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1	!> @file lpm_collision_kernels_mod.f90
2	!--------------------------------------------------------------------------------!
3	! This file is part of PALM.
4	!
5	! PALM is free software: you can redistribute it and/or modify it under the terms
6	! of the GNU General Public License as published by the Free Software Foundation,
7	! either version 3 of the License, or (at your option) any later version.
8	!
9	! PALM is distributed in the hope that it will be useful, but WITHOUT ANY
10	! WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
11	! A PARTICULAR PURPOSE. See the GNU General Public License for more details.
12	!
13	! You should have received a copy of the GNU General Public License along with
14	! PALM. If not, see <http://www.gnu.org/licenses/>.
15	!
16	! Copyright 1997-2016 Leibniz Universitaet Hannover
17	!--------------------------------------------------------------------------------!
18	!
19	! Current revisions:
20	! -----------------
21	!
22	!
23	! Former revisions:
24	! -----------------
25	! $Id: lpm_collision_kernels_mod.f90 1851 2016-04-08 13:32:50Z hoffmann $
26	!
27	! 1850 2016-04-08 13:29:27Z maronga
28	! Module renamed
29	!
30	!
31	! 1822 2016-04-07 07:49:42Z hoffmann
32	! PALM kernel has been deleted.
33	! Bugfix in the calculation of the turbulent enhancement factor of the
34	! collection efficiency.
35	!
36	! Unused variables removed.
37	!
38	! 1776 2016-03-02 17:54:58Z hoffmann
39	! Bugfix: Collection efficiencies must be calculated for the larger droplet.
40	!
41	! 1682 2015-10-07 23:56:08Z knoop
42	! Code annotations made doxygen readable
43	!
44	! 1519 2015-01-08 10:20:42Z hoffmann
45	! Bugfix: Using the new particle structure, particles are not sorted by size.
46	! Hence, computation of collision efficiencies must ensure that the ratio of
47	! two colliding droplets is < 1.
48	!
49	! 1359 2014-04-11 17:15:14Z hoffmann
50	! New particle structure integrated.
51	! Kind definition added to all floating point numbers.
52	!
53	! 1346 2014-03-27 13:18:20Z heinze
54	! Bugfix: REAL constants provided with KIND-attribute especially in call of
55	! intrinsic function like MAX, MIN, SIGN
56	!
57	! 1322 2014-03-20 16:38:49Z raasch
58	! REAL constants defined as wp_kind
59	!
60	! 1320 2014-03-20 08:40:49Z
61	! ONLY-attribute added to USE-statements,
62	! kind-parameters added to all INTEGER and REAL declaration statements,
63	! kinds are defined in new module kinds,
64	! revision history before 2012 removed,
65	! comment fields (!:) to be used for variable explanations added to
66	! all variable declaration statements
67	!
68	! 1092 2013-02-02 11:24:22Z raasch
69	! unused variables removed
70	!
71	! 1071 2012-11-29 16:54:55Z franke
72	! Bugfix: collision efficiencies for Hall kernel should not be < 1.0E-20
73	!
74	! 1036 2012-10-22 13:43:42Z raasch
75	! code put under GPL (PALM 3.9)
76	!
77	! 1007 2012-09-19 14:30:36Z franke
78	! converted all units to SI units and replaced some parameters by corresponding
79	! PALM parameters
80	! Bugfix: factor in calculation of enhancement factor for collision efficencies
81	! changed from 10. to 1.0
82	!
83	! 849 2012-03-15 10:35:09Z raasch
84	! routine collision_efficiency_rogers added (moved from former advec_particles
85	! to here)
86	!
87	! 835 2012-02-22 11:21:19Z raasch $
88	! Bugfix: array diss can be used only in case of Wang kernel
89	!
90	! 828 2012-02-21 12:00:36Z raasch
91	! code has been completely reformatted, routine colker renamed
92	! recalculate_kernel,
93	! routine init_kernels added, radius is now communicated to the collision
94	! routines by array radclass
95	!
96	! Bugfix: transformation factor for dissipation changed from 1E5 to 1E4
97	!
98	! 825 2012-02-19 03:03:44Z raasch
99	! routine renamed from wang_kernel to lpm_collision_kernels,
100	! turbulence_effects on collision replaced by wang_kernel
101	!
102	! 790 2011-11-29 03:11:20Z raasch
103	! initial revision
104	!
105	! Description:
106	! ------------
107	!> This module calculates collision efficiencies either due to pure gravitational
108	!> effects (Hall kernel, see Hall, 1980: J. Atmos. Sci., 2486-2507) or
109	!> including the effects of turbulence (Wang kernel, see Wang and
110	!> Grabowski, 2009: Atmos. Sci. Lett., 10, 1-8, and Ayala et al., 2008:
111	!> New J. Phys., 10, 075016). The original code has been
112	!> provided by L.-P. Wang but is substantially reformatted and speed optimized
113	!> here.
114	!------------------------------------------------------------------------------!
115	MODULE lpm_collision_kernels_mod
116
117
118	USE constants, &
119	ONLY: pi
120
121	USE kinds
122
123	USE particle_attributes, &
124	ONLY: collision_kernel, dissipation_classes, particles, &
125	radius_classes
126
127	USE pegrid
128
129
130	IMPLICIT NONE
131
132	PRIVATE
133
134	PUBLIC ckernel, init_kernels, rclass_lbound, rclass_ubound, &
135	recalculate_kernel
136
137	REAL(wp) :: epsilon !<
138	REAL(wp) :: rclass_lbound !<
139	REAL(wp) :: rclass_ubound !<
140	REAL(wp) :: urms !<
141
142	REAL(wp), DIMENSION(:), ALLOCATABLE :: epsclass !< dissipation rate class
143	REAL(wp), DIMENSION(:), ALLOCATABLE :: radclass !< radius class
144	REAL(wp), DIMENSION(:), ALLOCATABLE :: winf !<
145
146	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: ec !<
147	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: ecf !<
148	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: gck !<
149	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: hkernel !<
150	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: hwratio !<
151
152	REAL(wp), DIMENSION(:,:,:), ALLOCATABLE :: ckernel !<
153
154	SAVE
155
156	!
