Statistical equilibrium theory for axisymmetric flows: Kelvin’s variational principle and an explanation for the vortex ring pinch-off process

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Abstract
Thermodynamics of vorticity density fields (ω/r) in axisymmetric flows are considered, and the statistical equilibrium theories of Miller, Weichman, and Cross [Phys. Rev. A 45, 2328 (1992)], Robert and Sommeria [J. Fluid Mech. 229, 291 (1991)], and Turkington [Comm. Pure Appl. Math. 52, 781 (1999)] for the two-dimensional flows in Cartesian coordinates are extended to axisymmetric flows. It is shown that the statistical equilibrium of an axisymmetric flow is the state that maximizes an entropy functional with some constraints on the invariants of motion. A consequence of this argument is that only the linear functionals of vorticity density, e.g., energy and total circulation, are conserved during the evolution of an axisymmetric inviscid flow to the statistical equilibrium. Furthermore, it is shown that the final equilibrium state satisfies Kelvin’s variational principle; the mean field profiles maximize the energy compatible with the resulting dressed vorticity density. Finally, the vortex ring pinch-off process is explained through statistical equilibrium theories. It appears that only a few invariants of motion (the kinetic energy, total circulation, and impulse) are important in the pinch-off process, and the higher enstrophy densities do not play a significant role in this process.

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