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source: palm/trunk/SOURCE/diffusion_v.f90 @ 4651

Last change on this file since 4651 was 4583, checked in by raasch, 4 years ago
files re-formatted to follow the PALM coding standard
Property svn:keywords set to `Id`
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1	!> @file diffusion_v.f90
2	!--------------------------------------------------------------------------------------------------!
3	! This file is part of the PALM model system.
4	!
5	! PALM is free software: you can redistribute it and/or modify it under the terms of the GNU General
6	! Public License as published by the Free Software Foundation, either version 3 of the License, or
7	! (at your option) any later version.
8	!
9	! PALM is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the
10	! implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
11	! Public License for more details.
12	!
13	! You should have received a copy of the GNU General Public License along with PALM. If not, see
14	! <http://www.gnu.org/licenses/>.
15	!
16	! Copyright 1997-2020 Leibniz Universitaet Hannover
17	!--------------------------------------------------------------------------------------------------!
18	!
19	! Current revisions:
20	! -----------------
21	!
22	!
23	! Former revisions:
24	! -----------------
25	! $Id: diffusion_v.f90 4583 2020-06-29 12:36:47Z raasch $
26	! file re-formatted to follow the PALM coding standard
27	!
28	! 4360 2020-01-07 11:25:50Z suehring
29	! Introduction of wall_flags_total_0, which currently sets bits based on static topography
30	! information used in wall_flags_static_0
31	!
32	! 4329 2019-12-10 15:46:36Z motisi
33	! Renamed wall_flags_0 to wall_flags_static_0
34	!
35	! 4182 2019-08-22 15:20:23Z scharf
36	! Corrected "Former revisions" section
37	!
38	! 3655 2019-01-07 16:51:22Z knoop
39	! OpenACC port for SPEC
40	!
41	! Revision 1.1 1997/09/12 06:24:01 raasch
42	! Initial revision
43	!
44	!
45	! Description:
46	! ------------
47	!> Diffusion term of the v-component
48	!--------------------------------------------------------------------------------------------------!
49	MODULE diffusion_v_mod
50
51
52	PRIVATE
53	PUBLIC diffusion_v
54
55	INTERFACE diffusion_v
56	MODULE PROCEDURE diffusion_v
57	MODULE PROCEDURE diffusion_v_ij
58	END INTERFACE diffusion_v
59
60	CONTAINS
61
62
63	!--------------------------------------------------------------------------------------------------!
64	! Description:
65	! ------------
66	!> Call for all grid points
67	!--------------------------------------------------------------------------------------------------!
68	SUBROUTINE diffusion_v
69
70	USE arrays_3d, &
71	ONLY: ddzu, ddzw, drho_air, km, rho_air_zw, tend, u, v, w
72
73	USE control_parameters, &
74	ONLY: constant_top_momentumflux, use_surface_fluxes, use_top_fluxes
75
76	USE grid_variables, &
77	ONLY: ddx, ddy, ddy2
78
79	USE indices, &
80	ONLY: nxl, nxr, nyn, nysv, nzb, nzt, wall_flags_total_0
81
82	USE kinds
83
84	USE surface_mod, &
85	ONLY : surf_def_h, surf_def_v, surf_lsm_h, surf_lsm_v, surf_usm_h, surf_usm_v
86
87	IMPLICIT NONE
88
89	INTEGER(iwp) :: i !< running index x direction
90	INTEGER(iwp) :: j !< running index y direction
91	INTEGER(iwp) :: k !< running index z direction