157	!-- Public interfaces
158	INTERFACE init_kernels
159	MODULE PROCEDURE init_kernels
160	END INTERFACE init_kernels
161
162	INTERFACE recalculate_kernel
163	MODULE PROCEDURE recalculate_kernel
164	END INTERFACE recalculate_kernel
165
166
167	CONTAINS
168
169
170	!------------------------------------------------------------------------------!
171	! Description:
172	! ------------
173	!> Initialization of the collision efficiency matrix with fixed radius and
174	!> dissipation classes, calculated at simulation start only.
175	!------------------------------------------------------------------------------!
176
177	SUBROUTINE init_kernels
178
179	IMPLICIT NONE
180
181	INTEGER(iwp) :: i !<
182	INTEGER(iwp) :: j !<
183	INTEGER(iwp) :: k !<
184
185
186	!
187	!-- Calculate collision efficiencies for fixed radius- and dissipation
188	!-- classes
189	IF ( collision_kernel(6:9) == 'fast' ) THEN
190
191	ALLOCATE( ckernel(1:radius_classes,1:radius_classes, &
192	0:dissipation_classes), epsclass(1:dissipation_classes), &
193	radclass(1:radius_classes) )
194
195	!
196	!-- Calculate the radius class bounds with logarithmic distances
197	!-- in the interval [1.0E-6, 2.0E-4] m
198	rclass_lbound = LOG( 1.0E-6_wp )
199	rclass_ubound = LOG( 2.0E-4_wp )
200	radclass(1) = EXP( rclass_lbound )
201	DO i = 2, radius_classes
202	radclass(i) = EXP( rclass_lbound + &
203	( rclass_ubound - rclass_lbound ) * &
204	( i - 1.0_wp ) / ( radius_classes - 1.0_wp ) )
205	ENDDO
206
207	!
208	!-- Set the class bounds for dissipation in interval [0.0, 0.1] m2/s3
209	DO i = 1, dissipation_classes
210	epsclass(i) = 0.1_wp * REAL( i, KIND=wp ) / dissipation_classes
211	ENDDO
212	!
213	!-- Calculate collision efficiencies of the Wang/ayala kernel
214	ALLOCATE( ec(1:radius_classes,1:radius_classes), &
215	ecf(1:radius_classes,1:radius_classes), &
216	gck(1:radius_classes,1:radius_classes), &
217	winf(1:radius_classes) )
218
219	DO k = 1, dissipation_classes
220
221	epsilon = epsclass(k)
222	urms = 2.02_wp * ( epsilon / 0.04_wp )**( 1.0_wp / 3.0_wp )
223
224	CALL turbsd
225	CALL turb_enhance_eff
226	CALL effic
227
228	DO j = 1, radius_classes
229	DO i = 1, radius_classes
230	ckernel(i,j,k) = ec(i,j) * gck(i,j) * ecf(i,j)
231	ENDDO
232	ENDDO
233
234	ENDDO
235
236	!
237	!-- Calculate collision efficiencies of the Hall kernel
238	ALLOCATE( hkernel(1:radius_classes,1:radius_classes), &
239	hwratio(1:radius_classes,1:radius_classes) )
240
241	CALL fallg
242	CALL effic
243
244	DO j = 1, radius_classes
245	DO i = 1, radius_classes
246	hkernel(i,j) = pi * ( radclass(j) + radclass(i) )**2 &
247	* ec(i,j) * ABS( winf(j) - winf(i) )
248	ckernel(i,j,0) = hkernel(i,j) ! hall kernel stored on index 0
249	ENDDO
250	ENDDO
251
252	!
253	!-- Test output of efficiencies
254	IF ( j == -1 ) THEN
255
256	PRINT, '** Hall kernel'
257	WRITE ( ,'(5X,20(F4.0,1X))' ) ( radclass(i)1.0E6_wp, &
258	i = 1,radius_classes )
259	DO j = 1, radius_classes
260	WRITE ( *,'(F4.0,1X,20(F8.4,1X))' ) radclass(j), &
261	( hkernel(i,j), i = 1,radius_classes )
262	ENDDO
263
264	DO k = 1, dissipation_classes
265	DO i = 1, radius_classes
266	DO j = 1, radius_classes
267	IF ( hkernel(i,j) == 0.0_wp ) THEN
268	hwratio(i,j) = 9999999.9_wp
269	ELSE
270	hwratio(i,j) = ckernel(i,j,k) / hkernel(i,j)
271	ENDIF
272	ENDDO
273	ENDDO
274
275	PRINT, '** epsilon = ', epsclass(k)
276	WRITE ( ,'(5X,20(F4.0,1X))' ) ( radclass(i) 1.0E6_wp, &
277	i = 1,radius_classes )
278	DO j = 1, radius_classes
279	WRITE ( ,'(F4.0,1X,20(F8.4,1X))' ) radclass(j) 1.0E6_wp, &
280	( hwratio(i,j), i = 1,radius_classes )
281	ENDDO
282	ENDDO
283
284	ENDIF
285
286	DEALLOCATE( ec, ecf, epsclass, gck, hkernel, winf )
287
288	ELSEIF( collision_kernel == 'hall' .OR. collision_kernel == 'wang' ) &
289	THEN
290	!
291	!-- Initial settings for Hall- and Wang-Kernel
292	!-- To be done: move here parts from turbsd, fallg, ecoll, etc.
293	ENDIF
294
295	END SUBROUTINE init_kernels
296
297
298	!------------------------------------------------------------------------------!
299	! Description:
300	! ------------
301	!> Calculation of collision kernels during each timestep and for each grid box
302	!------------------------------------------------------------------------------!
303	SUBROUTINE recalculate_kernel( i1, j1, k1 )
304
305	USE arrays_3d, &
306	ONLY: diss
307
308	USE particle_attributes, &
309	ONLY: prt_count, radius_classes, wang_kernel
310
311	IMPLICIT NONE
312
313	INTEGER(iwp) :: i !<
314	INTEGER(iwp) :: i1 !<
315	INTEGER(iwp) :: j !<
316	INTEGER(iwp) :: j1 !<
317	INTEGER(iwp) :: k1 !<
318	INTEGER(iwp) :: pend !<
319	INTEGER(iwp) :: pstart !<
320
321
322	pstart = 1
323	pend = prt_count(k1,j1,i1)
324	radius_classes = prt_count(k1,j1,i1)
325
326	ALLOCATE( ec(1:radius_classes,1:radius_classes), &
327	radclass(1:radius_classes), winf(1:radius_classes) )
328
329	!
