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2. Statistical Mechanics of Strong and Weak Vortices in a Cylinder Oliver Bühler School of Mathematics University of St Andrews Scotland, United Kingdom

3. Outline of the talk • Statistical mechanics and fluid dynamics • Point vortex dynamics • Statistical mechanics suggestions • Numerical simulations • Detailed predictions

4. First of all... ... a paper to appear in Physics of Fluids, July 2002 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 numerically and compared with theoretical predictions based on statistical mechanics. The novel aspect considered here is that the vortices have greatly different circulation strengths. Specifically, there are four strong vortices and ninety-six weak vortices, the net circulation in 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)], .....

5. Pride Sloth Gluttony Envy Anger Covetousness Lust E.g. cosmic background radiation COBE data vs. statistical mechanics prediction. The error is less than the width of the curve. Statistical mechanics often works wonders in physics... ..but usually not in fluid dynamics! This motivated the present study..

6. eigenstate with energy E n Statistical mechanics usually requires an inert set of eigenstates ..and this usually fails in fluid dynamics E.g. quantum mechanics 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 wave function random coefficient, which in canonical statistical mechanics satisfies This kind of forced-dissipative equilibrium is difficult to model withstatistical mechanics E.g., this gives the COBE predictions. inverse “temperature”

7. 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 advance climate predictability: Moore is not enough! However: coherent structures often emerge in fluid dynamics. Their dynamics can sometimes be modelled using statistical mechanics. (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

8. A simple fluid model Can find a simple fluid dynamics model of coherent strucutures in which to test statistical mechanics: point vortices

9. 2 Hamiltonian formulation on R : y x 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 Point vortex dynamics Two-dimensional u = (u,v) vortex dynamics: x = (x,y) Hamiltonian H Interaction Energy Constant circulations stream function Point vortices: u = • N-body problem • same-signed vortices close together: high energy state • opposite-signed vortices close together: low energy state r velocity at jth vortex induced by the ith vortex >0 ith vortex PDE -> ODEs

10. ’  + Cylinder with radius R s energy decreases as s -> 0 Point vortices in a cylinder:a storm in a tea cup Need a single image to satisfy the boundary condition at wall New Hamiltonian: Image vortex with ’ = -  and radius r’= R*R/r sign-definite self-interaction Vicinity of wall is a low energy region Energy summary Energy range is doubly infinite s energy increases as s -> 0 s energy decreases as s -> 0 wall Note: there is also a second invariant, the angular momentum. Neglected here, but considered in paper.

11. Statistical mechanics based on Hamiltonian H “Principle of insufficient reason” (a better name for ignorance) Microcanonical statistical mechanics Canonical statistical mechanics for a system in contact with an infinite energy reservoir, or for a small part of a large isolated system. for an isolated system with fixed energy E The probability of any state with energy H is the same if The probability of any state with energy H is and 0 otherwise. Most relevant for comparison with direct numerical simulations. Usually easier to manipulate analytically. Focus here

12. the energy range is doubly infinite Onsager (1949) Onsager observed two key facts: the total phase space volume is finite. (cf. magnetic systems, Ising model) Together, these imply that states are scarce for both low andfor high energies! Unusual! is the phase space volume per unit E is the entropy at energy E is the inverse `temperature’ • Low: entropy increases with E • High: entropy decreases with E This has remarkable consequences.. Phase space volume per unit energy

13. Inhomogeneous maximum entropy states strong/weak cyclone with  > 0 strong/weak anticyclone with  < 0 clustering of same-signed vortices especially the strong ones! Can show that this is a consequence of an optimal trade-off between energy and entropy provided || is not constant! Expect inhomogeneous maximum entropy state Expect homogeneous disordered maximum entropy state

14. 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...............

15. Present work 4 strong cyclones/anticyclonees with  =  10, with zero net circulation. Consider: 96 weak cyclones/anticyclones with  =  2, with zero net circulation. N = 100 so need O(N*N) = 10000 logarithms to compute the energy... Expensive to increase N Three different energy cases: Low, Neutral, High Can you tell them apart with your naked eye? They will behave markedly different! L N H

16. Low energy case Expect clustering of oppositely- signed vortices. All 100 vortices are shown. The four strong vortices occupy symmetric, steady-state initial positions. ..difficult to see what’s going on..

17. Low energy case Expect clustering of oppositely- signed vortices. Only the 4 strong vortices are shown now. ..easy to see what’s going on.. Note sticking to the wall

18. Neutral energy case Expect clustering of oppositely-signed and of same-signed vortices. Only the 4 strong vortices are shown now.

19. High energy case Expect clustering of same-signed vortices. Only the 4 strong vortices are shown now.

20. Summary of direct numerical simulations (http://www-vortex.mcs.st-and.ac.uk/~obuhler/smvort.html) 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 N H L

21. How to make quantitative predictions ? Aim is to predict the behaviour of strong vortices embedded in a `sea’ of weak vortices. How can statistical mechanics be used to predict the average dynamics of the strong vortices ?

22. based on random samples with distribution Quantitative predictions How to use statistical mechanics theory into practice Use phase space measures such that For example: the uniform measure is Measures induce probability density functions (pdfs) for any state function taking real values according to Dirac delta function, picks up the shell These pdfs can be estimated numerically by forming histograms of Easier in practice is to use uniform random samples combined with histogram increments proportional to Monte-Carlo technique (Works well because phase space is bounded.)

23. Microcanonical measures The microcanonical measure based on the total energy E is and so (up to a normalization constant) the pdfs are Now split the coordinates of the strong and weak vortices and consider only functions of the strong vortices: Marginal measures for the strong vortices are induced as Marginal pdfs are “Only technical details missing.....”

24. The crucial finesse does not split; unlike energy of an ideal gas etc. Energy weak vortex energy strong vortex energy strong/weak interaction energy; crucial for dynamics In principle, this implies that must be fully tabulated. • requires a look-up table in 8 dimensions; impossible. Finesse: approximate by its average over all with the same strong vortex energy look-up table in one dimension; easy!

25. Algorithm in practice Once only pre-processing step: • 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 Thereafter: For each particular function compute a corresponding sample list from the stored coordinates 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 A single random sample covers all functions and all values of E !! Not obvious

26. Results 1: strong vortex energy Investigated function: Probability density functions • thick lines: statistical mechanics predictions • thin lines: direct numerical simulations • squares: crude guess based on uniform measure

27. Results 2: distances between strong vorticesof the same sign Investigated function: , where i and j are indices of same-signed strong vortices Probability density functions • thick lines: statistical mechanics predictions • thin lines: direct numerical simulations • squares: crude guess based on uniform measure clustering in high energy case

28. Results 3: distances between strong vorticesof the opposite sign Investigated function: , where i and j are indices of oppositely-signed strong vortices Probability density functions • thick lines: statistical mechanics predictions • thin lines: direct numerical simulations • squares: crude guess based on uniform measure Development with changing energy is well predicted

29. Results 4: distances of strong vorticesfrom the centre + Investigated function: Probability density functions “condensation” at the cylinder wall at low energies • thick lines: statistical mechanics predictions • thin lines: direct numerical simulations • squares: crude guess based on uniform measure

30. Concluding remarks Statistical mechanics can give detailed description of average strong vortex dynamics at 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 The holy grail of Applied Mathematics: “To discover the properties of solutions to differential equations without actually solving the equations”