Background Vorticity Distribution Research Articles

Quantized point vortex equilibria in a one-way interaction model with a Liouville-type background vorticity on a curved torus

We construct point vortex equilibria with strengths quantized by multiples of 2π in a fixed background vorticity field on the surface of a curved torus. The background vorticity consists of two terms: first, a term exponentially related to the stream function and a second term arising from the curvature of the torus, which leads to a Liouville-type equation for the stream function. By using a stereographic projection of the torus onto an annulus in a complex plane, the Liouville-type equation admits a class of exact solutions, given in terms of a loxodromic function on the annulus. We show that appropriate choices of the loxodromic function in the solution lead to stationary vortex patterns with 4n̂ point vortices of identical strengths, n̂∈N. The quantized point vortices are stationary in the sense that they are equilibria of a “one-way interaction” model where the evolution of point vortices is subject to the continuous background vorticity, while the background vorticity distribution is not affected by the velocity field induced by the point vortices. By choosing loxodromic functions continuously dependent on a parameter and taking appropriate limits with respect to this parameter, we show that there are solutions with inhomogeneous point vortex strengths, in which the exponential part of the background vorticity disappears. The point vortices are always located at the innermost and outermost rings of the torus owing to the curvature effects. The topological features of the streamlines are found to change as the modulus of the torus changes.

Controlled experiments on self-organization of ordered structures in a pure electron plasma

We review our recent studies on structure formation in pure electron plasmas confined in magneto-electric traps. Two-dimensional dynamics of strongly magnetized plasmas is studied in terms of a two-dimensional vortex composed of high-density clumps and a low-density distribution in the background (BG). By controlling the initial distributions we examine in detail the fundamental processes where the clumps form highly ordered geometrical configurations (vortex crystals). Experimental studies of the formation of three-dimensional density structures have been initiated with a newly built strong-magnetic-field (2.2 T) trap operatable with various potential profiles including a square-well potential (Malmberg trap) and harmonic potential (Mohri trap). In the Mohri trap, Penning-type distributions are observed to form as a result of interaction of string-shaped distributions of electrons. The density distributions observed in the relaxation process are quite different from those observed in a square-well potential most probably due to three-dimensional deformations and merger of the string distributions. The equilibrium distributions generally consist of sets of spheroidal distributions, a core distribution plus halo distributions. Observationally the halo distributions appear to be associated with the amount of excessive angular momentum provided through the injection of off-axial electron strings. Like the BG vorticity distribution in the two-dimensional merger of clumps, the halo may act as the absorber of momentum in the course of the merging process of strings to create the core distribution. In the Penning-like distributions we have also observed the accumulation of negative ions replacing the electrons in time scales much longer than thermal relaxation.

Numerical Simulation of Vortex Crystals and Merging inN-Point Vortex Systems with Circular Boundary

In two-dimensional (2D) inviscid incompressible flow, low background vorticity distribution accelerates intense vortices (clumps) to merge each other and to array in the symmetric pattern which is called ``vortex crystals''; they are observed in the experiments on pure electron plasma and the simulations of Euler fluid. Vortex merger is thought to be a result of negative ``temperature'' introduced by L. Onsager. Slight difference in the initial distribution from this leads to ``vortex crystals''. We study these phenomena by examining N-point vortex systems governed by the Hamilton equations of motion. First, we study a three-point vortex system without background distribution. It is known that a N-point vortex system with boundary exhibits chaotic behavior for N\geq 3. In order to investigate the properties of the phase space structure of this three-point vortex system with circular boundary, we examine the Poincar\'e plot of this system. Then we show that topology of the Poincar\'e plot of this system drastically changes when the parameters, which are concerned with the sign of ``temperature'', are varied. Next, we introduce a formula for energy spectrum of a N-point vortex system with circular boundary. Further, carrying out numerical computation, we reproduce a vortex crystal and a vortex merger in a few hundred point vortices system. We confirm that the energy of vortices is transferred from the clumps to the background in the course of vortex crystallization. In the vortex merging process, we numerically calculate the energy spectrum introduced above and confirm that it behaves as k^{-\alpha},(\alpha\approx 2.2-2.8) at the region 10^0<k<10^1 after the merging.

Merger and binary structure formation of two discrete vortices in a background vorticity distribution of a pure electron plasma

Observations have shown that two discrete vortices (clumps) immersed in a broad profile of background vorticity merge quickly or form a binary state that lasts for a long period. The different paths of the vortical evolution critically depend on slight differences in the initial background vorticity distribution (BGVD). By a fine control of the BGVD the asymptotic inter-clump distance is found to be a two-valued function of ∇ζb/ζb2, where ζb is the BGV at the initial location of the clumps, corresponding to the merger and the binary state. The multiplicity is removed by considering the degree of depletion of BGVD between two clumps at the time of their proximity in the initial phase.