330	!-- Store particle radii on the radclass array
331	radclass(1:radius_classes) = particles(pstart:pend)%radius
332
333	IF ( wang_kernel ) THEN
334	epsilon = diss(k1,j1,i1) ! dissipation rate in m2/s3
335	ELSE
336	epsilon = 0.0_wp
337	ENDIF
338	urms = 2.02_wp * ( epsilon / 0.04_wp )**( 0.33333333333_wp )
339
340	IF ( wang_kernel .AND. epsilon > 1.0E-7_wp ) THEN
341	!
342	!-- Call routines to calculate efficiencies for the Wang kernel
343	ALLOCATE( gck(1:radius_classes,1:radius_classes), &
344	ecf(1:radius_classes,1:radius_classes) )
345
346	CALL turbsd
347	CALL turb_enhance_eff
348	CALL effic
349
350	DO j = 1, radius_classes
351	DO i = 1, radius_classes
352	ckernel(pstart+i-1,pstart+j-1,1) = ec(i,j) * gck(i,j) * ecf(i,j)
353	ENDDO
354	ENDDO
355
356	DEALLOCATE( gck, ecf )
357
358	ELSE
359	!
360	!-- Call routines to calculate efficiencies for the Hall kernel
361	CALL fallg
362	CALL effic
363
364	DO j = 1, radius_classes
365	DO i = 1, radius_classes
366	ckernel(pstart+i-1,pstart+j-1,1) = pi * &
367	( radclass(j) + radclass(i) )**2 &
368	* ec(i,j) * ABS( winf(j) - winf(i) )
369	ENDDO
370	ENDDO
371
372	ENDIF
373
374	DEALLOCATE( ec, radclass, winf )
375
376	END SUBROUTINE recalculate_kernel
377
378
379	!------------------------------------------------------------------------------!
380	! Description:
381	! ------------
382	!> Calculation of effects of turbulence on the geometric collision kernel
383	!> (by including the droplets' average radial relative velocities and their
384	!> radial distribution function) following the analytic model by Aayala et al.
385	!> (2008, New J. Phys.). For details check the second part 2 of the publication,
386	!> page 37ff.
387	!>
388	!> Input parameters, which need to be replaced by PALM parameters:
389	!> water density, air density
390	!------------------------------------------------------------------------------!
391	SUBROUTINE turbsd
392
393	USE control_parameters, &
394	ONLY: g, molecular_viscosity
395
396	USE particle_attributes, &
397	ONLY: radius_classes
398
399	IMPLICIT NONE
400
401	INTEGER(iwp) :: i !<
402	INTEGER(iwp) :: j !<
403
404	REAL(wp) :: ao !<
405	REAL(wp) :: ao_gr !<
406	REAL(wp) :: bbb !<
407	REAL(wp) :: be !<
408	REAL(wp) :: b1 !<
409	REAL(wp) :: b2 !<
410	REAL(wp) :: ccc !<
411	REAL(wp) :: c1 !<
412	REAL(wp) :: c1_gr !<
413	REAL(wp) :: c2 !<
414	REAL(wp) :: d1 !<
415	REAL(wp) :: d2 !<
416	REAL(wp) :: eta !<
417	REAL(wp) :: e1 !<
418	REAL(wp) :: e2 !<
419	REAL(wp) :: fao_gr !<
420	REAL(wp) :: fr !<
421	REAL(wp) :: grfin !<
422	REAL(wp) :: lambda !<
423	REAL(wp) :: lambda_re !<
424	REAL(wp) :: lf !<
425	REAL(wp) :: rc !<
426	REAL(wp) :: rrp !<
427	REAL(wp) :: sst !<
428	REAL(wp) :: tauk !<
429	REAL(wp) :: tl !<
430	REAL(wp) :: t2 !<
431	REAL(wp) :: tt !<
432	REAL(wp) :: t1 !<
433	REAL(wp) :: vk !<
434	REAL(wp) :: vrms1xy !<
435	REAL(wp) :: vrms2xy !<
436	REAL(wp) :: v1 !<
437	REAL(wp) :: v1v2xy !<
438	REAL(wp) :: v1xysq !<
439	REAL(wp) :: v2 !<
440	REAL(wp) :: v2xysq !<
441	REAL(wp) :: wrfin !<
442	REAL(wp) :: wrgrav2 !<
443	REAL(wp) :: wrtur2xy !<
444	REAL(wp) :: xx !<
445	REAL(wp) :: yy !<
446	REAL(wp) :: z !<
447
448	REAL(wp), DIMENSION(1:radius_classes) :: st !< Stokes number
449	REAL(wp), DIMENSION(1:radius_classes) :: tau !< inertial time scale
450
451	lambda = urms * SQRT( 15.0_wp * molecular_viscosity / epsilon )
452	lambda_re = urms*2 SQRT( 15.0_wp / epsilon / molecular_viscosity )
453	tl = urms**2 / epsilon
454	lf = 0.5_wp * urms**3 / epsilon
455	tauk = SQRT( molecular_viscosity / epsilon )
456	eta = ( molecular_viscosity3 / epsilon )0.25_wp
457	vk = eta / tauk
458
459	ao = ( 11.0_wp + 7.0_wp * lambda_re ) / ( 205.0_wp + lambda_re )
460	tt = SQRT( 2.0_wp * lambda_re / ( SQRT( 15.0_wp ) * ao ) ) * tauk
461
462	!
463	!-- Get terminal velocity of droplets
464	CALL fallg
465
466	DO i = 1, radius_classes
467	tau(i) = winf(i) / g ! inertial time scale
468	st(i) = tau(i) / tauk ! Stokes number
469	ENDDO
470
471	!
