Three-dimensional stability of leapfrogging quantum vortex rings

TL;DR

This study uses numerical simulations to identify conditions under which two or three coaxial quantum vortex rings can be stable in three dimensions, revealing stable parameter regions linked to vortex size and wave numbers.

Contribution

It demonstrates the existence of stable regions for leapfrogging quantum vortex rings based on parameters like core width and radii, extending understanding of vortex stability in superfluids.

Findings

Stable bands of parameters exist for two or three rings.

Stability is most prominent for specific azimuthal wave numbers.

No stable domains are found for four or more rings at certain sizes.

Abstract

It is shown by numerical simulations within a regularized Biot-Savart law that dynamical systems of two or three leapfrogging coaxial quantum vortex rings having a core width $\xi$ and initially placed near a torus of radii $R_0$ and $r_0$, can be three-dimensionally (quasi-)stable in some regions of parameters $\Lambda=\ln(R_0/\xi)$ and $W=r_0/R_0$. At fixed $\Lambda$, stable bands on $W$ are intervals between non-overlapping main parametric resonances for different (integer) azimuthal wave numbers $m$. The stable intervals are most wide ($\Delta W\sim$ 0.01--0.05) between $m$-pairs $(1,2)$ and $(2,3)$ at $\Lambda\approx$ 4--12 thus corresponding to micro/mesoscopic sizes of vortex rings in the case of superfluid $^4$He. With four and more rings, at least for $W>0.1$, resonances overlap for all $\Lambda$ and no stable domains exist.