| 1 | SUBROUTINE lpm_droplet_condensation |
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| 2 | |
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| 3 | !------------------------------------------------------------------------------! |
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| 4 | ! Current revisions: |
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| 5 | ! ------------------ |
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| 6 | ! |
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| 7 | ! |
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| 8 | ! Former revisions: |
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| 9 | ! ----------------- |
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| 10 | ! $Id: lpm_droplet_condensation.f90 850 2012-03-15 12:09:25Z raasch $ |
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| 11 | ! |
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| 12 | ! 849 2012-03-15 10:35:09Z raasch |
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| 13 | ! initial revision (former part of advec_particles) |
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| 14 | ! |
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| 15 | ! |
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| 16 | ! Description: |
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| 17 | ! ------------ |
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| 18 | ! Calculates change in droplet radius by condensation/evaporation, using |
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| 19 | ! either an analytic formula or by numerically integrating the radius growth |
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| 20 | ! equation including curvature and solution effects using Rosenbrocks method |
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| 21 | ! (see Numerical recipes in FORTRAN, 2nd edition, p. 731). |
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| 22 | ! The analytical formula and growth equation follow those given in |
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| 23 | ! Rogers and Yau (A short course in cloud physics, 3rd edition, p. 102/103). |
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| 24 | !------------------------------------------------------------------------------! |
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| 25 | |
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| 26 | USE arrays_3d |
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| 27 | USE cloud_parameters |
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| 28 | USE constants |
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| 29 | USE control_parameters |
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| 30 | USE cpulog |
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| 31 | USE grid_variables |
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| 32 | USE interfaces |
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| 33 | USE lpm_collision_kernels_mod |
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| 34 | USE particle_attributes |
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| 35 | |
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| 36 | IMPLICIT NONE |
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| 37 | |
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| 38 | INTEGER :: i, internal_timestep_count, j, jtry, k, n |
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| 39 | |
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| 40 | INTEGER, PARAMETER :: maxtry = 40 |
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| 41 | |
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| 42 | REAL :: aa, afactor, arg, bb, cc, dd, ddenom, delta_r, drdt, drdt_ini, & |
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| 43 | drdt_m, dt_ros, dt_ros_last, dt_ros_next, dt_ros_sum, & |
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| 44 | dt_ros_sum_ini, d2rdtdr, d2rdt2, errmax, err_ros, g1, g2, g3, g4, & |
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| 45 | e_a, e_s, gg, new_r, p_int, pt_int, pt_int_l, pt_int_u, q_int, & |
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| 46 | q_int_l, q_int_u, ql_int, ql_int_l, ql_int_u, r_ros, r_ros_ini, & |
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| 47 | sigma, t_int, x, y |
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| 48 | |
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| 49 | ! |
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| 50 | !-- Parameters for Rosenbrock method |
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| 51 | REAL, PARAMETER :: a21 = 2.0, a31 = 48.0/25.0, a32 = 6.0/25.0, & |
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| 52 | a2x = 1.0, a3x = 3.0/5.0, b1 = 19.0/9.0, b2 = 0.5, & |
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| 53 | b3 = 25.0/108.0, b4 = 125.0/108.0, c21 = -8.0, & |
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| 54 | c31 = 372.0/25.0, c32 = 12.0/5.0, & |
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| 55 | c41 = -112.0/125.0, c42 = -54.0/125.0, & |
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| 56 | c43 = -2.0/5.0, c1x = 0.5, c2x= -3.0/2.0, & |
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| 57 | c3x = 121.0/50.0, c4x = 29.0/250.0, & |
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| 58 | errcon = 0.1296, e1 = 17.0/54.0, e2 = 7.0/36.0, & |
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| 59 | e3 = 0.0, e4 = 125.0/108.0, gam = 0.5, grow = 1.5, & |
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| 60 | pgrow = -0.25, pshrnk = -1.0/3.0, shrnk = 0.5 |
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| 61 | |
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| 62 | |
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| 63 | CALL cpu_log( log_point_s(42), 'lpm_droplet_condens', 'start' ) |
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| 64 | |
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| 65 | DO n = 1, number_of_particles |
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| 66 | ! |
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| 67 | !-- Interpolate temperature and humidity. |
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| 68 | !-- First determine left, south, and bottom index of the arrays. |
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| 69 | i = particles(n)%x * ddx |
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| 70 | j = particles(n)%y * ddy |
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| 71 | k = ( particles(n)%z + 0.5 * dz * atmos_ocean_sign ) / dz & |
