source: LMDZ6/trunk/libf/phylmd/lmdz_atke_exchange_coeff.F90 @ 5209

Last change on this file since 5209 was 5039, checked in by evignon, 2 months ago

ajout des tendances de tke de la routine ATKE dans les sorties
File size: 22.7 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,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
25USE lmdz_atke_turbulence_ini, ONLY : iflag_atke, kappa, l0, ric, cinf, rpi, rcpd, atke_ok_virtual, ri0, ri1
26USE lmdz_atke_turbulence_ini, ONLY : cepsilon, pr_slope, pr_asym, pr_neut, ctkes, rg, rd, rv, atke_ok_vdiff
27USE 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
60implicit none
61
62
63! Declarations:
64!=============
65
66INTEGER, INTENT(IN) :: ngrid ! number of horizontal index (flat grid)
67INTEGER, INTENT(IN) :: nlay  ! number of vertical index 
68
69REAL, INTENT(IN)    :: dtime ! physics time step (s)
70REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: wind_u   ! zonal velocity (m/s)
71REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: wind_v   ! meridional velocity (m/s)
72REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: temp     ! temperature (K)
73REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: qvap     ! specific humidity (kg/kg)
74REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: play     ! pressure (Pa)
75REAL, DIMENSION(ngrid,nlay+1), INTENT(IN)     :: pinterf  ! pressure at interfaces(Pa)
76REAL, DIMENSION(ngrid), INTENT(IN)            :: cdrag_uv ! surface drag coefficient for momentum
77
78REAL, DIMENSION(ngrid,nlay+1), INTENT(INOUT)  :: tke      ! turbulent kinetic energy at interface between layers (m2/s2)
79
80REAL, DIMENSION(ngrid,nlay+1), INTENT(OUT)    :: eps      ! output: TKE dissipation rate at interface between layers (m2/s3)
81REAL, DIMENSION(ngrid,nlay+1), INTENT(OUT)    :: tke_shear! output: TKE shear production rate (m2/s3)
82REAL, DIMENSION(ngrid,nlay+1), INTENT(OUT)    :: tke_buoy ! output: TKE buoyancy production rate (m2/s3)
83REAL, DIMENSION(ngrid,nlay+1), INTENT(OUT)    :: tke_trans! output: TKE transport (diffusion) term (m2/s3)
84REAL, DIMENSION(ngrid,nlay), INTENT(OUT)      :: Km_out   ! output: Exchange coefficient for momentum at interface between layers (m2/s)
85REAL, DIMENSION(ngrid,nlay), INTENT(OUT)      :: Kh_out   ! output: Exchange coefficient for heat flux at interface between layers (m2/s)
86
87! Local variables
88REAL, DIMENSION(ngrid,nlay+1) :: Km          ! Exchange coefficient for momentum at interface between layers
89REAL, DIMENSION(ngrid,nlay+1) :: Kh          ! Exchange coefficient for heat flux at interface between layers
90REAL, DIMENSION(ngrid,nlay)   :: theta       ! Potential temperature
91REAL, DIMENSION(ngrid,nlay+1) :: l_exchange  ! Length of exchange (at interface)
92REAL, DIMENSION(ngrid,nlay+1) :: z_interf    ! Altitude at the interface
93REAL, DIMENSION(ngrid,nlay)   :: z_lay       ! Altitude of layers
94REAL, DIMENSION(ngrid,nlay)   :: dz_interf   ! distance between two consecutive interfaces
95REAL, DIMENSION(ngrid,nlay)   :: dz_lay      ! distance between two layer middles (NB: first and last are half layers)
96REAL, DIMENSION(ngrid,nlay+1) :: N2          ! square of Brunt Vaisala pulsation (at interface)
97REAL, DIMENSION(ngrid,nlay+1) :: shear2      ! square of wind shear (at interface)
98REAL, DIMENSION(ngrid,nlay+1) :: Ri          ! Richardson's number (at interface)
99REAL, DIMENSION(ngrid,nlay+1) :: Prandtl     ! Turbulent Prandtl's number (at interface)
100REAL, DIMENSION(ngrid,nlay+1) :: Sm          ! Stability function for momentum (at interface)
101REAL, DIMENSION(ngrid,nlay+1) :: Sh          ! Stability function for heat (at interface)
102
103INTEGER :: igrid,ilay ! horizontal,vertical index (flat grid)
104REAL    :: preff      ! reference pressure for potential temperature calculations
105REAL    :: thetam     ! mean potential temperature at interface
106REAL    :: delta      ! discriminant of the second order polynomial
107REAL    :: qq         ! tke=qq**2/2
