 1 .AND. & gp_outside_of_building(8) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx + 0.5_wp * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = k * dz + 0.5_wp * dz ei(num_gp) = e(k+1,j+1,i) dissi(num_gp) = diss(k+1,j+1,i) de_dxi(num_gp) = 0.0_wp de_dyi(num_gp) = de_dy(k+1,j+1,i) de_dzi(num_gp) = de_dz(k+1,j+1,i) ENDIF IF ( gp_outside_of_building(8) == 1 .AND. & gp_outside_of_building(4) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx + 0.5_wp * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = k * dz + 0.5_wp * dz ei(num_gp) = e(k+1,j+1,i+1) dissi(num_gp) = diss(k+1,j+1,i+1) de_dxi(num_gp) = 0.0_wp de_dyi(num_gp) = de_dy(k+1,j+1,i+1) de_dzi(num_gp) = de_dz(k+1,j+1,i+1) ENDIF ! !-- If wall between gridpoint 3 and gridpoint 4, then !-- Neumann boundary condition has to be applied IF ( gp_outside_of_building(3) == 1 .AND. & gp_outside_of_building(4) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx location(num_gp,2) = j * dy + 0.5_wp * dy location(num_gp,3) = k * dz + 0.5_wp * dz ei(num_gp) = e(k+1,j,i) dissi(num_gp) = diss(k+1,j,i) de_dxi(num_gp) = de_dx(k+1,j,i) de_dyi(num_gp) = 0.0_wp de_dzi(num_gp) = de_dz(k+1,j,i) ENDIF IF ( gp_outside_of_building(4) == 1 .AND. & gp_outside_of_building(3) == 0 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx location(num_gp,2) = j * dy + 0.5_wp * dy location(num_gp,3) = k * dz + 0.5_wp * dz ei(num_gp) = e(k+1,j+1,i) dissi(num_gp) = diss(k+1,j+1,i) de_dxi(num_gp) = de_dx(k+1,j+1,i) de_dyi(num_gp) = 0.0_wp de_dzi(num_gp) = de_dz(k+1,j+1,i) ENDIF ! !-- If wall between gridpoint 1 and gridpoint 3, then !-- Neumann boundary condition has to be applied !-- (only one case as only building beneath is possible) IF ( gp_outside_of_building(1) == 0 .AND. & gp_outside_of_building(3) == 1 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx location(num_gp,2) = j * dy location(num_gp,3) = k * dz ei(num_gp) = e(k+1,j,i) dissi(num_gp) = diss(k+1,j,i) de_dxi(num_gp) = de_dx(k+1,j,i) de_dyi(num_gp) = de_dy(k+1,j,i) de_dzi(num_gp) = 0.0_wp ENDIF ! !-- If wall between gridpoint 5 and gridpoint 7, then !-- Neumann boundary condition has to be applied !-- (only one case as only building beneath is possible) IF ( gp_outside_of_building(5) == 0 .AND. & gp_outside_of_building(7) == 1 ) THEN num_gp = num_gp + 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = j * dy location(num_gp,3) = k * dz ei(num_gp) = e(k+1,j,i+1) dissi(num_gp) = diss(k+1,j,i+1) de_dxi(num_gp) = de_dx(k+1,j,i+1) de_dyi(num_gp) = de_dy(k+1,j,i+1) de_dzi(num_gp) = 0.0_wp ENDIF ! !-- If wall between gridpoint 2 and gridpoint 4, then !-- Neumann boundary condition has to be applied !-- (only one case as only building beneath is possible) IF ( gp_outside_of_building(2) == 0 .AND. & gp_outside_of_building(4) == 1 ) THEN num_gp = num_gp + 1 location(num_gp,1) = i * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = k * dz ei(num_gp) = e(k+1,j+1,i) dissi(num_gp) = diss(k+1,j+1,i) de_dxi(num_gp) = de_dx(k+1,j+1,i) de_dyi(num_gp) = de_dy(k+1,j+1,i) de_dzi(num_gp) = 0.0_wp ENDIF ! !-- If wall between gridpoint 6 and gridpoint 8, then !