source: LMDZ6/trunk/libf/phylmd/lmdz_atke_exchange_coeff.F90 @ 4688

Last change on this file since 4688 was 4688, checked in by evignon, 10 months ago

correction formulation nombre de Prandlt dans ATKE pour atelier de demain
File size: 16.6 KB
Line 
1module lmdz_atke_exchange_coeff
2
3implicit none
4
5contains
6
7subroutine atke_compute_km_kh(ngrid,nlay,dtime, &
8                        wind_u,wind_v,temp,qvap,play,pinterf,cdrag_uv, &
9                        tke,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) dry atmosphere
23! (2) horizontal homogeneity (Dx=Dy=0.)
24!=======================================================================
25
26
27
28USE lmdz_atke_turbulence_ini, ONLY : iflag_atke, kappa, l0, ric, cinf, rpi, rcpd, atke_ok_virtual
29USE lmdz_atke_turbulence_ini, ONLY : cepsilon, pr_slope, pr_asym, pr_neut, ctkes,rg, rd, rv, atke_ok_vdiff
30USE lmdz_atke_turbulence_ini, ONLY : viscom, viscoh, clmix, clmixshear, iflag_atke_lmix, lmin, smmin
31
32implicit none
33
34
35! Declarations:
36!=============
37
38INTEGER, INTENT(IN) :: ngrid ! number of horizontal index (flat grid)
39INTEGER, INTENT(IN) :: nlay  ! number of vertical index 
40
41REAL, INTENT(IN)    :: dtime ! physics time step (s)
42REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: wind_u   ! zonal velocity (m/s)
43REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: wind_v   ! meridional velocity (m/s)
44REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: temp   ! temperature (K)
45REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: qvap   ! specific humidity (kg/kg)
46REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: play   ! pressure (Pa)
47REAL, DIMENSION(ngrid,nlay+1), INTENT(IN)     :: pinterf   ! pressure at interfaces(Pa)
48REAL, DIMENSION(ngrid), INTENT(IN)            :: cdrag_uv   ! surface drag coefficient for momentum
49
50REAL, DIMENSION(ngrid,nlay+1), INTENT(INOUT)  :: tke  ! turbulent kinetic energy at interface between layers
51
52REAL, DIMENSION(ngrid,nlay), INTENT(OUT)      :: Km_out   ! output: Exchange coefficient for momentum at interface between layers
53REAL, DIMENSION(ngrid,nlay), INTENT(OUT)      :: Kh_out   ! output: Exchange coefficient for heat flux at interface between layers
54
55! Local variables
56REAL, DIMENSION(ngrid,nlay+1) :: Km          ! Exchange coefficient for momentum at interface between layers
57REAL, DIMENSION(ngrid,nlay+1) :: Kh          ! Exchange coefficient for heat flux at interface between layers
58REAL, DIMENSION(ngrid,nlay)   :: theta       ! Potential temperature
59REAL, DIMENSION(ngrid,nlay+1) :: l_exchange  ! Length of exchange (at interface)
60REAL, DIMENSION(ngrid,nlay+1) :: z_interf    ! Altitude at the interface
61REAL, DIMENSION(ngrid,nlay)   :: z_lay       ! Altitude of layers
62REAL, DIMENSION(ngrid,nlay)   :: dz_interf   ! distance between two consecutive interfaces
63REAL, DIMENSION(ngrid,nlay)   :: dz_lay      ! distance between two layer middles (NB: first and last are half layers)
64REAL, DIMENSION(ngrid,nlay+1) :: N2          ! square of Brunt Vaisala pulsation (at interface)
65REAL, DIMENSION(ngrid,nlay+1) :: shear2      ! square of wind shear (at interface)
