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lmdz_atke_exchange_coeff.f90 @ 5422

Last change on this file since 5422 was 5268, checked in by abarral, 2 months ago
.f90 <-> .F90 depending on cpp key use
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1	module lmdz_atke_exchange_coeff
2
3	implicit none
4
5	contains
6
7	subroutine atke_compute_km_kh(ngrid,nlay,dtime, &
8	wind_u,wind_v,temp,qvap,play,pinterf,cdrag_uv, &
9	tke,eps,tke_shear,tke_buoy,tke_trans,Km_out,Kh_out)
10
11	!========================================================================
12	! Routine that computes turbulent Km / Kh coefficients with a
13	! 1.5 order closure scheme (TKE) with or without stationarity assumption
14	!
15	! This parameterization has been constructed in the framework of a
16	! collective and collaborative workshop,
17	! the so-called 'Atelier TKE (ATKE)' with
18	! K. Arjdal, L. Raillard, C. Dehondt, P. Tiengou, A. Spiga, F. Cheruy, T Dubos,
19	! M. Coulon-Decorzens, S. Fromang, G. Riviere, A. Sima, F. Hourdin, E. Vignon
20	!
21	! Main assumptions of the model :
22	! (1) horizontal homogeneity (Dx=Dy=0.)
23	!=======================================================================
24
25	USE lmdz_atke_turbulence_ini, ONLY : iflag_atke, kappa, l0, ric, cinf, rpi, rcpd, atke_ok_virtual, ri0, ri1
26	USE lmdz_atke_turbulence_ini, ONLY : cepsilon, pr_slope, pr_asym, pr_neut, ctkes, rg, rd, rv, atke_ok_vdiff
27	USE lmdz_atke_turbulence_ini, ONLY : viscom, viscoh, clmix, clmixshear, iflag_atke_lmix, lmin, smmin, cn
28
29	!!-------------------------------------------------------------------------------------------------------------
30	! integer :: iflag_atke ! flag that controls options in atke_compute_km_kh
31	! integer :: iflag_atke_lmix ! flag that controls the calculation of mixing length in atke
32	! integer :: iflag_num_atke ! flag that controls the numerical treatment of diffusion coeffiient calculation
33
34	! logical :: atke_ok_vdiff ! activate vertical diffusion of TKE or not
35	! logical :: atke_ok_virtual ! account for vapor for flottability
36
37	! real :: kappa = 0.4 ! Von Karman constant
38	! real :: cn ! Sm value at Ri=0
39	! real :: cinf ! Sm value at Ri=-Inf
40	! real :: ri0 ! Richardson number near zero to guarantee continuity in slope of Sm (stability function) at Ri=0
41	! real :: ri1 ! Richardson number near zero to guarantee continuity in slope of Pr (Prandlt's number) at Ri=0
42	! real :: lmin ! minimum mixing length corresponding to the Kolmogov dissipation scale in planetary atmospheres (Chen et al 2016, JGR Atmos)
43	! real :: ctkes ! coefficient for surface TKE
44	! real :: clmixshear ! coefficient for mixing length depending on local wind shear
45
46	! Tunable parameters for the ATKE scheme and their range of values
47	!!-------------------------------------------------------------------------------------------------------------
48	! real :: cepsilon ! controls the value of the dissipation length scale, range [1.2 - 10]
