New Results about the Cloud-Top Entrainment Instability

Transcription

1 New Results about the Cloud-Top Entrainment Instability Juan Pedro Mellado, Bjorn Stevens Max Planck Institute Meteorology, Hamburg Heiko Schmidt Brandenburg University of Technology, Cottbus Norbert Peters RWTH Aachen University International MetStröm Conference, FU Berlin 6-10 June 2011 DFG programme 1276

2 Outline The Cloud-Top Entrainment Instability

3 Evaporative Cooling and Clouds Evaporation at cloud/free-atmosphere interface cooling convection: The Cloud-Top Entrainment Instability

4 Buoyancy Reversal The buoyancy may not follow a linear relation wrt conserved scalars: sign change buoyancy reversal The Cloud-Top Entrainment Instability

5 Stratocumulus Top In addition, inversion at the cloud boundary: adapted from D. Randall, J. Atmos. Sci., 1980 Cloud-Top Entrainment Instability (Randall, 1980; Deardor, 1980) The Cloud-Top Entrainment Instability

6 Outline The Cloud-Top Entrainment Instability

7 DNS, grid size (JPM, 2010; JPM et al., 2010)

8 Instantaneous cloud-top {x : χ(x, t) = χs }

9 Instantaneous cloud-top {x : χ(x, t) = χs } Cloud-top not broken and at: Two time scales associated w/ buoyancies b1, bs :D = bs /b1 1.

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11 Inversion Layer

12 Inversion Layer Molecular transport controls the system

13 Inversion Layer [ d ] (1 χ ) dz + w χ (z i, t) = κ χ dt z i z (z i, t) (1 χ i ) dz i dt

14 Inversion Layer [ d ] (1 χ ) dz + dt w χ (z i, t) = κ χ z i z (z i, t) (1 χ i ) dz i dt

15 Inversion Layer [ d ] dt (1 χ ) dz + w χ (z i, t) = κ χ z i z (z i, t) (1 χ i ) dz i dt Dominant balance w e = dz i dt κ h Constant thickness h = 1 χ i χ z (z i, t) Constant entrainment rate

16 Convection Layer

17 Convection Layer d zi χ dz = κ χ dt z (z i, t) w χ (z i, t) + χ i w e w e Then, w e determines also scaling inside the turbulent zone. From functional dependence between b and χ, reference buoyancy ux B s = w e b s /χ s = (0.1f 1 χ 2/3 c /χ s )(κb 4 s) 1/3 and then convection scales z = (1/B s ) Bdz, w = (z B s ) 1/3 characterize (some statistics of) the turbulent region (Deardor, 1980)

18 Turbulent Velocity Fluctuations Self-preservation. Anisotropy. Inhomogeneity.

19 Temporal Evolution of Convection Scales Integrating transport equation for q 2 /2, q 2 /2 t = T z + B ε,

20 Temporal Evolution of Convection Scales Integrating transport equation for q 2 /2, d dt (c qz w 2 ) = 2(1 c ε )w 3 c ε = 1 w 3 ε dz, c q = 1 w 2 z q 2 dz Constant c q (D, χ s ) Constant c ε (D, χ s ) With def. w 3 = B s z, closed system for w (t) and z (t)

21 Temporal Evolution of Convection Scales Solution: Scalings: [ z (t) = z t t 1 (t 1 ) 1 + (2f 2 /3) [z (t 1 ) 2 /B s ] 1/3 w (t) = (1/f 2 ) dz /dt, f ] 3/2 z t 3/2 w t 1/2 b t 1/2 Time scale: (z 2 /B s ) 1/3

22 Stratocumulus Turbulence does not break the cloud top, but enhance mixing up to a linear entrainment rate. Some numbers: w e 0.16 mm/s; h 0.1 m; B s 10 5 m 2 /s 3 ; z 2.5 m; w 30 mm/s. From previous growth rates, 100 m reached in about 45 min. Then, w 0.1 m/s; still measurements of 1 m/s (radiative forcing). Structure: Vertical interface displacement δ = w 2 /b 1 small and Ri = z /δ t 1/2. Internal Richardson number Ri (I) = h/δ t 1 decreases, but order 1 only after z = 300 m.

23 Conclusions The cloud-top entrainment instability cannot explain break-up Evaporative cooling eects are one order of magnitude too small. Buoyancy reversal w/o mean shear depends on molecular props.

24 Mean Vertical Shear {ν, κ, b, b s, χ s, u} {P r, D = b s / b, χ s, ( u) 3 /(ν b)}

25 Turbulent Inversion Layer

26 Turbulent Inversion Layer

27 Formulation Two-uid formulation (St 0.01, Sv 0.3, φ d 10 6 ) Mixture fraction χ (Albrecht et al. 1985; Bretherton 1987; JPM et al. 2010) q t q t,0 q t,1 q t,0 h h 0 h 1 h 0 0 q t q t,0 q t,1 q t,0 = h h 0 h 1 h 0 = χ

28 Governing Equations Boussinesq + Mixture Fraction χ + Non-Linear Eqn. State t u k = i (u k u i ) k p + ν i i u k + bδ k3, i u i = 0 t χ = i (χu i ) + κ i i χ b = b e (χ; b 1, b s, χ s ) Parameter space: {ν, κ, b 1, b s, χ s } {P r = 1, D = b s /b 1, χ s }

29 Mean Entrainment Rate Marginally stable thermal boundary layer χ c h: (χ c h)/(κ/h) ν/(χ c h b s ) 103 h (10/χ 2/3 c )(κ 2 / b s ) 1/3 f 1 = w e (χ 2/3 c /10)(κ b s ) 1/3 f 1 = f 1 (D, χ s )

30 Previous Work Siems et al. (1990), Shy and Breidenthal (1990), Siems et al. (1992) Tank experiments with liquid mixtures. Mechanically driven ICs. Denition of the problem in terms of D = b s /b 1 and χ s. Sims. Almost laminar behavior for D 0.04 (real conditions). Small reversal (D 1) cannot explain cloud break-up. Wunsch (2003) Stochastic models. Conrms previous results. Points to possible relevance of diusion at cloud interface. What is really going on at the interface?

31 Further Discussion Unsteady free convection; Nu (t)/(ra (t)) 1/3 = 0.1f 1 χc 2/3 const. Turbulent mixing across a density interface; Ri (t) increasing. Stratocumulus.