Spectrally condensed turbulence in two dimensions

Transcription

1 Spectrally condensed turbulence in two dimensions Hua Xia 1, Michael Shats 1, Gregory Falovich 1 The Australian National University, Canberra, Australia Weizmann Institute of Science, Rehovot, Israel Acnowledgements: H. Punzmann, D. Byrne TOY008: Summer school: Geophysical Turbulence, Boulder, Colorado

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3 Motivation Turbulence often coexists with coherent flow D turbulence is capable of generating such flows spectral condensation, crystallization Turbulence-condensate interplay dynamical steady-state energy transfer from turbulence to flows effects of shear flows on turbulence Practical applications atmospheric and oceanic processes, magnetically confined plasma, etc.

4 D turbulence and spectral condensation

5 Large coherent flows coexist with turbulence Antarctic Circumpolar Current Average volume transport ~ m 3 s 1 Jupiter s zonal flows GRS Planetary atmospheres are dominated by turbulent structures (cyclones, zonal winds, etc) Earth: atmospheric zonal winds Cassini spacecraft Courtesy NASA

6 1 8-3 n e (10 m ) Turbulence-driven structures in fusion plasma High 1.5 H n e Low L 0 Zonal flows in plasma Numerical simulations [Z. Lin, Science 1999] j r In magnetically confined plasma, turbulence-driven anisotropic flows develop, which inhibit radial transport of particles and energy

7 D turbulence 100 E Idealized homogeneous isotropic D turbulence Both energy and enstrophy are conserved /3 R. Kraichnan (1967) Dual cascade: Energy: inverse cascade, Enstrophy: forward cascade Energy cascade Enstrophy cascade f 1000 E -3 C /3 5/3 E C V /3 is the energy dissipation rate is the enstrophy dissipation rate 1 ω enstrophy 3 dv Opposite to 3D, energy flows from smaller to larger scales Basis for self-organization

8 Structure functions and Kolmogorov law ut1 1 Label an eddy by a velocity increment u l across a distance r: u L1 u 1 u r u x r u L u x Statistical moments of this increment are called structure functions of the n th order: S ( r) n u r n u x r u x n r u u T Kolmogorov law relates the third-order longitudinal structure function of turbulence to the mean energy dissipation per unit mass in D (e.g. [Lindborg 1999]): 3 3 S3 L ( r) VL r r [G. Boffetta, A. Celani, M. Vergassola, 000]

9 Spectral condensation of D turbulence The maximum of the energy spectrum lies in the low- range, at, in the absence of the energy dissipation at large scales can not be constant in time since it accumulates spectral energy f, t System size < dissipation scale -5/3 Dissipation at large scales (bottom damping) stabilizes the maximum of the spectrum at the scale 3 1/ Condensate Energy cascade -3 Kraichnan, 1967: predicted condensate L Enstrophy cascade At low dissipation in a bounded system, at << L spectral energy accumulates in a box-size coherent structure f

10 Spectral condensation of turbulence in thin layers Experiments: Sommeria (1986), Paret &Tabeling (1998), Shats et al (005, 007) Time evolution to condensed state [Shats et al (005)] Numerical simulations of D turbulence: Hossain (1983), Smith &Yahot (1993) van Heijst, Clercx, Molenaar ( ), Chertov et al. (007) Periodic boundary condition dipole No-slip boundary single vortex Vorticity of the condensate [M. Chertov et al. (007)]

11 Hi - res Camera Experimental setup Synchronization Pulse Generator 1 Laser 1 Pulse Generator Light fluid B Laser sheet Cylindrical lens Laser Heavy fluid Magnets Beam splitter j Electrodes Condensation Vertical magnetic field Vorticity field Bottom layer: isolator Fluorinert FC-77 (resist. = x10 15 Ohm cm; SG = 1.78) Top layer: electrolyte NaCl solution (SG = 1.04)

12 Condensed turbulence spectrum is robust E ( ) -3 Time-averaged L = 0.1 m strong condensate -5/3 turbulence -3 -Bottom drag, -boundary size, -forcing affect condensate strength and topology t f (3-4) at large scales (m ) -5/3 at meso-scales wea -(3-4) at small scales

13 Nastrom-Gage spectrum of atmospheric winds Atmospheric spectrum [Nastrom, Gage, Jasperson, Nature (1984) ] -3 and -5/3 ranges are present but in the reversed order compared to the Kraichnan theory E E C C /3 5/3 /3 What is the origin of -3 and -5/3 ranges in atmosphere? 3 at < f at > f Meso-scale -5/3 range can be due to 3D (downscale) direct energy cascade, D inverse (upscale) cascade Large-scale -3 range can be due to direct enstrophy cascade (large-scale forcing) spectral condensation Scales (m) Kinetic energy spectrum alone cannot resolve the question of the sources

14 Energy flux in atmospheric turbulence negative Third-order velocity moment gives the energy flux direction 3 S3( r) r Negative S 3 at scales up to 500 m interpreted as evidence against inverse energy cascade in the mesoscale range [Cho, Lindborg, J. Geophys. Res. (001) ]

15 Need to understand spectral flux in the presence of large coherent flow, which may affect higher moments Condensate coherent flow self-generated by turbulence In the lab can control strength and spectral extent of condensate (?)

