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arXiv:1706.03257 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Helicity conservation and twisted Seifert surfaces for superfluid vortices

Hayder Salman

Published Apr 1, 2017• 20 citations

Abstract

Starting from the continuum definition of helicity, we derive from first principles its different contributions for superfluid vortices. Our analysis shows that an internal twist contribution emerges naturally from the mathematical derivation. This reveals that the spanwise vector that is used to characterize the twist contribution must point in the direction of a surface of constant velocity potential. An immediate consequence of the Seifert framing is that the continuum definition of helicity for a superfluid is trivially zero at all times. It follows that the Gauss-linking number is a more appropriate definition of helicity for superfluids. Despite this, we explain how a quasi-classical limit can arise in a superfluid in which the continuum definition for helicity can be used. This provides a clear connection between a microscopic and a macroscopic description of a superfluid as provided by the Hall-Vinen-Bekarevich-Khalatnikov equations. This leads to consistency with the definition of helicity used for classical vortices.

DOI: 10.1098/rspa.2016.0853

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Starting from the continuum definition of helicity, we derive from first principles its different contributions for superfluid vortices. Our analysis shows that an internal twist contribution emerges naturally from the mathematical derivation. This reveals that the spanwise vector that is used to characterize the twist contribution must point in the direction of a surface of constant velocity potential. An immediate consequence of the Seifert framing is that the continuum definition of helicity for a superfluid is trivially zero at all times. It follows that the Gauss-linking number is a more appropriate definition of helicity for superfluids. Despite this, we explain how a quasi-classical limit can arise in a superfluid in which the continuum definition for helicity can be used. This provides a clear connection between a microscopic and a macroscopic description of a superfluid as provided by the Hall-Vinen-Bekarevich-Khalatnikov equations. This leads to consistency with the definition of helicity used for classical vortices.

Helicity - a quadratic invariant of the classical Euler equation like the energy - plays a fundamental role in turbulent flows, controlling the strength of the nonlinear interactions which cascade energy to smaller length scales. The growing interest in turbulent superfluid liquid helium, a disordered state of quantum matter consisting of a tangle of vortex lines - triggers a natural question: what is superfluid helicity ? The most used model of superfluid vortex lines is based on the Gross-Pitaevskii equation for a weakly interacting Bose gas. In this mean field model, unfortunately, helicity is ill-defined, as vorticity and velocity are singular on the centerline of the vortex. Here we show that by taking into account more realistic features of the vortex core arising from N-body quantum mechanics simulations, which account for the stronger atom interactions occurring in a liquid, the classical definition of helicity can be extended to superfluid helium. We also present results of numerical experiments which reveal the role and physical meaning of helicity in superfluid turbulence.

Helical waveguide filled by superfluid is shown to transform rotations of the reference frame

\vec Ω_{\oplus}

into linear displacements of the atomic ensemble and vise versa the linear displacements of the reference frame

\vec V

initialize rotations

\vec Ω_{\oplus}

of ensemble. In the mean-field Gross-Pitaevskii equation with a weakly modulated gravitation potential the exact solutions for macroscopic wavefunction

Ψ

demonstrate the emergence of

redshift

phase factor. For helical waveguides formed by phase-conjugated optical vortices with high angular momentum

L_z=\pm \ell \hbar \sim \pm \hbar \cdot 10^{3 ÷4}

one may expect the significant improvement of the phase-sensitivity

δϕ

by factor containing

2 \ell

compared to the conventional nonsingular matter-wave interferometry.

We study the effects of twisting on the low-energy excitations of superfluid

^{3}\mathit{A}

. They are small compared to the effects of bending, but they have a different symmetry which makes them important. The structure of the excitation wave functions has strong analogies with the eigenstates of a charged particle in a magnetic field. The group velocity is parallel to curl l^. The density of states at zero energy has a contribution proportional to twisting which is odd with respect to k^. Because of this term a density fluctuation produces a fluctuation of the current carried by the excitations. This current fluctuation turns out to be exactly equal to the fluctuation of the

{C}_{0}

term in the current. This shows that the

{C}_{0}

term fluctuation is carried by the normal liquid and leads us to believe that the

{C}_{0}

term as a whole is linked to the normal liquid. Since calculations making use of the Bardeen-Cooper-Schrieffer wave function consider only the condensate, our results offer a possible explanation for the ``angular-momentum paradox.'' We physically interpret the

{\mathrm{v}}_{n}

term in the Josephson equation at T=0 and show that it is readily obtained from the microscopic description of the normal liquid. In the same way we deduce from the momentum conservation law an equation of motion for

{\mathrm{v}}_{n}

and we show that it can be entirely understood from the microscopic framework.

Helicity is a topological invariant that measures the linkage and knottedness of lines, tubes, and ribbons. As such, it has found myriads of applications in astrophysics, fluid dynamics, atmospheric sciences, and biology. In quantum flows, where topology-changing reconnection events are a staple, helicity appears as a key quantity to study. However, the usual definition of helicity is not well posed in quantum vortices, and its computation based on counting links and crossings of centerline vorticity can be downright impossible to apply in complex and turbulent scenarios. We present a definition of helicity which overcomes these problems and which gives the expected result in the large-scale limit. With it, we show that certain reconnection events can excite Kelvin waves and other complex motions of the centerline vorticity, which slowly deplete helicity as they interact nonlinearly, thus linking the theory of vortex knots with observations of quantum fluids. This process also results in the depletion of helicity in a fully turbulent quantum flow, in a way reminiscent of the decay of helicity in classical fluids.