APPLICATION OF STATISTICAL MECHANICS TO THE MODELLING OF POTENTIAL VORTICITY AND DENSITY MIXING Jo l Sommeria CNRS-LE

1. APPLICATION OF STATISTICAL MECHANICS TO THE MODELLING OF POTENTIAL VORTICITY AND DENSITY MIXING Jo�l 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 vorticityOnsager (1949), Miller(1990), Robert (1990), Robert and Sommeria (1991) 2D Euler equations. - Conservation of the vorticity s(x,y) for fluid elements (Casimir constants) but extreme filamentation. - Statistical description by a local pdf: r(s,r) with local normalisation r(s,r) ds=1 - Maximisation of a mixing entropy: S=-?r lnr d2r with the constraint of energy conservation Energy is purely kinetic but can be expressed in terms of long range interactions: -Dy=s , the vorticity is a source of a long range stream function y energy E=?ys dxdy Mean field approximation (can be justified mathematicallly) -Dy=<s> (y smooth) E=?y<s >dxdy , with <s >= ?r(s,r) s ds

4. Statistical equilibrium <s>=-Dy=f(y), 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 s1 and s2 (patches) f(y)=(s1 + s2 )/2 + (s2 - s1 )/2 tanh(Ay+B) (s1 < f(y) < s2 : 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).

7. Extension to the QG model(Bouchet and Sommeria, JFM 2002)

8. Application to the Great Red Spot of Jupiter (Bouchet & Sommeria, JFM 2002)

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

11. Formal statistical equilibrium for density(Boussinesq approximation) s(z) is the density (-buoyancy) for fluid elements - Statistical description by a local pdf: r(s,r) with local normalisation ?r(s,r) ds=1 - Maximisation of a mixing entropy: S=-?r lnr d2r with the constraint of energy conservation Potential energy of gravity: E = g ? <s> z dz Equilibrium result: <s>~ tanh(-Az+B) (<s>~ exp(-Az) for molecules) see ref. Tabak & Tal, (2004) Comm. Pure Appl. Math.

12. Restratification by sedimentation

13. Competition of stirring and straining (cascade)

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 (s= scalar concentration variable) ?r(s) ds = 1 ?r(s) s ds = <s>=cte Time decay of min, max and variance (mixing)

16. Effect of strain on a scalar

17. Equation for the coarse-grained scalar pdf

18. Comparison with previous models: symmetric case

21. Full model for r(z,t)

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?