cursor

Oliver Bühler

University of St AndrewsSchool of Mathematics

Scotland, United Kingdom

Statistical Mechanics of Strong and WeakVortices in a Cylinder

Outline of the talk

• Statistical mechanics and fluid dynamics

• Point vortex dynamics

• Statistical mechanics suggestions

• Numerical simulations

• Detailed predictions

Statistical mechanics of strong and weak point vortices in a cylinder

By Oliver Bühler

The motion of one-hundred point vortices in a circular cylinder is simulated numericallyand compared with theoretical predictions based on statistical mechanics. The novelaspect considered here is that the vortices have greatly different circulation strengths.Specifically, there are four strong vortices and ninety-six weak vortices, the net circulationin either group is zero, and the strong circulations are five times larger than the weak circulations. As envisaged by Onsager [Nuovo Cimento 6 (suppl.), 279 (1949)], .....

... a paper to appear in Physics of Fluids, July 2002

First of all...

• Pride• Sloth• Gluttony• Envy• Anger• Covetousness• Lust

COBE data vs. statistical mechanics prediction.The error is less than the width of the curve.

E.g. cosmic background radiation

Statistical mechanics often works wonders

in physics...

..but usually not in fluid dynamics!

This motivated the present study..

E.g., this gives the COBE predictions.

E.g. quantum mechanics

Statistical mechanics usually requires an inert

set of eigenstates

..and this usually fails in fluid dynamics

ψ (x,t)= anψ n(x)exp(−iEnt /h)

n∑

wave function

random coefficient, which in canonicalstatistical mechanics satisfies

|an |2 ∝ e−βEn

eigenstate with energy En

β = 1kT

inverse “temperature”

E.g. 3-dimensional turbulence

• vigorous energy cascade from larger

scales to smaller scales

• no meaningful set of inert eigenstates

• statistically steady state requires

persistent forcing and dissipation

This kind of forced-dissipative equilibrium is difficult to model with

statistical mechanics

However: coherent structures often emerge in fluid dynamics.

Their dynamics can sometimes be modelled using statistical mechanics.

Examples of coherent structures• convection plumes• vortices in two dimensions• suspensions

Statistical/stochastic methods in climate research:• tropical convection• deep ocean convection• oceanic vortices• mixing in the stratosphere

Progress in this direction might be essential to advanceclimate predictability:

Moore is not enough!(It takes ~20 years to increase the numerical resolution of an unsteady 3-d model by a factor of 10, because cost increases by a factor of 10 000)

ie computer power doubles every 18 months

A simple fluid model

Can find a simple fluid dynamics modelof coherent strucutures in which to teststatistical mechanics:

point vortices

Point vortex dynamicsTwo-dimensional u = (u,v)vortex dynamics: x = (x,y)

xy∇ ⋅u=0, ∇ ×u=q, ∂q

∂t+u⋅∇q=0.

∂ψ∂x

=v, ∂ψ∂y

=−u ∴ ∇2ψ =q

stream function

Point vortices:

Constant circulations

Γi = qdxdyith vortex∫∫q= Γi δ(x-xi )

i=1

N∑

>0

u = Γ2πr

r

ith vortex

velocity at jth vortexinduced by the ith vortex

Hamiltonian formulation on R :2

Γidxi

dt=+∂H

∂yi, Γi

dyidt

=−∂H∂xi

dx1dy1....dxNdyN =invariant phase space volume element

Hamiltonian HInteraction Energy

H(x1,y1,...,xN,yN) =1

4π12 −ΓiΓj ln(xi −xj )2{ }

j=1j≠i

N∑i=1

N∑

• N-body problem• same-signed vortices close together: high energy state • opposite-signed vortices close together: low energy state

State of the system is described by the N vortex locations x (t). Vortex locations x (t) move with local fluid velocity u:i i

PDE -> ODEs

Point vortices in a cylinder:a storm in a tea cup

Image vortex with ’ = - and radius r’= R*R/r

Cylinder with radius R

+ ’

Need a single image to satisfy the boundary condition at wall

H(x1,y1,...,xN,yN) =1

4π12 −ΓiΓj ln(xi −xj )2 +ln R4 −2xi ⋅xjR

2+|xi |2|xj |2( ){ }j=1j≠i

N∑i=1

N∑

+ 14π

Γi2 ln R2−|xi |2( )

i=1

N∑ sign-definite self-interaction

New Hamiltonian:

Vicinity of wall is a low energy region

H ∈ −∞,+∞( )Energy range is doubly infinite

Energy summary

s energy decreases as s -> 0

s energy increases as s -> 0

s energy decreases as s -> 0wall

Note: there is also a second invariant, the angular momentum. Neglected here, but considered in paper.