108REAL    :: shear      ! wind shear
109REAL    :: lstrat     ! mixing length depending on local stratification
110REAL    :: taustrat   ! caracteristic timescale for turbulence in very stable conditions
111REAL    :: netloss    ! net loss term of tke
112REAL    :: netsource  ! net source term of tke
113REAL    :: ustar      ! friction velocity estimation
114REAL    :: invtau     
115REAL    :: rvap
116
117! Initializations:
118!================
119
120DO igrid=1,ngrid
121    dz_interf(igrid,1) = 0.0
122    z_interf(igrid,1) = 0.0
123END DO
124
125! Calculation of potential temperature: (if vapor -> virtual potential temperature)
126!=====================================
127
128preff=100000.
129! results should not depend on the choice of preff
130DO ilay=1,nlay
131    DO igrid = 1, ngrid
132        theta(igrid,ilay)=temp(igrid,ilay)*(preff/play(igrid,ilay))**(rd/rcpd)
133    END DO
134END DO
135
136! account for water vapor mass for buoyancy calculation
137IF (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
144ENDIF
145
146
147! Calculation of altitude of layers' middle and bottom interfaces:
148!=================================================================
149
150DO ilay=2,nlay+1
151    DO igrid=1,ngrid
152        dz_interf(igrid,ilay-1) = rd*temp(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
155ENDDO
156
157DO 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
161ENDDO
162
163
164! Computes the gradient Richardson's number and stability functions:
165!===================================================================
166
167DO 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_neut*exp(-pr_slope/pr_neut*Ri(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
193ENDDO
194
195
196! Computing the mixing length:
197!==============================================================
198
199
200IF (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) = kappa*l0*z_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) = kappa*l0*z_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) = kappa*l0*z_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=clmix*sqrt(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
245ELSE
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) = kappa*l0*z_interf(igrid,ilay) / (kappa*z_interf(igrid,ilay) + l0)
250        ENDDO
251    ENDDO
252ENDIF
253
254
255! Computing the TKE k>=2:
256!========================
257IF (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))/(cepsilon*l_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
272ELSE 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.)*cepsilon*l_exchange(igrid,ilay)/dtime)**2. &
280                    +4.*(2.*sqrt(2.)*cepsilon*l_exchange(igrid,ilay)/dtime*qq + &
281                    2.*l_exchange(igrid,ilay)*l_exchange(igrid,ilay)*cepsilon*Sm(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))/(cepsilon*l_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
294ELSE 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.+qq*dtime/(cepsilon*l_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))/(cepsilon*l_exchange(igrid,ilay))
310            ENDDO
311        ENDDO
312
313
314ELSE 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))/(cepsilon*l_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+qq*netsource)/(qq/dtime+netloss)
334            tke(igrid,ilay)=0.5*(qq**2)
335            ENDDO
336        ENDDO
337
338ELSE 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.+dtime*qq/(cepsilon*l_exchange(igrid,ilay)*2.*sqrt(2.)))) &
349                /(1.+2.*qq*dtime/(cepsilon*l_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))/(cepsilon*l_exchange(igrid,ilay))
353            ENDDO
354        ENDDO
355
356
357ELSE
358    call abort_physic("atke_compute_km_kh", &
359        'numerical treatment of TKE not possible yet', 1)
360
361END IF
362
363! We impose a 0 tke at nlay+1
364!==============================
365
366DO 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.