49	! real :: cke ! controls the value of the diffusion coefficient of TKE, range [1 - 5]
50	! real :: l0 ! asymptotic mixing length far from the ground [m] (Sun et al 2011, JAMC), range [15 - 75]
51	! real :: clmix ! controls the value of the mixing length in stratified conditions, range [0.1 - 2]
52	! real :: ric ! critical Richardson number controlling the slope of Sm in stable conditions, range [0.19 - 0.25]
53	! real :: smmin ! minimum value of Sm in very stable conditions (defined here as minsqrt(Ez/Ek)) at large Ri, range [0.025 - 0.1]
54	! real :: pr_neut ! neutral value of the Prandtl number (Ri=0), range [0.7 - 1]
55	! real :: pr_slope ! linear slope of Pr with Ri in the very stable regime, range [3 - 5]
56	! real :: cinffac ! ratio between cinf and cn controlling the convective limit of Sm, range [1.2 - 5.0]
57	! real :: pr_asym ! value of Prandlt in the convective limit(Ri=-Inf), range [0.3 - 0.5]
58	!!-------------------------------------------------------------------------------------------------------------
59
60	implicit none
61
62
63	! Declarations:
64	!=============
65
66	INTEGER, INTENT(IN) :: ngrid ! number of horizontal index (flat grid)
67	INTEGER, INTENT(IN) :: nlay ! number of vertical index
68
69	REAL, INTENT(IN) :: dtime ! physics time step (s)
70	REAL, DIMENSION(ngrid,nlay), INTENT(IN) :: wind_u ! zonal velocity (m/s)
71	REAL, DIMENSION(ngrid,nlay), INTENT(IN) :: wind_v ! meridional velocity (m/s)
72	REAL, DIMENSION(ngrid,nlay), INTENT(IN) :: temp ! temperature (K)
73	REAL, DIMENSION(ngrid,nlay), INTENT(IN) :: qvap ! specific humidity (kg/kg)
74	REAL, DIMENSION(ngrid,nlay), INTENT(IN) :: play ! pressure (Pa)
75	REAL, DIMENSION(ngrid,nlay+1), INTENT(IN) :: pinterf ! pressure at interfaces(Pa)
76	REAL, DIMENSION(ngrid), INTENT(IN) :: cdrag_uv ! surface drag coefficient for momentum
77
78	REAL, DIMENSION(ngrid,nlay+1), INTENT(INOUT) :: tke ! turbulent kinetic energy at interface between layers (m2/s2)
79
80	REAL, DIMENSION(ngrid,nlay+1), INTENT(OUT) :: eps ! output: TKE dissipation rate at interface between layers (m2/s3)
81	REAL, DIMENSION(ngrid,nlay+1), INTENT(OUT) :: tke_shear! output: TKE shear production rate (m2/s3)
82	REAL, DIMENSION(ngrid,nlay+1), INTENT(OUT) :: tke_buoy ! output: TKE buoyancy production rate (m2/s3)
83	REAL, DIMENSION(ngrid,nlay+1), INTENT(OUT) :: tke_trans! output: TKE transport (diffusion) term (m2/s3)
84	REAL, DIMENSION(ngrid,nlay), INTENT(OUT) :: Km_out ! output: Exchange coefficient for momentum at interface between layers (m2/s)
85	REAL, DIMENSION(ngrid,nlay), INTENT(OUT) :: Kh_out ! output: Exchange coefficient for heat flux at interface between layers (m2/s)
86
87	! Local variables
88	REAL, DIMENSION(ngrid,nlay+1) :: Km ! Exchange coefficient for momentum at interface between layers
89	REAL, DIMENSION(ngrid,nlay+1) :: Kh ! Exchange coefficient for heat flux at interface between layers
90	REAL, DIMENSION(ngrid,nlay) :: theta ! Potential temperature