-- Neumann boundary condition has to be applied !-- (only one case as only building beneath is possible) IF ( gp_outside_of_building(6) == 0 .AND. & gp_outside_of_building(8) == 1 ) THEN num_gp = num_gp + 1 location(num_gp,1) = (i+1) * dx location(num_gp,2) = (j+1) * dy location(num_gp,3) = k * dz ei(num_gp) = e(k+1,j+1,i+1) dissi(num_gp) = diss(k+1,j+1,i+1) de_dxi(num_gp) = de_dx(k+1,j+1,i+1) de_dyi(num_gp) = de_dy(k+1,j+1,i+1) de_dzi(num_gp) = 0.0_wp ENDIF ! !-- Carry out the interpolation IF ( num_gp == 1 ) THEN ! !-- If only one of the gridpoints is situated outside of the !-- building, it follows that the values at the particle !-- location are the same as the gridpoint values e_int(n) = ei(num_gp) diss_int(n) = dissi(num_gp) de_dx_int(n) = de_dxi(num_gp) de_dy_int(n) = de_dyi(num_gp) de_dz_int(n) = de_dzi(num_gp) ! !-- Set flag for stochastic equation which disables calculation !-- of drift and memory term near topography. term_1_2(n) = 0.0_wp ELSE IF ( num_gp > 1 ) THEN d_sum = 0.0_wp ! !-- Evaluation of the distances between the gridpoints !-- contributing to the interpolated values, and the particle !-- location DO agp = 1, num_gp d_gp_pl(agp) = ( particles(n)%x-location(agp,1) )**2 & + ( particles(n)%y-location(agp,2) )**2 & + ( zv(n)-location(agp,3) )**2 d_sum = d_sum + d_gp_pl(agp) ENDDO ! !-- Finally the interpolation can be carried out e_int(n) = 0.0_wp diss_int(n) = 0.0_wp de_dx_int(n) = 0.0_wp de_dy_int(n) = 0.0_wp de_dz_int(n) = 0.0_wp DO agp = 1, num_gp e_int(n) = e_int(n) + ( d_sum - d_gp_pl(agp) ) * & ei(agp) / ( (num_gp-1) * d_sum ) diss_int(n) = diss_int(n) + ( d_sum - d_gp_pl(agp) ) * & dissi(agp) / ( (num_gp-1) * d_sum ) de_dx_int(n) = de_dx_int(n) + ( d_sum - d_gp_pl(agp) ) * & de_dxi(agp) / ( (num_gp-1) * d_sum ) de_dy_int(n) = de_dy_int(n) + ( d_sum - d_gp_pl(agp) ) * & de_dyi(agp) / ( (num_gp-1) * d_sum ) de_dz_int(n) = de_dz_int(n) + ( d_sum - d_gp_pl(agp) ) * & de_dzi(agp) / ( (num_gp-1) * d_sum ) ENDDO ENDIF e_int(n) = MAX( 1E-20_wp, e_int(n) ) diss_int(n) = MAX( 1E-20_wp, diss_int(n) ) de_dx_int(n) = MAX( 1E-20_wp, de_dx_int(n) ) de_dy_int(n) = MAX( 1E-20_wp, de_dy_int(n) ) de_dz_int(n) = MAX( 1E-20_wp, de_dz_int(n) ) ! !-- Set flag for stochastic equation which disables calculation !-- of drift and memory term near topography. term_1_2(n) = 0.0_wp ENDIF ENDDO ENDDO ENDIF DO nb = 0,7 i = ip + block_offset(nb)%i_off j = jp + block_offset(nb)%j_off k = kp + block_offset(nb)%k_off DO n = start_index(nb), end_index(nb) ! !-- Vertical interpolation of the horizontally averaged SGS TKE and !-- resolved-scale velocity variances and use the interpolated values !-- to calculate the coefficient fs, which is a measure of the ratio !-- of the subgrid-scale turbulent kinetic energy to the total amount !