92	INTEGER(iwp) :: l !< running index of surface type, south- or north-facing wall
93	INTEGER(iwp) :: m !< running index surface elements
94	INTEGER(iwp) :: surf_e !< End index of surface elements at (j,i)-gridpoint
95	INTEGER(iwp) :: surf_s !< Start index of surface elements at (j,i)-gridpoint
96
97	REAL(wp) :: flag !< flag to mask topography grid points
98	REAL(wp) :: kmxm !< diffusion coefficient on leftward side of the v-gridbox - interpolated onto xu-yv grid
99	REAL(wp) :: kmxp !< diffusion coefficient on rightward side of the v-gridbox - interpolated onto xu-yv grid
100	REAL(wp) :: kmzm !< diffusion coefficient on bottom of the gridbox - interpolated onto yv-zw grid
101	REAL(wp) :: kmzp !< diffusion coefficient on top of the gridbox - interpolated onto yv-zw grid
102	REAL(wp) :: mask_bottom !< flag to mask vertical upward-facing surface
103	REAL(wp) :: mask_east !< flag to mask vertical surface south of the grid point
104	REAL(wp) :: mask_west !< flag to mask vertical surface north of the grid point
105	REAL(wp) :: mask_top !< flag to mask vertical downward-facing surface
106
107	!$ACC PARALLEL LOOP COLLAPSE(2) PRIVATE(i, j, k, l, m) &
108	!$ACC PRIVATE(surf_e, surf_s, flag, kmxm, kmxp, kmzm, kmzp) &
109	!$ACC PRIVATE(mask_bottom, mask_east, mask_west, mask_top) &
110	!$ACC PRESENT(wall_flags_total_0, km) &
111	!$ACC PRESENT(u, v, w) &
112	!$ACC PRESENT(ddzu, ddzw, drho_air, rho_air_zw) &
113	!$ACC PRESENT(surf_def_h(0:2), surf_def_v(2:3)) &
114	!$ACC PRESENT(surf_lsm_h, surf_lsm_v(2:3)) &
115	!$ACC PRESENT(surf_usm_h, surf_usm_v(0:3)) &
116	!$ACC PRESENT(tend)
117	DO i = nxl, nxr
118	DO j = nysv, nyn
119	!
120	!-- Compute horizontal diffusion
121	DO k = nzb+1, nzt
122
123	!
124	!-- Predetermine flag to mask topography and wall-bounded grid points.
125	!-- It is sufficient to masked only east- and west-facing surfaces, which need special
126	!-- treatment for the v-component.
127	flag = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k,j,i), 2 ) )
128	mask_east = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k,j,i+1), 2 ) )
129	mask_west = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k,j,i-1), 2 ) )
130	!
131	!-- Interpolate eddy diffusivities on staggered gridpoints
132	kmxp = 0.25_wp * ( km(k,j,i)+km(k,j,i+1)+km(k,j-1,i)+km(k,j-1,i+1) )
133	kmxm = 0.25_wp * ( km(k,j,i)+km(k,j,i-1)+km(k,j-1,i)+km(k,j-1,i-1) )
134
135	tend(k,j,i) = tend(k,j,i) + ( &
136	mask_east * kmxp * ( &
137	( v(k,j,i+1) - v(k,j,i) ) * ddx &
138	+ ( u(k,j,i+1) - u(k,j-1,i+1) ) * ddy &
139	) &
140	- mask_west * kmxm * ( &
141	( v(k,j,i) - v(k,j,i-1) ) * ddx &
142	+ ( u(k,j,i) - u(k,j-1,i) ) * ddy &
143	) &
144	) * ddx * flag &
145	+ 2.0_wp * ( &
146	km(k,j,i) * ( v(k,j+1,i) - v(k,j,i) ) &
147	- km(k,j-1,i) * ( v(k,j,i) - v(k,j-1,i) ) &
148	) * ddy2 * flag
149
150	ENDDO
151
152	!
153	!-- Add horizontal momentum flux v'u' at east- (l=2) and west-facing (l=3) surfaces. Note,
154	!-- in the the flat case, loops won't be entered as start_index > end_index. Furtermore,
155	!-- note, no vertical natural surfaces so far.
156	!-- Default-type surfaces
157	DO l = 2, 3
158	surf_s = surf_def_v(l)%start_index(j,i)
159	surf_e = surf_def_v(l)%end_index(j,i)
160	DO m = surf_s, surf_e
161	k = surf_def_v(l)%k(m)
162	tend(k,j,i) = tend(k,j,i) + surf_def_v(l)%mom_flux_uv(m) * ddx
163	ENDDO
164	ENDDO
165	!
