Stability of the physics-dynamics interface

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Solomon Harrington
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1 Stability of the physics-dynamics interface Rafiq Hamdi and Piet Termonia Royal Meteorological Institute of Belgium Stability of the physics-dynamics interface p.1/21

2 Contents METHODOLOGY: drastically compressed version of my lectures at the TCWGPDI in Prague CHOICES: modular criteria for stability and accuracy the organisation of the time step Stability of the physics-dynamics interface p.2/21

3 Room for coupling where on the semi-lagrangian trajectory? before or after the dynamics? parallel or sequential (= fractional)? Stability of the physics-dynamics interface p.3/21

4 Where on SL trajectory? t D t+ t Stability of the physics-dynamics interface p.4/21

5 Methodology simple system, but with EXCT solutions Stability of the physics-dynamics interface p.5/21

6 Methodology simple system, but with EXCT solutions Staniforth, Wood, and Côté, 2002 F t + U F x + iωf = βf + R ei[kx+ω] Stability of the physics-dynamics interface p.5/21

7 Methodology simple system, but with EXCT solutions Staniforth, Wood, and Côté, 2002 F t + U F x + iωf = βf + R ei[kx+ω] solution 1: free solution (homogeneous Eq.) F (x, t) = F free k e βt e i[kx (ω+ku)t] Stability of the physics-dynamics interface p.5/21

8 Methodology simple system, but with EXCT solutions Staniforth, Wood, and Côté, 2002 F t + U F x + iωf = βf + R ei[kx+ω] solution 2: forced regular solution R F (x, t) = β + i(ω + ku + Ω) ei[kx+ωt] β + i(ω + ku + Ω) 0 Stability of the physics-dynamics interface p.6/21

9 Methodology simple system, but with EXCT solutions Staniforth, Wood, and Côté, 2002 F t + U F x + iωf = βf + R ei[kx+ω] solution 3: forced resonant solution F (x, t) = R t e i[kx+ωt] β = 0 and ω + ku + Ω = 0 Stability of the physics-dynamics interface p.7/21

10 LDIN/RPEGE computation result 1 inv. FFT, inv. Legendre transformation F (t) 2 call physics (PLPR) Φ 3 update tendencies F = F (t) + tφ 4 compute departure(, middle) point (D, M) 5 interpolate to D(, M) FD 6 explicit part dynamics F exp 7 FFT, Legendre transformation 8 Helmholtz, Horizontal diffusion F + Stability of the physics-dynamics interface p.8/21

11 ECMWF: SLVEPP computation result 1 inv. FFT, inv. Legendre transformation F (t) 2 lin. terms, non-lin. R(t) and R(t t) L 0, R, R0 3 compute departure point (D) 4 interpolate to D L 0, R 0 = (2R R ) 5 adiabatic explicit tendencies at arrival point ( ) D 6 interpolate diab. tendencies of rad., conv. and cl. at t to D P 0 7 tendencies of parameterized processes P + (F (t), D, fractional) 8 add tendencies of adiabatic and diabatic processes FD L0 + t(r P 1 2 ) 9 FFT, Legendre transformation 10 Helmholtz, Horizontal diffusion F + Stability of the physics-dynamics interface p.9/21

12 lgorithmics: PRLLEL G α F 0 Φ (lev. I) = (1 ɛ t α )ξ α φ α [F 0, G α], α = 1,..., M Interface F = F 0 + t P M G α F 0 α=1 t GP Expl. Dyn. F exp = `1 iω t e 2 iku t F i 2 (ω ω ) tf (0) SP Φ (lev. II) Interface Impl. Dyn. Hor. Diff. Ders. Φ (lev. III) G exp exp α F G exp F dyn t = F exp = (1 ɛ α )(1 ξ α )(1 ν α )φ E [e iλαku t (F 0 ; F exp ; Gexp α α = 1,..., M + t P M α=1 G exp exp α F = h 1 + iω 2 t t P M α=1 (1 ɛ α)(1 ξ α )ν α φ I α p xf = (ik) p F G T T α GP Interface G T T Impl. Φ dyn F t = F dyn t = ɛ α (1 ζ α )φ E α [F 0 ; F dyn + t P M G T T α α=1 dyn F t F + = h 1 t P M α=1 ɛ αζ α φ I i 1 G T T i 1 G exp ; GT T α ], α = 1,..., M Stability of the physics-dynamics interface p.10/21

13 The choice? ν ζ ε Stability of the physics-dynamics interface p.11/21

14 The choice? ν ζ ε Stability of the physics-dynamics interface p.11/21

15 ctually, what is stability? Wish list: I have my physics wich I have thoroughly validated in dynamics X with time step organisation x and I want to plug it in dynamics Y. It is stable in X with time step organisation x so... it is a priori not clear it will be stable in another Φ D coupling need for more severe criteria, recall: stable, unconditionally stable,... Stability of the physics-dynamics interface p.12/21

16 3 steps toward more unconditionality absolute stability F + F 0 F + F, exact restriction to schemes where stability is independent of the dynamics (read ω ) compare to the exact solution, i.e. we want Φ to be stable modulo the stability of D, modular stability: F + F semi exact F semi exact e β t ( 1 i 2 ω) e iũ i 2 ( ω ω ) 1 + i 2 ω Stability of the physics-dynamics interface p.13/21

17 Stability independent of D Stability of the physics-dynamics interface p.14/21

18 bsolute stability Stability of the physics-dynamics interface p.15/21

19 Independent of D Stability of the physics-dynamics interface p.16/21

20 Relative stability Stability of the physics-dynamics interface p.17/21

21 ... in English there are two levels of where we get (de)stablization: in the interface, i.e. where we couple to the model below the interface, due to the amount of (split-) implicitness moving the interface later in the time step organisation has a stablizing effect Stability of the physics-dynamics interface p.18/21

22 ccuracy is take the coefficient of the second-order term in the expansion in t. For instance in the case ɛ = 0, ν = 0: ξp exp α 1 D exp (1 ξ)p imp α 1 D imp yields 1 βf [ 0 (β 2βξ + 2βξ 2 α 2 1 ) + i (ω ξω) ] t 2 ξ = 1 α 1 = 1 2 : RPEGE/LDIN ξ = 1 2 α 1 = 0: ECMWF compromise (Wedi paper) Stability of the physics-dynamics interface p.19/21

23 So... we don t want to jump to conclusions... BUT Stability of the physics-dynamics interface p.20/21

24 So... we don t want to jump to conclusions... BUT it seems as if RPEGE/LDIN uses the wrong choice...? Stability of the physics-dynamics interface p.20/21

25 So... we don t want to jump to conclusions... BUT it seems as if RPEGE/LDIN uses the wrong choice...? HOWEVER this knife cuts at 2 sides: Stability of the physics-dynamics interface p.20/21

26 So... we don t want to jump to conclusions... BUT it seems as if RPEGE/LDIN uses the wrong choice...? HOWEVER this knife cuts at 2 sides: this probably means that RPEGE/LDIN physics satisfies more severe unconditionally stability criteria! Stability of the physics-dynamics interface p.20/21

27 THNKS to the organisers for this opportunity to start paying back my debt to science... Stability of the physics-dynamics interface p.21/21

Overview of the Numerics of the ECMWF. Atmospheric Forecast Model

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