Time-stepping of the CROCO hydrostatic code and its impact on code structure (and on advection schemes)

Transcription

1 Time-stepping of the CROCO hydrostatic code and its impact on code structure (and on advection schemes) Florian Lemarié Inria (EPI AIRSEA), Laboratoire Jean Kuntzmann, Grenoble, France

2 General context Reynolds averaged primitive equations ( X = X + X ) D u h Dt + fk u h = hp ρ 0 zp = gρ u = 0 D T Dt D S Dt h u hu h z w u h = zqs ρ 0C p,o h u ht z w T = h u hs z w S ρ = ρ eos( T, S, z ) and the barotropic mode (with D = ζ + H and Dū = ζ u H h dz) tζ = h Dū t(dū) = Df v h (Dūū) gh xζ ( g /) xζ + DF (u) 3D t(d v) = Dfū h (D vū) gh yζ ( g /) yζ + DF (v) 3D F. Lemarié Croco time-stepping

3 Schematic view of the time-stepping Shchepetkin A. and J. McWilliams : The Regional Oceanic Modeling System: A split-explicit, free-surface, topography-following-coordinate ocean model, Ocean Model., 005 F. Lemarié Croco time-stepping 3

4 Croco source directory Inventory of source files actually involved in time-stepping computations Other files in OCEAN/ directory : diagnostics, I/Os, online interpolation, MPI exchanges, AGRIF,... F. Lemarié Croco time-stepping 4

5 Content 1. 3D momentum and tracers Time-stepping Tracers-momentum coupling. D barotropic mode Time-stepping Baroclinic-barotropic coupling 3. CROCO Step sequence 4. Elements on advection schemes 5. Future evolutions of the CROCO time-stepping F. Lemarié Croco time-stepping 5

6 1 3D momentum and tracers F. Lemarié Croco time-stepping 6

7 Equation Momentum du equation moment Equation Tracer desequation traceurs F. Lemarié Croco time-stepping 7

8 Time stepping (3D momentum and tracers) Predictor-corrector approach : Leapfrog (LF) predictor with 3rd-order Adams-Moulton (AM) interpolation LF-AM3 For a given quantity q q n+1, = q n 1 + t RHS {q n } (LF) q n+ 1 = ( 5 /1) q n+1, + ( /3) q n ( 1 /1) q n 1 (AM3) { } q n+1 = q n + t RHS q n+ 1 (corrector) Compact version used in the code : q n+ 1 = ( 1 / γ) q n 1 + ( 1 / + γ) q n + (1 γ) t RHS {q n } { } q n+1 = q n + t RHS q n+ 1 with γ = 1 /6 (cf pre step3d.f). F. Lemarié Croco time-stepping 8

9 Impact on the structure of the code Some terms are computed twice per time-steps - 3D Advection - Equation of state - Pressure gradient - Continuity equation - Coriolis term Some terms are computed once using an Euler step - Physical parameterization of vertical mixing - Rotated diffusion - Viscous terms and diffusion in general Structure of the step() subroutine call prestep3d thread() call stepd thread() call step3d uv thread() call step3d t thread() F. Lemarié Croco time-stepping 9

10 Structure of the arrays for state variables Need to store 3 time indices real u(global_d_array,n,3) real v(global_d_array,n,3) real t(global_d_array,n,3,nt) time indices Predictor Corrector nstp=1+mod(iic-ntstart,) <-- nstp = 1 or nrhs=nstp <-- n nnew=3 <-- n+1/ indx=3-nstp <-- n-1 nrhs=3 <-- n+1/ nnew=3-nstp <-- n+1 F. Lemarié Croco time-stepping 10

11 Tracers-momentum coupling momentum/ Couplage vitesses-traceurs tracers coupling F. Lemarié Croco time-stepping 11

12 Tracers-momentum coupling zw + xu = 0 zp + ρg = 0 tu + ( 1 /ρ 0 ) xp = 0 tρ + z(wρ) = 0 F. Lemarié Croco time-stepping 1

13 Time stepping (tracers-momentum coupling) Predictor xp n = g x ( 0 ) ρ n dz u n+1/ = (1/ γ) u n 1 + (1/ + γ) u n + (1 γ) t ρ 0 ( xp n ) z z w n = xu n dz ρ n+1/ = (1/ γ) ρ n 1 + (1/ + γ) ρ n + (1 γ) t z(w n ρ n ) H Corrector ( 0 ) xp n+1/ = g x ρ n+1/ dz u n+1 = u n + ( t/ρ 0 ) ( xp n+1/ ) z { z w n+1/ 3u n+1/ = x + un + u n+1 } dz H 4 8 ρ n+1 = ρ n + t z(w n+1/ ρ n+1/ ) Consequences : 3D-momentum integrated before the tracers in the corrector evaluations of the pressure gradient per time-step 3 evaluations of the continuity equation per time-step F. Lemarié Croco time-stepping 13

