Columnar Clouds and Internal Waves

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1 Columnar Clouds and Internal Waves Daniel Ruprecht 1, Andrew Majda 2, Rupert Klein 1 1 Mathematik & Informatik, Freie Universität Berlin 2 Courant Institute, NYU Tropical Meteorology Workshop, Banff April 27 May 1, 2009

2 Thanks to... Oswald Knoth Oliver Bühler (IfT, Leipzig) (Courant Institute, NYU)

3 Multiscale Modelling Framework Scalings and Expansion Scheme Exact Closure for the Small Scales Results Nonlinearity for Weak Undersaturation Conclusions

4 10 km / 20 min g p 1000 km / 2 days Winter (DJF) km / 1 season Scales

5 u t + u u + wu z + π = S u w t + u w + ww z + π z = θ + S w θ t + u θ + wθ z = S θ (ρ 0 u) + (ρ 0 w) z = 0 θ = 1 + ε 4 θ (x, z, t) + o(ε 4 ) Anelastic Boussinesque Model 10 km / 20 min ( τ + u (0) ) q = 0 ( ) q = ζ (0) + Ω 0 βη + Ω 0 ρ (0) ρ (0) z dθ/dz θ(3) ζ (0) = 2 π (3), θ (3) = π(3) z, u(0) = 1 Ω 0 k π (3) Quasi-geostrophic theory Q T + F t T = S T Q q t + F q = S q Ha Ha ( Qϕ = ρ ϕ dz, F ϕ = ρ u ϕ + (u ) ϕ ) + D ϕ dz, ( ϕ {T, q} ) 1000 km / 2 days zs T = Ts(t, x) + Γ(t, x) min(z, HT ) zs ( zs ) (, q = qs(t, x) exp z ) zs Hq ( ρ = ρ exp z ) (, p = p exp γz ) z T + p0(t, x) + gρ h sc h sc T 0 dz u = ug + ua, fρ k ug = xp uα = α p0 V. Petoukhov et al., CLIMBER-2..., Climate Dynamics, 16, (2000) EMIC - equations (CLIMBER-2) km / 1 season Scales

6 Three-dimensional compressible flow equations ρ t + (ρv) = 0 (ρv) t + (ρv v)+ p + Ω ρv = S ρv ρg k (ρe) t + (v [ρe + p]) = S ρe (ρy j ) t + ρy j v = S ρyj (ρe) = p γ ρv2 + ρ N j=1 Q j Y j How are the various reduced models related to this system? Motivation

7 Key ingredients 1. Identification of uniformly valid system scales non-dimensional parameters distinguished limits 2. Specializations of a multiple scales ansatz a = m Ω = / s g = 9.81 m/s 2 p ref = 10 5 kg/ms 2 T ref = 300 K θ = 50 K R = 287 m 2 /s 2 K γ = 1.4 nothingatall Dimensionless Parameters & Distinguished Limits

8 Key ingredients 1. Identification of c ref Ω a 0.5 uniformly valid system scales a Ω 2 g non-dimensional parameters distinguished limits θ T ref c ref = γrt ref 2. Specializations of a multiple scales ansatz nothingatall Dimensionless Parameters & Distinguished Limits

9 Key ingredients 1. Identification of c ref Ω a ε h sc a ε3 uniformly valid system scales non-dimensional parameters θ T ref ε distinguished limits (ε 0) 2. Specializations of a multiple scales ansatz c ref = p ref /ρ ref h sc = p ref /gρ ref nothingatall Dimensionless Parameters & Distinguished Limits

10 Scaled governing equations ρ t + (ρv) = 0 (ρv) t + (ρv v)+ 1 ε 4 p + ε Ω ρv = S ρv 1 ε 4 ρg k (ρe) t + (v [ρe + p]) = S ρe (ρy j ) t + ρy j v = ε µ is ρyj (ρe) = p γ 1 + ε4 2 ρv2 + ρ N j=1 ε ν j Q j Y j Ready for asymptotics in ε Dimensionless Parameters & Distinguished Limits