91	REAL, DIMENSION(ngrid,nlay+1) :: l_exchange ! Length of exchange (at interface)
92	REAL, DIMENSION(ngrid,nlay+1) :: z_interf ! Altitude at the interface
93	REAL, DIMENSION(ngrid,nlay) :: z_lay ! Altitude of layers
94	REAL, DIMENSION(ngrid,nlay) :: dz_interf ! distance between two consecutive interfaces
95	REAL, DIMENSION(ngrid,nlay) :: dz_lay ! distance between two layer middles (NB: first and last are half layers)
96	REAL, DIMENSION(ngrid,nlay+1) :: N2 ! square of Brunt Vaisala pulsation (at interface)
97	REAL, DIMENSION(ngrid,nlay+1) :: shear2 ! square of wind shear (at interface)
98	REAL, DIMENSION(ngrid,nlay+1) :: Ri ! Richardson's number (at interface)
99	REAL, DIMENSION(ngrid,nlay+1) :: Prandtl ! Turbulent Prandtl's number (at interface)
100	REAL, DIMENSION(ngrid,nlay+1) :: Sm ! Stability function for momentum (at interface)
101	REAL, DIMENSION(ngrid,nlay+1) :: Sh ! Stability function for heat (at interface)
102
103	INTEGER :: igrid,ilay ! horizontal,vertical index (flat grid)
104	REAL :: preff ! reference pressure for potential temperature calculations
105	REAL :: thetam ! mean potential temperature at interface
106	REAL :: delta ! discriminant of the second order polynomial
107	REAL :: qq ! tke=qq**2/2
108	REAL :: shear ! wind shear
109	REAL :: lstrat ! mixing length depending on local stratification
110	REAL :: taustrat ! caracteristic timescale for turbulence in very stable conditions
111	REAL :: netloss ! net loss term of tke
112	REAL :: netsource ! net source term of tke
113	REAL :: ustar ! friction velocity estimation
114	REAL :: invtau
115	REAL :: rvap
116
117	! Initializations:
118	!================
119
120	DO igrid=1,ngrid
121	dz_interf(igrid,1) = 0.0
122	z_interf(igrid,1) = 0.0
123	END DO
124
125	! Calculation of potential temperature: (if vapor -> virtual potential temperature)
126	!=====================================
127
128	preff=100000.
129	! results should not depend on the choice of preff
130	DO ilay=1,nlay
131	DO igrid = 1, ngrid
132	theta(igrid,ilay)=temp(igrid,ilay)(preff/play(igrid,ilay))*(rd/rcpd)
133	END DO
134	END DO
135
136	! account for water vapor mass for buoyancy calculation
137	IF (atke_ok_virtual) THEN
138	DO ilay=1,nlay
139	DO igrid = 1, ngrid
140	rvap=max(0.,qvap(igrid,ilay)/(1.-qvap(igrid,ilay)))
141	theta(igrid,ilay)=theta(igrid,ilay)*(1.+rvap/(RD/RV))/(1.+rvap)
142	END DO
143	END DO
144	ENDIF
145
146
147	! Calculation of altitude of layers' middle and bottom interfaces:
148	!=================================================================
149
150	DO ilay=2,nlay+1
151	DO igrid=1,ngrid
152	dz_interf(igrid,ilay-1) = rdtemp(igrid,ilay-1)/rg/play(igrid,ilay-1)(pinterf(igrid,ilay-1)-pinterf(igrid,ilay))
153	z_interf(igrid,ilay) = z_interf(igrid,ilay-1) + dz_interf(igrid,ilay-1)
154	ENDDO
155	ENDDO
156
157	DO ilay=1,nlay
158	DO igrid=1,ngrid
159	z_lay(igrid,ilay)=0.5*(z_interf(igrid, ilay+1) + z_interf(igrid, ilay))
160	ENDDO
161	ENDDO
162
163