472	!-- Calculate average radial relative velocity at contact (wrfin)
473	z = tt / tl
474	be = SQRT( 2.0_wp ) * lambda / lf
475	bbb = SQRT( 1.0_wp - 2.0_wp * be**2 )
476	d1 = ( 1.0_wp + bbb ) / ( 2.0_wp * bbb )
477	e1 = lf * ( 1.0_wp + bbb ) * 0.5_wp
478	d2 = ( 1.0_wp - bbb ) * 0.5_wp / bbb
479	e2 = lf * ( 1.0_wp - bbb ) * 0.5_wp
480	ccc = SQRT( 1.0_wp - 2.0_wp * z**2 )
481	b1 = ( 1.0_wp + ccc ) * 0.5_wp / ccc
482	c1 = tl * ( 1.0_wp + ccc ) * 0.5_wp
483	b2 = ( 1.0_wp - ccc ) * 0.5_wp / ccc
484	c2 = tl * ( 1.0_wp - ccc ) * 0.5_wp
485
486	DO i = 1, radius_classes
487
488	v1 = winf(i)
489	t1 = tau(i)
490
491	DO j = 1, i
492	rrp = radclass(i) + radclass(j)
493	v2 = winf(j)
494	t2 = tau(j)
495
496	v1xysq = b1 * d1 * phi_w(c1,e1,v1,t1) - b1 * d2 * phi_w(c1,e2,v1,t1) &
497	- b2 * d1 * phi_w(c2,e1,v1,t1) + b2 * d2 * phi_w(c2,e2,v1,t1)
498	v1xysq = v1xysq * urms**2 / t1
499	vrms1xy = SQRT( v1xysq )
500
501	v2xysq = b1 * d1 * phi_w(c1,e1,v2,t2) - b1 * d2 * phi_w(c1,e2,v2,t2) &
502	- b2 * d1 * phi_w(c2,e1,v2,t2) + b2 * d2 * phi_w(c2,e2,v2,t2)
503	v2xysq = v2xysq * urms**2 / t2
504	vrms2xy = SQRT( v2xysq )
505
506	IF ( winf(i) >= winf(j) ) THEN
507	v1 = winf(i)
508	t1 = tau(i)
509	v2 = winf(j)
510	t2 = tau(j)
511	ELSE
512	v1 = winf(j)
513	t1 = tau(j)
514	v2 = winf(i)
515	t2 = tau(i)
516	ENDIF
517
518	v1v2xy = b1 * d1 * zhi(c1,e1,v1,t1,v2,t2) - &
519	b1 * d2 * zhi(c1,e2,v1,t1,v2,t2) - &
520	b2 * d1 * zhi(c2,e1,v1,t1,v2,t2) + &
521	b2 * d2* zhi(c2,e2,v1,t1,v2,t2)
522	fr = d1 * EXP( -rrp / e1 ) - d2 * EXP( -rrp / e2 )
523	v1v2xy = v1v2xy * fr * urms**2 / tau(i) / tau(j)
524	wrtur2xy = vrms1xy2 + vrms2xy2 - 2.0_wp * v1v2xy
525	IF ( wrtur2xy < 0.0_wp ) wrtur2xy = 0.0_wp
526	wrgrav2 = pi / 8.0_wp * ( winf(j) - winf(i) )**2
527	wrfin = SQRT( ( 2.0_wp / pi ) * ( wrtur2xy + wrgrav2) )
528
529	!
530	!-- Calculate radial distribution function (grfin)
531	IF ( st(j) > st(i) ) THEN
532	sst = st(j)
533	ELSE
534	sst = st(i)
535	ENDIF
536
537	xx = -0.1988_wp * sst*4 + 1.5275_wp sst*3 - 4.2942_wp &
538	sst*2 + 5.3406_wp sst
539	IF ( xx < 0.0_wp ) xx = 0.0_wp
540	yy = 0.1886_wp * EXP( 20.306_wp / lambda_re )
541
542	c1_gr = xx / ( g / vk * tauk )**yy
543
544	ao_gr = ao + ( pi / 8.0_wp) * ( g / vk * tauk )**2
545	fao_gr = 20.115_wp * SQRT( ao_gr / lambda_re )
546	rc = SQRT( fao_gr * ABS( st(j) - st(i) ) ) * eta
547
548	grfin = ( ( eta2 + rc2 ) / ( rrp2 + rc2) )*( c1_gr0.5_wp )
549	IF ( grfin < 1.0_wp ) grfin = 1.0_wp
550
551	!
552	!-- Calculate general collection kernel (without the consideration of
553	!-- collection efficiencies)
554	gck(i,j) = 2.0_wp * pi * rrp*2 wrfin * grfin
555	gck(j,i) = gck(i,j)
556
557	ENDDO
558	ENDDO
559
560	END SUBROUTINE turbsd
561
562	REAL(wp) FUNCTION phi_w( a, b, vsett, tau0 )
563	!
564	!-- Function used in the Ayala et al. (2008) analytical model for turbulent
565	!-- effects on the collision kernel
566	IMPLICIT NONE
567
568	REAL(wp) :: a !<
569	REAL(wp) :: aa1 !<
570	REAL(wp) :: b !<
571	REAL(wp) :: tau0 !<
572	REAL(wp) :: vsett !<
573
574	aa1 = 1.0_wp / tau0 + 1.0_wp / a + vsett / b
575	phi_w = 1.0_wp / aa1 - 0.5_wp * vsett / b / aa1**2
576
577	END FUNCTION phi_w
578
579	REAL(wp) FUNCTION zhi( a, b, vsett1, tau1, vsett2, tau2 )
580	!