166	!-- Natural-type surfaces
167	DO l = 2, 3
168	surf_s = surf_lsm_v(l)%start_index(j,i)
169	surf_e = surf_lsm_v(l)%end_index(j,i)
170	DO m = surf_s, surf_e
171	k = surf_lsm_v(l)%k(m)
172	tend(k,j,i) = tend(k,j,i) + surf_lsm_v(l)%mom_flux_uv(m) * ddx
173	ENDDO
174	ENDDO
175	!
176	!-- Urban-type surfaces
177	DO l = 2, 3
178	surf_s = surf_usm_v(l)%start_index(j,i)
179	surf_e = surf_usm_v(l)%end_index(j,i)
180	DO m = surf_s, surf_e
181	k = surf_usm_v(l)%k(m)
182	tend(k,j,i) = tend(k,j,i) + surf_usm_v(l)%mom_flux_uv(m) * ddx
183	ENDDO
184	ENDDO
185	!
186	!-- Compute vertical diffusion. In case of simulating a surface layer, respective grid
187	!-- diffusive fluxes are masked (flag 10) within this loop, and added further below, else,
188	!-- simple gradient approach is applied. Model top is also mask if top-momentum flux is
189	!-- given.
190	DO k = nzb+1, nzt
191	!
192	!-- Determine flags to mask topography below and above. Flag 2 is used to mask
193	!-- topography in general, while flag 8 implies also information about
194	!-- use_surface_fluxes. Flag 9 is used to control momentum flux at model top.
195	mask_bottom = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k-1,j,i), 8 ) )
196	mask_top = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k+1,j,i), 8 ) ) * &
197	MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k+1,j,i), 9 ) )
198	flag = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k,j,i), 2 ) )
199	!
200	!-- Interpolate eddy diffusivities on staggered gridpoints
201	kmzp = 0.25_wp * ( km(k,j,i)+km(k+1,j,i)+km(k,j-1,i)+km(k+1,j-1,i) )
202	kmzm = 0.25_wp * ( km(k,j,i)+km(k-1,j,i)+km(k,j-1,i)+km(k-1,j-1,i) )
203
204	tend(k,j,i) = tend(k,j,i) &
205	& + ( kmzp * ( ( v(k+1,j,i) - v(k,j,i) ) * ddzu(k+1) &
206	& + ( w(k,j,i) - w(k,j-1,i) ) * ddy &
207	& ) * rho_air_zw(k) * mask_top &
208	& - kmzm * ( ( v(k,j,i) - v(k-1,j,i) ) * ddzu(k) &
209	& + ( w(k-1,j,i) - w(k-1,j-1,i) ) * ddy &
210	& ) * rho_air_zw(k-1) * mask_bottom &
211	& ) * ddzw(k) * drho_air(k) * flag
212	ENDDO
213
214	!
215	!-- Vertical diffusion at the first grid point above the surface, if the momentum flux at
216	!-- the bottom is given by the Prandtl law or if it is prescribed by the user.
217	!-- Difference quotient of the momentum flux is not formed over half of the grid spacing
218	!-- (2.0*ddzw(k)) any more, since the comparison with other (LES) models showed that the
219	!-- values of the momentum flux becomes too large in this case.
220	IF ( use_surface_fluxes ) THEN
221	!
222	!-- Default-type surfaces, upward-facing
223	surf_s = surf_def_h(0)%start_index(j,i)
224	surf_e = surf_def_h(0)%end_index(j,i)
225	DO m = surf_s, surf_e
226	k = surf_def_h(0)%k(m)
227
228	tend(k,j,i) = tend(k,j,i) &
229	+ ( - ( - surf_def_h(0)%vsws(m) ) ) * ddzw(k) * drho_air(k)
230	ENDDO
231	!
232	!-- Default-type surfaces, dowward-facing
233	surf_s = surf_def_h(1)%start_index(j,i)
234	surf_e = surf_def_h(1)%end_index(j,i)
235	DO m = surf_s, surf_e
236	k = surf_def_h(1)%k(m)
237
238	tend(k,j,i) = tend(k,j,i) &
239	+ ( - surf_def_h(1)%vsws(m) ) * ddzw(k) * drho_air(k)
240	ENDDO
241	!