11 Recovered classical single-scale models: U (i) = U (i) ( t ε, x, z ε ) Linear small scale internal gravity waves U (i) = U (i) (t, x,z) U (i) = U (i) (εt, ε 2 x,z) U (i) = U (i) (ε 2 t, ε 2 x,z) U (i) = U (i) (ε 2 t, ε 2 x,z) U (i) = U (i) (ε 2 t, ε 1 ξ(ε 2 x),z) U (i) = U (i) (ε 3/2 t, ε 5/2 x, ε 5/2 y, z) Anelastic & pseudo-incompressible models Linear large scale internal gravity waves Mid-latitude Quasi-Geostrophic Flow Equatorial Weak Temperature Gradients Semi-geostrophic flow Kelvin, Yanai, Rossby, and gravity Waves Asymptotic Expansions & Classical Results

12 [hsc/uref] 1/ 3 1/ 5/2 PG 1/ 2 QG 1/ 1 inertial waves advection Boussinesq WTG acoustic waves WTG +Coriolis HPE anelastic / pseudo-incompressible internal waves HPE +Coriolis Obukhov scale 1/ 5/2 bulk micro 1 1/ 1/ 2 1/ 3 convective meso synoptic planetary [h sc ] Scaling Regimes

13 Multiscale Modelling Framework Scalings and Expansion Scheme Exact Closure for the Small Scales Results Nonlinearity for Weak Undersaturation Conclusions

14 Scalings Characteristic (inverse) time scales dimensional dimensionless advection : u ref h sc 1 sound : internal waves : N = ghsc h sc g dθ θ dz ghsc u ref ghsc u ref = 1 ε 2 h sc dθ θ dz = 1 h sc ε 2 θ ε2dθ dz Scaling for the equatorial region: h sc θ dθ dz = O(ε2 ) implies t sound εt internal ε 2 t adv Majda & Klein, JAS, (2003)

15 Clouds and internal waves Columnar clouds / internal wave time scales general expansion scheme U(x,z,t; ε) = i ε i U (i) (η, x,z,τ) horizontal velocity scaling u (0) (η, x,z,τ) u(x,z,τ) η = x/ε τ = t/ε εh sc x = x h sc, t = t u ref h sc h sc Klein & Majda, TCFD, 2006

16 Clouds and internal waves Convective scale u τ + x π =0 H(q c ) 0 w τ + π z = θ θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling H(q c ) 1 C = H(q c ) C d +[1 H(q c )] C ev

17 Clouds and internal waves Saturated Air C d = Cd δq (n ) v q c = ( w + w) dq vs dz ( τ + u η ) q c = H(q c ) C d C cr q r q c ( τ + u η ) q r =0 Undersaturated Air C ev = C ev (q vs (z) q v ) q r 1/2 ( τ + u η ) q v =0 ( τ + u η ) q r =0

18 Multiscale Modelling Framework Scalings and Expansion Scheme Exact Closure for the Small Scales Results Nonlinearity for Weak Undersaturation Conclusions

19 Clouds and internal waves Convective scale (x,z,τ) u τ + x π =0 H(q c ) 0 w τ + π z = θ θ τ + wn 2 = ΓL C p 0 ρ 0 x u + (ρ 0 w) z =0 Cloud column scale (η, x, z, τ) ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling H(q c ) 1 C = H(q c ) C d +[1 H(q c )] C ev

20 Clouds and internal waves Convective scale u τ + x π =0 Analytical microscale closure I C ev = C ev (q vs (z) q v ) q r 1/2 w τ + π z = θ [1 H(q c )] C ev = C(x,z) θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev C d = ( w + w) dq vs dz H(q c ) C d = H(q c ) w + H(q c )w H(q c ) C d = ( w New averaged microscale variables w (x,z,τ) = H(q c ) w σ(x,z) = H(q c ) dqvs dz + σ(x,z) w ) dq vs dz

21 Clouds and internal waves Convective scale u τ + x π =0 Analytical microscale closure I C ev = C ev (q vs (z) q v ) q r 1/2 w τ + π z = θ [1 H(q c )] C ev = C(x,z) θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev C d = ( w + w) dq vs dz H(q c ) C d = H(q c ) w + H(q c )w H(q c ) C d = ( w New averaged microscale variables w (x,z,τ) = H(q c ) w σ(x,z) = H(q c ) dqvs dz + σ(x,z) w ) dq vs dz