581	!-- Function used in the Ayala et al. (2008) analytical model for turbulent
582	!-- effects on the collision kernel
583	IMPLICIT NONE
584
585	REAL(wp) :: a !<
586	REAL(wp) :: aa1 !<
587	REAL(wp) :: aa2 !<
588	REAL(wp) :: aa3 !<
589	REAL(wp) :: aa4 !<
590	REAL(wp) :: aa5 !<
591	REAL(wp) :: aa6 !<
592	REAL(wp) :: b !<
593	REAL(wp) :: tau1 !<
594	REAL(wp) :: tau2 !<
595	REAL(wp) :: vsett1 !<
596	REAL(wp) :: vsett2 !<
597
598	aa1 = vsett2 / b - 1.0_wp / tau2 - 1.0_wp / a
599	aa2 = vsett1 / b + 1.0_wp / tau1 + 1.0_wp / a
600	aa3 = ( vsett1 - vsett2 ) / b + 1.0_wp / tau1 + 1.0_wp / tau2
601	aa4 = ( vsett2 / b )2 - ( 1.0_wp / tau2 + 1.0_wp / a )2
602	aa5 = vsett2 / b + 1.0_wp / tau2 + 1.0_wp / a
603	aa6 = 1.0_wp / tau1 - 1.0_wp / a + ( 1.0_wp / tau2 + 1.0_wp / a) * &
604	vsett1 / vsett2
605	zhi = (1.0_wp / aa1 - 1.0_wp / aa2 ) * ( vsett1 - vsett2 ) * 0.5_wp / &
606	b / aa32 + ( 4.0_wp / aa4 - 1.0_wp / aa52 - 1.0_wp / aa1**2 ) &
607	* vsett2 * 0.5_wp / b /aa6 + ( 2.0_wp * ( b / aa2 - b / aa1 ) - &
608	vsett1 / aa22 + vsett2 / aa12 ) * 0.5_wp / b / aa3
609
610	END FUNCTION zhi
611
612
613	!------------------------------------------------------------------------------!
614	! Description:
615	! ------------
616	!> Parameterization of terminal velocity following Rogers et al. (1993, J. Appl.
617	!> Meteorol.)
618	!------------------------------------------------------------------------------!
619	SUBROUTINE fallg
620
621	USE particle_attributes, &
622	ONLY: radius_classes
623
624	IMPLICIT NONE
625
626	INTEGER(iwp) :: j !<
627
628	REAL(wp), PARAMETER :: k_cap_rog = 4.0_wp !< parameter
629	REAL(wp), PARAMETER :: k_low_rog = 12.0_wp !< parameter
630	REAL(wp), PARAMETER :: a_rog = 9.65_wp !< parameter
631	REAL(wp), PARAMETER :: b_rog = 10.43_wp !< parameter
632	REAL(wp), PARAMETER :: c_rog = 0.6_wp !< parameter
633	REAL(wp), PARAMETER :: d0_rog = 0.745_wp !< seperation diameter
634
635	REAL(wp) :: diameter !< droplet diameter in mm
636
637
638	DO j = 1, radius_classes
639
640	diameter = radclass(j) * 2000.0_wp
641
642	IF ( diameter <= d0_rog ) THEN
643	winf(j) = k_cap_rog * diameter * ( 1.0_wp - &
644	EXP( -k_low_rog * diameter ) )
645	ELSE
646	winf(j) = a_rog - b_rog * EXP( -c_rog * diameter )
647	ENDIF
648
649	ENDDO
650
651	END SUBROUTINE fallg
652
653
654	!------------------------------------------------------------------------------!
655	! Description:
656	! ------------
657	!> Interpolation of collision efficiencies (Hall, 1980, J. Atmos. Sci.)
658	!------------------------------------------------------------------------------!
659	SUBROUTINE effic
660
661	USE particle_attributes, &
662	ONLY: radius_classes
663
664	IMPLICIT NONE
665
666	INTEGER(iwp) :: i !<
667	INTEGER(iwp) :: iq !<
668	INTEGER(iwp) :: ir !<
669	INTEGER(iwp) :: j !<
670	INTEGER(iwp) :: k !<
671
672	INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: ira !<
673
674	LOGICAL, SAVE :: first = .TRUE. !<
675
676	REAL(wp) :: ek !<
677	REAL(wp) :: particle_radius !<
678	REAL(wp) :: pp !<
679	REAL(wp) :: qq !<
680	REAL(wp) :: rq !<
681
682	REAL(wp), DIMENSION(1:21), SAVE :: rat !<
683
684	REAL(wp), DIMENSION(1:15), SAVE :: r0 !<
685
686	REAL(wp), DIMENSION(1:15,1:21), SAVE :: ecoll !<
687
688	!
689	!-- Initial assignment of constants
690	IF ( first ) THEN
691
692	first = .FALSE.
693	r0 = (/ 6.0_wp, 8.0_wp, 10.0_wp, 15.0_wp, 20.0_wp, 25.0_wp, &
694	30.0_wp, 40.0_wp, 50.0_wp, 60.0_wp, 70.0_wp, 100.0_wp, &
695	150.0_wp, 200.0_wp, 300.0_wp /)
696
697	rat = (/ 0.00_wp, 0.05_wp, 0.10_wp, 0.15_wp, 0.20_wp, 0.25_wp, &
698	0.30_wp, 0.35_wp, 0.40_wp, 0.45_wp, 0.50_wp, 0.55_wp, &
699	0.60_wp, 0.65_wp, 0.70_wp, 0.75_wp, 0.80_wp, 0.85_wp, &
700	0.90_wp, 0.95_wp, 1.00_wp /)
701
702	ecoll(:,1) = (/ 0.001_wp, 0.001_wp, 0.001_wp, 0.001_wp, 0.001_wp, &
703	0.001_wp, 0.001_wp, 0.001_wp, 0.001_wp, 0.001_wp, &
704	0.001_wp, 0.001_wp, 0.001_wp, 0.001_wp, 0.001_wp /)
705	ecoll(:,2) = (/ 0.003_wp, 0.003_wp, 0.003_wp, 0.004_wp, 0.005_wp, &
706	0.005_wp, 0.005_wp, 0.010_wp, 0.100_wp, 0.050_wp, &
707	0.200_wp, 0.500_wp, 0.770_wp, 0.870_wp, 0.970_wp /)
