Universal statistics of point vortex turbulence

Abstract

A new methodology, based on the central limit theorem, is applied to describe the statistical mechanics of two-dimensional point vortex motion in a bounded container D, as the number of vortices N tends to infinity. The key to the approach is the identification of the normal modes of the system with the eigenfunction solutions of the so-called hydro- The pdf of the leading vorticity projection is of particular interest because it has a unimodal distribution at low energy and a bimodal distribution at high energy. This behaviour is indicative of a phase transition, known as Onsager-Kraichnan condensation in the literature, between low energy states with no mean flow in the domain and high energy states with a coherent mean flow. The critical temperature for the phase transition, which depends on the shape but not the size of D, and the associated critical energy are found. Finally the accuracy and the extent of the validity of the theory, at finite N , are explored using a Markov chain phase-space sampling method.

Giant vortex clusters in a two-dimensional quantum fluid

“…This efficiently injects many vortices into the BEC with minimal sound excitation [27,28]. We find the vortices rapidly organise into two Onsager vortex clusters that settle on the major axis [29] and are visible following a short 3 ms time-of-flight expansion ( Fig. 2B).…”

Theory of the vortex-clustering transition in a confined two-dimensional quantum fluid

“…The phenomenon stems from the bounded phase space, which supports negative temperature states that favor the spontaneous clustering of like-sign vortices. As an equilibrium statistical mechanics problem, cluster formation in 2D vortex systems has attracted much attention [5,[10][11][12][13][14][15][16][17][18][19][20]. Building on Joyce and Montgomery's formulation of vortex statistical mechanics [10], Smith and O'Neil [17,18] investigated the single-charge 2D plasma using the point-vortex model in a rotating container and showed that the formation of a nonaxisymmetric cluster resembles a second-order phase transition.…”

Two-dimensional isotropic turbulent inflow conditions for vortex particle method

“…6] induced by the point vortices raises no problem when used as the singularity distribution on the domain boundaries and/or as the circulation carrier at each timestep. However, Esler and Ashbee [16], and Kuvshinov and Schep [21] showed, in their work on point vortices to realize a white noise signal, that there exists a problem when approximating the vorticity with delta functions (point vortices), i.e., infinite spikes, which is an associated singularity in the streamfunction and velocity field. This singularity appears when the distance to the point vortex becomes very small, |r − r | → 0: both streamfunction and velocity become very large; the first blows up logarithmically fast and the other like r −1 .…”

“…Chapman [13], Benzi [14], and Elhmaidi [15] used a similar method to realize a two-dimensional synthetic turbulent inflow using point vortices. Esler [16] studied the behavior of a finite number of point vortices and examined the possibility of replacing a turbulent flow with a collection of point vortices. However, there is a problem with approximating the vorticity with point vortices (delta functions), i.e., infinite spikes, which introduces an associated singularity in the streamfunction and velocity fields.…”