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| 72 | + offset_ocean_nzt ! only exact if equidistant |
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| 73 | |
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| 74 | x = particles(n)%x - i * dx |
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| 75 | y = particles(n)%y - j * dy |
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| 76 | aa = x**2 + y**2 |
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| 77 | bb = ( dx - x )**2 + y**2 |
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| 78 | cc = x**2 + ( dy - y )**2 |
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| 79 | dd = ( dx - x )**2 + ( dy - y )**2 |
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| 80 | gg = aa + bb + cc + dd |
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| 81 | |
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| 82 | pt_int_l = ( ( gg - aa ) * pt(k,j,i) + ( gg - bb ) * pt(k,j,i+1) & |
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| 83 | + ( gg - cc ) * pt(k,j+1,i) + ( gg - dd ) * pt(k,j+1,i+1) & |
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| 84 | ) / ( 3.0 * gg ) |
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| 85 | |
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| 86 | pt_int_u = ( ( gg-aa ) * pt(k+1,j,i) + ( gg-bb ) * pt(k+1,j,i+1) & |
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| 87 | + ( gg-cc ) * pt(k+1,j+1,i) + ( gg-dd ) * pt(k+1,j+1,i+1) & |
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| 88 | ) / ( 3.0 * gg ) |
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| 89 | |
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| 90 | pt_int = pt_int_l + ( particles(n)%z - zu(k) ) / dz * & |
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| 91 | ( pt_int_u - pt_int_l ) |
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| 92 | |
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| 93 | q_int_l = ( ( gg - aa ) * q(k,j,i) + ( gg - bb ) * q(k,j,i+1) & |
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| 94 | + ( gg - cc ) * q(k,j+1,i) + ( gg - dd ) * q(k,j+1,i+1) & |
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| 95 | ) / ( 3.0 * gg ) |
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| 96 | |
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| 97 | q_int_u = ( ( gg-aa ) * q(k+1,j,i) + ( gg-bb ) * q(k+1,j,i+1) & |
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| 98 | + ( gg-cc ) * q(k+1,j+1,i) + ( gg-dd ) * q(k+1,j+1,i+1) & |
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| 99 | ) / ( 3.0 * gg ) |
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| 100 | |
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| 101 | q_int = q_int_l + ( particles(n)%z - zu(k) ) / dz * & |
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| 102 | ( q_int_u - q_int_l ) |
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| 103 | |
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| 104 | ql_int_l = ( ( gg - aa ) * ql(k,j,i) + ( gg - bb ) * ql(k,j,i+1) & |
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| 105 | + ( gg - cc ) * ql(k,j+1,i) + ( gg - dd ) * ql(k,j+1,i+1) & |
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| 106 | ) / ( 3.0 * gg ) |
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| 107 | |
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| 108 | ql_int_u = ( ( gg-aa ) * ql(k+1,j,i) + ( gg-bb ) * ql(k+1,j,i+1) & |
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| 109 | + ( gg-cc ) * ql(k+1,j+1,i) + ( gg-dd ) * ql(k+1,j+1,i+1) & |
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| 110 | ) / ( 3.0 * gg ) |
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| 111 | |
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| 112 | ql_int = ql_int_l + ( particles(n)%z - zu(k) ) / dz * & |
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| 113 | ( ql_int_u - ql_int_l ) |
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| 114 | |
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| 115 | ! |
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| 116 | !-- Calculate real temperature and saturation vapor pressure |
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| 117 | p_int = hyp(k) + ( particles(n)%z - zu(k) ) / dz * ( hyp(k+1)-hyp(k) ) |
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| 118 | t_int = pt_int * ( p_int / 100000.0 )**0.286 |
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| 119 | |
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| 120 | e_s = 611.0 * EXP( l_d_rv * ( 3.6609E-3 - 1.0 / t_int ) ) |
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| 121 | |
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| 122 | ! |
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| 123 | !-- Current vapor pressure |
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| 124 | e_a = q_int * p_int / ( 0.378 * q_int + 0.622 ) |
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| 125 | |
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| 126 | ! |
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| 127 | !-- Thermal conductivity for water (from Rogers and Yau, Table 7.1), |
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| 128 | !-- diffusivity for water vapor (after Hall und Pruppacher, 1976) |
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| 129 | thermal_conductivity_l = 7.94048E-05 * t_int + 0.00227011 |
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| 130 | diff_coeff_l = 0.211E-4 * ( t_int / 273.15 )**1.94 * & |
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| 131 | ( 101325.0 / p_int) |
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| 132 | ! |
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| 133 | !-- Change in radius by condensation/evaporation |
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| 134 | IF ( particles(n)%radius >= 1.0E-6 .OR. & |
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| 135 | .NOT. curvature_solution_effects ) THEN |
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| 136 | ! |
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| 137 | !-- Approximation for large radii, where curvature and solution |
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| 138 | !-- effects can be neglected |
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| 139 | arg = particles(n)%radius**2 + 2.0 * dt_3d * & |
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| 140 | ( e_a / e_s - 1.0 ) / & |
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| 141 | ( ( l_d_rv / t_int - 1.0 ) * l_v * rho_l / t_int / & |
