Motion of charged vortex rings in helium II

Abstract

We show that an ion carried through an electric field by a quantized vortex ring in helium II will cause a symmetric change in the radius of the ring, even though the ion is located asymmetrically on the core of the vortex ring. The localized force on the ion will combine with a specific series of vortex waves to give a uniform growth of the ring with a small component of drift velocity perpendicular to the applied electric field. This type of energy transfer may also play an important role in superfluid turbulence. Our calculations should be valid for localized forces on thin-core classical vortices as well.PACS numbers: 67.40.VsOne of the key investigations in the study of helium II was the motion of positive and negative ions at temperatures below 1 K. Rayfield and Reif demonstrated conclusively that both positive and negative ions moving through low-temperature helium nucleate quantized vortex rings and become attached to them 111. They were able to measure the dispersion curve of these ion-vortex complexes, establishing the motion unambiguously as that of vortex rings, they measured the quantum of circulation K^h/m, where h is Planck's constant and m is the mass of the helium atom, and they determined the core radius a of the vortex ring. At first it was not clear where the ion was located: Symmetry appears to demand that it be located in the center of the ring. It later became evident, however, that at the core of the ring there is an energy minimum for an ion [2], and that, in fact, the vortex ring begins its existence by "macroscopic" tunneling from one side of the ion [3]. The ion complex, however, moves through the fluid under the influence of an electric field, and it is not clear how the force on the ion can induce the growth and decay of the vortex ring from an asymmetric location. Early speculations suggested that the ion is delocalized in a hollow vortex core, or moves about the core by Brownian motion. We now know, however, that the mobility of the ion along the vortex core rapidly decreases with increasing temperature [4]. Assuming the ion diffuses around the vortex ring in a random walk, the ratio of the time / for the ion to diffuse around the ring to the time r for the ring to travel a distance equal to one circumference iswhere n is the ion mobility on the vortex line and e the ion charge. This ratio is greater than 1 for r>0.4 K and is 20 by T=0.6 K. Consequently, for temperatures greater than 0.4 K, the ion must be considered to be fixed in place. The great success of the Rayfield-Reif experiment rests on the idea that the vortex rings remain circular when an electric field is applied. The purpose of this Letter is to demonstrate how this paradoxical situation can be explained by modern vortex dynamics [5]. To the best of our knowledge there have been no studies of the effects of localized forces on classical vortex rings. These calculations should be applicable to classical thin core vortices of constant circulation as well as to quantized vortices.Consider a vortex ring in the x-y plane m...

Vortex nucleation limited mobility of free electron bubbles in the Gross-Pitaevskii model of a superfluid

“…We expect that these oscillations will be mediated by nonlinear interactions of Kelvin waves that act to transfer energy to smaller scales until they are dissipated through emission of phonons [34]. At later times, the size of the charged vortex ring continues to increase with its velocity asymptoting to the self-induced velocity of a circular vortex ring [35,36].…”

Motion of an object through a quantum fluid

“…2(a). One mode of oscillation dominates [12] and the frequency is independent of the applied force. As the fluid is compressible, an accelerating object creates sound waves which damp the motion.…”

Charged Tangles of Quantized Vortices in Superfluid 4He

“…There are three different types of perturbations, depending on the particular projection of the force on the directions of the line and its binormal (if it is bent at some radius R like, for example, a vortex ring). The component of the force that is normal to the line and directed along the binormal results in gradual self-similar ballooning out [40,41] (or in for the opposite direction of the binormal) of the segment; as a result the ion mainly drifts with the segment's selfinduced velocity. The component of the force that is normal to both vortex line and its binormal results in bending (rotating) the segment while no drift of the ion in the field direction occurs.…”

“…An isolated vortex ring has its energy nearly proportional to the radius while the velocity of self-induced motion inversely proportional to radius [15,20]. The presence of one a trapped ion in an electric field normal to the ring's plane, causes it to gain energy while maintaining nearly circular shape in a self-similar way [40,41]. Hence, thanks to the precise conservation of quantized circulation, κ = hm −1 4 , the energyvelocity relation becomes of type E ∝ v −1 [20].…”