On the regime of validity of sound-proof model equations for atmospheric flows

Transcription

1 On the regime of validity of sound-proof model equations for atmospheric flows Rupert Klein Mathematik & Informatik, Freie Universität Berlin ECMWF, Non-hydrostatic Modelling Workshop Reading, November 8, 2

2 Thanks to... Ulrich Achatz Didier Bresch Omar Knio Fabian Senf Piotr Smolarkiewicz (Goethe-Universität, Frankfurt) (Université de Savoie, Chambéry) (Johns Hopkins University, Baltimore) (IAP, Kühlungsborn) (NCAR, Boulder) MetStröm

3 Motivation... Numerics Why not simply solve the full compressible equations?.5 p T. Hundertmark and S. Reich wave fequency [s! ] 2!2!4 (a)! = min external Lz=8km Lz=8km Lz=8m Lz=8m x wave fequency [s! ] 2!2!4 (b)! = sec external Lz=8km Lz=8km Lz=8m Lz=8m!6 marked: regularized unmarked: exact dispersion 2 4 horizontal length scale [km]!6 marked: regularized unmarked: exact dispersion 2 4 horizontal length scale [km] adapted from Reich et al. (27)

4 Motivation... Numerics Why not simply solve the full compressible equations? Linear Acoustics, simple wave initial data, periodic domain (integration: implicit midpoint rule, staggered grid, 52 grid pts., CFL = ) p p t = x x p p t = 3 Ideas: x Slave short waves (c t/l > ) to long waves (c t/l ) with pseudo-incompressible limit behavior x super-implicit scheme non-standard multi grid projection method see, e.g., Reich et al. (27)

5 Regime(s) of validity of sound-proof models Background Stratification limit in the design-regime Wave-breaking regime with strong stratification Summary

6 Atmospheric Flow Regimes [hsc/uref] / 3 / 5/2 Scale-Dependent Models PG / 2 QG Classical length scales and dimensionless numbers / inertial waves advection Boussinesq WTG acoustic waves WTG +Coriolis HPE anelastic / pseudo-incompressible internal waves HPE L meso = ε +Coriolis h sc L Ro = ε 2 h sc L Ob = ε 5/2 h sc L p = ε 3 h sc Obukhov scale / 5/2 Fr int Ro hsc Ro LRo ε ε ε Ma ε 3/2 bulk micro / / 2 / 3 convective meso synoptic planetary [h sc ] R.K., Ann. Rev. F R.K., Ann. Rev. Fluid Mech, 42, 2

7 Sound-Proof Models Compressible flow equations ρ t + (ρv) = (ρu) t + (ρv u) + P π = (ρw) t + (ρvw) + P π z = ρg P t + (P v) = drop term for: anelastic (approx.) pseudo-incompressible hydrostatic-primitive P = p γ = ρθ, π = p/γp, Γ = c p /R, v = u + wk (u k ) Parameter range & length and time scales of validity? e.g. Lipps & Hemler, JAS, 29, (982) Durran, JAS, 46, (988)

8 Sound-Proof Models Compressible flow equations ρ t + (ρv) = (ρu) t + (ρv u) + P π = (ρw) t + (ρvw) + P π z = ρg P t + (P v) = drop term for: anelastic (approx.) pseudo-incompressible hydrostatic-primitive P = p γ = ρθ, π = p/γp, Γ = c p /R, v = u + wk (u k ) Parameter range & length and time scales of asymptotic validity? e.g. Lipps & Hemler, JAS, 29, (982) Durran, JAS, 46, (988)

9 Regime(s) of validity of sound-proof models Background Stratification limit in the design-regime Wave-breaking regime with strong stratification Summary K., Achatz, Bresch, Knio, Smolarkiewicz, JAS, 67, (2)

10 From here on: ε is the Mach number

11 Regimes of Validity... Design Regime Characteristic (inverse) time scales dimensional dimensionless advection : internal waves : N = u ref h sc g ghsc θ dθ dz u ref h sc dθ θ dz = ε h sc dθ θ dz sound : pref /ρ ref h sc = ghsc h sc ghsc u ref = ε Ogura & Phillips regime with two time scales θ = + ε 2 θ(z) +... h sc θ dθ dz = O(ε2 ) θ h sc z= < K

12 Regimes of Validity... Design Regime Characteristic (inverse) time scales dimensional dimensionless advection : internal waves : N = u ref h sc g ghsc θ dθ dz u ref h sc dθ θ dz = ε h sc d θ θ dz sound : pref /ρ ref h sc = ghsc h sc ghsc u ref = ε Ogura & Phillips regime with two time scales θ = + ε 2 θ(z) +... h sc θ dθ dz = O(ε2 ) θ h sc z= < K Ogura & Phillips (962)