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| 142 | thermal_conductivity_l + & |
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| 143 | rho_l * r_v * t_int / diff_coeff_l / e_s ) |
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| 144 | IF ( arg < 1.0E-16 ) THEN |
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| 145 | new_r = 1.0E-8 |
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| 146 | ELSE |
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| 147 | new_r = SQRT( arg ) |
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| 148 | ENDIF |
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| 149 | ENDIF |
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| 150 | |
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| 151 | IF ( curvature_solution_effects .AND. & |
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| 152 | ( ( particles(n)%radius < 1.0E-6 ) .OR. ( new_r < 1.0E-6 ) ) ) & |
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| 153 | THEN |
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| 154 | ! |
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| 155 | !-- Curvature and solutions effects are included in growth equation. |
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| 156 | !-- Change in Radius is calculated with 4th-order Rosenbrock method |
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| 157 | !-- for stiff o.d.e's with monitoring local truncation error to adjust |
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| 158 | !-- stepsize (see Numerical recipes in FORTRAN, 2nd edition, p. 731). |
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| 159 | !-- For larger radii the simple analytic method (see ELSE) gives |
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| 160 | !-- almost the same results. |
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| 161 | ! |
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| 162 | !-- Surface tension after (Straka, 2009) |
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| 163 | sigma = 0.0761 - 0.00155 * ( t_int - 273.15 ) |
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| 164 | |
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| 165 | r_ros = particles(n)%radius |
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| 166 | dt_ros_sum = 0.0 ! internal integrated time (s) |
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| 167 | internal_timestep_count = 0 |
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| 168 | |
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| 169 | ddenom = 1.0 / ( rho_l * r_v * t_int / ( e_s * diff_coeff_l ) + & |
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| 170 | ( l_v / ( r_v * t_int ) - 1.0 ) * & |
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| 171 | rho_l * l_v / ( thermal_conductivity_l * t_int )& |
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| 172 | ) |
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| 173 | |
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| 174 | afactor = 2.0 * sigma / ( rho_l * r_v * t_int ) |
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| 175 | |
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| 176 | IF ( particles(n)%rvar3 == -9999999.9 ) THEN |
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| 177 | ! |
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| 178 | !-- First particle timestep. Derivative has to be calculated. |
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| 179 | drdt_m = ddenom / r_ros * ( e_a / e_s - 1.0 - & |
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| 180 | afactor / r_ros + & |
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| 181 | bfactor / r_ros**3 ) |
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| 182 | ELSE |
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| 183 | ! |
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| 184 | !-- Take value from last PALM timestep |
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| 185 | drdt_m = particles(n)%rvar3 |
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| 186 | ENDIF |
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| 187 | ! |
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| 188 | !-- Take internal timestep values from the end of last PALM timestep |
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| 189 | dt_ros_last = particles(n)%rvar1 |
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| 190 | dt_ros_next = particles(n)%rvar2 |
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| 191 | ! |
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| 192 | !-- Internal timestep must not be larger than PALM timestep |
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| 193 | dt_ros = MIN( dt_ros_next, dt_3d ) |
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| 194 | ! |
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| 195 | !-- Integrate growth equation in time unless PALM timestep is reached |
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| 196 | DO WHILE ( dt_ros_sum < dt_3d ) |
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| 197 | |
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| 198 | internal_timestep_count = internal_timestep_count + 1 |
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| 199 | |
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| 200 | ! |
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| 201 | !-- Derivative at starting value |
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| 202 | drdt = ddenom / r_ros * ( e_a / e_s - 1.0 - afactor / r_ros + & |
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| 203 | bfactor / r_ros**3 ) |
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| 204 | drdt_ini = drdt |
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| 205 | dt_ros_sum_ini = dt_ros_sum |
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| 206 | r_ros_ini = r_ros |
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| 207 | |
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| 208 | ! |
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| 209 | !-- Calculate time derivative of dr/dt |
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| 210 | d2rdt2 = ( drdt - drdt_m ) / dt_ros_last |
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| 211 | |
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| 212 | ! |
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| 213 | !-- Calculate radial derivative of dr/dt |
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| 214 | d2rdtdr = ddenom * ( ( 1.0 - e_a/e_s ) / r_ros**2 + & |
