Comment on “Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation” [Phys. Fluids 25, 085101 (2013)]

Hietala, Niklas; Hänninen, Risto

doi:10.1063/1.4855296

Cited by 6 publications

(4 citation statements)

References 20 publications

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“…Van Gorder 21,22 employed this formulation to study a number of special case solutions. Limitations and benefits to this type of formulation were recently a topic of discussion in the sequence, [23][24][25] where (for the helical case) the direct approach using (10) when * x xx − x * xx is not a constant in x, the solution of (10) will be an approximation to the LIA, rather than an exact solution.…”

Section: Formulationmentioning

confidence: 99%

General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation

Gorder

2014

Physics of Fluids

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In his study of superfluid turbulence in the low-temperature limit, Svistunov [“Superfluid turbulence in the low-temperature limit,” Phys. Rev. B 52, 3647 (1995)] derived a Hamiltonian equation for the self-induced motion of a vortex filament. Under the local induction approximation (LIA), the Svistunov formulation is equivalent to a nonlinear dispersive partial differential equation. In this paper, we consider a family of rotating vortex filament solutions for the LIA reduction of the Svistunov formulation, which we refer to as the 2D LIA (since it permits a potential formulation in terms of two of the three Cartesian coordinates). This class of solutions holds the well-known Hasimoto-type planar vortex filament [H. Hasimoto, “Motion of a vortex filament and its relation to elastica,” J. Phys. Soc. Jpn. 31, 293 (1971)] as one reduction and helical solutions as another. More generally, we obtain solutions which are periodic in the space variable. A systematic analytical study of the behavior of such solutions is carried out. In the case where vortex filaments have small deviations from the axis of rotation, closed analytical forms of the filament solutions are given. A variety of numerical simulations are provided to demonstrate the wide range of rotating filament behaviors possible. Doing so, we are able to determine a number of vortex filament structures not previously studied. We find that the solution structure progresses from planar to helical, and then to more intricate and complex filament structures, possibly indicating the onset of superfluid turbulence.

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Section: Formulationmentioning

confidence: 99%

General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation

Gorder

2014

Physics of Fluids

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“…For a helical vortex, the dimensionless product Ak is an important parameter. It appears in the dispersion relation of a Kelvin wave of arbitrary amplitude (derived using the local induction approximation) 26,27…”

Section: Methodsmentioning

confidence: 99%

“…( 1). The inclusion of mutual friction or counterflow would change the dispersion relation 27 . The motion of a helical vortex is purely rotational when Ak → 0.…”

Section: Methodsmentioning

confidence: 99%

Leapfrogging Kelvin waves

et al. 2016

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Two vortex rings can form a localized configuration whereby they continually pass through one another in an alternating fashion. This phenomenon is called leapfrogging. Using parameters suitable for superfluid helium-4, we describe a recurrence phenomenon that is similar to leapfrogging, which occurs for two coaxial straight vortex filaments with the same Kelvin wave mode. For small-amplitude Kelvin waves we demonstrate that our full Biot-Savart simulations closely follow predictions obtained from a simplified model that provides an analytical approximation developed for nearly parallel vortices. Our results are also relevant to thin-cored helical vortices in classical fluids.

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“…For notation purposes, I assume such a filament takes the form r(x, t) = (x, A cos(kx − ωt + x 0 ), A sin(kx − ωt + x 0 )) which is equivalent to (x, t) = Ae i[kx−ωt+x 0 ] in potential form. In their comments, 2 Hietala and Hänninen give some criticisms of the work. In particular, they state (I) the local nonlinear equation (LNE) is not comparable to the LIA, since the LIA includes friction parameters α, α ; (II) the reduction of the LIA used in Ref.…”

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Response to “Comment on ‘Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation”’ [Phys. Fluids 26, 019101 (2014)]

Gorder

2014

Physics of Fluids

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I agree with the authors regarding their comments on the Donnelly-Glaberson instability for such helical filaments as those obtained in my paper. I also find merit in their derivation of the quantum LIA (local induction approximation) in the manner of the LIA of Boffetta et al. However, I disagree with the primary criticisms of Hietala and Hänninen. In particular, though they suggest LIA and local nonlinear equation modes are not comparable since the former class of models contains superfluid friction parameters, note that since these parameters are small one may take them to zero and consider a qualitative comparison of the models (which is what was done in my paper). Second, while Hietala and Hänninen criticize certain assumptions made in my paper (and the paper of Shivamoggi where the model comes from) since the results break-down when Ak → ∞, note that in my paper I state that any deviations from the central axis along which the filament is aligned must be sufficiently bounded in variation. Therefore, it was already acknowledged that Ak(=|Φx|) should be sufficiently bounded, precluding the Ak → ∞ case. I also show that, despite what Hietala and Hänninen claim, the dispersion relation obtained in my paper is consistent with LIA, where applicable. Finally, while Hietala and Hänninen claim that the dispersion parameter should be complex valued, I show that their dispersion relation is wrong, since it was derived incorrectly (they assume the complex modulus of the potential function is constant, yet then use this to obtain a potential function with non-constant modulus).

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Comment on “Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation” [Phys. Fluids 25, 085101 (2013)]

Cited by 6 publications

References 20 publications

General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation

General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation

Leapfrogging Kelvin waves

Response to “Comment on ‘Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation”’ [Phys. Fluids 26, 019101 (2014)]

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