164	! Computes the gradient Richardson's number and stability functions:
165	!===================================================================
166
167	DO ilay=2,nlay
168	DO igrid=1,ngrid
169	dz_lay(igrid,ilay)=z_lay(igrid,ilay)-z_lay(igrid,ilay-1)
170	thetam=0.5*(theta(igrid,ilay) + theta(igrid,ilay-1))
171	N2(igrid,ilay) = rg * (theta(igrid,ilay) - theta(igrid,ilay-1))/thetam / dz_lay(igrid,ilay)
172	shear2(igrid,ilay)= (((wind_u(igrid,ilay) - wind_u(igrid,ilay-1)) / dz_lay(igrid,ilay))**2 + &
173	((wind_v(igrid,ilay) - wind_v(igrid,ilay-1)) / dz_lay(igrid,ilay))**2 )
174	Ri(igrid,ilay) = N2(igrid,ilay) / MAX(shear2(igrid,ilay),1E-10)
175
176	IF (Ri(igrid,ilay) < 0.) THEN ! unstable cases
177	Sm(igrid,ilay) = 2./rpi * (cinf-cn) * atan(-Ri(igrid,ilay)/Ri0) + cn
178	Prandtl(igrid,ilay) = -2./rpi * (pr_asym - pr_neut) * atan(Ri(igrid,ilay)/Ri1) + pr_neut
179	ELSE ! stable cases
180	Sm(igrid,ilay) = max(smmin,cn*(1.-Ri(igrid,ilay)/Ric))
181	! prandlt expression from venayagamoorthy and stretch 2010, Li et al 2019
182	Prandtl(igrid,ilay) = pr_neutexp(-pr_slope/pr_neutRi(igrid,ilay)+Ri(igrid,ilay)/pr_neut) &
183	+ Ri(igrid,ilay) * pr_slope
184	IF (Ri(igrid,ilay) .GE. Prandtl(igrid,ilay)) THEN
185	call abort_physic("atke_compute_km_kh", &
186	'Ri>=Pr in stable conditions -> violates energy conservation principles, change pr_neut or slope', 1)
187	ENDIF
188	END IF
189
190	Sh(igrid,ilay) = Sm(igrid,ilay) / Prandtl(igrid,ilay)
191
192	ENDDO
193	ENDDO
194
195
196	! Computing the mixing length:
197	!==============================================================
198
199
200	IF (iflag_atke_lmix .EQ. 1 ) THEN
201	! Blackadar formulation (~kappa l) + buoyancy length scale (Deardoff 1980) for very stable conditions
202	DO ilay=2,nlay
203	DO igrid=1,ngrid
204	l_exchange(igrid,ilay) = kappal0z_interf(igrid,ilay) / (kappa*z_interf(igrid,ilay) + l0)
205	IF (N2(igrid,ilay) .GT. 0.) THEN
206	lstrat=clmix*sqrt(tke(igrid,ilay))/sqrt(N2(igrid,ilay))
207	lstrat=max(lstrat,lmin)
208	!Inverse interpolation, Van de Wiel et al. 2010
209	l_exchange(igrid,ilay)=(1./(l_exchange(igrid,ilay))+1./(lstrat))**(-1.0)
210	ENDIF
211	ENDDO
212	ENDDO
213
214	ELSE IF (iflag_atke_lmix .EQ. 2 ) THEN
215	! add effect of wind shear on lstrat following grisogono and belusic 2008, qjrms
216	! implies 2 tuning coefficients clmix and clmixshear
217	DO ilay=2,nlay
218	DO igrid=1,ngrid
219	l_exchange(igrid,ilay) = kappal0z_interf(igrid,ilay) / (kappa*z_interf(igrid,ilay) + l0)
220	IF (N2(igrid,ilay) .GT. 0. .AND. shear2(igrid,ilay) .GT. 0.) THEN
221	lstrat=min(clmix*sqrt(tke(igrid,ilay))/sqrt(N2(igrid,ilay)), &
222	clmixshear*sqrt(tke(igrid,ilay))/sqrt(shear2(igrid,ilay)))
223	lstrat=max(lstrat,lmin)
224	!Inverse interpolation, Van de Wiel et al. 2010
225	l_exchange(igrid,ilay)=(1./(l_exchange(igrid,ilay))+1./(lstrat))**(-1.0)
226	ENDIF
227	ENDDO
228	ENDDO
229
230	ELSE IF (iflag_atke_lmix .EQ. 3 ) THEN
231	! add effect of wind shear on lstrat following grisogono 2010, qjrms
