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Kinematic and thermodynamic stability of vortices in superfluids

David M. F. Chapman

Published Jan 1, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.14288/1.0093976

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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.
Abstract This chapter is devoted to discussing some key superfluid properties exhibited by dilute Bose gases confined in external traps. Key issues concern the behaviour of the critical velocity and the applicability of Landau’s criterion for superfluidity, the moment of inertia and its quenching caused by the irrotationality of the superfluid motion, the scissors mode, and the expansion of a rotating superfluid cloud. Rotational effects at high angular velocities, with the appearance of quantized vortices and vortex lattices, are also considered. The chapter also discusses the consequences for vortices on the collective oscillations of the trapped gas, the stability and the precession of vortex lines, and the nature of the superfluid flow in toroidal traps.
We show that in two dimensional superfluids a large number of quantum vortices with positive and negative circulations behave as an inviscid fluid on large scales. Two hydrodynamical velocities are introduced to describe this emergent binary vortex fluid, via vortex number current and vortex change current. The velocity field associated with the vortex number current evolves according to a hydrodynamic equation, subject to an anomalous stress absent from Euler's equation. In contrast to the chiral vortex fluid containing only like-sign vortices, the binary vortex fluid is compressible and the orbital angular momentum is not conserved, as characterized by an asymmetric Cauchy stress tensor. Dissipation effects due to thermal friction and vortex-sound interactions induce an effective damping rate, an anomalous viscous stress tensor, and a coefficient of second viscosity.
We give a general review of recent developments in the theory of vortices in superfluids and superconductors, discussing why the dynamics of vortices is important, and why some key results are still controversial. We discuss work that we have done on the dynamics of quantized vortices in a superfluid. Despite the fact that this problem has been recognized as important for forty years, there is still a lot of controversy about the forces on and masses of quantized vortices. We think that one can get unambiguous answers by considering a broken symmetry state that consists of one vortex in an infinite ideal system. We argue for a Magnus force that is proportional to the superfluid density, and we find that the effective mass density of a vortex in a neutral superfluid is divergent at low frequencies. We have generalized some of the results for a neutral superfluid to a charged system.
In this thesis we demonstrate the power of the Gross-Pitaevskii and the time-dependent Ginzburg-Landau equations by numerically solving them for various fundamental problems related to superfluidity and superconductivity. We start by studying the motion of a massive object through a quantum fluid modelled by the Gross-Pitaevskii equation. Below a critical velocity, the object does not exchange momentum or energy with the fluid. This is a manifestation of its superfluid nature. We discuss the effect of applying a constant force to the object and show that for small forces a vortex ring is created to which the object becomes attached. For a larger force the object detaches from the vortex ring and we observe periodic shedding of rings. All energy transfered to the system is contained within the vortex rings and the drag force on the object is due to the recoil of the vortex emission. If we exceed the speed of sound, there is an additional contribution to the drag from sound emission. To make a link to superconductivity, we then discuss vortex states in a rotating system. In the ground state, regular arrays of vortices are observed which, for systems containing many vortices, mimic solid-body rotation. In the second part of the thesis, we initially review solutions to the Ginzburg-Landau equations in an applied magnetic field. For superconducting disks we observe vortex arrays similar to those in rotating superfluids. Finally, we study an electrical current flow along a superconducting wire subject to an external magnetic field. We observe the motion of flux lines, and hence dissipation, due to the Lorentz force. We measure the V - I curve which is analogous to the drag force in a superfluid. With the introduction of impurities, flux lines become pinned which gives rise to an increased critical current. (author)
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.