16 Model of the spectrum In the inverse cascade, the turnover time of the eddy of scale l is /3 1/ 1/ 3 l S l t l C 1. Assume that the condensate (vortex) appears when the system size L is such that t L <1. Characterize the condensate amplitude by its mean velocity V. This velocity can be estimated from the energy balance, V which gives V / 3. We estimate that the condensate related velocity fluctuation on the scale l as Vl/L. Then we expect the nee of the spectrum to be at the scale l t defined by t l t L 3/ C / 3/ 4 1/ 4 This gives l t Vl L C 1/ t l t L 3/ C / 3/ 4 1/3 1/ 4

17 Knee of the spectrum shifts with and L t l t L 3/ C / 3/ 4 1/ E () 10-5 = 0.15 s -1 E () L = 0.35 m 10-6 L=0.15 m 10-6 = L=0.35 m -5/ = / (a) (m ) (b) t increases with the decrease in the boundary size L t increases with the decrease in the damping rate In the lab we can control strength, spectral extent of the condensate

18 Case of wea condensate E C /3 5/3 3 S ( r) V V V 3 L L T r S F S 3 / S S 4 / S 3/ E( ) (a) 1-7 S 3 (10 ) (b) 4 Flatness (c) Sewness / (m ) r (m) r (m) 0.1 Wea condensate case shows small differences with isotropic D turbulence ~ -5/3 spectrum in the energy range Kolmogorov law linear S 3 (r) dependence; Kolmogorov constant C 5.6 Sewness and flatness are close to their Gaussian values (S = 0, F = 3)

19 Case of stronger condensate E (m /s ) -3-5/3 (a) S 3 (10 m s ) turbulence condensate (b) 10-9 condensate turbulence t f (m ) 1000 Mean shear flow (condensate) V changes all velocity moments: V r (m) V ~ V V V ~ ~ V V V V 3 3 ~ ~ ~ 3 V 3 V V 3 V V V

20 Mean subtraction recovers isotropic turbulence 1.Compute time-average velocity field (N=400): V x y N N (, ) 1 n 1. Subtract V ( x, y) from N=400 instantaneous velocity fields 10-6 E () -5/ S 3 (10 m s ) V ( x, y, t n ) (m ) r (m) 0.1 Recover ~ -5/3 spectrum in the energy range S 3 (r) is positive recovered inverse energy cascade Kolmogorov law linear S 3 (r) dependence in the turbulence range ; Kolmogorov constant C 7

21 condensate turbulence turbulence condensate Strong condensate: effect of mean subtraction E (m /s ) 7 6 F S Before: / (c) t -3 3 (d) (m ) r t r (m) E (m /s ) -5/3 7 F S 0.6 After: (e) t f (m ) r r (m) 0.01 t 0.1 Normalized moments ~ scale-independent Flatness is higher than in isotropic turbulence (f)

22 Similarity with atmospheric turbulence [Cho, Lindborg, J. Geophys. Res. (001) ] negative Laboratory experiment Stronger condensate, no mean subtraction 1E-6 S 3L 1E-7 Mean shear flows present in the atmosphere affect velocity moments, similarly to laboratory experiments 1E-8 negative 1E-9 r (m)

23 Different moments affected in different ranges Second moment Large-scale flow is spatially smooth: V sr, V s r Small-scale velocity fluctuations in turbulence v C r /3 Small-scale fluctuations dominate at scales smaller than l l t C 3/ 4 s 3/ 1/ Third moment Large-scale flow: V 3 V v src r /3 Small-scale fluctuations: v 3 r Large-scale flow dominates 3 rd moment in a range to much smaller scales: l 3/ * C s 3/ 1/, since C > 1 Condensate imposes different scales on different moments

24 The strongest condensate can suppress turbulence (a) Self-generated condensate (b) Externally imposed flow

25 10-6 Shear E() / (m ) [M.G. Shats, H. Xia, H. Punzmann and G. Falovich (007)]

26 N S Sweeping of the forcing scale S N S N N S E () fl ~ -5 / 3 with external flow no flow ( m -1 ) ~ -5 / S3 ( 10 m / s ) f no flow 1 with external flow l (m)

27 Sweeping due to mean flow Imposed flow affects scales down to the forcing scale Mean flow sweeps forcing scale vortices relative to magnets Sweeping becomes important when sweeping parameter sw V sw e S 1 e Eddy life time l / u ( l) Sweeping is more efficient on small scales: sw sw e V S 1 l 1/3

28 Summary Spectral condensation leads to the generation of mean flow coherent across the system size Spectrum of condensed turbulence shows 3 power laws: ~ -3 at large scales; -5/3 in the meso-scales, -(3-4) at small scales Condensate modifies statistical moments of velocity fluctuations Different moments are affected in different ranges of scales Velocity moments of condensed turbulence similar to those in the atmosphere Coherent shear flow suppresses turbulence which generates it via shearing and sweeping