Vortex nucleation by a moving ion in a Bose condensate

The nonlinear Schrödinger equation is used to analyze the superfluid flow around an ion and to elucidate the vortex nucleation process. Asymptotic expansion for the flow is used to find the critical velocity of the ion for vortex production. 3D numerical calculations demonstrate, that if the axisymmetry of the flow is broken by introducing a solid boundary several healing length from the sphere, vortex loops rather than vortex rings may be formed. These loops may detach from the ion and attach themselves to the wall. In the presence of small 3D random noise, the system still favors the creation of vortex rings.  2000 Elsevier Science B.V. All rights reserved. PACS: 47.20.Ky; 67.55.Fa; 02.60.Cb; 05.45.-a The deliberately introduced impurity can be a fruitful experimental probe of the structure and behavior of superfluid helium. These impurities are: 3He atoms of radius 4 Å, electrons that by their motion create a bubble of about 16 Å radius, and 4He2 positive ions of radius 8 Å. Vortex nucleation by an ion moving in superfluid helium at low temperature has been studied experimentally and theoretically (see [1] for a review) and has led to a number of interesting results. The superfluid offers no resistance to the ion provided that its speed, v, relative to the ion is less than a certain critical value, vc . At speeds greater than vc, the ion continually sheds vortex rings and these create a time-varying drag on the ion [2]. The critical speed vc may be estimated by modeling the ion as a solid sphere and noting that the maximum relative velocity, umax, between fluid and sphere is greatest on the equator of the sphere (defined relative to the direction Oz of motion as poE-mail address: [email protected] (N.G. Berloff). lar axis), and is approximately 3v/2, assuming that v is small enough for compressibility to be negligible. Muirhead et al. [3] developed the theory of vortex nucleation using a semiclassical (hydrodynamic) approach; the vortex was taken in the form derived in [4] from the Gross–Pitaevskii (GP) condensate model [5]. They analyzed two scenarios for the vortex shedding that occur for v > vc . In the first, a fully-formed ring detaches, simultaneously and as a whole, from the equator of the sphere. We call this the ‘complete ring’ scenario to distinguish it from the second or ‘vortex loop’ scenario, in which a vortex loop grows from the ion’s equator, is stretched by the flow, and later detaches from the sphere to become eventually a circular vortex ring also. Muirhead et al. [3], using an energy barrier argument, conclude that loop nucleation is favored over ring nucleation. The principal aim of this Letter is to compare the two scenarios by numerical simulations using the Bose condensate model. It is well known that, when the Reynolds number is large enough, the viscous boundary layer in contact 0375-9601/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S0375-9601(00) 00 71 55 N.G. Berloff / Physics Letters A 277 (2000) 240–244 241 with a moving sphere separates, so creating vorticity in the wake of the sphere and an associated drag. Helmholtz’s theorem would forbid this process in a classical fluid that is strictly inviscid. The superfluid can, however, defeat Helmholtz’s by the separation (breakdown) of the healing layer on the surface of the ion [6]. In [6] we used the GP model to investigate the complete ring scenario. If v > vc (umax > c, where c is the local speed of sound) circular vortices are emitted at or near the equator of the ion, defined by the direction of its motion. The flow created by this flow, when combined with the flow over the surface of the ion, at first reduces umax below c. The self-induced speed of the ring is, however, less than v, so that the ring falls increasingly far behind the ion, and its effect on the ion diminishes. Eventually criticality again occurs and another ring is emitted. Frisch et al. [7] and Winiecki et al. [8] have demonstrated analogous phenomena for the emission of vortex pairs from a cylinder moving with speed v > vc ≈ 2c. The drag on the cylinder created by the vortices in its wake was evaluated in [8]. In terms of the single-particle wavefunction ψ(x, t) for the N bosons of mass M , the time-dependent selfconsistent field equation of the GP model is [5]