Statistical mechanics based on Hamiltonian H

Microcanonical statistical mechanicsfor an isolated system with fixed energy E

H ∈ E,E +dE( )

“Principle of insufficient reason”

Canonical statistical mechanicsfor a system in contact with an infinite energy reservoir, or for a small part of a large isolated system.

Aexp−βH( )

(a better name for ignorance)

The probability of any state with energy H is the same if

and 0 otherwise.

The probability of any state withenergy H is

Most relevant for comparison with direct numerical simulations.

Usually easier to manipulate analytically.

Focus here

Onsager (1949)Onsager observed two key facts:

the total phase space volume dx1dy1...dxNdyN = πR2( )N∫ is finite.

the energy range is doubly infinite H ∈ −∞,+∞( )Together, these imply that states are scarce for both low and for high energies! Unusual!

Phase space volume per unit energy

p0(E) is the phase space volume per unit E

lnp0 =S is the entropy at energy EdSdE

=1T

=β is the inverse `temperature’

β >0 β <0

(cf. magnetic systems,Ising model)

• Low: entropy increases with E

• High: entropy decreases with E

This has remarkable consequences..

Inhomogeneous maximum entropy statesstrong/weak cyclone with > 0

strong/weak anticyclone with < 0

β >0 β <0

Expect homogeneous disorderedmaximum entropy state

Expect inhomogeneous maximum entropy state

clustering of same-signed vortices

especially thestrong ones!

Can show that this is a consequence of an optimal trade-off between energy and entropy

provided || is not constant!

Some relevant earlier work Montgomery & Joyce (1973-4) Devised mean-field theory for many vortices of equal strengths ||.

Pointin & Lundgren (1976) Refined the mean-field theory, considered cylinder explicitly.

Weiss, McWilliams, Provenzale (1991,1998) Direct numerical simulations of point vortices to test ergodic behaviour and investigate velocity statistics.

Caglioti, Lions PL, Marchioro, Pulvirenti (1992) Rigorous mean-field theory for identical = 1. (Some errors to do with self-interaction effects at wall.)

Lions PL & Majda (2000) Consider = 1 case for slightly undulated 3d vortex tubes.

{Several authors (eg Sommeria, Roberts, Turkington, Majda,...) have attempted extensions of the theory to continuous vorticity distributions.The main unanswered question here is whether the resulting flows exhibit ergodic behaviour over physically meaningful time scales.}

No results for variable || and outside mean-field theory ...................aim of present work...............

Present work4 strong cyclones/anticyclonees with = 10,with zero net circulation.

96 weak cyclones/anticyclones with = 2,with zero net circulation.

Consider:

N = 100so need O(N*N) = 10000 logarithms to compute the energy...

Expensive to increase NThree different energy cases: Low, Neutral, High

L N H

Can you tell them apart with your naked eye?They will behave markedly different!

β <0β ≈0β >0

Low energy case

QuickTime™ and aGIF decompressor

are needed to see this picture.

All 100 vorticesare shown.

The four strongvortices occupysymmetric,steady-state initial positions.

Expect clustering of oppositely-signed vortices.

..difficultto see what’s going on..

Low energy case

Only the 4 strongvorticesare shownnow.

Expect clustering of oppositely-signed vortices.

..easyto see what’s going on..



Note stickingto the wall

Neutral energy case


Expect clustering of oppositely-signed and of same-signedvortices.



High energy case


Expect clustering of same-signed vortices.



Summary of direct numerical simulations

Observed strong vortex behaviour compatiblewith expectations from statistical mechanics

Marked transition as energy is increased:

preferred clustering of oppositely-signed vortices

(and sticking to the wall)

clustering of oppositely-signed and of same-signed vortices

preferred clustering of same-signed vortices

L N H

(http://www-vortex.mcs.st-and.ac.uk/~obuhler/smvort.html)

How to make quantitative predictions ?