-- of turbulent kinetic energy. IF ( k == 0 ) THEN e_mean_int = hom(0,1,8,0) ELSE e_mean_int = hom(k,1,8,0) + & ( hom(k+1,1,8,0) - hom(k,1,8,0) ) / & ( zu(k+1) - zu(k) ) * & ( zv(n) - zu(k) ) ENDIF kw = kp - 1 IF ( k == 0 ) THEN aa = hom(k+1,1,30,0) * ( zv(n) / & ( 0.5_wp * ( zu(k+1) - zu(k) ) ) ) bb = hom(k+1,1,31,0) * ( zv(n) / & ( 0.5_wp * ( zu(k+1) - zu(k) ) ) ) cc = hom(kw+1,1,32,0) * ( zv(n) / & ( 1.0_wp * ( zw(kw+1) - zw(kw) ) ) ) ELSE aa = hom(k,1,30,0) + ( hom(k+1,1,30,0) - hom(k,1,30,0) ) * & ( ( zv(n) - zu(k) ) / ( zu(k+1) - zu(k) ) ) bb = hom(k,1,31,0) + ( hom(k+1,1,31,0) - hom(k,1,31,0) ) * & ( ( zv(n) - zu(k) ) / ( zu(k+1) - zu(k) ) ) cc = hom(kw,1,32,0) + ( hom(kw+1,1,32,0)-hom(kw,1,32,0) ) * & ( ( zv(n) - zw(kw) ) / ( zw(kw+1)-zw(kw) ) ) ENDIF vv_int = ( 1.0_wp / 3.0_wp ) * ( aa + bb + cc ) ! !-- Needed to avoid NaN particle velocities. The value of 1.0 is just !-- an educated guess for the given case. IF ( vv_int + ( 2.0_wp / 3.0_wp ) * e_mean_int == 0.0_wp ) THEN fs_int(n) = 1.0_wp ELSE fs_int(n) = ( 2.0_wp / 3.0_wp ) * e_mean_int / & ( vv_int + ( 2.0_wp / 3.0_wp ) * e_mean_int ) ENDIF ENDDO ENDDO DO nb = 0, 7 DO n = start_index(nb), end_index(nb) rg(n,1) = random_gauss( iran_part, 5.0_wp ) rg(n,2) = random_gauss( iran_part, 5.0_wp ) rg(n,3) = random_gauss( iran_part, 5.0_wp ) ENDDO ENDDO DO nb = 0, 7 DO n = start_index(nb), end_index(nb) ! !-- Calculate the Lagrangian timescale according to Weil et al. (2004). lagr_timescale = ( 4.0_wp * e_int(n) + 1E-20_wp ) / & ( 3.0_wp * fs_int(n) * c_0 * diss_int(n) + 1E-20_wp ) ! !-- Calculate the next particle timestep. dt_gap is the time needed to !-- complete the current LES timestep. dt_gap = dt_3d - particles(n)%dt_sum dt_particle(n) = MIN( dt_3d, 0.025_wp * lagr_timescale, dt_gap ) ! !-- The particle timestep should not be too small in order to prevent !-- the number of particle timesteps of getting too large IF ( dt_particle(n) < dt_min_part .AND. dt_min_part < dt_gap ) THEN dt_particle(n) = dt_min_part ENDIF ! !-- Calculate the SGS velocity components IF ( particles(n)%age == 0.0_wp ) THEN ! !-- For new particles the SGS components are derived from the SGS !-- TKE. Limit the Gaussian random number to the interval !-- [-5.0*sigma, 5.0*sigma] in order to prevent the SGS velocities !-- from becoming unrealistically large. particles(n)%rvar1 = SQRT( 2.0_wp * sgs_wf_part * e_int(n) & + 1E-20_wp ) * & ( rg(n,1) - 1.0_wp ) particles(n)%rvar2 = SQRT( 2.0_wp * sgs_wf_part * e_int(n) & + 1E-20_wp ) * & ( rg(n,2) - 1.0_wp ) particles(n)%rvar3 = SQRT( 2.0_wp * sgs_wf_part * e_int(n) & + 1E-20_wp ) * & ( rg(n,3) - 1.0_wp ) ELSE ! !-- Restriction of the size of the new timestep: compared to the !-- previous timestep the increase must not exceed 200%. First, !-- check if age > age_m, in order to prevent that particles get zero !-- timestep. dt_particle_m = MERGE( dt_particle(n), & particles(n)%age - particles(n)%age_m, & particles(n)%age - particles(n)%age_m < & 1E-8_wp ) IF ( dt_particle(n) > 2.0_wp * dt_particle_m ) THEN dt_particle(n) = 2.0_wp * dt_particle_m ENDIF ! !-- For old particles the SGS components are correlated with the !-- values from the previous timestep. Random numbers have also to !-- be limited (see above). !-- As negative values for the subgrid TKE are not allowed, the !-- change of the subgrid TKE with time cannot be smaller than !-- -e_int(n)/dt_particle. This value is used as a lower boundary !