Motion of isolated open vortex filaments evolving under the truncated local induction approximation

Abstract

The study of nonlinear waves along open vortex filaments continues to be an area of active research. While the local induction approximation (LIA) is attractive due to locality compared with the nonlocal Biot-Savart formulation, it has been argued that LIA appears too simple to model some relevant features of Kelvin wave dynamics, such as Kelvin wave energy transfer. Such transfer of energy is not feasible under the LIA due to integrability, so in order to obtain a non-integrable model, a truncated LIA, which breaks the integrability of the classical LIA, has been proposed as a candidate model with which to study such dynamics. Recently Laurie et al. ["Interaction of Kelvin waves and nonlocality of energy transfer in superfluids," Phys. Rev. B 81, 104526 (2010)] derived truncated LIA systematically from Biot-Savart dynamics. The focus of the present paper is to study the dynamics of a section of common open vortex filaments under the truncated LIA dynamics. We obtain the analog of helical, planar, and more general filaments which rotate without a change in form in the classical LIA, demonstrating that while quantitative differences do exist, qualitatively such solutions still exist under the truncated LIA. Conversely, solitons and breather solutions found under the LIA should not be expected under the truncated LIA, as the existence of such solutions relies on the existence of an infinite number of conservation laws which is violated due to loss of integrability. On the other hand, similarity solutions under the truncated LIA can be quite different to their counterparts found for the classical LIA, as they must obey a t 1/3 type scaling rather than the t 1/2 type scaling commonly found in the LIA and Biot-Savart dynamics. This change in similarity scaling means that Kelvin waves are radiated at a slower rate from vortex kinks formed after reconnection events. The loss of soliton solutions and the difference in similarity scaling indicate that dynamics emergent under the truncated LIA can indeed differ a great deal from those previously studied under the classical LIA.

Non-Hamiltonian Kelvin wave generation on vortices in Bose-Einstein condensates

“…Currently, there are two derivations of this result which predict cascade features and time scalings arising from the kinetics of Kelvin waves. [12,13,14] While the exact source of driven Kelvin wave motion is an open topic [15,16,17], there is little doubt that vortex plucking through reconnection generates helical wave motion. [18] Our results do not seek to model the inception of localized regions of curvature and are instead focused on how the fluid responds to curved abnormalities on the vortex and are based on the interplay between the characteristic length scales of curvature, arclength and core size, which are highly constrained in the LIA.…”