13 Regimes of Validity... Design Regime Characteristic (inverse) time scales dimensional dimensionless advection : internal waves : N = u ref h sc g ghsc θ dθ dz u ref h sc dθ θ dz = ε h sc d θ θ dz sound : pref /ρ ref h sc = ghsc h sc ghsc u ref = ε Ogura & Phillips regime with two time scales θ = + ε 2 θ(z) +... h sc θ dθ dz = O(ε2 ) θ h sc z= < K Ogura & Phillips (962)

14 Regimes of Validity... Design Regime Desirable:. Sound-proof model which 2. accurately represents the (fast) internal waves, and 3. remains accurate over advective time scales.

15 Regimes of Validity... Design Regime Characteristic (inverse) time scales dimensional dimensionless advection : internal waves : N = u ref h sc g ghsc θ dθ dz u ref h sc dθ θ dz = ε ν h sc d θ θ dz sound : pref /ρ ref h sc = ghsc h sc ghsc u ref = ε Realistic regime with three time scales θ = + ε µ θ(z) +... h sc θ dθ dz = O(εµ ) (ν = µ/2)

16 Regimes of Validity... Design Regime θ τ + ε ν ṽ τ + ε ν π τ d θ w = ṽ θ dz θ θ k + ε θ π = ṽ ṽ ε ν θ π + ( γγπ ṽ + w dπ ) = ṽ π γγ π ṽ ε dz. For the linear variable coefficient system: Conservation of weighted quadratic energy Control of time derivatives by initial data (τ = O())... consider internal wave scalings for τ = O(ε ν ): ϑ = τ ε, ν π = ε ν π, Klainerman, Majda (982)... Dutrifoy, Majda, Schochet (26,28)

17 Regimes of Validity... Design Regime Fast linear compressible / pseudo-incompressible modes θ ϑ + w dθ dz = ṽ ϑ + θ θ k + θ π = ( ε µ πϑ + γγπ ṽ + w dπ ) = dz Vertical mode expansion (separation of variables) θ ũ w π (ϑ, x, z) = Θ U W Π (z) exp (i [ωϑ λ x])

18 Regimes of Validity... Design Regime Relation between compressible and pseudo-incompressible vertical modes d dz ( ε µω2 /λ 2 c 2 θ P ) dw dz + λ2 θ P W = ω 2 λ 2 N 2 θ P W ε µ = : pseudo-incompressible case regular Sturm-Liouville problem for internal wave modes (rigid lid) ε µ > : compressible case nonlinear Sturm-Liouville problem... ω 2 /λ 2 c 2 = O() : perturbations of pseudo-incompressible modes & EVals

19 Regimes of Validity... Design Regime d dz ( ε µω2 /λ 2 c 2 θ P ) dw dz + λ2 θ P W = ω 2 λ 2 N 2 θ P W Internal wave modes ( ω 2 /λ 2 c 2 ) = O() pseudo-incompressible modes/evals = compressible modes/evals + O(ε µ ) phase errors remain small over advection time scales for µ > 2 3 The anelastic and pseudo-incompressible models remain relevant for stratifications θ dθ dz < O(ε2/3 ) θ h sc < 4 K not merely up to O(ε 2 ) as in Ogura-Phillips (962) rigorous proof with D. Bresch

20 Regimes of Validity... Design Regime A typical vertical structure function (L πh sc 3 km) 3 2 anelastic pseudo-inc compressible w ˇ - -2 λ = z/h sc thanks to Dr. V. LeDoux, Ghent, for the SL-solver MATSLISE!

21 Regimes of Validity... Design Regime Relative eigenvalue errors EV sproof EVcompr EVcompr -3 5 anelastic pseudo-inc -4 5 anelastic pseudo-inc Δ EV / EV compr -5 - λ =.5 Δ EV / EV compr 5 λ = mode no mode no. thanks to Dr. V. LeDoux, Ghent, for the SL-solver MATSLISE!

