Boundary conditions control in ORCA2

Transcription

1 Boundary conditions control in ORCA2 Eugene Kazantsev To cite this version: Eugene Kazantsev. Boundary conditions control in ORCA2. Journée thématique - Que peuvent attendre les modélisateurs de l assimilation?, Feb 2013, Paris, France <hal > HAL Id: hal Submitted on 8 Jan 2014 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Boundary conditions control in ORCA2 Eugene Kazantsev INRIA, Moise Journée thématique Que peuvent attendre les modélisateurs de l assimilation de données? Paris, le 12 février 2013 Eugene Kazantsev Boundary conditions control for ORCA2 page 1 of 19

3 Model configuration ORCA-2 configuration of NEMO nodes in curvilinear (x,y) coordinates; z levels with partial steps at the bottom; leap-frog scheme with Asselin filter; implicit surface pressure gradient with External Gravity Waves filter; implicit vertical diffusion with TKE Turbulent Closure Scheme; Solar Radiation + Geothermal Heating + BBL + Surface evaporation/precipitation; surface wind stress. Eugene Kazantsev Boundary conditions control for ORCA2 page 2 of 19

4 Space discretization on the C-grid u t T t ( S xu = S xs yv )S 2 +S yv 2 ) y(ω +f) D x S z (S xwd zu +D xa h u 2 ξ +DyAh u ω + z + g D xs zρ(x,y,ζ)dζ +D zz(a z uu)+gd x(η +T cφ) 0 = D x(us xt) D y(vs yt) D z(ws zt)+a h T ( ) D xd xt +D yd yt + + D zz(a z TT)+Solar Radiation+Geothermal Heating+BBL z ξ = D xu+d yv, ω = D yu D xv, w = ξ(x,y,ζ)dζ;w(x,y,h) = 0 H Interpolations and Derivatives (Sw) k+1/2 (DT) k = w k+1 +w k 2 = T k+1/2 T k 1/2 h k = 1,...,K 1 k = 1,...,K 1 Eugene Kazantsev Boundary conditions control for ORCA2 page 3 of 19

5 Space discretization on the C-grid u t T t ( S xu = S xs yv )S 2 +S yv 2 ) y(ω +f) D x S z (S xwd zu +D xa h u 2 ξ +DyAh u ω + z + g D xs zρ(x,y,ζ)dζ +D zz(a z uu)+gd x(η +T cφ) 0 = D x(us xt) D y(vs yt) D z(ws zt)+a h T ( ) D xd xt +D yd yt + + D zz(a z TT)+Solar Radiation+Geothermal Heating+BBL z ξ = D xu+d yv, ω = D yu D xv, w = ξ(x,y,ζ)dζ;w(x,y,h) = 0 H Interpolations and Derivatives Modified Near the boundary (Sw) k+1/2 (DT) k = w k+1 +w k, k = 1,...,K 2, (Sw) 1/2 = α S 0 2 +αs 1 w 0 +α S 2 w 1 = T k+1/2 T k 1/2, i = 2,...,K 2, (DT) 1 = α D 0 h + αd 1 T 1/2 +α D 2 T 3/2 h T 1/2 T 3/2 T 5/2 T 7/2 T T T K 5/2 K 3/2 K 1/2 w 0 w 1 w 2 w 3 w K 3 w K 2 w K 1 w K Eugene Kazantsev Boundary conditions control for ORCA2 page 3 of 19

6 Space discretization on the C-grid u t T t ( S xu = S xs yv )S 2 +S yv 2 ) y(ω +f) D x S z (S xwd zu +D xa h 2 uξ +D ya h uω + z + g D xs zρ(x,y,ζ)dζ +D zz(a z uu)+gd x(η +T cφ) 0 = D x(us xt) D y(vs yt) D z(ws zt)+a h T ( ) D xd xt +D yd yt + + D zz(a z TT)+Solar Radiation+Geothermal Heating+BBL z ξ = D xu+d yv, ω = D yu D xv, w = ξ(x,y,ζ)dζ;w(x,y,h) = α 0 (x,y) H Vertical velocity w i,j,k 1 = α0 wb α wb 1 hz i,j,k 1/2 ξ i,j,k 1/2 w i,j,k 1 = w i,j,k hz i,j,k 1/2 ξ i,j,k 1/2 k : 2 k K 1 w i,j,0 = w i,j,1 +α ws 0 α ws 1 hz i,j,1/2ξ i,j,1/2 α w 1 hz 1/2 α w 1 α w {}}{ hz K 1/2+α w 0 {}}{ 0 w 0 w 1 w 2 w 3 w K 3 w K 2 w K 1 w K Eugene Kazantsev Boundary conditions control for ORCA2 page 3 of 19