22 Clouds and internal waves Convective scale u τ + x π =0 Analytical microscale closure I C ev = C ev (q vs (z) q v ) q r 1/2 w τ + π z = θ [1 H(q c )] C ev = C(x,z) θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev C d = ( w + w) dq vs dz H(q c ) C d = H(q c ) w + H(q c )w H(q c ) C d = ( w New averaged microscale variables w (x,z,τ) = H(q c ) w σ(x,z) = H(q c ) dqvs dz + σ(x,z) w ) dq vs dz

23 Clouds and internal waves Convective scale u τ + x π =0 Analytical microscale closure I C ev = C ev (q vs (z) q v ) q r 1/2 w τ + π z = θ [1 H(q c )] C ev = C(x,z) θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev C d = ( w + w) dq vs dz H(q c ) C d = H(q c ) w + H(q c )w H(q c ) C d = ( w New averaged microscale variables w (x,z,τ) = H(q c ) w σ(x,z) = H(q c ) dqvs dz + σ(x,z) w ) dq vs dz

24 Clouds and internal waves Convective scale u τ + x π =0 Analytical microscale closure I C ev = C ev (q vs (z) q v ) q r 1/2 w τ + π z = θ [1 H(q c )] C ev = C(x,z) θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev C d = ( w + w) dq vs dz H(q c ) C d = H(q c ) w + H(q c )w H(q c ) C d = ( w New averaged microscale variables w (x,z,τ)=h(q c ) w σ(x,z)=h(q c ) dqvs dz + σ(x,z) w ) dq vs dz

25 Clouds and internal waves Convective scale u τ + x π =0 w τ + π z = θ θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ Analytical microscale closure II ( τ + u η ) H(q c )=0 ( τ + u η ) H(q c ) w = H(q c ) θ (H(q c ) w) τ = H(q c ) θ w τ = θ analogously θτ + σw N 2 = σ (1 σ)wn 2 + C ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev where θ (x,z,τ) = H(q c ) θ w (x,z,τ) = H(q c ) w

26 Clouds and internal waves Convective scale u τ + x π =0 w τ + π z = θ θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ Analytical microscale closure II ( τ + u η ) H(q c )=0 ( τ + u η ) H(q c ) w = H(q c ) θ (H(q c ) w) τ = H(q c ) θ w τ = θ analogously θτ + σw N 2 = σ (1 σ)wn 2 + C ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev where θ (x,z,τ)=h(q c ) θ w (x,z,τ)=h(q c ) w

27 Clouds and internal waves Coupled micro-macro dynamics on convective scales (with mean advection) u τ + x π =0 w τ + π z = θ θ τ +(1 σ)wn 2 = w N 2 C ρ 0 x u +(ρ 0 w) z =0 w τ = θ where θ τ + σw N 2 = σ(1 σ)wn 2 + σc. D τ = τ + u x and σ(x,z), C(x,z),N(z) are prescribed

28 Clouds and internal waves Coupled micro-macro dynamics on convective scales (with mean advection) D τ u + x π =0 D τ w + π z = θ D τ θ +(1 σ)wn 2 = w N 2 C ρ 0 x u +(ρ 0 w) z =0 D τ w = θ D τ θ + σw N 2 = σ(1 σ)wn 2 + σc. where D τ = τ + u x and σ(x,z), C(x,z),N(z) are prescribed

29 Multiscale Modelling Framework Scalings and Expansion Scheme Exact Closure for the Small Scales Results Nonlinearity for Weak Undersaturation Conclusions

30 Clouds and internal waves Clouds may narrow the spectrum of lee waves Background Wind Stratified Atmosphere Topography without cloud k up = N u with cloud k up = N u and k low = σ N u