-- value for the change of TKE de_dt_min = - e_int(n) / dt_particle(n) de_dt = ( e_int(n) - particles(n)%e_m ) / dt_particle_m IF ( de_dt < de_dt_min ) THEN de_dt = de_dt_min ENDIF CALL weil_stochastic_eq(particles(n)%rvar1, fs_int(n), e_int(n),& de_dx_int(n), de_dt, diss_int(n), & dt_particle(n), rg(n,1), term_1_2(n) ) CALL weil_stochastic_eq(particles(n)%rvar2, fs_int(n), e_int(n),& de_dy_int(n), de_dt, diss_int(n), & dt_particle(n), rg(n,2), term_1_2(n) ) CALL weil_stochastic_eq(particles(n)%rvar3, fs_int(n), e_int(n),& de_dz_int(n), de_dt, diss_int(n), & dt_particle(n), rg(n,3), term_1_2(n) ) ENDIF u_int(n) = u_int(n) + particles(n)%rvar1 v_int(n) = v_int(n) + particles(n)%rvar2 w_int(n) = w_int(n) + particles(n)%rvar3 ! !-- Store the SGS TKE of the current timelevel which is needed for !-- for calculating the SGS particle velocities at the next timestep particles(n)%e_m = e_int(n) ENDDO ENDDO ELSE ! !-- If no SGS velocities are used, only the particle timestep has to !-- be set dt_particle = dt_3d ENDIF dens_ratio = particle_groups(particles(1:number_of_particles)%group)%density_ratio IF ( ANY( dens_ratio == 0.0_wp ) ) THEN DO nb = 0, 7 DO n = start_index(nb), end_index(nb) ! !-- Particle advection IF ( dens_ratio(n) == 0.0_wp ) THEN ! !-- Pure passive transport (without particle inertia) particles(n)%x = xv(n) + u_int(n) * dt_particle(n) particles(n)%y = yv(n) + v_int(n) * dt_particle(n) particles(n)%z = zv(n) + w_int(n) * dt_particle(n) particles(n)%speed_x = u_int(n) particles(n)%speed_y = v_int(n) particles(n)%speed_z = w_int(n) ELSE ! !-- Transport of particles with inertia particles(n)%x = particles(n)%x + particles(n)%speed_x * & dt_particle(n) particles(n)%y = particles(n)%y + particles(n)%speed_y * & dt_particle(n) particles(n)%z = particles(n)%z + particles(n)%speed_z * & dt_particle(n) ! !-- Update of the particle velocity IF ( cloud_droplets ) THEN ! !-- Terminal velocity is computed for vertical direction (Rogers et !-- al., 1993, J. Appl. Meteorol.) diameter = particles(n)%radius * 2000.0_wp !diameter in mm IF ( diameter <= d0_rog ) THEN w_s = k_cap_rog * diameter * ( 1.0_wp - EXP( -k_low_rog * diameter ) ) ELSE w_s = a_rog - b_rog * EXP( -c_rog * diameter ) ENDIF ! !-- If selected, add random velocities following Soelch and Kaercher !-- (2010, Q. J. R. Meteorol. Soc.) IF ( use_sgs_for_particles ) THEN lagr_timescale = km(kp,jp,ip) / MAX( e(kp,jp,ip), 1.0E-20_wp ) RL = EXP( -1.0_wp * dt_3d / lagr_timescale ) sigma = SQRT( e(kp,jp,ip) ) rg1 = random_gauss( iran_part, 5.0_wp ) - 1.0_wp rg2 = random_gauss( iran_part, 5.0_wp ) - 1.0_wp rg3 = random_gauss( iran_part, 5.0_wp ) - 1.0_wp particles(n)%rvar1 = RL * particles(n)%rvar1 + & SQRT( 1.0_wp - RL**2 ) * sigma * rg1 particles(n)%rvar2 = RL * particles(n)%rvar2 + & SQRT( 1.0_wp - RL**2 ) * sigma * rg2 particles(n)%rvar3 = RL * particles(n)%rvar3 + & SQRT( 1.0_wp - RL**2 ) * sigma * rg3 particles(n)%speed_x = u_int(n) + particles(n)%rvar1 particles(n)%speed_y = v_int(n) + particles(n)%rvar2 particles(n)%speed_z = w_int(n) + particles(n)%rvar3 - w_s ELSE particles(n)%speed_x = u_int(n) particles(n)%speed_y = v_int(n) particles(n)%speed_z = w_int(n) - w_s ENDIF ELSE IF ( use_sgs_for_particles ) THEN exp_arg = particle_groups(particles(n)%group)%exp_arg exp_term = EXP( -exp_arg * dt_particle(n) ) ELSE exp_arg = particle_groups(particles(n)%group)%exp_arg exp_term = particle_groups(particles(n)%group)%exp_term ENDIF particles(n)%speed_x = particles(n)%speed_x * exp_term + & u_int(n) * ( 1.0_wp - exp_term ) particles(n)%speed_y = particles(n)%speed_y * exp_term + & v_int(n) * ( 1.0_wp - exp_term ) particles(n)%speed_z = particles(n)%speed_z * exp_term + & ( w_int(n) - ( 1.0_wp - dens_ratio(n) ) * & g / exp_arg ) * ( 1.0_wp - exp_term ) ENDIF ENDIF ENDDO ENDDO ELSE DO nb = 0, 7 DO n = start_index(nb), end_index(nb) ! !-- Transport of particles with inertia particles(n)%x = xv(n) + particles(n)%speed_x * dt_particle(n) particles(n)%y = yv(n) + particles(n)%speed_y * dt_particle(n) particles(n)%z = zv(n) + particles(n)%speed_z * dt_particle(n) ! !-- Update of the particle velocity IF ( cloud_droplets ) THEN ! !-- Terminal velocity is computed for vertical direction (Rogers et al., !-- 1993, J. Appl. Meteorol.) diameter = particles(n)%radius * 2000.0_wp !diameter in mm IF ( diameter <= d0_rog ) THEN w_s = k_cap_rog * diameter * ( 1.0_wp - EXP( -k_low_rog * diameter ) ) ELSE w_s = a_rog - b_rog * EXP( -c_rog * diameter ) ENDIF ! !-- If selected, add random velocities following Soelch and Kaercher !-- (2010, Q. J. R. Meteorol. Soc.) IF ( use_sgs_for_particles ) THEN lagr_timescale = km(kp,jp,ip) / MAX( e(kp,jp,ip), 1.0E-20_wp ) RL = EXP( -1.0_wp * dt_3d / lagr_timescale ) sigma = SQRT( e(kp,jp,ip) ) rg1 = random_gauss( iran_part, 5.0_wp ) - 1.0_wp rg2 = random_gauss( iran_part, 5.0_wp ) - 1.0_wp rg3 = random_gauss( iran_part, 5.0_wp ) - 1.0_wp particles(n)%rvar1 = RL * particles(n)%rvar1 + & SQRT( 1.0_wp - RL**2 ) * sigma * rg1 particles(n)%rvar2 = RL * particles(n)%rvar2 + & SQRT( 1.0_wp - RL**2 ) * sigma * rg2 particles(n)%rvar3 = RL * particles(n)%rvar3 + & SQRT( 1.0_wp - RL**2 ) * sigma * rg3 particles(n)%speed_x = u_int(n) + particles(n)%rvar1 particles(n)%speed_y = v_int(n) + particles(n)%rvar2 particles(n)%speed_z = w_int(n) + particles(n)%rvar3 - w_s ELSE particles(n)%speed_x = u_int(n) particles(n)%speed_y = v_int(n) particles(n)%speed_z = w_int(n) - w_s ENDIF ELSE IF ( use_sgs_for_particles ) THEN exp_arg = particle_groups(particles(n)%group)%exp_arg exp_term = EXP( -exp_arg * dt_particle(n) ) ELSE exp_arg = particle_groups(particles(n)%group)%exp_arg exp_term = particle_groups(particles(n)%group)%exp_term ENDIF particles(n)%speed_x = particles(n)%speed_x * exp_term + & u_int(n) * ( 1.0_wp - exp_term ) particles(n)%speed_y = particles(n)%speed_y * exp_term + & v_int(n) * ( 1.0_wp - exp_term ) particles(n)%speed_z = particles(n)%speed_z * exp_term + & ( w_int(n) - ( 1.0_wp - dens_ratio(n) ) * g / & exp_arg ) * ( 1.0_wp - exp_term ) ENDIF ENDDO ENDDO ENDIF ! !-- Store the old age of the particle ( needed to prevent that a !-- particle crosses several PEs during one timestep, and for the !