7 Space discretization on the C-grid Vertical diffusion ( ) Dzzu i,j,1/2 ( ) Dzzu i,j,k 1/2 ( ) Dzzu i,j,k 1/2 = = = u z Az u z is replaced by (A z u ) 1 (α DzzUs 2 u 3/2 α DzzUs 1 u 1/2 ) hz 1 hz 1/2 1 hz k 1/2 1 [ hz K 1/2 ( (A z u ) k (u k+1/2 u k 1/2 ) (A z u ) ) k 1 (u k 1/2 u k 3/2 ) hz k hz k 1 α DzzUb 2 (A z u ) K 1 u K 1/2 α DzzUb ( (A z u ) K 1 + (A hz K 1 hz K z u ) K 1 hz K 1 k : 2 ) u K 3/ u = α DzzUs 0 + τx, z w0 hz 1 ρ 0 v = α DzzUs 0 + τy, z w0 hz 1 ρ 0 T = S = α DzzTs 0 z w0 z w0 u bottom = v bottom = α DzzUb 0 hz 1/2 {}}{ hz 3/2 hz 5/2 {}}{{}}{ u 1/2 u 3/2 u 5/2 T bottom = S bottom = α DzzTb 0 hz K 3/2 hz K 1/2 u 7/2 u u K 5/2 K 3/2 (1) {}}{{}}{ u K 1/2 }{{} hz 1 } {{ } hz 2 } {{ } hz 3 } {{ } hz K 2 } {{ } hz K 1 Eugene Kazantsev Boundary conditions control for ORCA2 page 3 of 19

8 Adjoint The models solution depend on initial and boundary conditions : ( D xxt+d yyt T t = Dx(uSxT) Dy(vSyT) D(α) z (ws z (α) T)+A h T ) +Dzz (α) (A z T )T The model x(t) = M 0,t (x 0,α) We calculate the derivatives and their adjoints with respect to x 0, α by TAPENADE 3.6 (Tropics team, INRIA) that allows us to avoid a HUGE development/coding (a double of the classical one, at least) to obtain immediately the derivative with respect to any parameter we want. Eugene Kazantsev Boundary conditions control for ORCA2 page 4 of 19

9 Adjoint search for push/pop TAPENADE 3.6 (Tropics team, INRIA) with the Memory Usage Optimization: CALL PUSHREAL8ARRAY(sold, nx*ny*nz) CALL PUSHREAL8ARRAY(told, nx*ny*nz) CALL PUSHREAL8ARRAY(vold, nx*ny*nz) CALL PUSHREAL8ARRAY(uold, nx*ny*nz) CALL PUSHREAL8ARRAY(ssh, nx*ny) CALL PUSHREAL8ARRAY(s, nx*ny*nz) CALL PUSHREAL8ARRAY(t, nx*ny*nz) CALL PUSHREAL8ARRAY(v, nx*ny*nz) CALL PUSHREAL8ARRAY(u, nx*ny*nz) replace by call push uvts(u,v,t,s,ssh) Procedure push/pop uvts(u,v,t,s,ssh): does not push n 1 step and pops appropriate values (divides the required memory by 2) does not push u,v,t,s in lower level routines does not push values on continents (divides by 2) pushes values in Real*4 format (divides by 2) eventually pushes only odd timesteps and interpolate when poping (divides by 2) Total reduction of required memory is up to 25 times. 10 hours window = 10 days window. Eugene Kazantsev Boundary conditions control for ORCA2 page 4 of 19

10 Data ECMWF data issued from Jason-1 and Envisat altimetric missions and ENACT/ENSEMBLES data banque. January, 1, Difference between observations and background during the 1st of January. Eugene Kazantsev Boundary conditions control for ORCA2 page 5 of 19

11 Cost function The model: x N = M 0,N (x 0,α) with x = (u,v,t,s,ssh) T Cost function J J = x 0 x bgr 2 B 1 + α α bgr 2 B N t n HM 0,n (x 0,α) y n 2 R 1 n=0 Matrices: B 1 = diag(10 4 ), R 1 = diag(1/σ u,1/σ v,1/σ T,1/σ S,1/σ ssh ) where σ 2 u = 1 N obs (uobs u bgr ) 2 Minimization is performed by M1QN3 (JC Gilbert, C.Lemarechal) Data Assimilation Forecast Assimilation window 10 days (Jan. 1-10, 2006), Test time 20 or 30 days (Jan. 1-31, 2006). Eugene Kazantsev Boundary conditions control for ORCA2 page 6 of 19