31 Lee waves over sin(x)+sin(2x) topography

32 Cloud meets lee wave Vertical velocity at t = 5.0, U=0.5 z x 0.005

33 Cloud meets lee wave Vertical velocity at t = 10.0, U=

34 Cloud meets lee wave Vertical velocity at t = 15.0, U=

35 Cloud meets lee wave Vertical velocity at t = 17.5, U=

36 Cloud meets lee wave Vertical velocity at t = 20.0, U=

37 Cloud meets lee wave Vertical velocity at t = 22.5, U=

38 Cloud meets lee wave Vertical velocity at t = 5.0, U=0.5 z x 0.005

39 Clouds and internal waves Dry flow over a hill, w ASAM Flow code, courtesy Oswald Knoth, IFT, Leipzig, Germany

40 Clouds and internal waves Moist flow over a hill, q c ASAM Flow code, courtesy Oswald Knoth, IFT, Leipzig, Germany

41 Clouds and internal waves Moist flow over a hill, w ASAM Flow code, courtesy Oswald Knoth, IFT, Leipzig, Germany

42 Multiscale Modelling Framework Scalings and Expansion Scheme Exact Closure for the Small Scales Results Nonlinearity for Weak Undersaturation Conclusions

43 Clouds and nonlinear internal waves Original regime: subsaturation O(1) saturated

44 Clouds and nonlinear internal waves Original regime: subsaturation O(1) saturated

45 Clouds and nonlinear internal waves New regime: subsaturation O(ε) saturated

46 Clouds and nonlinear internal waves New regime: subsaturation O(ε) saturated

47 Clouds and nonlinear internal waves Convective scale u τ + x π =0 Analytical microscale closure I C ev = C ev (q vs (z) q v ) q r 1/2 w τ + π z = θ [1 H(q c )] C ev = C(x,z) θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev C d = ( w + w) dq vs dz H(q c ) C d = H(q c ) w + H(q c )w H(q c ) C d = ( w New averaged microscale variables w (x,z,τ)=h(q c ) w σ(x,z)=h(q c ) dqvs dz + σ(x,z) w ) dq vs dz

48 Clouds and nonlinear internal waves Convective scale u τ + x π =0 w τ + π z = θ θ τ + wn 2 = ΓL C p 0 ρ 0 x u +(ρ 0 w) z =0 Cloud column scale ( τ + u η ) w = θ Analytical microscale closure II ( τ + u η ) H(q c ) = 0 ( τ + u η ) H(q c ) w =?? (H(q c ) w) τ =!! w τ = θ analogously θτ + σw N 2 = σ (1 σ)wn 2 + C ( τ + u η ) θ + wn 2 = ΓL p 0 C. Moisture coupling C = H(q c ) C d +[1 H(q c )] C ev where θ (x,z,τ) = H(q c ) θ w (x,z,τ) = H(q c ) w

49 Clouds and nonlinear internal waves New formulation for the saturation indicator function H(q c ): Total moisture conservation ( τ + u η ) H(q c )q c (1) +(1 H(q c ))q v (1) (0) dq(0) vs + w dz =0 ζ: First-order vertical displacement with ( τ + u η ) ζ = w (0) By choice of initial data: H(q c ) H(ζ) and σ H(ζ) saturated

50 Clouds and nonlinear internal waves H(q c ) w = area-integral of w over saturated domain. As in finite volumes with moving boundary: H(q c ) w = H(q c) θ + τ Observation for undersaturated regions A wv n dσ ( τ + u η ) w = θ ( τ + u η ) θ + wn 2 = C d (τ,x,z) Suggestive assumption: w A w us (τ,x,z)

51 Clouds and nonlinear internal waves Coupled micro-macro dynamics on convective scales (with mean advection) u τ + x π =0 w τ + π z = θ θ τ +(1 σ)wn 2 = w N 2 ρ 0 x u +(ρ 0 w) z =0 θ τ w us, θ us are particular solutions of wτ = θ σ + w us τ + σw N 2 = σ(1 σ)wn 2 + θ σ us τ. σ τ =(w + w us) σ 0 ζ (ζ,x,z) w τ = θ θ τ + wn 2 = σ w dq vs dz

52 Multiscale Modelling Framework Scalings and Expansion Scheme Exact Closure for the Small Scales Results Nonlinearity for Weak Undersaturation Conclusions

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

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