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Quantum vortex dynamics under the tangent representation of the local induction approximation

Published online by Cambridge University Press:  10 January 2014

Robert A. Van Gorder
Robert A. Van Gorder*
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
*
†Email address for correspondence: rav@knights.ucf.edu
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Abstract

We derive the local induction approximation (LIA) for a quantum vortex filament in the arclength coordinate frame where the tangent vector is the unknown function. The equation for the tangent vector to the filament is then converted to a potential form, which ends up being a type of nonlinear Schrödinger equation that governs the tangential LIA model (T-LIA). Such a formulation was previously derived by Umeki for the standard fluid model (in the absence of superfluid friction terms). While it is challenging to generalize many of the exact solutions found for the standard LIA to the quantum LIA model, we demonstrate that the T-LIA model facilitates this generalization nicely. Indeed, under the T-LIA model, we are able to construct a variety of solutions. The Hasimoto solution related to elastica is one of the fundamental solutions present for the standard fluid model; however, using the T-LIA model, we are able to demonstrate the existence of such a solution, thereby extending the Hasimoto solution to the superfluid case. In the zero-temperature limit, purely self-similar solutions are shown to exist for the T-LIA model. As the superfluid warms (so that the influence of the normal flow is no longer negligible), the analogue to the self-similar solution is a new class of solutions, which depend on the similarity variable as well as a time-dependent additive scaling. In other words, the self-similar structures gradually deform as the magnitude of the normal-fluid velocity increases, which makes complete physical sense. When dealing with small deviations from the central axis of alignment, we can describe such solutions analytically. There exists a family of helical vortex filaments in the presence of a normal fluid impinging on the vortex, in complete agreement with the previously studied results for the LIA model. Finally, a number of soliton solutions are shown to exist in different regimes of the T-LIA model.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows Complex Fluids: Quantum fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 740 , 10 February 2014 , pp. 5 - 16
Copyright
©2014 Cambridge University Press 

