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Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

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Laurie, Jason, L'vov, V. S. (Viktor Sergeevich), 1942- , Nazarenko, Sergey and Rudenko, Oleksii. (2010) Interaction of Kelvin waves and nonlocality of energy transfer in superfluids. Physical Review B (Condensed Matter and Materials Physics), Vol.81 (No.10). ISSN 1098-0121

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Official URL: http://dx.doi.org/10.1103/PhysRevB.81.104526

Abstract

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Waves, Superfluidity, Turbulence
Journal or Publication Title:	Physical Review B (Condensed Matter and Materials Physics)
Publisher:	American Physical Society
ISSN:	1098-0121
Date:	26 March 2010
Volume:	Vol.81
Number:	No.10
Number of Pages:	14
Identification Number:	10.1103/PhysRevB.81.104526
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] E. Kozik and B. Svistunov, Phys. Rev. Lett. 92, 035301 (2004), DOI: 10.1103/PhysRevLett.92.035301. [2] E. Kozik and B. Svistunov, Journal of Low Temp. Phys. 156, 215-267 (2009), DOI: 10.1007/s10909-009-9914-y. [3] S. Nazarenko, JETP Letters 83, 198-200 (2006), DOI: 10.1134/S0021364006050031. [4] V. S. L’vov, S. V. Nazarenko and O. Rudenko, Phys. Rev. B 76, 024520 (2007), DOI: 10.1103/Phys- RevB.76.024520. [5] V. S. L’vov, S. V. Nazarenko and O. Rudenko), Journal of Low Temp. Phys., 153, 140-161 (2008). [6] E.V. Kozik, B.V. Svistunov, Phys. Rev. B 77, 060502(R) (2008) [7] R. J. Donnelly, Quantized Vortices in He II (Cambridge University Press, Cambridge, 1991) [8] Quantized Vortex Dynamics and Superfluid Turbulence, ed. by C. F. Barenghi et al., Lecture Notes in Physics 571 (Springer-Verlag, Berlin, 2001) [9] K.W. Schwarz, Phys. Rev. B 31, 5782 (1985), DOI: 10.1103/PhysRevB.31.5782, and 38, 2398 (1988), DOI: 10.1103/PhysRevB.38.2398. [10] B.V. Svistunov, Phys. Rev. B 52, 3647 (1995), DOI: 10.1103/PhysRevB.52.3647. [11] R.J. Arms and F.R. Hama, Phys. Fluids 8, 553 (1965). [12] H. Hasimoto, Journal of Fluid Mech. 51, 477-485 (1972), DOI: 10.1017/S0022112072002307. [13] G. Boffetta, A. Celani, D. Dezzani, J. Laurie and S. Nazarenko, Journal of Low Temp. Phys. 156, 193-214 (2009), DOI: 10.1007/s10909-009-9895-x. [14] V.E. Zakharov, V.S. L’vov and G.E. Falkovich. Kol- mogorov Spectra of Turbulence, (Springer-Verlag, 1992). [15] R. Kraichnan, Phys. Fluids, 10 1417 (1967), DOI: 10.1063/1.1762301. [16] R. Kraichnan, J. Fluid Mech, 47 525 (1971), DOI: 10.1017/S0022112071001216. [17] S.V.Nazarenko, JETP Letters 84, 585-587 (2007), DOI: 10.1134/S0021364006230032. [18] V.E. Zakharov and E.I. Schulman, Physica D: Nonlin- ear Phenomena 4, 270-274 (1982), DOI: 10.1016/0167- 2789(82)90068-9. [19] I. Gradstein and I. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York (1980). [20] The limit of three small wave numbers is not allowed by the resonance conditions. Indeed, putting three wave numbers to zero, we get a 1 <-> 2 process which is not allowed in 1D for Aw - k2. [21] Here, we evoke a quantum mechanical analogy as an elegant shortcut, allowing us to introduce KE and the re- spective solutions easily. However, the reader should not get confused with this analogy and understand that our KW system is purely classical. In particular, the Plank’s constant ~ is irrelevant outside of this analogy, and should be simply replaced by 1. [22] It is evident for the approximation Eq. (31). For the full expression Eq. (29a) it was confirmed by symbolic computation with the help of Mathematica. [23] It is appropriate to remind the reader, that we use the bold face notation of the one-dimensional wave vector for convenience only. Indeed, such a vector is just a real number, k 2 R.
URI:	http://wrap.warwick.ac.uk/id/eprint/3959