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arXiv:2407.04177 · Scientific Reports

Crow instability of vortex lines in dipolar superfluids

Srivatsa B. Prasad, Nick G. Parker, Andrew William Baggaley

Published Sep 29, 2025• 0 citations
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Abstract

In classical inviscid fluids, antiparallel vortices perturbed by Kelvin waves exhibit the Crow instability, where the mutual interaction of the Kelvin modes renders them dynamically unstable. This results in the approach and reconnection of the vortices, leading to a cascaded decay into ever-smaller vortex loops. Through mean-field simulations we study the Crow instability of quantum vortex lines in a superfluid whose atoms are subject to the anisotropic, long-ranged dipole-dipole interaction. We observe that the direction of dipole polarization plays a crucial role in determining the dynamically favored Kelvin modes. The subsequent rate of the instability is linked to the mediation of the vortex curvature by the effective dipole-dipole interaction between the vortices themselves. The vortex curvature is strongly suppressed and modes of lower wavenumber are preferred when the dipole polarization is parallel to the vortices, whereas the curvature is maximized for polarizations along the vortices' separation axis. For polarizations along the binormal axis, modes of higher wavenumber are favorable but the instability rate is considerably inhibited. This paves the way to a deeper understanding of vortex reconnections, vortex loop cascades and turbulence in dipolar superfluids.

DOI
10.1038/s41598-025-17373-8

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In classical inviscid fluids, antiparallel vortices perturbed by Kelvin waves exhibit the Crow instability, where the mutual interaction of the Kelvin modes renders them dynamically unstable. This results in the approach and reconnection of the vortices, leading to a cascaded decay into ever-smaller vortex loops. Through mean-field simulations we study the Crow instability of quantum vortex lines in a superfluid whose atoms are subject to the anisotropic, long-ranged dipole-dipole interaction. We observe that the direction of dipole polarization plays a crucial role in determining the dynamically favored Kelvin modes. The subsequent rate of the instability is linked to the mediation of the vortex curvature by the effective dipole-dipole interaction between the vortices themselves. The vortex curvature is strongly suppressed and modes of lower wavenumber are preferred when the dipole polarization is parallel to the vortices, whereas the curvature is maximized for polarizations along the vortices' separation axis. For polarizations along the binormal axis, modes of higher wavenumber are favorable but the instability rate is considerably inhibited. This paves the way to a deeper understanding of vortex reconnections, vortex loop cascades and turbulence in dipolar superfluids.
We show theoretically that elongated vortex-antivortex dipoles can be created controllably in trapped Bose-Einstein condensates, using known experimental techniques. Vortex dipoles of sufficient length are unstable and cascade into slow vortex rings which ultimately decay via sound emission. This instability of antiparallel vortex line elements, which self-generates Kelvin waves on vortex loops in trapped atomic gases, may play a role in bridging the Kelvin-wave and Kolmogorov-Richardson cascades of quantum turbulence.
The static and dynamic properties of vortices in dipolar Bose-Einstein condensates (dBECs) can be considerably modified relative to their nondipolar counterparts by the anisotropic and long-ranged nature of the dipole-dipole interaction. Working in a uniform dBEC, we analyze the structure of single vortices and the dynamics of vortex pairs, investigating the deviations from the nondipolar paradigm. For a straight vortex line, we find that the induced dipolar interaction potential is axially anisotropic when the dipole moments have a nonzero projection orthogonal to the vortex line. This results in a corresponding elongation of the vortex core along this projection as well as an anisotropic superfluid phase and enhanced compressibility in the vicinity of the vortex core. Consequently, the trajectories of like-signed vortex pairs are described by a family of elliptical and oval-like curves rather than the familiar circular orbits. Similarly for opposite-signed vortex pairs their translation speeds along the binormal are found to be dipole-interaction dependent. We expect that these findings will shed light on the underlying mechanisms of many-vortex phenomena in dBECs such as quantum turbulence, vortex reconnections, and vortex lattices.
The static and dynamic properties of vortices in dipolar Bose-Einstein condensates (dBECs) can be considerably modified relative to their nondipolar counterparts by the anisotropic and long-ranged nature of the dipole-dipole interaction. Working in a uniform dBEC, we analyze the structure of single vortices and the dynamics of vortex pairs, investigating the deviations from the nondipolar paradigm. For a straight vortex line, we find that the induced dipolar interaction potential is axially anisotropic when the dipole moments have a nonzero projection orthogonal to the vortex line. This results in a corresponding elongation of the vortex core along this projection as well as an anisotropic superfluid phase and enhanced compressibility in the vicinity of the vortex core. Consequently, the trajectories of like-signed vortex pairs are described by a family of elliptical and oval-like curves rather than the familiar circular orbits. Similarly for opposite-signed vortex pairs their translation speeds along the binormal are found to be dipole-interaction dependent. We expect that these findings will shed light on the underlying mechanisms of many-vortex phenomena in dBECs such as quantum turbulence, vortex reconnections, and vortex lattices. Published by the American Physical Society 2024
The Gross–Pitaevskii (GP) equation admits a two-dimensional solitary wave solution representing two mutually self-propelled, antiparallel straight line vortices. The complete sequence of such solitary wave solutions has been computed by Jones and Roberts (Jones C A and Roberts P H 1982 J. Phys. A: Math. Gen. 15 2599 ). These solutions are unstable with respect to three-dimensional perturbations (the Crow instability). The most unstable mode has a wavelength along the direction of the vortices of the same order as their separation. The growth rate associated with this mode is evaluated here and it is found to increase very rapidly with decreasing separation. It is shown, through numerical integrations of the GP equation that, as the perturbations grow to finite amplitude, the lines reconnect to produce a sequence of almost circular vortex rings.
The Gross-Pitaevskii (GP) equation admits a two-dimensional solitary wave solution representing two mutually self-propelled, anti-parallel straight line vortices. The complete sequence of such solitary wave solutions has been computed by Jones and Roberts (J. Phys. A, 15, 2599, 1982). These solutions are unstable with respect to three-dimensional perturbations (the Crow instability). The most unstable mode has a wavelength along the direction of the vortices of the same order as their separation. The growth rate associated with this mode is evaluated here, and it is found to increase very rapidly with decreasing separation. It is shown, through numerical integrations of the GP equation that, as the perturbations grow to finite amplitude, the lines reconnect to produce a sequence of almost circular vortex rings. 1
In a recent experiment, Finne et al. discovered an intrinsic condition for the onset of quantum turbulence in 3 He-B, that q = α/(1 − α') < 1.3, where α and α' are mutual friction parameters. The authors put forward a qualitative argument that q is the ratio of dissipative and inertial forces on the superfluid, so for q < 1 inertial forces should overcome the dissipative forces and cause turbulence. Thus 1/ q would play, for a quantum fluid, the same role played in classical fluid dynamics by the Reynolds number (the ratio of inertial forces and dissipative forces in the Navier-Stokes equation). The aim of this work is to supplement this qualitative condition q = 1 with a quantitative calculation. By analysing both axisymmetric and non-axisymmetric modes of a continuum of vortices in a rotating superfluid, we find that in the long axial wavelength limit the condition q = 1 is the crossover between damped and propagating Kelvin waves; thus, for q > 1, perturbations on the vortices are unlikely to cause vortex reconnections and turbulence. Besides the relevance to the experiment of Finne et al. , the spectrum of oscillations which we find is relevant to the study of torsional oscillations of a rotating superfluid and generalises to three dimensions the spectrum of Kelvin waves on an isolated vortex line.