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References

Alamri, S. Z., Youd, A. J. & Barenghi, C. F. 2008 Reconnection of superfluid vortex bundles. Phys. Rev. Lett. 101, 215302.Google Scholar
Araki, T., Tsubota, M. & Nemirovskii, S. K. 2002 Energy spectrum of superfluid turbulence with no normal-fluid component. Phys. Rev. Lett. 89, 145301.Google Scholar
Arms, R. J. & Hama, F. R. 1965 Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553.Google Scholar
Baggaley, A. W., Laurie, J. & Barenghi, C. F. 2012 Vortex-density fluctuations, energy spectra, and vortical regions in superfluid turbulence. Phys. Rev. Lett. 109, 205304.Google Scholar
Bekarevich, I. L. & Khalatnikov, I. M. 1961 Phenomenological derivation of the equations of vortex motion in He II. Sov. Phys. JETP 13, 643.Google Scholar
Boffetta, G., Celani, A., Dezzani, D., Laurie, J. & Nazarenko, S. 2009 Modelling Kelvin wave cascades in superfluid Helium. J. Low Temp. Phys. 156, 193.Google Scholar
Da Rios, L. S. 1906 Sul moto d’un liquido indefinite con un filetto vorticoso di forma qualunque. Rend. Circ. Mat. Palermo 22, 117.CrossRefGoogle Scholar
Eltsov, V. B., de Graaf, R., Hänninen, R., Krusius, M. & Solntsev, R. E. 2008 Experiments on the twisted vortex state in superfluid ${\text{} }^{3} \mathrm{He} $-B. J. Low Temp. Phys. 150, 373.Google Scholar
Fisher, S. N. & Pickett, G. R. 2009 Quantum turbulence in superfluid ${\text{} }^{3} \mathrm{He} $ at very low temperatures. Prog. Low Temp. Phys. 16, 147–194.Google Scholar
Fonda, E., Meichle, D. P., Ouellette, N. T., Hormoz, S., Sreenivasan, K. R. & Lathrop, D. P. 2012 Visualization of Kelvin waves on quantum vortices, arXiv:1210.5194.Google Scholar
Gutiérrez, S., Rivas, J. & Vega, L. 2003 Formation of singularities and self-similar vortex motion under the localized induction approximation. Commun. Pure Diff. Equ. 28, 927.CrossRefGoogle Scholar
Hall, H. E. & Vinen, W. F. 1956a The rotation of liquid helium II. I. Experiments on the propagation of second sound in uniformly rotating helium II. Proc. R. Soc. Lond. A 238, 204.Google Scholar
Hall, H. E. & Vinen, W. F. 1956b The rotation of liquid helium II. II. The theory of mutual friction in uniformly rotating helium II. Proc. R. Soc. Lond. A 238, 215.Google Scholar
Hasimoto, H. 1971 Motion of a vortex filament and its relation to elastica. J. Phys. Soc. Japan 31, 293.Google Scholar
Hasimoto, H. 1972 A soliton on a vortex filament. J. Fluid Mech. 51, 477.Google Scholar
Kida, S. 1981 A vortex filament moving without change of form. J. Fluid Mech. 112, 397.Google Scholar
Kida, S. 1982 Stability of a steady vortex filament. J. Phys. Soc. Japan 51, 1655.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Addison and Wesley.Google Scholar
Lipniacki, T. 2003a Shape-preserving solutions for quantum vortex motion under localized induction approximation. Phys. Fluids 15, 1381.Google Scholar
Lipniacki, T. 2003b Quasi-static solutions for quantum vortex motion under the localized induction approximation. J. Fluid Mech. 477, 321.Google Scholar
Nemirovskii, S. K. 2013 Quantum turbulence: theoretical and numerical problems. Phys. Rep. 524, 85 2.5.1.Google Scholar
Ricca, R. L. 1991 Rediscovery of Da Rios equations. Nature 352, 561.Google Scholar
Schwarz, K. W. 1985 Three-dimensional vortex dynamics in superfluid ${\text{} }^{4} \mathrm{He} $: line–line and line–boundary interactions. Phys. Rev. B 31, 5782.Google Scholar
Shivamoggi, B. K. 2011 Vortex motion in superfluid ${\text{} }^{4} \mathrm{He} $: reformulation in the extrinsic vortex-filament coordinate space. Phys. Rev. B 84, 012506.Google Scholar
Shivamoggi, B. K. 2013 Vortex motion in superfluid ${\text{} }^{4} \mathrm{He} $: effects of normal fluid flow. Eur. Phys. J. B 86, 275.Google Scholar
Sonin, E. B. 2012 Dynamics of helical vortices and helical-vortex rings. Eur. Phys. Lett. 97, 46002.CrossRefGoogle Scholar
Svistunov, B. 1995 Superfluid turbulence in the low-temperature limit. Phys. Rev. B 52, 3647.Google Scholar
Umeki, M. 2010 A locally induced homoclinic motion of a vortex filament. Theor. Comput. Fluid Dyn. 24, 383.Google Scholar
Umeki, M. 2013 A real-space representation of a locally induced vortex filament. Theor. Appl. Mech. Japan 61, 195.Google Scholar
Van Gorder, R. A. 2012a Fully nonlinear local induction equation describing the motion of a vortex filament in superfluid ${\text{} }^{4} \mathrm{He} $. J. Fluid Mech. 707, 585.Google Scholar
Van Gorder, R. A. 2012b Exact solution for the self-induced motion of a vortex filament in the arclength representation of the local induction approximation. Phys. Rev. E 86, 057301.Google Scholar
Van Gorder, R. A. 2013a Motion of a helical vortex filament in superfluid ${\text{} }^{4} \mathrm{He} $ under the extrinsic form of the local induction approximation. Phys. Fluids 25, 085101.Google Scholar
Van Gorder, R. A. 2013b Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation. Phys. Rev. E 87, 043203.Google Scholar
Van Gorder, R. A. 2013c Orbital stability for rotating planar vortex filaments in the Cartesian and arclength forms of the local induction approximation. J. Phys. Soc. Japan 82, 094005.Google Scholar
Van Gorder, R. A. 2013d Self-similar vortex dynamics in superfluid ${\text{} }^{4} \mathrm{He} $ under the Cartesian representation of the Hall–Vinen model including superfluid friction. Phys. Fluids 25, 095105.Google Scholar
Vinen, W. F. 2001 Decay of superfluid turbulence at a very low temperature: the radiation of sound from a Kelvin wave on a quantized vortex. Phys. Rev. B 64, 134520.Google Scholar