14 D barotropic mode F. Lemarié Croco time-stepping 14

15 Barotropic mode Barotropic Mode barotrope mode F. Lemarié Croco time-stepping 15

16 Time-stepping (barotropic mode) Generalized forward-backward (predictor-corrector) AB3-type extrapolation D m+1/ = H + ( 3 / + β) ζ m ( 1 / + β) ζ m-1 + βζ m- u m+1/ = ( 3 / + β) u m ( 1 / + β) u m-1 + βu m- Integration of ζ ζ m+1 = ζ m τ x(d m+1/ u m+1/ ) AM4 interpolation ζ = ( 1 / + γ + ε)ζ m+1 + ( 1 / γ 3ε)ζ m + γζ m-1 + εζ m- Integration of ū u m+1 = 1 [ ] D m u m + τ RHSD(D m+1/, u m+1/, ζ ) D m+1 F. Lemarié Croco time-stepping 16

17 Barotropic variables real zeta(global D ARRAY,4) real ubar(global D ARRAY,4) real vbar(global D ARRAY,4) Time indices kold <-- m- kbak <-- m-1 kstp <-- m knew <-- m+1 F. Lemarié Croco time-stepping 17

18 Baroclinic-barotropic coupling Couplage Baroclinic/ barocline-barotrop barotropic coupling F. Lemarié Croco time-stepping 18

19 Baroclinic-barotropic coupling [M FILTER POWER] Barotropic integration from n to n + M τ (M 1.5M) Predictor-corrector integration barotropic filters are needed - ζ n+1 update of the vertical grid - U n+1 correction of baroclinic velocities at time n U n+1/ correction of baroclinic velocities at time n + 1/ Slow forcing term of the barotrope by the barocline is extrapolated { } n+1/ F n+1/ 3D = rhs(u, v)dz rhsd(ū, v) = Extrap(F3D n, F n 1 3D, F n 3D ) F. Lemarié Croco time-stepping 19

20 A new alternative : [M FILTER NONE] Demange J., L. Debreu, P. Marchesiello, F. Lemarié, E. Blayo : Stability analysis of split-explicit ocean models, almost submitted Motivation: Averaging filters excessive dissipation in the barotropic mode Objective: put the minimum amount of dissipation to stabilize the splitting Barotropic mode amplification factor for N = 10 ) 5 Maximum of free surface elevation Flat weights Cosine weights θ = Reference (non splitting) Flat filter over [t : t + t] Cosine filter Power law Filter Forward Backward θ = 0.14 Modal decomposition α0π 3α0π µ Time in hours diffusion is introduced within the barotropic time-stepping rather than averaging filters May require to increase NDTFAST = t 3D/ t D Systematically more efficient than averaging filters F. Lemarié Croco time-stepping 0

21 3CROCO Step sequence F. Lemarié Croco time-stepping 1

22 Complete CROCO time-stepping flowchart can be downloaded from CROCO time stepping F. Lemarié Croco time-stepping

23 4Elements on advection schemes F. Lemarié Croco time-stepping 3

24 Available advection schemes Equation horizontal vertical 3D momentum UV HADV UP3 UV VADV SPLINES UV HADV C4 UV VADV C UV HADV C Tracers TS HADV C4 TS VADV SPLINES TS HADV UP5 TS VADV AKIMA TS HADV WENO5 TS VADV C TS HADV C6 TS VADV WENO5 TS HADV RSUP3 TS HADV RSUP5 D momentum M HADV UP3 - M HADV C - F. Lemarié Croco time-stepping 4

25 Advection problem = interpolation problem Tracers Need Besoin to evaluate d évaluer q q at cell auxinterfaces v i,j+ 1 Momentum u i 1,j q i,j u i+ 1,j Need Besointo d évaluer evaluate uet and v v at aucell centre centers et aux andcoins corners des mailles v i,j+ 1 v i+1,j+ 1 y v i,j 1 x q i,j u i+ 1,j+1 q i+1,j y v i,j 1 v i+1,j 1 x F. Lemarié Croco time-stepping 5

26 Horizontal advection [HADV C, HADV UP3, HADV C4, HADV UP5, HADV C6] eq i 1 u i 1 q i 3 q i q i 1 q i q i+1 q i+ q C i 1/ = q i + q i 1 x(uq) x=xi = 1 x i { ui+1/ q i+1/ u i 1/ q i 1/ } q C4 i 1/ = (7/6) q C i 1/ ( 1/1)(q i+1 + q i ) q UP3 i 1/ = q C4 i 1/ + sign( 1/1, u i 1/ )(q i+1 3q i + 3q i 1 q i ) q C6 i 1/ = (8/5) q C4 i 1/ ( 19/60) q C i 1/ + ( 1/60)(q i+ + q i 3 ) q UP5 i 1/ = q C6 i 1/ sign( 1/60, u i 1/ )(q i+ 5q i q i 10q i 1 + 5q i q i 3 ) F. Lemarié Croco time-stepping 6