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| 215 | 2.0 * afactor / r_ros**3 - & |
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| 216 | 4.0 * bfactor / r_ros**5 ) |
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| 217 | ! |
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| 218 | !-- Adjust stepsize unless required accuracy is reached |
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| 219 | DO jtry = 1, maxtry+1 |
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| 220 | |
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| 221 | IF ( jtry == maxtry+1 ) THEN |
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| 222 | message_string = 'maxtry > 40 in Rosenbrock method' |
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| 223 | CALL message( 'lpm_droplet_condensation', 'PA0347', 2, 2, & |
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| 224 | 0, 6, 0 ) |
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| 225 | ENDIF |
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| 226 | |
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| 227 | aa = 1.0 / ( gam * dt_ros ) - d2rdtdr |
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| 228 | g1 = ( drdt_ini + dt_ros * c1x * d2rdt2 ) / aa |
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| 229 | r_ros = r_ros_ini + a21 * g1 |
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| 230 | drdt = ddenom / r_ros * ( e_a / e_s - 1.0 - & |
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| 231 | afactor / r_ros + & |
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| 232 | bfactor / r_ros**3 ) |
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| 233 | |
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| 234 | g2 = ( drdt + dt_ros * c2x * d2rdt2 + c21 * g1 / dt_ros )& |
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| 235 | / aa |
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| 236 | r_ros = r_ros_ini + a31 * g1 + a32 * g2 |
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| 237 | drdt = ddenom / r_ros * ( e_a / e_s - 1.0 - & |
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| 238 | afactor / r_ros + & |
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| 239 | bfactor / r_ros**3 ) |
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| 240 | |
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| 241 | g3 = ( drdt + dt_ros * c3x * d2rdt2 + & |
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| 242 | ( c31 * g1 + c32 * g2 ) / dt_ros ) / aa |
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| 243 | g4 = ( drdt + dt_ros * c4x * d2rdt2 + & |
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| 244 | ( c41 * g1 + c42 * g2 + c43 * g3 ) / dt_ros ) / aa |
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| 245 | r_ros = r_ros_ini + b1 * g1 + b2 * g2 + b3 * g3 + b4 * g4 |
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| 246 | |
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| 247 | dt_ros_sum = dt_ros_sum_ini + dt_ros |
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| 248 | |
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| 249 | IF ( dt_ros_sum == dt_ros_sum_ini ) THEN |
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| 250 | message_string = 'zero stepsize in Rosenbrock method' |
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| 251 | CALL message( 'lpm_droplet_condensation', 'PA0348', 2, 2, & |
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| 252 | 0, 6, 0 ) |
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| 253 | ENDIF |
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| 254 | ! |
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| 255 | !-- Calculate error |
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| 256 | err_ros = e1*g1 + e2*g2 + e3*g3 + e4*g4 |
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| 257 | errmax = 0.0 |
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| 258 | errmax = MAX( errmax, ABS( err_ros / r_ros_ini ) ) / eps_ros |
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| 259 | ! |
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| 260 | !-- Leave loop if accuracy is sufficient, otherwise try again |
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| 261 | !-- with a reduced stepsize |
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| 262 | IF ( errmax <= 1.0 ) THEN |
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| 263 | EXIT |
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| 264 | ELSE |
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| 265 | dt_ros = SIGN( MAX( ABS( 0.9 * dt_ros * errmax**pshrnk ), & |
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| 266 | shrnk * ABS( dt_ros ) ), dt_ros ) |
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| 267 | ENDIF |
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| 268 | |
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| 269 | ENDDO ! loop for stepsize adjustment |
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| 270 | |
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| 271 | ! |
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| 272 | !-- Calculate next internal timestep |
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| 273 | IF ( errmax > errcon ) THEN |
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| 274 | dt_ros_next = 0.9 * dt_ros * errmax**pgrow |
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| 275 | ELSE |
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| 276 | dt_ros_next = grow * dt_ros |
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| 277 | ENDIF |
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| 278 | |
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| 279 | ! |
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| 280 | !-- Estimated timestep is reduced if the PALM time step is exceeded |
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| 281 | dt_ros_last = dt_ros |
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| 282 | IF ( ( dt_ros_next + dt_ros_sum ) >= dt_3d ) THEN |
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| 283 | dt_ros = dt_3d - dt_ros_sum |
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| 284 | ELSE |
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| 285 | dt_ros = dt_ros_next |
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| 286 | ENDIF |
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| 287 | |
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| 288 | drdt_m = drdt |
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| 289 | |
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| 290 | ENDDO |
