3-Fold Decomposition EFB Closure for Convective Turbulence and Organized Structures

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ROGACHEVSKII and Nathan KLEEORIN Ben-Gurion

1 3-Fold Decomposition EFB Closure for Convective Turbulence and Organized Structures Igor ROGACHEVSKII and Nathan KLEEORIN Ben-Gurion University of the Negev, Beer-Sheva, Israel N.I. Lobachevsky State University of Nizhni Novgorod, Russia Sergej ZILITINKEVICH Finnish Meteorological Institute and University of Helsinki, Finland N.I. Lobachevsky State University of Nizhni Novgorod, Russia

2 Outline Introduction Mechanism of formation of cloud cells in shear-free convection EFB closure for shear-free convection Mechanism of formation of cloud streets in sheared convection EFB closure for sheared convection Conclusions and future studies

Two types of the semi-organized structures: cloud streets cloud cells The life-times and spatial scales of the semi-organized structures are

3 Atmospheric Turbulent Convection The atmospheric turbulent convection: the fully organized large-scale flow (the mean flow or mean wind) the small-scale turbulent fluctuations, long-lived large-scale semi-organized (coherent) structures. Two types of the semi-organized structures: cloud streets cloud cells The life-times and spatial scales of the semi-organized structures are much larger than the turbulent scales. Etling, D. and Brown, R. A., Boundary-Layer Meteorol., 65, Atkinson, B. W. and Wu Zhang, J., Reviews of Geophysics, 34,

4 Closed cloud cells over the Atlantic Ocean

5 Open cloud cells over the Pacific Ocean

6 Cloud streets over Indian ocean

7 Cloud streets over the Amazon River

8 Laboratory Turbulent Convection In laboratory turbulent convection several organized features of motion, such as plumes, jets, and largescale circulation patterns are observed. The large-scale circulation in a closed box with the heated bottom (in the Rayleigh-Benard apparatus) is often called the "mean wind". There are several unsolved theoretical questions concerning these flows: How do they arise? What are their characteristics and dynamics?

9 Coherent Structures (Mean Wind) in Laboratory Turbulent Convection

10 Preferential Coherent structures for A=2 Preferential Coherent structures for A=4

11 Unforced Convection: A = 1

12 Unforced Convection: A=2

The effective Rayleigh numbers based on the turbulent transport coefficients (the turbulent viscosity and turbulent diffusivity) are not

13 The Rayleigh numbers based on the molecular transport coefficients are very large: Problems This corresponds to fully developed turbulent convection in atmospheric and laboratory flows. The effective Rayleigh numbers based on the turbulent transport coefficients (the turbulent viscosity and turbulent diffusivity) are not high. They are less than the critical Rayleigh numbers required for the excitation of large-scale convection. Hence the emergence of large-scale convective flows (which are observed in the atmospheric and laboratory flows) seems puzzling.

14 Turbulent objects Interaction between mean-flow and turbulent objects Mean-flow objects Regular flow Semi-organized structures Tangling True Turbulence (Kolmogorov cascade) Tangling Turbulence (no cascade) Inverse cascade through large-scale instability

15 Tangling turbulence in sheared mean flow Sheared mean flow Kolmogorov turbulence Tangling turbulence Lumley (1967)

16 Heat flux

17 Redistribution of a homogeneous vertical turbulent heat flux by a converging horizontal mean flow

18 Heat flux Traditional parameterization Modification of the heat flux by tangling turbulence

19 Mean field equations is the heat flux are the Reynolds stresses

20 Method of Derivation Equations for the correlation functions for: The velocity fluctuations The temperature fluctuations The heat flux The spectral t-approximation (the third-order closure procedure)

21 Heat flux Traditional parameterization Modification of the heat flux by tangling turbulence

22 The growth rate of convective wind instability t 0 t 0 L l 0 L z L

23 Convective-wind instability The range of parameters for which the convective-wind instability occurs for different anisotropy of turbulence.