27 Properties of linear schemes 1.0 wavelength 50 x 10 x 4 x x 1.0 wavelength 50 x 10 x 4 x x Amplification factor C UP3 C4 UP5 C6 0 π 4 π k x 3π 4 π Phase error C UP3 C4 UP5 C6 0 π 4 π k x 3π 4 π Figure : Amplification error (left) and phase error (right) for linear advection schemes of order to 6. explicit diffusion/viscosity is needed with even-ordered schemes F. Lemarié Croco time-stepping 7

28 Rotated Split UP3 [TS HADV RSUP3] q UP3 i 1/ = q C4 i 1/ + sign( 1 /1, u i 1/ )(q i+1 3q i + 3q i 1 q i ) Split-UP3 q C4 i 1/ treated in the LF-AM3 The red term is treated as an explicit diffusion (i.e. with Euler forward) Rotated Split-UP3 The diffusive term is rotated in the isoneutral direction - small slope assumption - non-monotonic diffusion (additional dispersive cross-terms) - assumes low frequency evolution of isopycnals - a rotation along geopotentials could be a good alternative for high-resolution applications F. Lemarié Croco time-stepping 8

29 WENO5 Scheme (e.g. Acker et al., 016) [TS HADV WENO5] Nonlinear weighting between 3 evaluations of interfacial values q i 3 q i eq k 1 q i q i+1 q k 1/ = w 0 q (0) k 1/ + w 1 q (1) k 1/ + w q () k 1/ x i 3 q i 1 x i x i 1 x i 1 x i x i+1 1. Convexity ( j=0 w j = 1). ENO property (essentially non-scillatory) 3. fifth-order if q(x) is smooth The scheme satisfies the ENO property (Total Variation Bounded property) Warning: no monotonicity-preserving property!!! Best choice for biogeochemical tracers F. Lemarié Croco time-stepping 9

30 Vertical advection : spline reconstruction [TS VADV SPLINES, UV VADV SPLINES] The flux are obtained as a solution of a tridiagonal problem Hz k+1 q k 1/ + (Hz k + Hz k+1 ) q k+1/ + Hz k q k+3/ +hk/ hk/ h k+1 h k q( ) q k+1 q k eq k+ 1 eq k 1 = 3(Hz k q k+1 + Hz k+1 q k ) Amplification factor Splines UP5 C π wavelength 50 x 10 x 4 x x π k x wavelength 50 x 10 x 4 x x 3π 4 π h k 1 q k 1 Phase error q( ) =q k + eq k+ 1 eq k 1 hk eq k eq k 1 q k! h k Splines UP5 C π 4 π k x 3π 4 π F. Lemarié Croco time-stepping 30

31 Vertical advection : semi-implicit formulation [VADV ADAPT IMP] Idea : split the vertical velocity Ω in an explicit and implicit contribution [Shchepetkin, 015] Ω = Ω (e) + Ω (i), Ω (e) = { Ω 1, α z adv α max f(α z adv, αmax), f(αz adv, αmax) = α/α max, α z adv > αmax Ω (e) integrated with an explicit scheme (with CFL α max) Ω (i) integrated with an implicit upwind Euler scheme Configuration Resolution old t new t BENGUELA [Penven et al.] 5 km 6300 s 7140 s OMAN [Vic et al.] km 160 s 470 s Implemented only with Spline reconstruction for the explicit part F. Lemarié Croco time-stepping 31

32 5Future evolutions of the CROCO time-stepping F. Lemarié Croco time-stepping 3

33 Critical comments on current time-stepping LFAM3 significant loss of stability when using odd-ordered advection schemes inadequate for the implicit treatment of bottom drag AM3 interpolation has negative weights no positive-definiteness - pb for layer thicknesses advection and wetting & drying - pb for biogeochemical tracers Existence of a computational mode which is not always efficiently damped Stability with lagrangian treatment of the vertical direction (remapping schemes) Complex interfacing with external modules computation of horizontal pressure gradient Complicated to ensure perfect restarts Generalized Forward-Backward Cold restart at each barotropic integration F. Lemarié Croco time-stepping 33

34 Future evolutions Completion phase for mode splitting analysis and implementation Transition toward an RK framework - Absence of computational mode - RK + forward-backward for internal waves (3rd-order accurate in time) - Easier implementation of time refinement with Agrif - Compatible with TVD or monotonicity-preserving advection schemes - Possibility to derive RK-based coupled space-time schemes for advection Design of an Arbitrary Lagrangian Eulerian vertical coordinate F. Lemarié Croco time-stepping 34