TURBULENCE IN STRATIFIED ROTATING FLUIDS Joel Sommeria, Coriolis-LEGI Grenoble

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1 TURBULENCE IN STRATIFIED ROTATING FLUIDS Joel Sommeria, Coriolis-LEGI Grenoble Collaborations: Olivier Praud, Toulouse P.H Chavanis, Toulouse F. Bouchet, INLN Nice A. Venaille, PhD student LEGI

2 OVERVIEW 1) Laboratory study of stratified rotating turbulence 2) Statistical mechanics and the emergence of organised vortices 3) Competition between advection and straining

3 Antagonistic effects stratification rotation z Froude number: Fr=U/(NL) N 2 =-g(d / dz) Rossby number: Ro=U/(fL) f=2

4 Fr << 1 Ro << 1 Quasi-geostrophic model Non-divergent horizontal flow at each z, PV=- h -(N/f) 2 2 / z 2 Formal analogy with two dimensional turbulence (conservation of PV) Charney (1971) Non dissipative dynamics Emergence of coherent structures

5 Jupiter atmosphere

6 Coriolis rotating platform

7 Grid turbulence CCD camera U Laser sheet (z) 0.9 m 8 m Rake of vertical flat-plates U 0.9 m Laser sheet Mesh 4 m Measurement area. 4 m 2.48 m 8 m

8 Experimental parameters (no rotation) Praud, Fincham, Sommeria, (2005), J. Fluid Mech.

9 t ad = m Influence of rotation t ad = 6 Re M =4500 Fr M = m f = 0 f / N = 1.2 t ad = 37 t ad = 37

10 Strong rotation f / N =1.2

11 f(z) Vertical correlation Vertical correlation function 1.2 Re M =9000, t ad = f/n= f/n=0.6 The vertical scale increases with f/n f/n=0 f/n=0.15 f/n=0.3 Geostrophic adjustment of the vortices z (cm)

12 Aspect ratio

13 E / U 2 Inhibition of the energy decay Kinetic energy decay = Inhibition of the kinetic energy decay with rotation 0.01 =0.56 Conservation for Ro < 0.2 Little influence of stratification 1E-3 f = Time * U / M

14 Energy spectra Horizontal energy spectra 10 t ad = k h -3 energy spectra in agreement with Charney s predictions E(k h ) / U The flow dynamics is quasigeostrophic 0.01 =0.14, Fr M =0.17 =0.14, Fr M =0.09 =0.14, Fr M =0.04 =0.07, Fr M =0.09 =0.07, Fr M =0.04 1E k h * M

15 L h (cm) Horizontal length scale Horizontal integral scale L h 100 =0.00, Fr M =0.04 =0.07, Fr M =0.04 =0.14, Fr M =0.17 =0.28, Fr M =0.04 =0.28, Fr M =0.08 =0.56, Fr M =0.08 Very little influence of rotation and stratification on the horizontal length scale 0.4 Re M = Time * U / M

16 Emergence of vortices t=377 s t=619 s t=915 s t=3000 s

17 Ku Intermittency Ku =< ω z 4 ω z 2 2 = 0.07 f = Strong intermittency for Ro < 0.2 Quasi Gaussian distribution in absence of rotation Time * U / M

18 Intermittency and symmetry P.D.F. of vertical vorticity 1 = = E-3 1E E-3 1E-4 Intermittency and symmetry for < 0.2 1E = E = 0.56 Domination of cyclones for > 0.2 Anticyclones limited by Ro= E-3 1E-3 1E-4 1E-4 1E z / < z 2 > 1/2 1E z / < z 2 > 1/2

19 Conclusions Stratified turbulence without rotation has a 3D dynamics (strong decay) Stratified turbulence submitted to strong rotation (Ro < 0.2) has a quasi-geostrophic dynamics: Energy conservation Energy spectra in k -3 Formation of coherent vortices (with symmetry cycloneanticyclone) (similar to 2D turbulence, but with a z dependency) For moderate rotation (Ro close to 1): departure from quasi-geostrophic model: - Suppression of anticyclonic vortices such that /f <-1 -Other possible ageostrophic effects: front formation, wave generation?

20 Statistical mechanics of vorticity Miller(1989), Robert and Sommeria (1990) Properties of the 2D Euler equations: -Conservation of vorticity (x,y) for fluid elements -Long range interactions: - =, energy E= dxdy Mean field statistical equilibrium: - =f( ), selects a steady solution The function f depends on the global pdf of vorticity (given by the initial condition): sinh for intense vortices tanh for wide patches

21 Z. Yin, D.C. Montgomery, and H.J.H. Clercx "Alternative statistical-mechanical descriptions of decaying twodimensional turbulence in terms of 'patches' and 'points'" Physics of Fluids 15, (2003).

22 Application to the quasi-geostrophic model (Great Red Spot) Relaxation toward statisitical equilibrium R. Robert and J. Sommeria (1991). F. Bouchet and T.Dumont (2004): QG model for a shallow layer over a deep imposed shear flow

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24 Statistical equilibrium, with the assumption h smooth: B=B( ) q ( +2 )/h=-db/d Steady solution of the shallow water equations Fundamental question: excitation of waves thermodynamically possible

25 Competition with vorticity straining Scale l~ L 0 exp(-st), s rate of strain viscous time l 2 /n = (L 2 0/n) exp(-2st), viscous effect ~ advection time -1 for t=ln(l 2 0n/ )/(2 s) ~ ln (Re) Navier-Stokes converges to Euler very slowly with increasing Re

26 Effect of strain on a scalar time t d d s 1 s 2 Venaille and Sommeria, Phys. Fluids 2006, submitted PRL 2007 rate of strain s time t+ ln2 /s reduction factor 2 (s 1 +s 2 )/2 (s,t+ln2 /s) = 2 (s-d, s+d, t) dd 2 : joint probability for pairs separated by d Closure: independence of fluctuations assumed (s,t+ln2 /s) =2 (s,t) (s-s,t) ds (self-convolution) Laplace transform:

27 Equation for the coarse-grained scalar pdf -n self-convolutions: transformed in a product by Laplace transform -infinitesimal limit n =1+e: relaxation toward a Gaussian with decreasing variance (symmetric case), or through gamma pdf.

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29 Self-convolution model vs experiment

30 SOME CONCEPTUAL QUESTIONS Is the statistical equilibrium the true asymptotic limit of the 2D Euler equations? How the theory extends to balanced states beyond the quasi-geostrophic model? Can we derive a kinetic model combining all the requested fundamental properties? (relaxation to statistical equilibrium, galilean invariance )