242	!-- Natural-type surfaces, upward-facing
243	surf_s = surf_lsm_h%start_index(j,i)
244	surf_e = surf_lsm_h%end_index(j,i)
245	DO m = surf_s, surf_e
246	k = surf_lsm_h%k(m)
247
248	tend(k,j,i) = tend(k,j,i) &
249	+ ( - ( - surf_lsm_h%vsws(m) ) ) * ddzw(k) * drho_air(k)
250
251	ENDDO
252	!
253	!-- Urban-type surfaces, upward-facing
254	surf_s = surf_usm_h%start_index(j,i)
255	surf_e = surf_usm_h%end_index(j,i)
256	DO m = surf_s, surf_e
257	k = surf_usm_h%k(m)
258
259	tend(k,j,i) = tend(k,j,i) &
260	+ ( - ( - surf_usm_h%vsws(m) ) ) * ddzw(k) * drho_air(k)
261
262	ENDDO
263	ENDIF
264	!
265	!-- Add momentum flux at model top
266	IF ( use_top_fluxes .AND. constant_top_momentumflux ) THEN
267	surf_s = surf_def_h(2)%start_index(j,i)
268	surf_e = surf_def_h(2)%end_index(j,i)
269	DO m = surf_s, surf_e
270
271	k = surf_def_h(2)%k(m)
272
273	tend(k,j,i) = tend(k,j,i) &
274	+ ( - surf_def_h(2)%vsws(m) ) * ddzw(k) * drho_air(k)
275	ENDDO
276	ENDIF
277
278	ENDDO
279	ENDDO
280
281	END SUBROUTINE diffusion_v
282
283
284	!--------------------------------------------------------------------------------------------------!
285	! Description:
286	! ------------
287	!> Call for grid point i,j
288	!--------------------------------------------------------------------------------------------------!
289	SUBROUTINE diffusion_v_ij( i, j )
290
291	USE arrays_3d, &
292	ONLY: ddzu, ddzw, drho_air, km, tend, u, v, w, rho_air_zw
293
294	USE control_parameters, &
295	ONLY: constant_top_momentumflux, use_surface_fluxes, use_top_fluxes
296
297	USE grid_variables, &
298	ONLY: ddx, ddy, ddy2
299
300	USE indices, &
301	ONLY: nzb, nzt, wall_flags_total_0
302
303	USE kinds
304
305	USE surface_mod, &
306	ONLY : surf_def_h, surf_def_v, surf_lsm_h, surf_lsm_v, surf_usm_h, surf_usm_v
307
308	IMPLICIT NONE
309
310
311	INTEGER(iwp) :: i !< running index x direction
312	INTEGER(iwp) :: j !< running index y direction
313	INTEGER(iwp) :: k !< running index z direction
314	INTEGER(iwp) :: l !< running index of surface type, south- or north-facing wall
315	INTEGER(iwp) :: m !< running index surface elements
316	INTEGER(iwp) :: surf_e !< End index of surface elements at (j,i)-gridpoint
317	INTEGER(iwp) :: surf_s !< Start index of surface elements at (j,i)-gridpoint
318
319	REAL(wp) :: flag !< flag to mask topography grid points
320	REAL(wp) :: kmxm !< diffusion coefficient on leftward side of the v-gridbox - interpolated onto xu-yv grid
321	REAL(wp) :: kmxp !< diffusion coefficient on rightward side of the v-gridbox - interpolated onto xu-yv grid
322	REAL(wp) :: kmzm !< diffusion coefficient on bottom of the gridbox - interpolated onto xu-zw grid
323	REAL(wp) :: kmzp !< diffusion coefficient on top of the gridbox - interpolated onto xu-zw grid
324	REAL(wp) :: mask_bottom !< flag to mask vertical upward-facing surface
325	REAL(wp) :: mask_east !< flag to mask vertical surface south of the grid point
326	REAL(wp) :: mask_west !< flag to mask vertical surface north of the grid point
327	REAL(wp) :: mask_top !< flag to mask vertical downward-facing surface
328
329	!