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| 291 | ! |
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| 292 | !-- Store derivative and internal timestep values for next PALM step |
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| 293 | particles(n)%rvar1 = dt_ros_last |
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| 294 | particles(n)%rvar2 = dt_ros_next |
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| 295 | particles(n)%rvar3 = drdt_m |
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| 296 | |
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| 297 | new_r = r_ros |
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| 298 | ! |
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| 299 | !-- Radius should not fall below 1E-8 because Rosenbrock method may |
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| 300 | !-- lead to errors otherwise |
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| 301 | new_r = MAX( new_r, 1.0E-8 ) |
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| 302 | |
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| 303 | ENDIF |
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| 304 | |
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| 305 | delta_r = new_r - particles(n)%radius |
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| 306 | |
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| 307 | ! |
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| 308 | !-- Sum up the change in volume of liquid water for the respective grid |
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| 309 | !-- volume (this is needed later in lpm_calc_liquid_water_content for |
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| 310 | !-- calculating the release of latent heat) |
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| 311 | i = ( particles(n)%x + 0.5 * dx ) * ddx |
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| 312 | j = ( particles(n)%y + 0.5 * dy ) * ddy |
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| 313 | k = particles(n)%z / dz + 1 + offset_ocean_nzt_m1 |
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| 314 | ! only exact if equidistant |
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| 315 | |
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| 316 | ql_c(k,j,i) = ql_c(k,j,i) + particles(n)%weight_factor * & |
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| 317 | rho_l * 1.33333333 * pi * & |
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| 318 | ( new_r**3 - particles(n)%radius**3 ) / & |
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| 319 | ( rho_surface * dx * dy * dz ) |
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| 320 | IF ( ql_c(k,j,i) > 100.0 ) THEN |
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| 321 | WRITE( message_string, * ) 'k=',k,' j=',j,' i=',i, & |
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| 322 | ' ql_c=',ql_c(k,j,i), ' &part(',n,')%wf=', & |
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| 323 | particles(n)%weight_factor,' delta_r=',delta_r |
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| 324 | CALL message( 'lpm_droplet_condensation', 'PA0143', 2, 2, -1, 6, 1 ) |
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| 325 | ENDIF |
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| 326 | |
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| 327 | ! |
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| 328 | !-- Change the droplet radius |
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| 329 | IF ( ( new_r - particles(n)%radius ) < 0.0 .AND. new_r < 0.0 ) & |
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| 330 | THEN |
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| 331 | WRITE( message_string, * ) '#1 k=',k,' j=',j,' i=',i, & |
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| 332 | ' e_s=',e_s, ' e_a=',e_a,' t_int=',t_int, & |
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| 333 | ' &delta_r=',delta_r, & |
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| 334 | ' particle_radius=',particles(n)%radius |
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| 335 | CALL message( 'lpm_droplet_condensation', 'PA0144', 2, 2, -1, 6, 1 ) |
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| 336 | ENDIF |
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| 337 | |
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| 338 | ! |
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| 339 | !-- Sum up the total volume of liquid water (needed below for |
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| 340 | !-- re-calculating the weighting factors) |
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| 341 | ql_v(k,j,i) = ql_v(k,j,i) + particles(n)%weight_factor * new_r**3 |
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| 342 | |
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| 343 | particles(n)%radius = new_r |
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| 344 | |
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| 345 | ! |
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| 346 | !-- Determine radius class of the particle needed for collision |
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| 347 | IF ( ( hall_kernel .OR. wang_kernel ) .AND. use_kernel_tables ) & |
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| 348 | THEN |
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| 349 | particles(n)%class = ( LOG( new_r ) - rclass_lbound ) / & |
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| 350 | ( rclass_ubound - rclass_lbound ) * & |
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| 351 | radius_classes |
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| 352 | particles(n)%class = MIN( particles(n)%class, radius_classes ) |
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| 353 | particles(n)%class = MAX( particles(n)%class, 1 ) |
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| 354 | ENDIF |
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| 355 | |
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| 356 | ENDDO |
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| 357 | |
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| 358 | CALL cpu_log( log_point_s(42), 'lpm_droplet_condens', 'stop' ) |
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| 359 | |
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| 360 | |
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| 361 | END SUBROUTINE lpm_droplet_condensation |
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