Unifying scaling theory for vortex dynamics in two-dimensional turbulence

Nonequilibrium Statistical Mechanics of Two-Dimensional Vortices

It has been observed empirically that two-dimensional vortices tend to cluster, forming a giant vortex. To account for this observation, Onsager introduced the concept of negative absolute temperature in equilibrium statistical mechanics. In this Letter, we show that in the thermodynamic limit a system of interacting vortices does not relax to the thermodynamic equilibrium but becomes trapped in a nonequilibrium stationary state. We show that the vortex distribution in this nonequilibrium stationary state has a characteristic core-halo structure, which can be predicted a priori. All the theoretical results are compared with explicit molecular dynamics simulations.

Ergodicity and spectral cascades in point vortex flows on the sphere

We present results for the equilibrium statistics and dynamic evolution of moderately large [n=O(10(2)-10(3))] numbers of interacting point vortices on the sphere under the constraint of zero mean angular momentum. For systems with equal numbers of positive and negative identical circulations, the density of rescaled energies, p(E), converges rapidly with n to a function with a single maximum with maximum entropy. Ensemble-averaged wave-number spectra of the nonsingular velocity field induced by the vortices exhibit the expected k(-1) behavior at small scales for all energies. Spectra at the largest scales vary continuously with the inverse temperature of the system. For positive temperatures, spectra peak at finite intermediate wave numbers; for negative temperatures, spectra decrease everywhere. Comparisons of time and ensemble averages, over a large range of energies, strongly support ergodicity in the dynamics even for highly atypical initial vortex configurations. Crucially, rapid relaxation of spectra toward the microcanonical average implies that the direction of any spectral cascade process depends only on the relative difference between the initial spectrum and the ensemble mean spectrum at that energy, not on the energy, or temperature, of the system.

Effective merging dynamics of two and three fluid vortices: application to two-dimensional decaying turbulence

We present a kinetic theory of two-dimensional decaying turbulence in the context of two-body and three-body vortex merging processes. By introducing the equations of motion for two or three vortices in the effective noise due to all the other vortices, we demonstrate analytically that a two-body mechanism becomes inefficient at low vortex density n<<1. When the more efficient three-body vortex mergings are considered (involving vortices of different signs), we show that n~t(-ξ), with ξ=1. We generalize this argument to three-dimensional geostrophic turbulence, finding ξ=5/4, in excellent agreement with direct Navier-Stokes simulations [McWilliams et al., J. Fluid Mech. 401, 1 (1999)].

Generic mechanism of microturbulence suppression by vortex flows

The interaction between two-dimensional vortex flows and microturbulence is studied numerically using gyrofluid simulations. It is shown that, qualitatively different from usual mean flows, vortex flows can dramatically suppress microturbulence even with weak flow shear. A generic suppression mechanism is identified as the multiplied effect of both radial and poloidal mode couplings, which induce the formation of a new global mode. Furthermore, an oscillatory zonal flow is found to form through interaction between the vortex flows and microturbulence.