66REAL, DIMENSION(ngrid,nlay+1) :: Ri          ! Richardson's number (at interface)
67REAL, DIMENSION(ngrid,nlay+1) :: Prandtl     ! Turbulent Prandtl's number (at interface)
68REAL, DIMENSION(ngrid,nlay+1) :: Sm          ! Stability function for momentum (at interface)
69REAL, DIMENSION(ngrid,nlay+1) :: Sh          ! Stability function for heat (at interface)
70
71INTEGER :: igrid,ilay ! horizontal,vertical index (flat grid)
72REAL    :: cn,Ri0,Ri1    ! parameter for Sm stability function and Prandlt
73REAL    :: preff      ! reference pressure for potential temperature calculations
74REAL    :: thetam     ! mean potential temperature at interface
75REAL    :: delta      ! discriminant of the second order polynomial
76REAL    :: qq         ! tke=qq**2/2
77REAL    :: shear      ! wind shear
78REAL    :: lstrat     ! mixing length depending on local stratification
79REAL    :: taustrat   ! caracteristic timescale for turbulence in very stable conditions
80REAL    :: netloss    ! net loss term of tke
81REAL    :: netsource  ! net source term of tke
82REAL    :: ustar      ! friction velocity estimation
83REAL    :: invtau     
84REAL    :: rvap
85
86! Initializations:
87!================
88
89DO igrid=1,ngrid
90    dz_interf(igrid,1) = 0.0
91    z_interf(igrid,1) = 0.0
92END DO
93
94! Calculation of potential temperature: (if vapor -> virtual potential temperature)
95!=====================================
96
97preff=100000.
98! results should not depend on the choice of preff
99DO ilay=1,nlay
100     DO igrid = 1, ngrid
101        theta(igrid,ilay)=temp(igrid,ilay)*(preff/play(igrid,ilay))**(rd/rcpd)
102     END DO
103END DO
104
105! account for water vapor mass for buoyancy calculation
106IF (atke_ok_virtual) THEN
107  DO ilay=1,nlay
108     DO igrid = 1, ngrid
109        rvap=max(0.,qvap(igrid,ilay)/(1.-qvap(igrid,ilay)))
110        theta(igrid,ilay)=theta(igrid,ilay)*(1.+rvap/(RD/RV))/(1.+rvap)
111     END DO
112  END DO
113ENDIF
114
115
116! Calculation of altitude of layers' middle and bottom interfaces:
117!=================================================================
118
119DO ilay=2,nlay+1
120    DO igrid=1,ngrid
121        dz_interf(igrid,ilay-1) = rd*temp(igrid,ilay-1)/rg/play(igrid,ilay-1)*(pinterf(igrid,ilay-1)-pinterf(igrid,ilay))
122        z_interf(igrid,ilay) = z_interf(igrid,ilay-1) + dz_interf(igrid,ilay-1)
123    ENDDO
124ENDDO
125
126DO ilay=1,nlay
127   DO igrid=1,ngrid
128      z_lay(igrid,ilay)=0.5*(z_interf(igrid, ilay+1) + z_interf(igrid, ilay))
129   ENDDO
130ENDDO
131
132
133! Computes the gradient Richardson's number and stability functions:
134!===================================================================
135
136! calculation of cn = Sm value at Ri=0
137! direct dependance on cepsilon to guarantee Fm=1 (first-order like stability function) at Ri=0
138cn=(1./sqrt(cepsilon))**(2/3)
139! calculation of Ri0 such that continuity in slope of Sm at Ri=0
140Ri0=2./rpi*(cinf - cn)*ric/cn
141! calculation of Ri1 to guarantee continuity in slope of Prandlt number at Ri=0
142Ri1 = -2./rpi * (pr_asym - pr_neut)
143
144
145DO ilay=2,nlay
146    DO igrid=1,ngrid
147        dz_lay(igrid,ilay)=z_lay(igrid,ilay)-z_lay(igrid,ilay-1)