Formation of a vortex crystal cell assisted by a background vorticity distribution.

A vortex crystal is a quasistationary, symmetric array of intense vortices (clumps). A low level of background vorticity is experimentally observed to assist three clumps in forming an equilateral triangle starting from initial positions on a linear array. The triangle constitutes a unit cell of a crystal in a many-vortex system. The background vortex curbs the orbital motion of the clumps with unequal strengths to arrest them at the vertices of an equilateral triangle by wrapping them with different sized belts of depleted vorticity (ring holes). We characterize the contributions of a low-level background vorticity distribution on the formation of ordered states of clumps in light of the experimental results and existing theories.

Strong Effects of Background Vorticity Distribution on the Dynamics of Coherent Vortices in Electron Plasma

Two-dimensional vortex dynamics is characterized by motion of coherent vortices or clumps in a rugged distribution of background vortices. We study eminent features in this process by using a magnetized pure electron plasma starting from highly controlled initial density distribution. The topics included are (1) relaxation of many-clump dynamics states to crystallization, (2) single clump dynamics in a background vorticity distribution, (3) two-clump interaction in a low-level background vorticity distribution, and (4) accelerated crystallization of three clumps in a low-level background vorticity distribution.

Theory and simulations of two-dimensional vortex motion driven by a background vorticity gradient

This paper examines two-dimensional vortex motion in a shear-flow with nonuniform vorticity. Typically, a vortex travels to an extremum in the background vorticity distribution. In general, the rate of this migration increases with the magnitude of the background vorticity gradient; however, a retrograde vortex, which rotates against the local shear, moves orders of magnitude faster than a prograde vortex of equal strength. Retrograde and prograde vortices travel at different speeds because they perturb the background vorticity differently. Linearized equations accurately describe the background vorticity perturbation that is created by a weak retrograde vortex, whereas nonlinear effects dominate for a prograde vortex of any strength. An analytic theory is developed for the velocity of a retrograde vortex, based on the linearized equations. The velocity of a prograde vortex is obtained by a simple “mix-and-move” estimate, which takes into account the nonlinear trapping of fluid around the vortex. Both velocity predictions are verified by vortex-in-cell simulations. If the ratio of background shear to background vorticity gradient exceeds a critical level, there is no vortex motion up or down the background vorticity gradient. Estimates of the critical shear are obtained for both prograde and retrograde vortices. These estimates compare favorably to vortex-in-cell simulations.

Two-Dimensional Vortex Dynamics in a Potential-Confined Electron Plasma

We study dynamics of discrete vortex strings interacting in a background vorticity distribution by using a magnetized electron plasma confined by a potential distribution in a cylindrical trap, and demonstrate relevant 2-dimensional features characterizing the incompressible dissipation-free guiding-center fluid. Some specific elementary processes are examined and compared with a recently proposed theoretical model that describes a motion of a point vortex moving up a hill of background vorticity distribution. Radial transport of particles is observed to be very limited and associated with several conserved parameters.

Dynamics, Statistics and Vortex Crystals in the Relaxation of 2D Turbulence

Magnetically confined electron columns evolve in (r, ) as essentially inviscid, incompressible 2D fluids with a single sign of vorticity. Turbulent initial states with 50–100 vortices relax due to vortex merger and filamentation, in general agreement with a recent dynamical scaling theory. However, this relaxation sometimes halts when 3–20 vortices “anneal” into a fixed pattern, or “vortex crystal.” A new “regional maximum fluid entropy” theory predicts the crystal patterns and background vorticity distribution, by assuming conservation of the robust flow invariants and preservation of the intense vortices. However, simulations show that the character of the relaxed state generally depends strongly on initial conditions and dynamics.

Relaxation of 2D turbulence of vortex crystals

A magnetically confined electron column evolves in ( r, θ) as an essentially inviscid, incompressible 2D fluid with a single sign of vorticity. Turbulent initial states with 50–100 vortices relax due to vortex merger and filamentation, in general agreement with recent scaling theories. However, this relaxation is sometimes halted when 3–20 vortices “anneal” into a fixed pattern, or “vortex crystal”. 2D vortex-indashcell simulations reproduce this effect, demonstrating that the vortex “cooling” is independent of fine-scale viscosity, but strongly dependent on the strength of the weak background vorticity. A new “restricted maximum fluid entropy” theory predicts the crystal patterns and background vorticity distribution, by assuming conservation of the robust flow invariants and preservation of the intense vortices.