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.

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

Statistical mechanics of vorticityOnsager (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

normalisationrd- Maximisation of a mixing entropy: S∫lnd2r 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 ∫rd

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

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=3.8Domain area/patch area=100

~ point vortices

bar dipole

x

y

x

y

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

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

q = -+/R2

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)

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

1

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

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

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

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∫rd- Maximisation of a mixing entropy: S∫lnd2r

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.

Restratification by sedimentation

Initial profile

Equilibrium profile

<>

<>

z

z

<>

z

Competition of stirring and straining (cascade)

Scale l~ L0exp(-st), s rate of strain (for 2D Euler)

viscous time l2/ = (L20/) exp(-2st),

viscous effect ~ advection time -1

for t=ln(L20/)/(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

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)

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)

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 2

1 +2)/2

time t

time t+ ln2 /s

reduction factor 2

d

d

Venaille and Sommeria, Phys. Fluids 2007, PRL 2008

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

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 <2(t)>=<2(0)>exp[ -∫s(t’)dt’]

Re=16 10^3(about 8 times Rec)

Taylor scale: 0.5 mmRe=5

Self-convolution model vs experiment

Full model for (z,t)

diffusion sedimentation

Div of flux cascade

Self-convolution (cascade):

Turbulent energy (like k-epsilon models)

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?