The Library

Modeling Kelvin wave cascades in superfluid helium

Tools

Boffetta, G., Celani, A., Dezzani, D., Laurie, Jason and Nazarenko, Sergey. (2009) Modeling Kelvin wave cascades in superfluid helium. Journal of Low Temperature Physics, Vol.156 (No.3-6). pp. 193-214. ISSN 0022-2291

Preview

PDF
WRAP_Nazarenko_modeling_kelvin_waves.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (892Kb)

Official URL: http://dx.doi.org/10.1007/s10909-009-9895-x

Abstract

We study two different types of simplified models for Kelvin wave turbulence on quantized vortex lines in superfluids near zero temperature. Our first model is obtained from a truncated expansion of the Local Induction Approximation (Truncated-LIA) and it is shown to possess the same scalings and the essential behaviour as the full Biot-Savart model, being much simpler than the later and, therefore, more amenable to theoretical and numerical investigations. The Truncated-LIA model supports six-wave interactions and dual cascades, which are clearly demonstrated via the direct numerical simulation of this model in the present paper. In particular, our simulations confirm presence of the weak turbulence regime and the theoretically predicted spectra for the direct energy cascade and the inverse wave action cascade. The second type of model we study, the Differential Approximation Model (DAM), takes a further drastic simplification by assuming locality of interactions in k-space via using a differential closure that preserves the main scalings of the Kelvin wave dynamics. DAMs are even more amenable to study and they form a useful tool by providing simple analytical solutions in the cases when extra physical effects are present, e.g. forcing by reconnections, friction dissipation and phonon radiation. We study these models numerically and test their theoretical predictions, in particular the formation of the stationary spectra, and closeness of numerics for the higher-order DAM to the analytical predictions for the lower-order DAM.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Turbulence -- Mathematical models, Liquid helium, Superfluidity, Quantum statistics, Hydrodynamics
Journal or Publication Title:	Journal of Low Temperature Physics
Publisher:	Springer New York LLC
ISSN:	0022-2291
Date:	September 2009
Volume:	Vol.156
Number:	No.3-6
Number of Pages:	22
Page Range:	pp. 193-214
Identification Number:	10.1007/s10909-009-9895-x
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] W.F. Vinen, Phys. Rev. B 64, 134520 (2001). [2] B.V. Svistunov, Phys. Rev. B 52, 3647 (1995). [3] W.F. Vinen, Phys. Rev. B 61, 1410 (2000). [4] E. Kozik and B. Svistunov, Phys. Rev. Lett. 92, 03501 (2004). [5] S. Nazarenko, JETP Lett. 83, No 5, 198 (2006). [6] D. Kivotides, J.C. Vassilicos, D.C. Samuels and C.F. Barenghi, Phys. Rev. Lett. 86, 3080 (2001). [7] W.F. Vinen, M. Tsubota and A. Mitani, Phys. Rev. Lett. 91, 135301 (2003). [8] E. Kozik and B. Svistunov, Phys. Rev. Lett. 94, 025301 (2005). [9] K. W. Schwarz, Phys. Rev. B 31, 5782, (1985). [10] P.M. Walmsey, A.I. Golov, H.E. Hall, A.A. Levchenko and W.F. Vinen, Phys. Rev. Lett. 99, 265302 (2007). [11] R.J. Arms and F.R. Hama, Phys. Fluids 8, 553 (1965). [12] H. Hasimoto, J. Fluid Mech. 51, 477 (1972). [13] R.H. Kraichnan and D. Montgomery, Rep. Prog. Phys., 43, 547 (1980). [14] V. Lebedev, private communication. [15] C. Leith, Phys. Fluids 10, 1409 (1967); Phys. Fluids 11, 1612 (1968), [16] S. Hasselmann and K. Hasselmann, J. Phys. Oceanogr. 15, 1369 (1985), [17] R.S. Iroshnikov, Sov. Phys. Dokl. 30, 126 (1985), [18] V.E. Zakharov and A.N. Pushkarev, Nonlinear Proc. Geophys. 6 (1), 1 (1999), [19] C. Connaughton and S. Nazarenko, Phys. Rev. Lett. 92, 044501 (2004), [20] V.S. Lvov, S. Nazarenko and G. Volovik, JETP Lett. 80, 535 (2004), [21] V.S. Lvov, S.V. Nazarenko and L. Skrbek, JLTP, 145 (1-4), pp. 125-142 (Nov 2006). [22] R.J. Donnelly, Quantized Vortices in Helium II, Cambridge Studies in Low Temperature Physics, Cambridge University Press (1991). [23] V. Zakharov, V. L'vov and G. Falkovich, Kolmogorov Spectra of Turbulence, Nonlinear Dynamics, Springer-Verlag (1992). [24] V.E. Zakharov and E.I. Schulman, Phys. D 270-274 (1982). [25] R. Fj�rtoft, Tellus 5, 225 (1953). [26] L. Smith and V. Yakhot, J. Fluid Mech. 274, 115 (1994). [27] M.J. Lighthill, Proc. R. Soc. London, Ser. A 211, 564 (1952). [28] E. Kozik and B. Svistunov, Phys. Rev. B 72, 172505 (2005). [29] I. Gradstein and I. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York (1980). [30] Weakness of waves implies that there are no vortex line reconnections. On the other hand, for strong waves the reconnections can occur and they could qualitatively be described by the self-crossing solutions of LIA [2]. At self-crossing events, the LIA model fails, but it can be \reset" via an ad hoc reconnection procedure [9].
URI:	http://wrap.warwick.ac.uk/id/eprint/2295