232	! keeping a single tuning coefficient clmix
233	DO ilay=2,nlay
234	DO igrid=1,ngrid
235	l_exchange(igrid,ilay) = kappal0z_interf(igrid,ilay) / (kappa*z_interf(igrid,ilay) + l0)
236	IF (N2(igrid,ilay) .GT. 0. .AND. shear2(igrid,ilay) .GT. 0.) THEN
237	lstrat=clmixsqrt(tke(igrid,ilay))/sqrt(shear2(igrid,ilay))(1.0+Ri(igrid,ilay)/(2.*Prandtl(igrid,ilay)))
238	lstrat=max(lstrat,lmin)
239	!Inverse interpolation, Van de Wiel et al. 2010
240	l_exchange(igrid,ilay)=(1./(l_exchange(igrid,ilay))+1./(lstrat))**(-1.0)
241	ENDIF
242	ENDDO
243	ENDDO
244
245	ELSE
246	! default Blackadar formulation: neglect effect of local stratification and shear
247	DO ilay=2,nlay+1
248	DO igrid=1,ngrid
249	l_exchange(igrid,ilay) = kappal0z_interf(igrid,ilay) / (kappa*z_interf(igrid,ilay) + l0)
250	ENDDO
251	ENDDO
252	ENDIF
253
254
255	! Computing the TKE k>=2:
256	!========================
257	IF (iflag_atke == 0) THEN
258
259	! stationary solution (dtke/dt=0)
260
261	DO ilay=2,nlay
262	DO igrid=1,ngrid
263	tke(igrid,ilay) = cepsilon * l_exchange(igrid,ilay)*2 Sm(igrid,ilay) * &
264	shear2(igrid,ilay) * (1. - Ri(igrid,ilay) / Prandtl(igrid,ilay))
265	eps(igrid,ilay) = (tke(igrid,ilay)*(3./2))/(cepsilonl_exchange(igrid,ilay))
266	tke_shear(igrid,ilay)=l_exchange(igrid,ilay)Sm(igrid,ilay)sqrt(tke(igrid,ilay))*shear2(igrid,ilay)
267	tke_buoy(igrid,ilay)=-l_exchange(igrid,ilay)Sm(igrid,ilay)sqrt(tke(igrid,ilay))*shear2(igrid,ilay) &
268	*(Ri(igrid,ilay) / Prandtl(igrid,ilay))
269	ENDDO
270	ENDDO
271
272	ELSE IF (iflag_atke == 1) THEN
273
274	! full implicit scheme resolved with a second order polynomial equation
275	! default solution which shows fair convergence properties
276	DO ilay=2,nlay
277	DO igrid=1,ngrid
278	qq=max(sqrt(2.*tke(igrid,ilay)),1.e-10)
279	delta=(2.sqrt(2.)cepsilonl_exchange(igrid,ilay)/dtime)*2. &
280	+4.(2.sqrt(2.)cepsilonl_exchange(igrid,ilay)/dtime*qq + &
281	2.l_exchange(igrid,ilay)l_exchange(igrid,ilay)cepsilonSm(igrid,ilay) &
282	shear2(igrid,ilay) (1. - Ri(igrid,ilay) / Prandtl(igrid,ilay)))
283	qq=(-2.sqrt(2.)cepsilon*l_exchange(igrid,ilay)/dtime + sqrt(delta))/2.
284	qq=max(0.,qq)
285	tke(igrid,ilay)=0.5(qq*2)
286	eps(igrid,ilay) = (tke(igrid,ilay)*(3./2))/(cepsilonl_exchange(igrid,ilay))
287	tke_shear(igrid,ilay)=l_exchange(igrid,ilay)Sm(igrid,ilay)sqrt(tke(igrid,ilay))*shear2(igrid,ilay)
288	tke_buoy(igrid,ilay)=-l_exchange(igrid,ilay)Sm(igrid,ilay)sqrt(tke(igrid,ilay))*shear2(igrid,ilay) &
289	*(Ri(igrid,ilay) / Prandtl(igrid,ilay))
290	ENDDO
291	ENDDO
292
293
294	ELSE IF (iflag_atke == 2) THEN
295
296	! semi implicit scheme when l does not depend on tke
297	! positive-guaranteed if pr slope in stable condition >1
298
299	DO ilay=2,nlay
300	DO igrid=1,ngrid
301	qq=max(sqrt(2.*tke(igrid,ilay)),1.e-10)
302	qq=(qq+l_exchange(igrid,ilay)Sm(igrid,ilay)dtime/sqrt(2.) &
303	shear2(igrid,ilay)(1.-Ri(igrid,ilay)/Prandtl(igrid,ilay))) &
304	/(1.+qqdtime/(cepsilonl_exchange(igrid,ilay)2.sqrt(2.)))