24 Critical Rayleigh Number In laminar convection:

25 Closed Cloud Cells over the Atlantic Ocean

26 Cloud cells Observations Theory Several hours

27 Budget Equations

28 Cloud Cells

29 Solution for Cloud Cells

30 Budget Equations

31 Dimensionless Ratios for Cloud Cells implies the averaging over the volume of the semi-organized structure.

32 Energies of Coherent Structures The kinetic energy of the semi-organized structures (cloud cells): The thermal energy of the semi-organized structures (cloud cells): The mean vertical temperature gradient:

33 The vertical flux of entropy transported by the cloud cells The vertical flux of entropy transported by the semi-organized structures: The ratio of fluxes of entropy :

34 Mechanism of convective-shear instability U (0) U W W U W U U W

35 Convective-shear waves (0) U y ( z) z y x wave

36 Counter wind flux z (0) U Tangling fluctuations U (0) ( z)

37 Convective-shear instability The range of parameters for which the convective-shear instability occurs for different values of shear and anisotropy.

38 Convective-shear instability The growth rate of the convective-shear instability and frequencies of the generated convective-shear waves.

39 Convective-shear instability z y x (0) U y ( z) K

40 Maximum growth rate t inst 1.0 l l z l l z l l z The growth rate of the convective shear instability for different thermal anisotropy

41 Conditions for the instability L z /L L L L z 0 K y sin K 6 Instability 3 2 L/l 0 The range of parameters and for which the convective shear instability occurs

42 Thermal anisotropy u i ( x) ( x r) ( ) l r l Column -like thermal structure ( ) ( ) l z l ( ) ( ) l z Pancake -like thermal structure

43 Cloud streets over the Amazon River

44 Cloud streets Observations Theory

45 Budget Equations

46 Cloud Streets. Sheared Convection The shear velocity: The solution of linearized equations:

47 Vorticily of Cloud Streets A large-scale helicity produced by the semi-organized structures:

48 Budget Equations

49 Production in Sheared Convection The production of turbulence is caused by three sources: a) the shear of the semi-organized structures: b) the background wind shear: c) the buoyancy:

50 Energies of Coherent Structures The kinetic energy of the semi-organized structures (cloud streets): The thermal energy of the semi-organized structures (cloud streets): The Deardorff velocity scale: The vertical turbulent flux of entropy:

References Elperin T., Kleeorin N., Rogachevskii I., and Zilitinkevich S., Formation of largescale semi-organized structures in turbulent convection, Phys. Rev. E 66, 066305 (1 15), (2002). Elperin T., Kleeorin N., Rogachevskii I., and Zilitinkevich S., Tangling turbulence and semi-organized structures in convective boundary layers, Boundary-Layer Meteorol.

, Kleeorin N., and Rogachevskii I., Large-scale instabilities in turbulent convection, Phys. Fluids, 18, 126601 (1 11), (2006). Bukai M., Eidelman A., Elperin T., Kleeorin N., Rogachevskii I.

51 References Elperin T., Kleeorin N., Rogachevskii I., and Zilitinkevich S., Formation of largescale semi-organized structures in turbulent convection, Phys. Rev. E 66, (1 15), (2002). Elperin T., Kleeorin N., Rogachevskii I., and Zilitinkevich S., Tangling turbulence and semi-organized structures in convective boundary layers, Boundary-Layer Meteorol., 119, (2006). Eidelman A., Elperin T., Kleeorin N., Markovich A., and Rogachevskii I., Hysteresis phenomenon in turbulent convection, Experiments in Fluids, 40, (2006). Elperin T., Golubev I., Kleeorin N., and Rogachevskii I., Large-scale instabilities in turbulent convection, Phys. Fluids, 18, (1 11), (2006). Bukai M., Eidelman A., Elperin T., Kleeorin N., Rogachevskii I., Sapir-Katiraie I., Effect of Large-Scale Coherent Structures on Turbulent Convection, Phys. Rev. E 79, (1-9), (2009). Bukai M., Eidelman A., Elperin T., Kleeorin N., Rogachevskii I. and Sapir- Katiraie I., "Transition Phenomena in Unstably Stratified Turbulent Flows", Phys. Rev. E 83, (1-12), (2011). New, in preparation,

52 THE END

Sergej S. Zilitinkevich 1,2,3. Helsinki 27 May 1 June Division of Atmospheric Sciences, University of Helsinki, Finland 2

Sergej S. Zilitinkevich 1,2,3. Helsinki 27 May 1 June Division of Atmospheric Sciences, University of Helsinki, Finland 2 Atmospheric Planetary Boundary Layers (ABLs / PBLs) in stable, neural and unstable stratification: scaling, data, analytical models and surface-flux algorithms Sergej S. Zilitinkevich 1,2,3 1 Division