148        thetam=0.5*(theta(igrid,ilay) + theta(igrid,ilay-1))
149        N2(igrid,ilay) = rg * (theta(igrid,ilay) - theta(igrid,ilay-1))/thetam / dz_lay(igrid,ilay)
150        shear2(igrid,ilay)= (((wind_u(igrid,ilay) - wind_u(igrid,ilay-1)) / dz_lay(igrid,ilay))**2 + &
151            ((wind_v(igrid,ilay) - wind_v(igrid,ilay-1)) / dz_lay(igrid,ilay))**2 )
152        Ri(igrid,ilay) = N2(igrid,ilay) / MAX(shear2(igrid,ilay),1E-10)
153       
154        IF (Ri(igrid,ilay) < 0.) THEN ! unstable cases
155            Sm(igrid,ilay) = 2./rpi * (cinf-cn) * atan(-Ri(igrid,ilay)/Ri0) + cn
156            Prandtl(igrid,ilay) = -2./rpi * (pr_asym - pr_neut) * atan(Ri(igrid,ilay)/Ri1) + pr_neut
157        ELSE ! stable cases
158            Sm(igrid,ilay) = max(smmin,cn*(1.-Ri(igrid,ilay)/Ric))
159            ! prandlt expression from venayagamoorthy and stretch 2010, Li et al 2019
160            Prandtl(igrid,ilay) = pr_neut*exp(-pr_slope/pr_neut*Ri(igrid,ilay)+Ri(igrid,ilay)/pr_neut) &
161                                + Ri(igrid,ilay) * pr_slope
162            IF (Ri(igrid,ilay) .GE. Prandtl(igrid,ilay)) THEN
163               call abort_physic("atke_compute_km_kh", &
164               'Ri>=Pr in stable conditions -> violates energy conservation principles, change pr_neut or slope', 1)
165            ENDIF
166        END IF
167       
168        Sh(igrid,ilay) = Sm(igrid,ilay) / Prandtl(igrid,ilay)
169
170    ENDDO
171ENDDO
172
173
174! Computing the mixing length:
175!==============================================================
176
177
178IF (iflag_atke_lmix .EQ. 1 ) THEN
179
180   DO ilay=2,nlay
181      DO igrid=1,ngrid
182          l_exchange(igrid,ilay) = kappa*l0*z_interf(igrid,ilay) / (kappa*z_interf(igrid,ilay) + l0)
183          IF (N2(igrid,ilay) .GT. 0.) THEN
184             lstrat=clmix*sqrt(tke(igrid,ilay))/sqrt(N2(igrid,ilay))
185             lstrat=max(lstrat,lmin)
186             !Inverse interpolation, Van de Wiel et al. 2010
187             l_exchange(igrid,ilay)=(1./(l_exchange(igrid,ilay))+1./(lstrat))**(-1.0)
188          ENDIF
189      ENDDO
190   ENDDO
191
192ELSE IF (iflag_atke_lmix .EQ. 2 ) THEN
193! add effect of wind shear on lstrat following grisogono and belusic 2008, qjrms
194DO ilay=2,nlay
195      DO igrid=1,ngrid
196          l_exchange(igrid,ilay) = kappa*l0*z_interf(igrid,ilay) / (kappa*z_interf(igrid,ilay) + l0)
197          IF (N2(igrid,ilay) .GT. 0. .AND. shear2(igrid,ilay) .GT. 0.) THEN
198             lstrat=min(clmix*sqrt(tke(igrid,ilay))/sqrt(N2(igrid,ilay)), &
199                    clmixshear*sqrt(tke(igrid,ilay))/sqrt(shear2(igrid,ilay)))
200             lstrat=max(lstrat,lmin)
201             !Inverse interpolation, Van de Wiel et al. 2010   
202             l_exchange(igrid,ilay)=(1./(l_exchange(igrid,ilay))+1./(lstrat))**(-1.0)
203          ENDIF
204      ENDDO
205   ENDDO
206
207
208
209ELSE
210! default: neglect effect of local stratification and shear
211
212   DO ilay=2,nlay+1
213      DO igrid=1,ngrid
214          l_exchange(igrid,ilay) = kappa*l0*z_interf(igrid,ilay) / (kappa*z_interf(igrid,ilay) + l0)
215      ENDDO
216
217   ENDDO
218ENDIF
219
220
221! Computing the TKE k>=2:
