Vortex density waves and high-frequency second sound in superfluid turbulence hydrodynamics

Abstract

In this Letter we show that a recent hydrodynamical model of superfluid turbulence describes vortex density waves and their effects on the speed of high-frequency second sound. In this frequency regime, the vortex dynamics is not purely diffusive, as for low frequencies, but exhibits ondulatory features, whose influence on the second sound is here explored.Hydrodynamics of superfluid turbulence, characterized by a tangle of quantized vortex lines [1,2], aims to describe the couplings between pressure and heat perturbations, and the vortex density dynamics [3][4][5][6][7]. Such hydrodynamics is a lively topic, with recent emphasis, for instance, on nonlinear features such as the influence of intense heat pulses on the vortex tangle [8-10], or in multi-scale formulations allowing to eliminate the fast processes to derive evolution equations for the slow processes [11]. In the present Letter we will focus our attention on a different topic, namely, the behavior of linear heat waves and vortex density waves in the highfrequency domain. This has not received much previous attention, but we think it is worthwhile because, whereas at low frequencies the behavior of the vortex distortions is mainly diffusive, at high enough frequencies, the vortex lines behave as an elastic medium, and are able to propagate waves by themselves, i.e. they behave like a viscoelastic medium: diffusive at low frequencies, elastic at high frequencies. Their effect on second sound propagation is of much interest to provide a suitable physical interpretation of the experimental data based on second sound.Collective vortex waves in rotating superfluids have been studied in depth for a long time [12], and the so-called Tkachenko transverse elastic waves in the vortex arrays with crystalline order arising in rotating cylinders have been theoretically discussed since 1966. However, vortex density waves in counterflow situations have not been studied, up to our knowledge.Usually, one considers homogeneous vortex tangles in counterflow situation (i.e. under vanishing barycentric speed) and the evolution of L, the vortex-line density, is assumed to be the well-known Vinen's equation [1,2,6,13] (1)where q is the absolute value of the heat flux q, A and B parameters linked to the dimensionless coefficients appearing in Vinen's equation by the relations A = α v /ρ s T s and B = κβ v [7], ρ s being the density mass of the superfluid component, T the absolute

A mathematical model of counterflow superfluid turbulence describing heat waves and vortex-density waves

“…The latter result is logic because when the vortex wave is propagated in the superfluid helium, temperature T and heat flux q 1 cannot remain constant. This behavior is different from that obtained in [9], because using that model in the second sound also the line density L vibrates. In fact, there the flux of vortices J was chosen proportional to q, so that vibrations in the heat flux (second sound) produce vibrations in the vortex tangle.…”

“…In particular, high-frequency second sound may be of special interest to probe small length scales in the tangle, which is necessary in order to explore, for instance, the statistical properties of the vortex loops of several sizes. In fact, the reduction of the size of space averaging is one of the active frontiers in second sound techniques applied to turbulence, but at high-frequencies, the response of the tangle to the second sound is expected to be qualitatively different than at low frequencies, as its perturbations may change from diffusive to propagative behavior [4]- [9].…”

“…In a previous paper [4] a thermodynamical model of inhomogeneous superfluid turbulence was built up with the aim to study the mutual interactions between second sound and the vortex tangle. The fundamental fields of the model were the density ρ, the velocity v, the internal energy density E, the heat flux q and the average vortex line length per unit volume L. In a successive paper [9], starting from this model, a semiquantitative expression for the vortex diffusion coefficient was obtained and the interaction between second sound and the tangle in the high-frequency regime was studied. In both these works, for the sake of simplicity, the diffusion flux of vortices J was considered as a dependent variable, collinear with the heat flux q, which is proportional to the counterflow velocity V, defined as V = v n − v s , v n and v s being the velocities of the normal and superfluid components, respectively.…”

“…The mathematical formalism is physically motivated to explore the interactions between heat waves and vortex-density waves. In Section 4, the physical meaning of the several coefficients appearing in the model are examined, and in Section 5 wave propagation in this more complete model in uncoupled and coupled situations is studied and the results are compared with those obtained in [4,9].…”

Hydrodynamic equations of anisotropic, polarized and inhomogeneous superfluid vortex tangles

“…In Ref. [34], because we were interested to study wall effects on the evolution of L, the following extension of equation (4.14) was written …”