330	!-- Compute horizontal diffusion
331	DO k = nzb+1, nzt
332	!
333	!-- Predetermine flag to mask topography and wall-bounded grid points.
334	!-- It is sufficient to masked only east- and west-facing surfaces, which need special
335	!-- treatment for the v-component.
336	flag = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k,j,i), 2 ) )
337	mask_east = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k,j,i+1), 2 ) )
338	mask_west = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k,j,i-1), 2 ) )
339	!
340	!-- Interpolate eddy diffusivities on staggered gridpoints
341	kmxp = 0.25_wp * ( km(k,j,i)+km(k,j,i+1)+km(k,j-1,i)+km(k,j-1,i+1) )
342	kmxm = 0.25_wp * ( km(k,j,i)+km(k,j,i-1)+km(k,j-1,i)+km(k,j-1,i-1) )
343
344	tend(k,j,i) = tend(k,j,i) + ( &
345	mask_east * kmxp * ( &
346	( v(k,j,i+1) - v(k,j,i) ) * ddx &
347	+ ( u(k,j,i+1) - u(k,j-1,i+1) ) * ddy &
348	) &
349	- mask_west * kmxm * ( &
350	( v(k,j,i) - v(k,j,i-1) ) * ddx &
351	+ ( u(k,j,i) - u(k,j-1,i) ) * ddy &
352	) &
353	) * ddx * flag &
354	+ 2.0_wp * ( &
355	km(k,j,i) * ( v(k,j+1,i) - v(k,j,i) ) &
356	- km(k,j-1,i) * ( v(k,j,i) - v(k,j-1,i) ) &
357	) * ddy2 * flag
358	ENDDO
359
360	!
361	!-- Add horizontal momentum flux v'u' at east- (l=2) and west-facing (l=3) surfaces. Note, in the
362	!-- the flat case, loops won't be entered as start_index > end_index. Furtermore, note, no
363	!-- vertical natural surfaces so far.
364	!-- Default-type surfaces
365	DO l = 2, 3
366	surf_s = surf_def_v(l)%start_index(j,i)
367	surf_e = surf_def_v(l)%end_index(j,i)
368	DO m = surf_s, surf_e
369	k = surf_def_v(l)%k(m)
370	tend(k,j,i) = tend(k,j,i) + surf_def_v(l)%mom_flux_uv(m) * ddx
371	ENDDO
372	ENDDO
373	!
374	!-- Natural-type surfaces
375	DO l = 2, 3
376	surf_s = surf_lsm_v(l)%start_index(j,i)
377	surf_e = surf_lsm_v(l)%end_index(j,i)
378	DO m = surf_s, surf_e
379	k = surf_lsm_v(l)%k(m)
380	tend(k,j,i) = tend(k,j,i) + surf_lsm_v(l)%mom_flux_uv(m) * ddx
381	ENDDO
382	ENDDO
383	!
384	!-- Urban-type surfaces
385	DO l = 2, 3
386	surf_s = surf_usm_v(l)%start_index(j,i)
387	surf_e = surf_usm_v(l)%end_index(j,i)
388	DO m = surf_s, surf_e
389	k = surf_usm_v(l)%k(m)
390	tend(k,j,i) = tend(k,j,i) + surf_usm_v(l)%mom_flux_uv(m) * ddx
391	ENDDO
392	ENDDO
393	!
394	!-- Compute vertical diffusion. In case of simulating a surface layer, respective grid diffusive
395	!-- fluxes are masked (flag 8) within this loop, and added further below, else, simple gradient
396	!-- approach is applied. Model top is also mask if top-momentum flux is given.
397	DO k = nzb+1, nzt
398	!
399	!-- Determine flags to mask topography below and above. Flag 2 is used to mask topography in
400	!-- general, while flag 10 implies also information about use_surface_fluxes. Flag 9 is used
401	!-- to control momentum flux at model top.
402	mask_bottom = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k-1,j,i), 8 ) )
403	mask_top = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k+1,j,i), 8 ) ) * &
404	MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k+1,j,i), 9 ) )
405	flag = MERGE( 1.0_wp, 0.0_wp, BTEST( wall_flags_total_0(k,j,i), 2 ) )
406	!