305	tke_shear(igrid,ilay)=l_exchange(igrid,ilay)Sm(igrid,ilay)qq/sqrt(2.)*shear2(igrid,ilay)
306	tke_buoy(igrid,ilay)=-l_exchange(igrid,ilay)Sm(igrid,ilay)qq/sqrt(2.)*shear2(igrid,ilay) &
307	*(Ri(igrid,ilay) / Prandtl(igrid,ilay))
308	tke(igrid,ilay)=0.5(qq*2)
309	eps(igrid,ilay) = (tke(igrid,ilay)*(3./2))/(cepsilonl_exchange(igrid,ilay))
310	ENDDO
311	ENDDO
312
313
314	ELSE IF (iflag_atke == 3) THEN
315	! numerical resolution adapted from that in MAR (Deleersnijder 1992)
316	! positively defined by construction
317
318	DO ilay=2,nlay
319	DO igrid=1,ngrid
320	eps(igrid,ilay) = (tke(igrid,ilay)*(3./2))/(cepsilonl_exchange(igrid,ilay))
321	qq=max(sqrt(2.*tke(igrid,ilay)),1.e-10)
322	tke_shear(igrid,ilay)=l_exchange(igrid,ilay)Sm(igrid,ilay)qq/sqrt(2.)*shear2(igrid,ilay)
323	tke_buoy(igrid,ilay)=-l_exchange(igrid,ilay)Sm(igrid,ilay)qq/sqrt(2.)*shear2(igrid,ilay) &
324	*(Ri(igrid,ilay) / Prandtl(igrid,ilay))
325	IF (Ri(igrid,ilay) .LT. 0.) THEN
326	netloss=qq/(2.sqrt(2.)cepsilon*l_exchange(igrid,ilay))
327	netsource=l_exchange(igrid,ilay)Sm(igrid,ilay)/sqrt(2.)shear2(igrid,ilay)*(1.-Ri(igrid,ilay)/Prandtl(igrid,ilay))
328	ELSE
329	netloss=qq/(2.sqrt(2.)cepsilon*l_exchange(igrid,ilay))+ &
330	l_exchange(igrid,ilay)Sm(igrid,ilay)/sqrt(2.)N2(igrid,ilay)/Prandtl(igrid,ilay)
331	netsource=l_exchange(igrid,ilay)Sm(igrid,ilay)/sqrt(2.)shear2(igrid,ilay)
332	ENDIF
333	qq=((qq*2)/dtime+qqnetsource)/(qq/dtime+netloss)
334	tke(igrid,ilay)=0.5(qq*2)
335	ENDDO
336	ENDDO
337
338	ELSE IF (iflag_atke == 4) THEN
339	! semi implicit scheme from Arpege (V. Masson methodology with
340	! Taylor expansion of the dissipation term)
341	DO ilay=2,nlay
342	DO igrid=1,ngrid
343	qq=max(sqrt(2.*tke(igrid,ilay)),1.e-10)
344	tke_shear(igrid,ilay)=l_exchange(igrid,ilay)Sm(igrid,ilay)qq/sqrt(2.)*shear2(igrid,ilay)
345	tke_buoy(igrid,ilay)=-l_exchange(igrid,ilay)Sm(igrid,ilay)qq/sqrt(2.)*shear2(igrid,ilay) &
346	*(Ri(igrid,ilay) / Prandtl(igrid,ilay))
347	qq=(l_exchange(igrid,ilay)Sm(igrid,ilay)/sqrt(2.)shear2(igrid,ilay)*(1.-Ri(igrid,ilay)/Prandtl(igrid,ilay)) &
348	+qq(1.+dtimeqq/(cepsilonl_exchange(igrid,ilay)2.*sqrt(2.)))) &
349	/(1.+2.qqdtime/(cepsilonl_exchange(igrid,ilay)2.*sqrt(2.)))
