1. APPLICATION OF STATISTICAL MECHANICS TO THE MODELLING OF POTENTIAL VORTICITY AND DENSITY MIXING Joël Sommeria CNRS-LEGI Grenoble, France Newton’s Institute, December 11th 2008.

2. OVERVIEW Statistical equilibrium for the 2D Euler equations Link with PV mixing in the limit of small Rossby radius of deformation Similarity with vertical density mixing. Competition with local straining and cascade

3. Statistical mechanics of vorticity Onsager (1949), Miller(1990), Robert (1990), Robert and Sommeria (1991) 2D Euler equations. - Conservation of the vorticity  (x,y) for fluid elements (Casimir constants) but extreme filamentation. - Statistical description by a local pdf:  r  with local normalisation  r  d  - Maximisation of a mixing entropy: S  ∫  ln  d 2 r with the constraint of energy conservation -Energy is purely kinetic but can be expressed in terms of long range interactions:  the vorticity is a source of a long range stream function  energy  ∫  dxdy -Mean field approximation (can be justified mathematicallly)  smooth   ∫  dxdy, with  ∫  r  d 

4. Statistical equilibrium  =f(  ), The locally averaged field is a steady solution of the Euler equation. -The function f is a monotonic. It depends on the energy and the global pdf of vorticity (given by the initial condition): -For two vorticity levels  1 and  2 (patches) f(  )=(  1 +  2 )/2 + (  2 -  1 )/2 tanh(A  +B) (  1 < f(  ) <  2 : represents mixing of the two initial levels

5. Dipole vs bar in the doubly- periodic domain Z. Yin, D.C. Montgomery, and H.J.H. Clercx, Phys. Fluids 15, 1937-1953 (2003). Domain area/patch area=3.8 Domain area/patch area=100 ~ point vortices bar dipole x y x y

6. Z. Yin, D.C. Montgomery, and H.J.H. Clercx "Alternative statistical- mechanical descriptions of decaying two-dimensional turbulence in terms of 'patches' and 'points'" Phys. Fluids (2003). Numerical test

7. Extension to the QG model (Bouchet and Sommeria, JFM 2002) q = -  +  /R 2 x y Shallow layer, R=Rossby radius of deformation Energy Asymmetry vortex zonal jets Limit of small R, large E: coexistence of two phases with uniform PV (PV staircase)

8. Application to the Great Red Spot of Jupiter (Bouchet & Sommeria, JFM 2002) 11 22 Velocity measured from cloud motion (Dowling and Ingersoll 1989) Prediction: -The jet width is of the order of the radius of deformation -The elongated shape is controlled by the deep zonal shear flow R

9. Statistical equilibrium, with the assumption h smooth: B=B(  ) (B=Bernouilli function) q≡(  +2  )/h=-dB/d  = stream function for hu  Steady solution of the shallow water equations Generalisation to multi-layer hydrostatic models formally straightforward

10. Vertical mixing in a stratified fluid (cf. A. Venaille, PhD thesis Grenoble)

11. Formal statistical equilibrium for density (Boussinesq approximation)  (z) is the density (-buoyancy) for fluid elements - Statistical description by a local pdf:  r  with local normalisation  ∫  r  d  - Maximisation of a mixing entropy: S  ∫  ln  d 2 r with the constraint of energy conservation Potential energy of gravity: E  g ∫  z dz  Equilibrium result: ~ tanh(-Az+B) ( ~ exp(-Az) for molecules) see ref. Tabak & Tal, (2004) Comm. Pure Appl. Math.

12. Restratification by sedimentation Initial profile Equilibrium profile <><> <><> z z <><> z

13. Competition of stirring and straining (cascade) Scale l~ L 0 exp(-st), s rate of strain (for 2D Euler) viscous time l 2 / = (L 2 0 / ) exp(-2st), viscous effect ~ advection time  -1 for t=ln( L 2 0 /  )/(2 s) ~ ln (Re) Navier-Stokes converges to Euler very slowly with increasing Re Strain leads to local mixing : reduction of the pdf to its mean

14. Previous models for local cascade Intermittency for a scalar in the turbulent cascade at high Re: delta-correlated velocity (Kraichnan model), steady regimes. Linear mean square estimate (LMSE), O’Brian (1980): no evolution of the pdf shape Coalescence-dispersion: Curl (1963), Pope (1982), Villermaux and Duplat (2003)

15. Requested properties for the pdf Conservation of the normalisation and mean (  = scalar concentration variable) ∫  (  ) d  = 1 ∫  (  )  d  = =cte Time decay of min, max and variance (mixing)

16. Effect of strain on a scalar rate of strain s  ( ,t+ln2 /s) = ∫   ( , , t) d   2 : joint probability for pairs separated by d Closure: independence of fluctuations assumed  ( ,t+ln2 /s) =2 ∫  (  ’,t)  (  -  ’,t) d  ’ (self-convolution) Laplace transform: 11 22  1 +  2 )/2 time t time t+ ln2 /s reduction factor 2 d d Venaille and Sommeria, Phys. Fluids 2007, PRL 2008

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

18. Comparison with previous models: symmetric case Rmq: Villermaux and Duplat(2003) does not apply to this initial condition Phys. Fluids(1988) Fitting parameter: Scalar variance = exp[ -∫s(t’)dt’]

19. Re=16 10^3 (about 8 times Rec) Taylor scale: 0.5 mm Re =5

20. Self-convolution model vs experiment

21. Full model for  (z,t) diffusionsedimentation Div of fluxcascade Self-convolution (cascade): Turbulent energy (like k-epsilon models)

22. Conclusions Mixing can be described as the increase of a mixing entropy Energy conservation is a constraint: -> vortex or jet formation in QG turbulence -> restratification for density Effect of local cascade toward dissipative scales must be also taken into account. Self-convolution provides a good approach. Application to stratified turbulence: work in progress Possibility of modelling source of PV by density mixing?