371END 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)
377DO 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.
383END DO
384
385
386! vertical diffusion of TKE
387!==========================
388tke_trans(:,:)=0.
389IF (atke_ok_vdiff) THEN
390    CALL atke_vdiff_tke(ngrid,nlay,dtime,z_lay,z_interf,temp,play,l_exchange,Sm,tke,tke_trans)
391ENDIF
392
393
394! Computing eddy diffusivity coefficients:
395!========================================
396DO 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
402END DO
403
404! for output:
405!===========
406Km_out(1:ngrid,2:nlay)=Km(1:ngrid,2:nlay)
407Kh_out(1:ngrid,2:nlay)=Kh(1:ngrid,2:nlay)
408
409end subroutine atke_compute_km_kh
410
411!===============================================================================================
412subroutine 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
418USE lmdz_atke_turbulence_ini, ONLY : rd, cke, viscom
419
420
421INTEGER, INTENT(IN) :: ngrid ! number of horizontal index (flat grid)
422INTEGER, INTENT(IN) :: nlay  ! number of vertical index 
423
424REAL, INTENT(IN)    :: dtime ! physics time step (s)
425REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: z_lay   ! altitude of mid-layers (m)
426REAL, DIMENSION(ngrid,nlay+1), INTENT(IN)       :: z_interf   ! altitude of bottom interfaces (m)
427REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: temp   ! temperature (K)
428REAL, DIMENSION(ngrid,nlay), INTENT(IN)       :: play   ! pressure (Pa)
429REAL, DIMENSION(ngrid,nlay+1), INTENT(IN)     :: l_exchange     ! mixing length at interfaces between layers
430REAL, DIMENSION(ngrid,nlay+1), INTENT(IN)     :: Sm     ! stability function for eddy diffusivity for momentum at interface between layers
431
432REAL, DIMENSION(ngrid,nlay+1), INTENT(INOUT)  :: tke    ! turbulent kinetic energy at interface between layers
433REAL, DIMENSION(ngrid,nlay+1), INTENT(INOUT)  :: tke_trans ! turbulent kinetic energy transport term (m2/s3)
434
435
436INTEGER                                       :: igrid,ilay
437REAL, DIMENSION(ngrid,nlay+1)                 :: Ke     ! eddy diffusivity for TKE
438REAL, DIMENSION(ngrid,nlay+1)                 :: dtke
439REAL, DIMENSION(ngrid,nlay+1)                 :: ak, bk, ck, CCK, DDK
440REAL                                          :: gammak,Kem,KKb,KKt
441
442
443! Few initialisations
444CCK(:,:)=0.
445DDK(:,:)=0.
446dtke(:,:)=0.
447
448
449! Eddy diffusivity for TKE
450
451DO 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
455ENDDO
456! at the top of the atmosphere set to 0
457Ke(:,nlay+1)=0.
458! at the surface, set it equal to that at the first model level
459Ke(:,1)=Ke(:,2)
460
461
462! calculate intermediary variables
463
464DO 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)=-gammak*dtime*KKb
472    ck(igrid,ilay)=-gammak*dtime*KKt
473    bk(igrid,ilay)=1.+gammak*dtime*(KKt+KKb)
474    ENDDO
475ENDDO
476
477! calculate CCK and DDK coefficients
478! downhill phase
479
480DO igrid=1,ngrid
481    CCK(igrid,nlay)=tke(igrid,nlay)/bk(igrid,nlay)
482    DDK(igrid,nlay)=-ak(igrid,nlay)/bk(igrid,nlay)
483ENDDO
484
485
486DO 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
492ENDDO
493
494! calculate TKE
495! uphill phase
496
497DO 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
501ENDDO
502
503! update TKE
504tke(:,:)=tke(:,:)+dtke(:,:)
505tke_trans(:,:)=dtke(:,:)/dtime
506
507
508end subroutine atke_vdiff_tke
509
510
511
512end module lmdz_atke_exchange_coeff
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