350	qq=max(0.,qq)
351	tke(igrid,ilay)=0.5(qq*2)
352	eps(igrid,ilay) = (tke(igrid,ilay)*(3./2))/(cepsilonl_exchange(igrid,ilay))
353	ENDDO
354	ENDDO
355
356
357	ELSE
358	call abort_physic("atke_compute_km_kh", &
359	'numerical treatment of TKE not possible yet', 1)
360
361	END IF
362
363	! We impose a 0 tke at nlay+1
364	!==============================
365
366	DO igrid=1,ngrid
367	tke(igrid,nlay+1)=0.
368	eps(igrid,nlay+1)=0.
369	tke_shear(igrid,nlay+1)=0.
370	tke_buoy(igrid,nlay+1)=0.
371	END DO
372
373
374	! Calculation of surface TKE (k=1)
375	!=================================
376	! surface TKE calculation inspired from what is done in Arpege (see E. Bazile note)
377	DO igrid=1,ngrid
378	ustar=sqrt(cdrag_uv(igrid)(wind_u(igrid,1)2+wind_v(igrid,1)*2))
379	tke(igrid,1)=ctkes(ustar*2)
380	eps(igrid,1)=0. ! arbitrary as TKE is not properly defined at the surface
381	tke_shear(igrid,1)=0.
382	tke_buoy(igrid,1)=0.
383	END DO
384
385
386	! vertical diffusion of TKE
387	!==========================
388	tke_trans(:,:)=0.
389	IF (atke_ok_vdiff) THEN
390	CALL atke_vdiff_tke(ngrid,nlay,dtime,z_lay,z_interf,temp,play,l_exchange,Sm,tke,tke_trans)
391	ENDIF
392
393
394	! Computing eddy diffusivity coefficients:
395	!========================================
396	DO ilay=2,nlay ! TODO: also calculate for nlay+1 ?
397	DO igrid=1,ngrid
398	! we add the molecular viscosity to Km,h
399	Km(igrid,ilay) = viscom + l_exchange(igrid,ilay) * Sm(igrid,ilay) * tke(igrid,ilay)**0.5
400	Kh(igrid,ilay) = viscoh + l_exchange(igrid,ilay) * Sh(igrid,ilay) * tke(igrid,ilay)**0.5
401	END DO
402	END DO
403
404	! for output:
405	!===========
406	Km_out(1:ngrid,2:nlay)=Km(1:ngrid,2:nlay)
407	Kh_out(1:ngrid,2:nlay)=Kh(1:ngrid,2:nlay)
408
409	end subroutine atke_compute_km_kh
410
411	!===============================================================================================
412	subroutine atke_vdiff_tke(ngrid,nlay,dtime,z_lay,z_interf,temp,play,l_exchange,Sm,tke,tke_trans)
413
414	! routine that computes the vertical diffusion of TKE by the turbulence
415	! using an implicit resolution (See note by Dufresne and Ghattas (2009))
416	! E Vignon, July 2023
417
418	USE lmdz_atke_turbulence_ini, ONLY : rd, cke, viscom
419
420
421	INTEGER, INTENT(IN) :: ngrid ! number of horizontal index (flat grid)
422	INTEGER, INTENT(IN) :: nlay ! number of vertical index
423
424	REAL, INTENT(IN) :: dtime ! physics time step (s)
425	REAL, DIMENSION(ngrid,nlay), INTENT(IN) :: z_lay ! altitude of mid-layers (m)
426	REAL, DIMENSION(ngrid,nlay+1), INTENT(IN) :: z_interf ! altitude of bottom interfaces (m)
427	REAL, DIMENSION(ngrid,nlay), INTENT(IN) :: temp ! temperature (K)
428	REAL, DIMENSION(ngrid,nlay), INTENT(IN) :: play ! pressure (Pa)
429	REAL, DIMENSION(ngrid,nlay+1), INTENT(IN) :: l_exchange ! mixing length at interfaces between layers
430	REAL, DIMENSION(ngrid,nlay+1), INTENT(IN) :: Sm ! stability function for eddy diffusivity for momentum at interface between layers