-- evaluation of the subgrid particle velocity fluctuations ) particles(1:number_of_particles)%age_m = particles(1:number_of_particles)%age DO nb = 0, 7 DO n = start_index(nb), end_index(nb) ! !-- Increment the particle age and the total time that the particle !-- has advanced within the particle timestep procedure particles(n)%age = particles(n)%age + dt_particle(n) particles(n)%dt_sum = particles(n)%dt_sum + dt_particle(n) ! !-- Check whether there is still a particle that has not yet completed !-- the total LES timestep IF ( ( dt_3d - particles(n)%dt_sum ) > 1E-8_wp ) THEN dt_3d_reached_l = .FALSE. ENDIF ENDDO ENDDO CALL cpu_log( log_point_s(44), 'lpm_advec', 'pause' ) END SUBROUTINE lpm_advec ! Description: ! ------------ !> Calculation of subgrid-scale particle speed using the stochastic model !> of Weil et al. (2004, JAS, 61, 2877-2887). !------------------------------------------------------------------------------! SUBROUTINE weil_stochastic_eq( v_sgs, fs_n, e_n, dedxi_n, dedt_n, diss_n, & dt_n, rg_n, fac ) USE kinds USE particle_attributes, & ONLY: c_0, sgs_wf_part IMPLICIT NONE REAL(wp) :: a1 !< dummy argument REAL(wp) :: dedt_n !< time derivative of TKE at particle position REAL(wp) :: dedxi_n !< horizontal derivative of TKE at particle position REAL(wp) :: diss_n !< dissipation at particle position REAL(wp) :: dt_n !< particle timestep REAL(wp) :: e_n !< TKE at particle position REAL(wp) :: fac !< flag to identify adjacent topography REAL(wp) :: fs_n !< weighting factor to prevent that subgrid-scale particle speed becomes too large REAL(wp) :: sgs_w !< constant (1/3) REAL(wp) :: rg_n !< random number REAL(wp) :: term1 !< memory term REAL(wp) :: term2 !< drift correction term REAL(wp) :: term3 !< random term REAL(wp) :: v_sgs !< subgrid-scale velocity component !-- At first, limit TKE to a small non-zero number, in order to prevent !-- the occurrence of extremely large SGS-velocities in case TKE is zero, !-- (could occur at the simulation begin). e_n = MAX( e_n, 1E-20_wp ) ! !-- Please note, terms 1 and 2 (drift and memory term, respectively) are !-- multiplied by a flag to switch of both terms near topography. !-- This is necessary, as both terms may cause a subgrid-scale velocity build up !-- if particles are trapped in regions with very small TKE, e.g. in narrow street !-- canyons resolved by only a few grid points. Hence, term 1 and term 2 are !-- disabled if one of the adjacent grid points belongs to topography. !-- Moreover, in this case, the previous subgrid-scale component is also set !-- to zero. a1 = fs_n * c_0 * diss_n ! !-- Memory term term1 = - a1 * v_sgs * dt_n / ( 4.0_wp * sgs_wf_part * e_n + 1E-20_wp ) & * fac ! !-- Drift correction term term2 = ( ( dedt_n * v_sgs / e_n ) + dedxi_n ) * 0.5_wp * dt_n & * fac ! !-- Random term term3 = SQRT( MAX( a1, 1E-20 ) ) * ( rg_n - 1.0_wp ) * SQRT( dt_n ) ! !-- In cese one of the adjacent grid-boxes belongs to topograhy, the previous !-- subgrid-scale velocity component is set to zero, in order to prevent a !-- velocity build-up. !-- This case, set also previous subgrid-scale component to zero. v_sgs = v_sgs * fac + term1 + term2 + term3 END SUBROUTINE weil_stochastic_eq