12 Distance Model-Observations The model: x(t) = M 0,t (x 0,α) with x = (u,v,t,s,ssh) T t Distance: ξ(t) = HM 0,n (x 0,α) y n R 1 n=0 Convergence of J and evolution of ξ 20 Cost function calls with T = 5 days and 40 calls with T = 10 days. Eugene Kazantsev Boundary conditions control for ORCA2 page 7 of 19

13 Optimal IC and Optimal BCz SSH, North Atlantic, January, Optimal IC Optimal BCz Eugene Kazantsev Boundary conditions control for ORCA2 page 8 of 19

14 Optimal IC and Optimal BCz SSH, North Pacific, January, Optimal IC Optimal BCz Eugene Kazantsev Boundary conditions control for ORCA2 page 9 of 19

15 BC for the vertical velocity Modified formula w i,j,k 1 = α0 wb α wb 2 hz i,j,k 1/2ξ i,j,k 1/2 w i,j,k 1 = w i,j,k hz i,j,k 1/2 ξ i,j,k 1/2 k : 1 k K 2 w i,j,0 = w i,j,1 +α ws 0 α ws 2 hz i,j,1/2 ξ i,j,1/2 Eugene Kazantsev Boundary conditions control for ORCA2 page 10 of 19

16 BC for the vertical velocity α for the vertical velocity w. North Atlantic. α 0 on the surface α 2 on the surface α 0 on the bottom α 2 on the bottom Eugene Kazantsev Boundary conditions control for ORCA2 page 10 of 19

17 Vertical velocity North Atlantic, January, 30, 2006, surface Original model Optimal BCz Eugene Kazantsev Boundary conditions control for ORCA2 page 11 of 19

18 Vertical velocity North Atlantic, January, 30, 2006, y z section Original model Optimal BCz Eugene Kazantsev Boundary conditions control for ORCA2 page 11 of 19

19 Vertical velocity North Atlantic, January, 30, 2006, x z section Original model Optimal BCz Eugene Kazantsev Boundary conditions control for ORCA2 page 11 of 19

20 Tourbillon Levels z = 28 and z = 29 Velocity u Velocity v Velocity w Velocity u Velocity v Velocity w Eugene Kazantsev Boundary conditions control for ORCA2 page 12 of 19

21 α 0 for the operator D zz u u z w0 = α DzzUs 0 + τx hz 1 ρ 0, v z = α0 DzzUs + τy, w0 hz 1 ρ 0 u bottom = v bottom = α DzzUb 0 North Atlantic Surface Bottom Eugene Kazantsev Boundary conditions control for ORCA2 page 13 of 19

22 Velocity components North Atlantic, January, 30, 2006, Velocity u, y z section Original model Optimal BCz Eugene Kazantsev Boundary conditions control for ORCA2 page 14 of 19

23 It is not an artefact. North Atlantic Modification of the SSH in the North Atlantic is strongly related to the boundary conditions of u and v especially on the bottom. Eugene Kazantsev Boundary conditions control for ORCA2 page 15 of 19

24 Restrained control Only α 0 on the Bottom for u and v, only in the Vertical diffusion u z Az u z is replaced by ( ) Dzzu i,j,1/2 = (A z u ) 1 hz 1 hz 1/2 (u 3/2 u 1/2 ) ( ) Dzzu i,j,k 1/2 = 1 hz k 1/2 ( (A z u ) k (u k+1/2 u k 1/2 ) (A z u ) ) k 1 (u k 1/2 u k 3/2 ) hz k hz k 1 k : 2 ( ) Dzzu i,j,k 1/2 = 1 [ (A z u ) ( K 1 (A z u ) K u K 1/2 + (A z u ) ] K 1 )u K 3/2 hz K 1/2 hz K 1 hz K hz K 1 u bottom = α u 0 v bottom = α v 0 (2) Control space dimension Initial conditions: Full vertical boundary: Only bottom: Eugene Kazantsev Boundary conditions control for ORCA2 page 16 of 19

25 SSH in the restrained control experiment North Atlantic Eugene Kazantsev Boundary conditions control for ORCA2 page 17 of 19

26 SSH in the restrained control experiment North Pacific Eugene Kazantsev Boundary conditions control for ORCA2 page 18 of 19

27 Que peuvent attendre les modélisateurs de l assimilation de données? Extending the set of control parameters we can find a way to compensate model errors showing the most influent parameter and the most important geographical regions. Automatic adjoint code generation helps us to generate TLM/AM almost immediately, to avoid a HUGE development/coding, to obtain immediately the derivative with respect to any parameter we want. Eugene Kazantsev Boundary conditions control for ORCA2 page 19 of 19