431
432	REAL, DIMENSION(ngrid,nlay+1), INTENT(INOUT) :: tke ! turbulent kinetic energy at interface between layers
433	REAL, DIMENSION(ngrid,nlay+1), INTENT(INOUT) :: tke_trans ! turbulent kinetic energy transport term (m2/s3)
434
435
436	INTEGER :: igrid,ilay
437	REAL, DIMENSION(ngrid,nlay+1) :: Ke ! eddy diffusivity for TKE
438	REAL, DIMENSION(ngrid,nlay+1) :: dtke
439	REAL, DIMENSION(ngrid,nlay+1) :: ak, bk, ck, CCK, DDK
440	REAL :: gammak,Kem,KKb,KKt
441
442
443	! Few initialisations
444	CCK(:,:)=0.
445	DDK(:,:)=0.
446	dtke(:,:)=0.
447
448
449	! Eddy diffusivity for TKE
450
451	DO ilay=2,nlay
452	DO igrid=1,ngrid
453	Ke(igrid,ilay)=(viscom+l_exchange(igrid,ilay)Sm(igrid,ilay)sqrt(tke(igrid,ilay)))*cke
454	ENDDO
455	ENDDO
456	! at the top of the atmosphere set to 0
457	Ke(:,nlay+1)=0.
458	! at the surface, set it equal to that at the first model level
459	Ke(:,1)=Ke(:,2)
460
461
462	! calculate intermediary variables
463
464	DO ilay=2,nlay
465	DO igrid=1,ngrid
466	Kem=0.5*(Ke(igrid,ilay+1)+Ke(igrid,ilay))
467	KKt=Kem*play(igrid,ilay)/rd/temp(igrid,ilay)/(z_interf(igrid,ilay+1)-z_interf(igrid,ilay))
468	Kem=0.5*(Ke(igrid,ilay)+Ke(igrid,ilay-1))
469	KKb=Kem*play(igrid,ilay-1)/rd/temp(igrid,ilay-1)/(z_interf(igrid,ilay)-z_interf(igrid,ilay-1))
470	gammak=1./(z_lay(igrid,ilay)-z_lay(igrid,ilay-1))
471	ak(igrid,ilay)=-gammakdtimeKKb
472	ck(igrid,ilay)=-gammakdtimeKKt
473	bk(igrid,ilay)=1.+gammakdtime(KKt+KKb)
474	ENDDO
475	ENDDO
476
477	! calculate CCK and DDK coefficients
478	! downhill phase
479
480	DO igrid=1,ngrid
481	CCK(igrid,nlay)=tke(igrid,nlay)/bk(igrid,nlay)
482	DDK(igrid,nlay)=-ak(igrid,nlay)/bk(igrid,nlay)
483	ENDDO
484
485
486	DO ilay=nlay-1,2,-1
487	DO igrid=1,ngrid
488	CCK(igrid,ilay)=(tke(igrid,ilay)/bk(igrid,ilay)-ck(igrid,ilay)/bk(igrid,ilay)*CCK(igrid,ilay+1)) &
489	/ (1.+ck(igrid,ilay)/bk(igrid,ilay)*DDK(igrid,ilay+1))
490	DDK(igrid,ilay)=-ak(igrid,ilay)/bk(igrid,ilay)/(1+ck(igrid,ilay)/bk(igrid,ilay)*DDK(igrid,ilay+1))
491	ENDDO
492	ENDDO
493
494	! calculate TKE
495	! uphill phase
496
497	DO ilay=2,nlay+1
498	DO igrid=1,ngrid
499	dtke(igrid,ilay)=CCK(igrid,ilay)+DDK(igrid,ilay)*tke(igrid,ilay-1)-tke(igrid,ilay)
500	ENDDO
501	ENDDO
502
503	! update TKE
504	tke(:,:)=tke(:,:)+dtke(:,:)
505	tke_trans(:,:)=dtke(:,:)/dtime
506
507
508	end subroutine atke_vdiff_tke
509
510
511
512	end module lmdz_atke_exchange_coeff

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