222!========================
223IF (iflag_atke == 0) THEN
224
225! stationary solution (dtke/dt=0)
226
227   DO ilay=2,nlay
228        DO igrid=1,ngrid
229            tke(igrid,ilay) = cepsilon * l_exchange(igrid,ilay)**2 * Sm(igrid,ilay) * &
230            shear2(igrid,ilay) * (1. - Ri(igrid,ilay) / Prandtl(igrid,ilay))
231        ENDDO
232    ENDDO
233
234ELSE IF (iflag_atke == 1) THEN
235
236! full implicit scheme resolved with a second order polynomial equation
237
238    DO ilay=2,nlay
239        DO igrid=1,ngrid
240           qq=max(sqrt(2.*tke(igrid,ilay)),1.e-10)
241           delta=(2.*sqrt(2.)*cepsilon*l_exchange(igrid,ilay)/dtime)**2. &
242                 +4.*(2.*sqrt(2.)*cepsilon*l_exchange(igrid,ilay)/dtime*qq + &
243                 2.*l_exchange(igrid,ilay)*l_exchange(igrid,ilay)*cepsilon*Sm(igrid,ilay) &
244                 *shear2(igrid,ilay) * (1. - Ri(igrid,ilay) / Prandtl(igrid,ilay)))
245           qq=(-2.*sqrt(2.)*cepsilon*l_exchange(igrid,ilay)/dtime + sqrt(delta))/2.
246           qq=max(0.,qq)
247           tke(igrid,ilay)=0.5*(qq**2)
248        ENDDO
249    ENDDO
250
251
252ELSE IF (iflag_atke == 2) THEN
253
254! semi implicit scheme when l does not depend on tke
255! positive-guaranteed if pr slope in stable condition >1
256
257   DO ilay=2,nlay
258        DO igrid=1,ngrid
259           qq=max(sqrt(2.*tke(igrid,ilay)),1.e-10)
260           qq=(qq+l_exchange(igrid,ilay)*Sm(igrid,ilay)*dtime/sqrt(2.)      &
261               *shear2(igrid,ilay)*(1.-Ri(igrid,ilay)/Prandtl(igrid,ilay))) &
262               /(1.+qq*dtime/(cepsilon*l_exchange(igrid,ilay)*2.*sqrt(2.)))
263           tke(igrid,ilay)=0.5*(qq**2)
264        ENDDO
265    ENDDO
266
267
268ELSE IF (iflag_atke == 3) THEN
269! numerical resolution adapted from that in MAR (Deleersnijder 1992)
270! positively defined by construction
271
272    DO ilay=2,nlay
273        DO igrid=1,ngrid
274           qq=max(sqrt(2.*tke(igrid,ilay)),1.e-10)
275           IF (Ri(igrid,ilay) .LT. 0.) THEN
276              netloss=qq/(2.*sqrt(2.)*cepsilon*l_exchange(igrid,ilay))
277              netsource=l_exchange(igrid,ilay)*Sm(igrid,ilay)/sqrt(2.)*shear2(igrid,ilay)*(1.-Ri(igrid,ilay)/Prandtl(igrid,ilay))
278           ELSE
279              netloss=qq/(2.*sqrt(2.)*cepsilon*l_exchange(igrid,ilay))+ &
280                      l_exchange(igrid,ilay)*Sm(igrid,ilay)/sqrt(2.)*N2(igrid,ilay)/Prandtl(igrid,ilay)
281              netsource=l_exchange(igrid,ilay)*Sm(igrid,ilay)/sqrt(2.)*shear2(igrid,ilay)
282           ENDIF
283           qq=((qq**2)/dtime+qq*netsource)/(qq/dtime+netloss)
284           tke(igrid,ilay)=0.5*(qq**2)
285        ENDDO
286    ENDDO
287
288ELSE IF (iflag_atke == 4) THEN
289! semi implicit scheme from Arpege (V. Masson methodology with
290! Taylor expansion of the dissipation term)
291    DO ilay=2,nlay
292        DO igrid=1,ngrid
293           qq=max(sqrt(2.*tke(igrid,ilay)),1.e-10)
294           qq=(l_exchange(igrid,ilay)*Sm(igrid,ilay)/sqrt(2.)*shear2(igrid,ilay)*(1.-Ri(igrid,ilay)/Prandtl(igrid,ilay)) &
295             +qq*(1.+dtime*qq/(cepsilon*l_exchange(igrid,ilay)*2.*sqrt(2.)))) &
296             /(1.+2.*qq*dtime/(cepsilon*l_exchange(igrid,ilay)*2.*sqrt(2.)))
