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Physical Review

Vortex Rings in a Bose Fluid

D. Amit, Eugene P. Gross

Published May 6, 1966• 97 citations
Publisher

Abstract

Vortex rings are considered as flow patterns of a phenomenological Hartree-type fluid with parameters fitted to the observed phonon-roton spectrum. The basis of the calculations is variational, and density patterns which are constant except in vortex cores are employed. The energy of vortex lines and rings is expressed in terms of a core radius $c$, defined as the distance from the singular line of the vortex to the point of half the asymptotic density. The core size is determined by minimizing the energy and is found to be independent of the ring radius when the latter is larger than ten times the core radius. Comparing the results with the data of Rayfield and Reif, one finds $c=1.7$ \AA{}, while the phenomenological model yields half this value. The critical velocity associated with ring formation in a channel is considered following the lines of Fetter's earlier calculation. In the present calculation it is possible to incorporate the Fineman-Chase analysis of channel wall effects. Agreement with experiment is considerably improved.

DOI
10.1103/physrev.145.130

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Vortex rings are considered as flow patterns of a phenomenological Hartree-type fluid with parameters fitted to the observed phonon-roton spectrum. The basis of the calculations is variational, and density patterns which are constant except in vortex cores are employed. The energy of vortex lines and rings is expressed in terms of a core radius ccc, defined as the distance from the singular line of the vortex to the point of half the asymptotic density. The core size is determined by minimizing the energy and is found to be independent of the ring radius when the latter is larger than ten times the core radius. Comparing the results with the data of Rayfield and Reif, one finds c=1.7c=1.7c=1.7 \AA{}, while the phenomenological model yields half this value. The critical velocity associated with ring formation in a channel is considered following the lines of Fetter's earlier calculation. In the present calculation it is possible to incorporate the Fineman-Chase analysis of channel wall effects. Agreement with experiment is considerably improved.
A theoretical study of rectilinear vortices in an imperfect Bose gas shows a close correspondence with classical hydrodynamics. The energy and momentum of a vortex pair in an unbounded fluid are calculated. The similarity between a vortex pair and a vortex ring leads to an estimate of the critical velocity vc{v}_{c}vc​ of liquid He II in a tube of radius RRR that includes the effect of the walls ${v}_{c}=\frac{C\ensuremath{\hbar}}{2mR}$, where CCC is a constant of order unity. A variational treatment of a system of many identical vortices in a container shows that the energy is lowest for a uniform distribution, and that the number of vortices per unit area $\ensuremath{\nu}$ agrees with Feyman's result $\ensuremath{\nu}=\frac{2m\ensuremath{\omega}}{h}$. In the classical limit ($h\ensuremath{\rightarrow}0$), the angular momentum and energy approach the values for solid-body rotation.
Nonlocal nonlinear Schrödinger equations are considered as models of liquid helium II. The models contain a nonlocal interaction potential that leads to a phonon-roton-like dispersion relation. Also, a higher-order term in the local density approximation for the correlation energy is introduced into the model to overcome nonphysical mass concentrations. These equations are solved for straight-line vortices. It is demonstrated that the parameters of the equation can be chosen to bring into agreement the vortex core parameter and the healing length. The structure of vortex rings of large radius is studied. The family of the vortex rings of different radii propagating with different velocities is found numerically. As the velocity of the vortex ring reaches the Landau critical velocity the sequence of rings terminates.
We present a numerical method for generating vortex rings in Bose-Einstein condensates confined in axially symmetric traps. The vortex ring is generated using the line-source approximation for the vorticity, i.e., the curl of the superfluid velocity field is different from zero only on a circumference of a given radius located on a plane perpendicular to the symmetry axis and coaxial with it. The particle density is obtained by solving a modified Gross-Pitaevskii equation that incorporates the effect of the velocity field. We discuss the appearance of density profiles, the vortex core structure, and the vortex nucleation energy, i.e., the energy difference between vortical and ground-state configurations. This is used to present a qualitative description of the vortex dynamics.
Model many-body wave functions are constructed for a quantized vortex line and ring. The corresponding variational energies are calculated using an approximate integral equation derived by Percus and Yevick for classical fluids. The energy of the line is close to that found in the Hartree theory. However, the density distribution in the core region is markedly different, and the core size is somewhat smaller, being of the order of 1 \AA{}. The velocity of translation of the vortex ring is calculated by a new method, and excellent agreement is obtained with the experimental results of Rayfield and Reif. The condensed-state wave function is computed numerically, and it is found that in the core of the vortex there are about 20% fewer particles in the condensed state than in the zero-momentum state when the system is in equilibrium.
The translational velocity of a vortex with circulation hm\frac{h}{m}mh​ in an imperfect Bose gas is calculated by considering the time dependence of the condensate wave function corresponding to a given initial configuration of vortices. Each vortex in a system of rectilinear vortices is shown to move with the local fluid velocity at its core; a vortex ring of radius RRR is shown to move with velocity $(\frac{h}{4\ensuremath{\pi}\mathrm{mR}})\mathrm{ln}(\frac{8R}{\ensuremath{\alpha}})$, where $\ensuremath{\alpha}$ is a length the order of the core size. Both results agree with the predictions of classical hydrodynamics.
A procedure is given which in principle permits the exact evaluation of any term in the asymptotic expansion in a/c of the velocity U o and energy E of a circular vortex line of radius c in a Bose condensate (where a is the healing length). The procedure is used to obtain the two leading terms (previously only inaccurately determined by a variational calculation), namely U 0 = kappa /4 pi c(In(8c/a)-0.615); E= 1 / 2 rho kappa 2 c(1n(8c/a)-1.615); where kappa denotes the quantum of circulation, and rho denotes the fluid density at infinity.