407	!-- Interpolate eddy diffusivities on staggered gridpoints
408	kmzp = 0.25_wp * ( km(k,j,i)+km(k+1,j,i)+km(k,j-1,i)+km(k+1,j-1,i) )
409	kmzm = 0.25_wp * ( km(k,j,i)+km(k-1,j,i)+km(k,j-1,i)+km(k-1,j-1,i) )
410
411	tend(k,j,i) = tend(k,j,i) &
412	& + ( kmzp * ( ( v(k+1,j,i) - v(k,j,i) ) * ddzu(k+1) &
413	& + ( w(k,j,i) - w(k,j-1,i) ) * ddy &
414	& ) * rho_air_zw(k) * mask_top &
415	& - kmzm * ( ( v(k,j,i) - v(k-1,j,i) ) * ddzu(k) &
416	& + ( w(k-1,j,i) - w(k-1,j-1,i) ) * ddy &
417	& ) * rho_air_zw(k-1) * mask_bottom &
418	& ) * ddzw(k) * drho_air(k) * flag
419	ENDDO
420
421	!
422	!-- Vertical diffusion at the first grid point above the surface, if the momentum flux at the
423	!-- bottom is given by the Prandtl law or if it is prescribed by the user.
424	!-- Difference quotient of the momentum flux is not formed over half of the grid spacing
425	!-- (2.0*ddzw(k)) any more, since the comparison with other (LES) models showed that the values
426	!-- of the momentum flux becomes too large in this case.
427	IF ( use_surface_fluxes ) THEN
428	!
429	!-- Default-type surfaces, upward-facing
430	surf_s = surf_def_h(0)%start_index(j,i)
431	surf_e = surf_def_h(0)%end_index(j,i)
432	DO m = surf_s, surf_e
433	k = surf_def_h(0)%k(m)
434
435	tend(k,j,i) = tend(k,j,i) + ( - ( - surf_def_h(0)%vsws(m) ) ) * ddzw(k) * drho_air(k)
436	ENDDO
437	!
438	!-- Default-type surfaces, dowward-facing
439	surf_s = surf_def_h(1)%start_index(j,i)
440	surf_e = surf_def_h(1)%end_index(j,i)
441	DO m = surf_s, surf_e
442	k = surf_def_h(1)%k(m)
443
444	tend(k,j,i) = tend(k,j,i) + ( - surf_def_h(1)%vsws(m) ) * ddzw(k) * drho_air(k)
445	ENDDO
446	!
447	!-- Natural-type surfaces, upward-facing
448	surf_s = surf_lsm_h%start_index(j,i)
449	surf_e = surf_lsm_h%end_index(j,i)
450	DO m = surf_s, surf_e
451	k = surf_lsm_h%k(m)
452
453	tend(k,j,i) = tend(k,j,i) + ( - ( - surf_lsm_h%vsws(m) ) ) * ddzw(k) * drho_air(k)
454
455	ENDDO
456	!
457	!-- Urban-type surfaces, upward-facing
458	surf_s = surf_usm_h%start_index(j,i)
459	surf_e = surf_usm_h%end_index(j,i)
460	DO m = surf_s, surf_e
461	k = surf_usm_h%k(m)
462
463	tend(k,j,i) = tend(k,j,i) + ( - ( - surf_usm_h%vsws(m) ) ) * ddzw(k) * drho_air(k)
464
465	ENDDO
466	ENDIF
467	!
468	!-- Add momentum flux at model top
469	IF ( use_top_fluxes .AND. constant_top_momentumflux ) THEN
470	surf_s = surf_def_h(2)%start_index(j,i)
471	surf_e = surf_def_h(2)%end_index(j,i)
472	DO m = surf_s, surf_e
473
474	k = surf_def_h(2)%k(m)
475
476	tend(k,j,i) = tend(k,j,i) + ( - surf_def_h(2)%vsws(m) ) * ddzw(k) * drho_air(k)
477	ENDDO
478	ENDIF
479
480	END SUBROUTINE diffusion_v_ij
481
482	END MODULE diffusion_v_mod

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