708	ecoll(:,3) = (/ 0.007_wp, 0.007_wp, 0.007_wp, 0.008_wp, 0.009_wp, &
709	0.010_wp, 0.010_wp, 0.070_wp, 0.400_wp, 0.430_wp, &
710	0.580_wp, 0.790_wp, 0.930_wp, 0.960_wp, 1.000_wp /)
711	ecoll(:,4) = (/ 0.009_wp, 0.009_wp, 0.009_wp, 0.012_wp, 0.015_wp, &
712	0.010_wp, 0.020_wp, 0.280_wp, 0.600_wp, 0.640_wp, &
713	0.750_wp, 0.910_wp, 0.970_wp, 0.980_wp, 1.000_wp /)
714	ecoll(:,5) = (/ 0.014_wp, 0.014_wp, 0.014_wp, 0.015_wp, 0.016_wp, &
715	0.030_wp, 0.060_wp, 0.500_wp, 0.700_wp, 0.770_wp, &
716	0.840_wp, 0.950_wp, 0.970_wp, 1.000_wp, 1.000_wp /)
717	ecoll(:,6) = (/ 0.017_wp, 0.017_wp, 0.017_wp, 0.020_wp, 0.022_wp, &
718	0.060_wp, 0.100_wp, 0.620_wp, 0.780_wp, 0.840_wp, &
719	0.880_wp, 0.950_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
720	ecoll(:,7) = (/ 0.030_wp, 0.030_wp, 0.024_wp, 0.022_wp, 0.032_wp, &
721	0.062_wp, 0.200_wp, 0.680_wp, 0.830_wp, 0.870_wp, &
722	0.900_wp, 0.950_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
723	ecoll(:,8) = (/ 0.025_wp, 0.025_wp, 0.025_wp, 0.036_wp, 0.043_wp, &
724	0.130_wp, 0.270_wp, 0.740_wp, 0.860_wp, 0.890_wp, &
725	0.920_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
726	ecoll(:,9) = (/ 0.027_wp, 0.027_wp, 0.027_wp, 0.040_wp, 0.052_wp, &
727	0.200_wp, 0.400_wp, 0.780_wp, 0.880_wp, 0.900_wp, &
728	0.940_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
729	ecoll(:,10) = (/ 0.030_wp, 0.030_wp, 0.030_wp, 0.047_wp, 0.064_wp, &
730	0.250_wp, 0.500_wp, 0.800_wp, 0.900_wp, 0.910_wp, &
731	0.950_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
732	ecoll(:,11) = (/ 0.040_wp, 0.040_wp, 0.033_wp, 0.037_wp, 0.068_wp, &
733	0.240_wp, 0.550_wp, 0.800_wp, 0.900_wp, 0.910_wp, &
734	0.950_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
735	ecoll(:,12) = (/ 0.035_wp, 0.035_wp, 0.035_wp, 0.055_wp, 0.079_wp, &
736	0.290_wp, 0.580_wp, 0.800_wp, 0.900_wp, 0.910_wp, &
737	0.950_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
738	ecoll(:,13) = (/ 0.037_wp, 0.037_wp, 0.037_wp, 0.062_wp, 0.082_wp, &
739	0.290_wp, 0.590_wp, 0.780_wp, 0.900_wp, 0.910_wp, &
740	0.950_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
741	ecoll(:,14) = (/ 0.037_wp, 0.037_wp, 0.037_wp, 0.060_wp, 0.080_wp, &
742	0.290_wp, 0.580_wp, 0.770_wp, 0.890_wp, 0.910_wp, &
743	0.950_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
744	ecoll(:,15) = (/ 0.037_wp, 0.037_wp, 0.037_wp, 0.041_wp, 0.075_wp, &
745	0.250_wp, 0.540_wp, 0.760_wp, 0.880_wp, 0.920_wp, &
746	0.950_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
747	ecoll(:,16) = (/ 0.037_wp, 0.037_wp, 0.037_wp, 0.052_wp, 0.067_wp, &
748	0.250_wp, 0.510_wp, 0.770_wp, 0.880_wp, 0.930_wp, &
749	0.970_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
750	ecoll(:,17) = (/ 0.037_wp, 0.037_wp, 0.037_wp, 0.047_wp, 0.057_wp, &
751	0.250_wp, 0.490_wp, 0.770_wp, 0.890_wp, 0.950_wp, &
752	1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp, 1.000_wp /)
753	ecoll(:,18) = (/ 0.036_wp, 0.036_wp, 0.036_wp, 0.042_wp, 0.048_wp, &
754	0.230_wp, 0.470_wp, 0.780_wp, 0.920_wp, 1.000_wp, &
755	1.020_wp, 1.020_wp, 1.020_wp, 1.020_wp, 1.020_wp /)
756	ecoll(:,19) = (/ 0.040_wp, 0.040_wp, 0.035_wp, 0.033_wp, 0.040_wp, &
757	0.112_wp, 0.450_wp, 0.790_wp, 1.010_wp, 1.030_wp, &
758	1.040_wp, 1.040_wp, 1.040_wp, 1.040_wp, 1.040_wp /)
759	ecoll(:,20) = (/ 0.033_wp, 0.033_wp, 0.033_wp, 0.033_wp, 0.033_wp, &
760	0.119_wp, 0.470_wp, 0.950_wp, 1.300_wp, 1.700_wp, &
761	2.300_wp, 2.300_wp, 2.300_wp, 2.300_wp, 2.300_wp /)
762	ecoll(:,21) = (/ 0.027_wp, 0.027_wp, 0.027_wp, 0.027_wp, 0.027_wp, &
763	0.125_wp, 0.520_wp, 1.400_wp, 2.300_wp, 3.000_wp, &
764	4.000_wp, 4.000_wp, 4.000_wp, 4.000_wp, 4.000_wp /)
765	ENDIF
766
767	!
768	!-- Calculate the radius class index of particles with respect to array r
769	!-- Radius has to be in microns
770	ALLOCATE( ira(1:radius_classes) )
771	DO j = 1, radius_classes
772	particle_radius = radclass(j) * 1.0E6_wp
773	DO k = 1, 15
774	IF ( particle_radius < r0(k) ) THEN
775	ira(j) = k
776	EXIT
777	ENDIF
778	ENDDO
779	IF ( particle_radius >= r0(15) ) ira(j) = 16
780	ENDDO
781
782	!
783	!-- Two-dimensional linear interpolation of the collision efficiency.