22 Regimes of Validity... Design Regime Remarks Estimates are uniform in the horizontal long-wave limit (Coriolis not yet included) Regime of validity includes isothermal stratification if.286 γ γ (Newtonian Limit ) ε 2/3.25 (ε = /8) K., Majda, TCFD, 2, (26)

23 Regimes of Validity... Design Regime Remarks Estimates are uniform in the horizontal long-wave limit (Coriolis not yet included) Regime of validity includes isothermal stratification if.286 γ γ (Newtonian Limit ) ε 2/3.25 (ε = /8) K., Majda, TCFD, 2, (26)

24 Regime(s) of validity of sound-proof models Background Stratification limit in the design-regime Wave-breaking regime with strong stratification Summary Achatz, K., Senf, JFM, 663, 2 47 (2)

25 6 km Breaking wave-test for anelastic models (Smolarkiewicz & Margolin (997)) km -6 km km 6 km

26 Time scale gap for short wave lengths L 2π km wave fequency [s ] vertical acoustics internal waves Lamb wave external Lz=8km Lz=8km Lz=8m Lz=8m 6 exact dispersion 2 4 horizontal length scale [km]

27 Wave breaking regime, strong stratification WKB theory: 2π km wave packets modulated over km distances θ stratification of order O() scalings allow for overturning of θ-contours Expansion scheme: U(t, x, z; ε) = U(z) + U () exp (i ϕε ε ) + ε 2 n= U () n ) exp (in ϕε ε ϕ ε = ϕ () + εϕ () + o(ε) ( U (i) n, ϕ (i) ) ( U (i) n, ϕ (i) ) (t, x, z)

28 Wave breaking regime, strong stratification Leading order: classical Boussinesq / ray tracing theory iˆω ik iˆω N im N iˆω ik im }{{} Û () Ŵ () ˆΘ () ˆθ () () ˆΠ(2) N ˆθ = where ˆω = ϕ() t k = m = ϕ() x ϕ() z ku () M(ˆω, k, m)

29 Wave breaking regime, strong stratification First order: pseudo-incompressible corrections M () U W Θ N θ θπ () + M () U W Θ N θ θπ () = U () τ N [ U () W () τ τ U () χ ( U () χ W () U () ) Θ () θ () () W ζ W () χ U () κ κ U () ζ θ() Π(2) ( χ θ () Π(2) χ ζ )] Θ () θ () W () π () π () ζ First-order Hamilton-Jacobi-eqn. for ϕ () exp ( i n ϕ () /ε ) Explicit solutions for all higher-oder modes exp ( i n ϕ () /ɛ ), (n =, 2,...)

30 Wave breaking regime, strong stratification First order: phase corrections M () U W Θ N θ θπ () + M () U W Θ N θ θπ () = U () τ N [ U () W () τ τ U () χ ( U () χ W () U () ) Θ () θ () () W ζ W () χ U () κ κ U () ζ θ() Π(2) ( χ θ () Π(2) χ ζ )] Θ () θ () W () π () π () ζ First-order Hamilton-Jacobi-eqn. for ϕ () exp ( i n ϕ () /ε ) Explicit solutions for all higher-oder modes exp ( i n ϕ () /ɛ ), (n =, 2,...)

31 Wave breaking regime, strong stratification First order: pseudo-incompressible corrections M () U W Θ N θ θπ () + M () U W Θ N θ θπ () = U () τ N [ U () W () τ τ () U χ ( U () χ W () U () ) Θ () θ () W ζ () W () χ U () U () ζ θ() Π(2) ( χ θ () Π(2) χ ζ )] Θ () θ () κ W () π () κ π () ζ pseudo-incompressible wave action conservation law ϕ () exp ( i n ϕ () /ε ) Explicit solutions for all higher-oder modes exp ( i n ϕ () /ɛ ), (n =, 2,...)

32 Wave breaking regime, strong stratification First order: higher harmonics M () U W Θ N θ θπ () + M () U W Θ N θ θπ () = U () τ N [ U () W () τ τ U () χ ( U () χ W () U () ) Θ () θ () () W ζ W () χ U () κ κ U () ζ θ() Π(2) ( χ θ () Π(2) χ ζ )] Θ () θ () W () π () π () ζ + Nonlinear Effects: Explicit solutions for all higher-oder modes exp ( i n ϕ () /ε ), (n =, 2,...)

33 Regime(s) of validity of sound-proof models Background Stratification limit in the design-regime Wave-breaking regime with strong stratification Summary

34 Summary Compressible flow equations ρ t + (ρv) = (ρu) t + (ρv u) + P π = (ρw) t + (ρvw) + P π z = ρg drop term for: anelastic (approx.) pseudo-incompressible P t + (P v) = P = p γ = ρθ, π = p/γp, Γ = c p /R, v = u + wk, (u k )

35 Summary dθ θ dz wave breaking regime Pseudo-incompressible WKB - Extension to Pot-Temp Scale Height Remarks: ε 2/3 Anelastic Isothermal stratification captured for γ γ ) = O (ε 2/3 ε 2 Ogura & Phillips (962) Uniform approximation in the long wave limit ε 2πH/L Pseudo-incompressible model wins by small margin