297           qq=max(0.,qq)
298           tke(igrid,ilay)=0.5*(qq**2)
299        ENDDO
300    ENDDO
301
302
303ELSE
304   call abort_physic("atke_compute_km_kh", &
305        'numerical treatment of TKE not possible yet', 1)
306
307END IF
308
309! We impose a 0 tke at nlay+1
310!==============================
311
312DO igrid=1,ngrid
313 tke(igrid,nlay+1)=0.
314END DO
315
316
317! Calculation of surface TKE (k=1)
318!=================================
319! surface TKE calculation inspired from what is done in Arpege (see E. Bazile note)
320DO igrid=1,ngrid
321 ustar=sqrt(cdrag_uv(igrid)*(wind_u(igrid,1)**2+wind_v(igrid,1)**2))
322 tke(igrid,1)=ctkes*(ustar**2)
323END DO
324
325
326! vertical diffusion of TKE
327!==========================
328IF (atke_ok_vdiff) THEN
329   CALL atke_vdiff_tke(ngrid,nlay,dtime,z_lay,z_interf,temp,play,l_exchange,Sm,tke)
330ENDIF
331
332
333! Computing eddy diffusivity coefficients:
334!========================================
335DO ilay=2,nlay ! TODO: also calculate for nlay+1 ?
336    DO igrid=1,ngrid
337        ! we add the molecular viscosity to Km,h
338        Km(igrid,ilay) = viscom + l_exchange(igrid,ilay) * Sm(igrid,ilay) * tke(igrid,ilay)**0.5
339        Kh(igrid,ilay) = viscoh + l_exchange(igrid,ilay) * Sh(igrid,ilay) * tke(igrid,ilay)**0.5
340    END DO
341END DO
342
343! for output:
344!===========
345Km_out(1:ngrid,2:nlay)=Km(1:ngrid,2:nlay)
346Kh_out(1:ngrid,2:nlay)=Kh(1:ngrid,2:nlay)
347
348end subroutine atke_compute_km_kh
349
350!===============================================================================================
351subroutine atke_vdiff_tke(ngrid,nlay,dtime,z_lay,z_interf,temp,play,l_exchange,Sm,tke)
352
353! routine that computes the vertical diffusion of TKE by the turbulence
354! using an implicit resolution (See note by Dufresne and Ghattas (2009))
355! E Vignon, July 2023
356
357USE lmdz_atke_turbulence_ini, ONLY : rd, cke, viscom
358
359
360INTEGER, INTENT(IN) :: ngrid ! number of horizontal index (flat grid)
361INTEGER, INTENT(IN) :: nlay  ! number of vertical index 
362
363REAL, INTENT(IN)    :: dtime ! physics time step (s)
364REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: z_lay   ! altitude of mid-layers (m)
365REAL, DIMENSION(ngrid,nlay+1), INTENT(IN)       :: z_interf   ! altitude of bottom interfaces (m)
366REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: temp   ! temperature (K)
367REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: play   ! pressure (Pa)
368REAL, DIMENSION(ngrid,nlay+1), INTENT(IN)     :: l_exchange     ! mixing length at interfaces between layers
369REAL, DIMENSION(ngrid,nlay+1), INTENT(IN)     :: Sm     ! stability function for eddy diffusivity for momentum at interface between layers
370
371REAL, DIMENSION(ngrid,nlay+1), INTENT(INOUT)  :: tke    ! turbulent kinetic energy at interface between layers
372
373
374
375INTEGER                                       :: igrid,ilay
376REAL, DIMENSION(ngrid,nlay+1)                 :: Ke     ! eddy diffusivity for TKE
377REAL, DIMENSION(ngrid,nlay+1)                 :: dtke
378REAL, DIMENSION(ngrid,nlay+1)                 :: ak, bk, ck, CCK, DDK
379REAL                                          :: gammak,Kem,KKb,KKt
380
381
382! Few initialisations
383CCK(:,:)=0.
