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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?