784	!-- Radius has to be in microns
785	DO j = 1, radius_classes
786	DO i = 1, j
787
788	ir = ira(j)
789	rq = MIN( radclass(i) / radclass(j), radclass(j) / radclass(i) )
790	iq = INT( rq * 20 ) + 1
791	iq = MAX( iq , 2)
792
793	IF ( ir < 16 ) THEN
794	IF ( ir >= 2 ) THEN
795	pp = ( ( MAX( radclass(j), radclass(i) ) * 1.0E6_wp ) - &
796	r0(ir-1) ) / ( r0(ir) - r0(ir-1) )
797	qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
798	ec(j,i) = ( 1.0_wp - pp ) * ( 1.0_wp - qq ) &
799	* ecoll(ir-1,iq-1) &
800	+ pp * ( 1.0_wp - qq ) * ecoll(ir,iq-1) &
801	+ qq * ( 1.0_wp - pp ) * ecoll(ir-1,iq) &
802	+ pp * qq * ecoll(ir,iq)
803	ELSE
804	qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
805	ec(j,i) = ( 1.0_wp - qq ) * ecoll(1,iq-1) + qq * ecoll(1,iq)
806	ENDIF
807	ELSE
808	qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
809	ek = ( 1.0_wp - qq ) * ecoll(15,iq-1) + qq * ecoll(15,iq)
810	ec(j,i) = MIN( ek, 1.0_wp )
811	ENDIF
812
813	IF ( ec(j,i) < 1.0E-20_wp ) ec(j,i) = 0.0_wp
814
815	ec(i,j) = ec(j,i)
816
817	ENDDO
818	ENDDO
819
820	DEALLOCATE( ira )
821
822	END SUBROUTINE effic
823
824
825	!------------------------------------------------------------------------------!
826	! Description:
827	! ------------
828	!> Interpolation of turbulent enhancement factor for collision efficencies
829	!> following Wang and Grabowski (2009, Atmos. Sci. Let.)
830	!------------------------------------------------------------------------------!
831	SUBROUTINE turb_enhance_eff
832
833	USE particle_attributes, &
834	ONLY: radius_classes
835
836	IMPLICIT NONE
837
838	INTEGER(iwp) :: i !<
839	INTEGER(iwp) :: iq !<
840	INTEGER(iwp) :: ir !<
841	INTEGER(iwp) :: j !<
842	INTEGER(iwp) :: k !<
843	INTEGER(iwp) :: kk !<
844
845	INTEGER(iwp), DIMENSION(:), ALLOCATABLE :: ira !<
846
847	LOGICAL, SAVE :: first = .TRUE. !<
848
849	REAL(wp) :: particle_radius !<
850	REAL(wp) :: pp !<
851	REAL(wp) :: qq !<
852	REAL(wp) :: rq !<
853	REAL(wp) :: y1 !<
854	REAL(wp) :: y2 !<
855	REAL(wp) :: y3 !<
856
857	REAL(wp), DIMENSION(1:11), SAVE :: rat !<
858	REAL(wp), DIMENSION(1:7), SAVE :: r0 !<
859
860	REAL(wp), DIMENSION(1:7,1:11), SAVE :: ecoll_100 !<
861	REAL(wp), DIMENSION(1:7,1:11), SAVE :: ecoll_400 !<
862
863	!
864	!-- Initial assignment of constants
865	IF ( first ) THEN
866
867	first = .FALSE.
868
869	r0 = (/ 10.0_wp, 20.0_wp, 30.0_wp, 40.0_wp, 50.0_wp, 60.0_wp, &
870	100.0_wp /)
871
872	rat = (/ 0.0_wp, 0.1_wp, 0.2_wp, 0.3_wp, 0.4_wp, 0.5_wp, 0.6_wp, &
873	0.7_wp, 0.8_wp, 0.9_wp, 1.0_wp /)
874	!
875	!-- Tabulated turbulent enhancement factor at 100 cm2/s3
876	ecoll_100(:,1) = (/ 1.74_wp, 1.74_wp, 1.773_wp, 1.49_wp, &
877	1.207_wp, 1.207_wp, 1.0_wp /)
878	ecoll_100(:,2) = (/ 1.46_wp, 1.46_wp, 1.421_wp, 1.245_wp, &
879	1.069_wp, 1.069_wp, 1.0_wp /)
880	ecoll_100(:,3) = (/ 1.32_wp, 1.32_wp, 1.245_wp, 1.123_wp, &
881	1.000_wp, 1.000_wp, 1.0_wp /)
882	ecoll_100(:,4) = (/ 1.250_wp, 1.250_wp, 1.148_wp, 1.087_wp, &
883	1.025_wp, 1.025_wp, 1.0_wp /)
884	ecoll_100(:,5) = (/ 1.186_wp, 1.186_wp, 1.066_wp, 1.060_wp, &
885	1.056_wp, 1.056_wp, 1.0_wp /)
886	ecoll_100(:,6) = (/ 1.045_wp, 1.045_wp, 1.000_wp, 1.014_wp, &
887	1.028_wp, 1.028_wp, 1.0_wp /)
888	ecoll_100(:,7) = (/ 1.070_wp, 1.070_wp, 1.030_wp, 1.038_wp, &
889	1.046_wp, 1.046_wp, 1.0_wp /)
890	ecoll_100(:,8) = (/ 1.000_wp, 1.000_wp, 1.054_wp, 1.042_wp, &
891	1.029_wp, 1.029_wp, 1.0_wp /)
892	ecoll_100(:,9) = (/ 1.223_wp, 1.223_wp, 1.117_wp, 1.069_wp, &
893	1.021_wp, 1.021_wp, 1.0_wp /)
894	ecoll_100(:,10) = (/ 1.570_wp, 1.570_wp, 1.244_wp, 1.166_wp, &
895	1.088_wp, 1.088_wp, 1.0_wp /)
896	ecoll_100(:,11) = (/ 20.3_wp, 20.3_wp, 14.6_wp, 8.61_wp, &
897	2.60_wp, 2.60_wp, 1.0_wp /)
898	!