384DDK(:,:)=0.
385dtke(:,:)=0.
386
387
388! Eddy diffusivity for TKE
389
390DO ilay=2,nlay
391    DO igrid=1,ngrid
392       Ke(igrid,ilay)=(viscom+l_exchange(igrid,ilay)*Sm(igrid,ilay)*sqrt(tke(igrid,ilay)))*cke
393    ENDDO
394ENDDO
395! at the top of the atmosphere set to 0
396Ke(:,nlay+1)=0.
397! at the surface, set it equal to that at the first model level
398Ke(:,1)=Ke(:,2)
399
400
401! calculate intermediary variables
402
403DO ilay=2,nlay
404    DO igrid=1,ngrid
405    Kem=0.5*(Ke(igrid,ilay+1)+Ke(igrid,ilay))   
406    KKt=Kem*play(igrid,ilay)/rd/temp(igrid,ilay)/(z_interf(igrid,ilay+1)-z_interf(igrid,ilay))
407    Kem=0.5*(Ke(igrid,ilay)+Ke(igrid,ilay-1))
408    KKb=Kem*play(igrid,ilay-1)/rd/temp(igrid,ilay-1)/(z_interf(igrid,ilay)-z_interf(igrid,ilay-1))
409    gammak=1./(z_lay(igrid,ilay)-z_lay(igrid,ilay-1))
410    ak(igrid,ilay)=-gammak*dtime*KKb
411    ck(igrid,ilay)=-gammak*dtime*KKt
412    bk(igrid,ilay)=1.+gammak*dtime*(KKt+KKb)
413    ENDDO
414ENDDO
415
416! calculate CCK and DDK coefficients
417! downhill phase
418
419DO igrid=1,ngrid
420  CCK(igrid,nlay)=tke(igrid,nlay)/bk(igrid,nlay)
421  DDK(igrid,nlay)=-ak(igrid,nlay)/bk(igrid,nlay)
422ENDDO
423
424
425DO ilay=nlay-1,2,-1
426    DO igrid=1,ngrid
427        CCK(igrid,ilay)=(tke(igrid,ilay)/bk(igrid,ilay)-ck(igrid,ilay)/bk(igrid,ilay)*CCK(igrid,ilay+1)) &
428                       / (1.+ck(igrid,ilay)/bk(igrid,ilay)*DDK(igrid,ilay+1))
429        DDK(igrid,ilay)=-ak(igrid,ilay)/bk(igrid,ilay)/(1+ck(igrid,ilay)/bk(igrid,ilay)*DDK(igrid,ilay+1))
430    ENDDO
431ENDDO
432
433! calculate TKE
434! uphill phase
435
436DO ilay=2,nlay+1
437    DO igrid=1,ngrid
438        dtke(igrid,ilay)=CCK(igrid,ilay)+DDK(igrid,ilay)*tke(igrid,ilay-1)-tke(igrid,ilay)
439    ENDDO
440ENDDO
441
442! update TKE
443tke(:,:)=tke(:,:)+dtke(:,:)
444
445
446end subroutine atke_vdiff_tke
447
448
449
450end module lmdz_atke_exchange_coeff
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