David Anton Schecter

Inviscid damping of asymmetries on a two-dimensional vortex

The inviscid damping of an asymmetric perturbation on a two-dimensional circular vortex is examined theoretically, and with an electron plasma experiment. In the experiment, an elliptical perturbation is created by an external impulse. After the impulse, the ellipticity (quadrupole moment) of the vortex exhibits an early stage of exponential decay. The measured decay rate is in good agreement with theory, in which the perturbation is governed by the linearized Euler equations. Often, the exponential decay of ellipticity is slow compared to a vortex rotation period, due to the excitation of a quasimode. A quasimode is a vorticity perturbation that behaves like a single azimuthally propagating wave, which is weakly damped by a resonant interaction with corotating fluid. Analytically, the quasimode appears as a wave packet of undamped continuum modes, with a sharply peaked frequency spectrum, and it decays through interference as the modes disperse. When the exponential decay rate of ellipticity is comparabl...

Vortex crystals from 2D Euler flow: Experiment and simulation

Vortex-in-cell simulations that numerically integrate the 2D Euler equations are compared directly to experiments on magnetized electron columns [K. S. Fine, A. C. Cass, W. G. Flynn, and C. F. Driscoll, “Relaxation of 2D turbulence to vortex crystals,” Phys. Rev. Lett. 75, 3277 (1995)], where turbulent flows relax to metastable vortex crystals. A vortex crystal is a lattice of intense small diameter vortices that rotates rigidly in a lower vorticity background. The simulations and experiments relax at the same rates to vortex crystals with similar vorticity distributions. The relaxation is caused by mixing of the background by the intense vortices: the relaxation rate is peaked when the background circulation is 0.2–0.4 times the total circulation. Close quantitative agreement between experiment and simulation provides strong evidence that vortex crystals can be explained without incorporating physics beyond 2D Euler theory, despite small differences between a magnetized electron column and an ideal 2D fluid.

Vortex motion driven by a background vorticity gradient

The motion of self-trapped vortices on a background vorticity gradient is examined numerically and analytically. The vortices act to level the local background vorticity gradient. Conservation of momentum dictates that positive vortices (“clumps”) and negative vortices (“holes”) react oppositely: clumps move up the gradient whereas holes move down the gradient. A linear analysis gives the trajectory of small clumps and holes that rotate against the local shear. Prograde clumps and holes are always nonlinear, and move along the gradient at a slower rate. This rate vanishes when the background shear is sufficiently large.

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

Two-dimensional ~2D! shear-flows, from laboratory to atmospheric scales, typically contain long-lived vortices. Such vortices are carried along by the shear-flow, but they can also drift in the transverse direction. This transverse drift is generally toward an extremum in the vorticity distribution of the shear-flow, i.e., a peak or trough in the ‘‘background’’ vorticity. In this article, we derive simple expressions for the rate at which a vortex drifts transverse to the shear-flow, up or down a background vorticity gradient. These analytic results are found to agree with vortex-in-cell simulations of the 2D Euler equations. We focus on the regime where the vortex is point-like, and the background flow has strong shear. In this regime, we find that the vortex speed increases with the magnitude of the local background vorticity gradient, whereas the vortex speed decreases as the local background shear intensifies. When the shear-flow is reversed, the vortex speed changes by orders of magnitude. We also demonstrate that there is a critical level of background shear, above which the transverse vortex motion is suppressed. A brief account of some of these results has been published in a previous letter. 1 One motivation for this article is a recent electron plasma experiment 2 on the free relaxation of an unstable cylindrical shear-flow. In this experiment, a Kelvin‐Helmholtz instability generates multiple vortices within the shear-flow. These vortices then ‘‘creep’’ radially outward, down a background vorticity gradient. The outward radial drift causes the

Vortex dynamics of 2D electron plasmas

Abstract Electron columns confined magnetically in vacuum evolve in ( r , θ ) as N ∼10 9 field-aligned rods of charge, or point vortices. Neglecting discreteness, the column evolves as would vorticity in an inviscid, incompressible fluid, governed by the Euler equations. The macroscopic flow dynamics is readily imaged, including effects such as surface waves and inviscid damping, two vortex merger, and gradient-driven vortex motion. Turbulent initial states are observed to relax to “vortex crystal” meta-equlibria, due to vortex “cooling” from entropic mixing of background vorticity; and characteristics of this process are predicted by theory. The microscopic discreteness gives rise to point-vortex diffusion, which is strongly affected by the overall flow shear. Macroscopically and microscopically, the vortex dynamics depends critically on whether the vortex is prograde or retrograde with respect to the flow shear.

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.

Inviscid damping of elliptical perturbations on a 2D vortex

The inviscid damping of an elliptical perturbation on a 2D vortex is examined experimentally and theoretically. The perturbation is generated by an impulse at the wall. Initially, the quadrupole moment (ellipticity) of the perturbation decays exponentially. This result is significant, since arbitrary perturbations need not decay exponentially. The decay rate is given by a “Landau pole” of the equilibrium profile. When the Landau damping is weak, the vorticity perturbation, in addition to the quadrupole moment, behaves like an exponentially damped mode. This “quasi-mode” is actually a wave-packet of exceptional continuum modes that decays as the continuum modes disperse.