899	!-- Tabulated turbulent enhancement factor at 400 cm2/s3
900	ecoll_400(:,1) = (/ 4.976_wp, 4.976_wp, 3.593_wp, 2.519_wp, &
901	1.445_wp, 1.445_wp, 1.0_wp /)
902	ecoll_400(:,2) = (/ 2.984_wp, 2.984_wp, 2.181_wp, 1.691_wp, &
903	1.201_wp, 1.201_wp, 1.0_wp /)
904	ecoll_400(:,3) = (/ 1.988_wp, 1.988_wp, 1.475_wp, 1.313_wp, &
905	1.150_wp, 1.150_wp, 1.0_wp /)
906	ecoll_400(:,4) = (/ 1.490_wp, 1.490_wp, 1.187_wp, 1.156_wp, &
907	1.126_wp, 1.126_wp, 1.0_wp /)
908	ecoll_400(:,5) = (/ 1.249_wp, 1.249_wp, 1.088_wp, 1.090_wp, &
909	1.092_wp, 1.092_wp, 1.0_wp /)
910	ecoll_400(:,6) = (/ 1.139_wp, 1.139_wp, 1.130_wp, 1.091_wp, &
911	1.051_wp, 1.051_wp, 1.0_wp /)
912	ecoll_400(:,7) = (/ 1.220_wp, 1.220_wp, 1.190_wp, 1.138_wp, &
913	1.086_wp, 1.086_wp, 1.0_wp /)
914	ecoll_400(:,8) = (/ 1.325_wp, 1.325_wp, 1.267_wp, 1.165_wp, &
915	1.063_wp, 1.063_wp, 1.0_wp /)
916	ecoll_400(:,9) = (/ 1.716_wp, 1.716_wp, 1.345_wp, 1.223_wp, &
917	1.100_wp, 1.100_wp, 1.0_wp /)
918	ecoll_400(:,10) = (/ 3.788_wp, 3.788_wp, 1.501_wp, 1.311_wp, &
919	1.120_wp, 1.120_wp, 1.0_wp /)
920	ecoll_400(:,11) = (/ 36.52_wp, 36.52_wp, 19.16_wp, 22.80_wp, &
921	26.0_wp, 26.0_wp, 1.0_wp /)
922
923	ENDIF
924
925	!
926	!-- Calculate the radius class index of particles with respect to array r0
927	!-- The droplet radius has to be given in microns.
928	ALLOCATE( ira(1:radius_classes) )
929
930	DO j = 1, radius_classes
931	particle_radius = radclass(j) * 1.0E6_wp
932	DO k = 1, 7
933	IF ( particle_radius < r0(k) ) THEN
934	ira(j) = k
935	EXIT
936	ENDIF
937	ENDDO
938	IF ( particle_radius >= r0(7) ) ira(j) = 8
939	ENDDO
940
941	!
942	!-- Two-dimensional linear interpolation of the turbulent enhancement factor.
943	!-- The droplet radius has to be given in microns.
944	DO j = 1, radius_classes
945	DO i = 1, j
946
947	ir = ira(j)
948	rq = MIN( radclass(i) / radclass(j), radclass(j) / radclass(i) )
949
950	DO kk = 2, 11
951	IF ( rq <= rat(kk) ) THEN
952	iq = kk
953	EXIT
954	ENDIF
955	ENDDO
956
957	y1 = 1.0_wp ! turbulent enhancement factor at 0 m2/s3
958
959	IF ( ir < 8 ) THEN
960	IF ( ir >= 2 ) THEN
961	pp = ( MAX( radclass(j), radclass(i) ) * 1.0E6_wp - &
962	r0(ir-1) ) / ( r0(ir) - r0(ir-1) )
963	qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
964	y2 = ( 1.0_wp - pp ) * ( 1.0_wp - qq ) * ecoll_100(ir-1,iq-1) + &
965	pp * ( 1.0_wp - qq ) * ecoll_100(ir,iq-1) + &
966	qq * ( 1.0_wp - pp ) * ecoll_100(ir-1,iq) + &
967	pp * qq * ecoll_100(ir,iq)
968	y3 = ( 1.0-pp ) * ( 1.0_wp - qq ) * ecoll_400(ir-1,iq-1) + &
969	pp * ( 1.0_wp - qq ) * ecoll_400(ir,iq-1) + &
970	qq * ( 1.0_wp - pp ) * ecoll_400(ir-1,iq) + &
971	pp * qq * ecoll_400(ir,iq)
972	ELSE
973	qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
974	y2 = ( 1.0_wp - qq ) * ecoll_100(1,iq-1) + qq * ecoll_100(1,iq)
975	y3 = ( 1.0_wp - qq ) * ecoll_400(1,iq-1) + qq * ecoll_400(1,iq)
976	ENDIF
977	ELSE
978	qq = ( rq - rat(iq-1) ) / ( rat(iq) - rat(iq-1) )
979	y2 = ( 1.0_wp - qq ) * ecoll_100(7,iq-1) + qq * ecoll_100(7,iq)
980	y3 = ( 1.0_wp - qq ) * ecoll_400(7,iq-1) + qq * ecoll_400(7,iq)
981	ENDIF
982	!
983	!-- Linear interpolation of turbulent enhancement factor
984	IF ( epsilon <= 0.01_wp ) THEN
985	ecf(j,i) = ( epsilon - 0.01_wp ) / ( 0.0_wp - 0.01_wp ) * y1 &
986	+ ( epsilon - 0.0_wp ) / ( 0.01_wp - 0.0_wp ) * y2
987	ELSEIF ( epsilon <= 0.06_wp ) THEN
988	ecf(j,i) = ( epsilon - 0.04_wp ) / ( 0.01_wp - 0.04_wp ) * y2 &
989	+ ( epsilon - 0.01_wp ) / ( 0.04_wp - 0.01_wp ) * y3
990	ELSE
991	ecf(j,i) = ( 0.06_wp - 0.04_wp ) / ( 0.01_wp - 0.04_wp ) * y2 &
992	+ ( 0.06_wp - 0.01_wp ) / ( 0.04_wp - 0.01_wp ) * y3
993	ENDIF
994
995	IF ( ecf(j,i) < 1.0_wp ) ecf(j,i) = 1.0_wp
996
997	ecf(i,j) = ecf(j,i)
998
999	ENDDO
1000	ENDDO
1001
1002	END SUBROUTINE turb_enhance_eff
1003
1004	END MODULE lpm_collision_kernels_mod

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