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

2016

Phys. Rev. E

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We demonstrate the existence of localized structures along quantized vortex filaments in superfluid helium under the quantum form of the local induction approximation (LIA), which includes mutual friction and normal fluid effects. For small magnitude normal fluid velocities, the dynamics are dissipative under mutual friction. On the other hand, when normal fluid velocities are sufficiently large, we observe parametric amplification of the localized disturbances along quantized vortex filaments, akin to the Donnelly-Glaberson instability for regular Kelvin waves. As the waves amplify they will eventually cause breakdown of the LIA assumption (and perhaps the vortex filament itself), and we derive a characteristic time for which this breakdown occurs under our model. More complicated localized waves are shown to occur, and we study these asymptotically and through numerical simulations. Such solutions still exhibit parametric amplification for large enough normal fluid velocities, although this amplification may be less uniform than would be seen for more regular filaments such as those corresponding to helical curves. We find that large rotational velocities or large wave speeds of nonlinear waves along the filaments will result in more regular and stable structures, while small rotational velocities and wave speeds will permit far less regular dynamics.

Section: Numerical Simulation Of Localized Wave Solutionssupporting

confidence: 90%

“…Note that when χ = 0, we would obtain a planar vortex filament, but only in the case where α = 0. If α > 0, such a planar solution is not possible, due to induced torsion from the mutual friction terms [31]. Therefore, χ = 0 and α > 0 will result in nonplanar solutions.…”

Section: Numerical Simulation Of Localized Wave Solutionsmentioning

confidence: 99%

Localized nonlinear waves on quantized superfluid vortex filaments in the presence of mutual friction and a driving normal fluid flow

Shah

2016

Phys. Rev. E

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“…Translational effects for such solutions are no longer zero, but they can be shown to be small (of the order α 2 , where α is the larger of the two small mutual friction parameters). Therefore, between [23] and the present paper, it is reasonable to expect that analogous solutions (which are planar at t = 0 and gradually evolve into non-planar structures) can be found in the non-local form of the quantum model, where LIA terms are replaced by Biot-Savart integrals.…”

Section: Discussionsupporting

confidence: 53%

“…The quantum analogue to the LIA was provided by Schwarz [10]. It was shown in Van Gorder [23] that an exact planar solution does not exist under the quantum LIA (unless there are extraordinary special experimental conditions) due to torsion effects. (The planar filament is a torsionless solution.)…”

Section: Discussionmentioning

confidence: 99%

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

2015

Physics of Fluids

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While the Hasimoto planar vortex filament is one of few exact solutions to the local induction approximation (LIA) approximating the self-induced motion of a vortex filament, it is natural to wonder whether such a vortex filament solution would exist for the non-local Biot-Savart dynamics exactly governing the filament motion, and if so, whether the non-local effects would drastically modify the solution properties. Both helical vortex filaments and vortex rings are known to exist under both the LIA and non-local Biot-Savart dynamics, however the planar filament is a bit more complicated. In the present paper, we demonstrate that a planar vortex filament solution does exist for the non-local Biot-Savart formulation, provided that a specific non-linear integral equation (governing the spatial structure of such a filament) has a non-trivial solution. By using the Poincaré-Lindstedt method, we are able to obtain an accurate analytical approximation to the solution of this integral equation under physically reasonable assumptions. To obtain these solutions, we approximate local effects near the singularity of the integral equation using the LIA, and non-local effects using the Biot-Savart formulation. Mathematically, the results constitute an analytical solution to an interesting nonlinear singular integro-differential equation in space and time variables. Physically, these results show that planar vortex filaments exist and maintain their form under the non-local Biot-Savart formulation, as one would hope. Due to the regularization approach utilized, we are able to compare the structure of the planar filaments obtained under both LIA and Biot-Savart formulations in a rather straightforward manner, in order to determine the role of the non-locality on the structure of the planar filament.

“…into account in the Schwarz model [35]. It was shown in [36] that purely planar filaments should be expected in the classical LIA, or in the zero-temperature limit for the quantum model, while they will not exist in the presence of mutual friction and normal fluid flow due to the induced torsion due to these small terms. The result will be a twisted planar filament.…”

Section: A the Planar Vortex Filament Reductionmentioning

confidence: 99%

Solitons and nonlinear waves along quantum vortex filaments under the low-temperature two-dimensional local induction approximation

2016

Phys. Rev. E

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Very recent experimental work has demonstrated the existence of Kelvin waves along quantized vortex filaments in superfluid helium. The possible configurations and motions of such filaments is of great physical interest, and Svistunov previously obtained a Hamiltonian formulation for the dynamics of quantum vortex filaments in the low-temperature limit under the assumption that the vortex filament is essentially aligned along one axis, resulting in a two-dimensional (2D) problem. It is standard to approximate the dynamics of thin filaments by employing the local induction approximation (LIA), and we show that by putting the two-dimensional LIA into correspondence with the first equation in the integrable Wadati-Konno-Ichikawa-Schimizu (WKIS) hierarchy, we immediately obtain solutions to the two-dimensional LIA, such as helix, planar, and self-similar solutions. These solutions are obtained in a rather direct manner from the WKIS equation and then mapped into the 2D-LIA framework. Furthermore, the approach can be coupled to existing inverse scattering transform results from the literature in order to obtain solitary wave solutions including the analog of the Hasimoto one-soliton for the 2D-LIA. One large benefit of the approach is that the correspondence between the 2D-LIA and the WKIS allows us to systematically obtain vortex filament solutions directly in the Cartesian coordinate frame without the need to solve back from curvature and torsion. Implications of the results for the physics of experimentally studied solitary waves, Kelvin waves, and postvortex reconnection events are mentioned.