Non-local dynamics governing the self-induced motion of a planar vortex filament

2014

Proc. R. Soc. A.

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The Hasimoto planar vortex filament is one of the rare exact solutions to the classical local induction approximation (LIA). This solution persists in the absence of friction or other disturbances, and it maintains its form over time. As such, the dynamics of such a filament have not been extended to more complicated physical situations. We consider the planar vortex filament under the quantum LIA, which accounts for mutual friction and the velocity of a normal fluid impinging on the filament. We show that, for most interesting situations, a filament which is planar in the absence of mutual friction at zero temperature will gradually deform owing to friction effects and the normal fluid flow corresponding to warmer temperatures. The influence of friction is to induce torsion, so the filaments bend as they rotate. Furthermore, the flow of a normal fluid along the vortex filament length will result in a growth in space of the initial planar perturbations of a line filament. For warmer temperatures, these effects increase in magnitude, since the growth in space scales with the mutual friction coefficient. A number of nice qualitative results are analytical in nature, and these results are verified numerically for physically interesting cases.

Section: Resultsmentioning

confidence: 99%

Dynamics of a planar vortex filament under the quantum local induction approximation

2014

Proc. R. Soc. A.

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“…38 studied the influence of background flows on planar filaments. The planar vortex filament solution was also recently shown to exist 40 for the non-local Biot-Savart dynamics, which is more complicated yet more realistic formulation for the motion of classical vortex filaments. Analytical perturbation solutions were also constructed in Ref.…”

mentioning

confidence: 99%

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

2017

Physics of Fluids

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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.

Self-similar vortex filament motion under the non-local Biot–Savart model

2016

J. Fluid Mech.

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One type of thin vortex filament structure that has attracted interest in recent years is that which obeys self-similar scaling. Among various applications, these filaments have been used to model the motion of quantized vortex filaments in superfluid helium after reconnection events. While similarity solutions have been described analytically and numerically using the local induction approximation (LIA), they have not been studied (or even shown to exist) under the non-local Biot–Savart model. In this present paper, we show not only that self-similar vortex filament solutions exist for the non-local Biot–Savart model, but that such solutions are qualitatively similar to their LIA counterparts. This suggests that the various LIA solutions found previously should be valid physically (at least in the small amplitude regime), since they agree well with the more accurate Biot–Savart model.