Aim is to predict the behaviour of strong vorticesembedded in a `sea’ of weak vortices.

How can statistical mechanics be used to predictthe average dynamics of the strong vortices ?

Quantitative predictionsHow to use statistical mechanics theory into practice

Use phase space measures ρ(x1,...,xN) such that ρdx1...dxN =∫ ρ dx( )N∫ =1

For example: the uniform measure is ρ0 = πR2( )−N

Measures induce probability density functions (pdfs) for any state function

Ψ(x1,...,xN) taking real values φ

according to p(φ) = δ(Ψ −φ)ρ dx( )N∫These pdfs can be estimated numerically by forming histograms of Ψ(x1,...,xN)

based on random samples with distribution ρ(x1,...,xN)

Easier in practice is to use uniform random samples combined with histogramincrements proportional to ρ(x1,...,xN)

Dirac delta function, picks up the shell

(Works well because phase space is bounded.)Monte-Carlo technique

Microcanonical measuresThe microcanonical measure based on the total energy E is ρE ∝δ(H −E)

pE(φ) ∝ δ(Ψ −φ)δ(H −E) dx( )N∫and so (up to a normalizationconstant) the pdfs are

Now split the coordinates of the strong and weak vortices and consider only functions of the strong vortices:

XA = x1,y1,...,x4,y4{ }XB = x5,y5,...,x100,y100{ }

dXAdXB = dx( )N

Marginal measures for the strong vortices are induced as

ρE(XA)∝ δ(Η −Ε)dXB∫Marginal pdfs are

pE(φ)∝ δ(Ψ(XA) −φ)ρE(XA ) dXA∫“Only technical details missing.....”

The crucial finesse

Energy H =HA(XA)+HB(XB)+HI (XA,XB) does not split; unlike energy of an ideal gas etc.

strong/weak interaction energy;crucial for dynamics

ρE(XA)In principle, this implies that must be fully tabulated.

• requires a look-up table in 8 dimensions; impossible.

Finesse:approximate ρE(XA) by its average over all XA with the same

strong vortex energy

weak vortex energy

strong vortex energy HA

ρE(XA) ρE HA(XA)[ ] look-up table inone dimension; easy!

Algorithm in practice

• generate a sample of 100 000 random vortex configurations from the uniform distribution (ie put 100 vortices anywhere in the cylinder)• compute the energy H and its strong/weak components for each member of the sample (requires computing 1 000 000 000 logarithms; this is the expensive bit). • Store the results together with the sample coordinates of the strong vortices

XA

Once only pre-processing step:

For each particular function compute a corresponding sample listfrom the stored coordinates

Ψ(XA)XA

Thereafter:

Finally, form a histogram for based on the approximated for the current value of the energy E. This produces the pdf belonging to the function

Ψ(XA) ρE(HA)

Ψ(XA) A single random sample covers all functions and all values of E !!

Not obvious

Results 1: strong vortex energy Investigated function:

Probability density functions

HA(XA) = 14π

12 −ΓiΓj ln(xi −xj)2 +ln R4 −2xi ⋅xjR

2+|xi |2|xj |2( ){ }j=1j≠i

4∑i=1

4∑

+ 14π

Γi2 ln R2−|xi |2( )

i=1

4∑

• thick lines: statistical mechanics predictions• thin lines: direct numerical simulations• squares: crude guess based on uniform measure

Results 2: distances between strong vorticesof the same sign

Investigated function:


rij =|xi −xj |

• thick lines: statistical mechanics predictions• thin lines: direct numerical simulations • squares: crude guess based on uniform measure

, where i and j are indices of same-signed strong vortices

clustering in high energy case

rij

Results 3: distances between strong vorticesof the opposite sign



rij =|xi −xj |


, where i and j are indices of oppositely-signed strong vortices

Development with changing energy is well predicted

rij

Results 4: distances of strong vorticesfrom the centre



ri = xi2 +yi

2


“condensation” at the cylinder wallat low energies

ri+

Concluding remarksStatistical mechanics can give detailed description of average strong vortex dynamicsat a fraction of the computational cost

An alternative to running many numerical initial-value problems in order to

explore the phase space by trajectories.

Computational Statistical Mechanics

“To discover the properties of solutions to differential equations without actually solving the equations”

cursor

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