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Distribution of Circles on a Circle and Correlation Between Vortex Rings of Superfluids

ERKOÇ, Mehmet Onur FEN and Şakir

Published Mar 1, 2007• 0 citations
OpenAlex

Abstract

Superfluids are characterized by absence of viscosity. When superfluids are rotated, differently from normal fluids, they form more than one vortex in the containers where they are placed. The number of vortices change as the rotation velocity changes, but this change is not linear. M.W. Zwierlein et al. observed the vortices in experiments, observing up to a number of 80. Experiments also showed that the vortex distributions cannot include large spaces. By using experimental data, we noticed that when we think of vortices as vortex rings, their centers are at the same geometric location and these geometric locations are concentric circles. We generalized the distribution of these geometric places and formulized it. Our formula includes the magic circle numbers. When the number of vortices reach these magic numbers, the number of geometric locations increase by 1.

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Superfluids are characterized by absence of viscosity. When superfluids are rotated, differently from normal fluids, they form more than one vortex in the containers where they are placed. The number of vortices change as the rotation velocity changes, but this change is not linear. M.W. Zwierlein et al. observed the vortices in experiments, observing up to a number of 80. Experiments also showed that the vortex distributions cannot include large spaces. By using experimental data, we noticed that when we think of vortices as vortex rings, their centers are at the same geometric location and these geometric locations are concentric circles. We generalized the distribution of these geometric places and formulized it. Our formula includes the magic circle numbers. When the number of vortices reach these magic numbers, the number of geometric locations increase by 1.
We numerically model experiments in which large-scale vortex rings---bundles of quantized vortex loops---are created in superfluid helium by a piston-cylinder arrangement. We show that the presence of a normal-fluid vortex ring together with the quantized vortices is essential to explain the coherence of these large-scale vortex structures at nonzero temperatures, as observed experimentally. Finally we argue that the interaction of superfluid and normal-fluid vortex bundles is relevant to recent investigations of superfluid turbulence.
We numerically model experiments in which large-scale vortex rings-bundles of quantized vortex loops-are created in superfluid helium by a piston-cylinder arrangement. We show that the presence of a normal-fluid vortex ring together with the quantized vortices is essential to explain the coherence of these large-scale vortex structures at nonzero temperatures, as observed experimentally. Finally we argue that the interaction of superfluid and normal-fluid vortex bundles is relevant to recent investigations of superfluid turbulence.
The equations of motion of vortices in an ideal two-dimensional fluid are derived, and the laws of conservation associated with the motion are discussed. Two regions of flow are considered, an infinite region and a region bounded on the outside by a circle. The kinetic energy and angular momentum of a vortex fluid in these two regions are calculated. A Lagrangian formalism is introduced in order to discuss the symmetry transformations of vortex systems and their associated conservation laws. The kinematic stability of rigidly rotating polygonal configurations of quantized vortices is determined, and the states of thermodynamic equilibrium of a rotating superfluid are found for low angular velocities, resulting in the calculation of the spectrum of critical angular velocities for the creation of one, two, and three vortices.
In this work, we study Bose–Einstein condensation in a system, where the rotation frequency decreases radially from the center. The system is described by a modified Gross–Pitaevskii equation that includes a nonuniform rotation term controlled by a parameter [Formula: see text], with the length scale of the rotation variation fixed relative to the healing length. The results show behavior distinctly different from the uniformly rotating condensate. In the uniform case, a triangular vortex lattice forms, whereas here we observe that above a critical value [Formula: see text] the vortices arrange themselves into a circular ring-shaped pattern near the boundary. Below [Formula: see text], the vortex configuration is disordered, lacking both triangular symmetry and a clear ring structure. The energy and chemical potential exhibit a sharp change at [Formula: see text], signaling a structural phase transition. A phase diagram is presented for the transition from the disordered to the ring-shaped vortex arrangement. The findings are discussed in the context of current experiments on synthetic gauge fields and structured rotation.
In this thesis, we analytically and numerically study two different scenarios of vortex dynamics in a two-dimensional superfluid. The first system we explore is an expanding cluster of like-sign vortices, which recently suggested it could explain observations in a superfluid optomechanics experiment. We employ a recent theoretical model that approximates large collections of vortices as a fluid and present an analytic solution for its dynamics. Analysis suggests that as a cluster expands due to thermal friction, it forms a Rankine vortex (i.e., a top-hat distribution) that expands self-similarly in time regardless of initial vorticity distribution. Numerical simulations of point vortices confirm this universal expanding regime and further demonstrates vortex lattice formation that goes beyond the vortex fluid theory. We analyse correlations that emerge in the resulting lattices and observe decaying disorder as the cluster expands. Furthermore, we analyse experimental data from a quasi-two-dimensional 87Rb Bose-Einstein condensate, which supports the vortex fluid theory’s predictions.The second scenario considers the dynamics of a scattering event where a single quantum vortex interacts with a uniform, circular pinning potential in a steady two-dimensional superfluid flow. We first develop an approximate hydrodynamical model to understand the interaction between the vortex and the pinning potential in the conservative regime and find it is in good agreement with the mean-field Gross-Pitaevskii equation in the low-velocity limit. We further solve the Gross-Pitaevskii equation for higher background velocities to study how the vortex moves around pinning sites, interacts with it, and ultimately becomes pinned. We uncover four different regimes of vortex pinning, each marked with a characteristic emission of sound energy. Finally, we study how vortices become unpinned from a pinning site, revealing large vortex dipoles form to stabilise the system.Our work in this thesis takes significant steps towards advancing knowledge of both the collective dynamics of vortices and their individual interaction with defects upon surface substrates. Our numerical and analytical results opens up new avenues of exploration be it further understanding dissipation mechanisms in two-dimensional superfluids, or elucidating the nature of vortex pinning in systems such as neutron stars.
The time evolution and scattering interactions of superfluid vortex rings are studied using the Gross-Pitaevskii model, in order to consider the maintenance and generation of vorticity in superfluid flows, and to compare ring dynamics in Navier-Stokes and superfluids. For a single ring, we verify previous analytic results on stability and the value of the translation velocity. When two vortex rings scatter, we generically observe merger and subsequent breakup in cases of close approach. Depending on the initial conditions, the final state can contain zero, two, or more final rings.