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Spreading of superfluid vorticity clouds in normal-fluid turbulence

Published online by Cambridge University Press:  16 December 2010

DEMOSTHENES KIVOTIDES
DEMOSTHENES KIVOTIDES*
Affiliation:
Department of Chemical Engineering, University of California, Santa Barbara, CA 93117, USA
*
†Email address for correspondence: demos@engineering.ucsb.edu
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Abstract

In this paper, we formulate a self-consistent model of thermal superfluid dynamics. By solving it, we analyse the problem of superfluid vorticity cloud propagation in normal-fluid turbulence. We show that superfluid cloud expansion is driven by pattern-forming superfluid vortex instabilities taking place in the interface layer between the cloud's bulk and the outer undisturbed normal-fluid turbulence. The radius of the cloud increases linearly with time. Mutual friction transfers energy from the normal-fluid turbulence to the superfluid cloud, whilst damping the smallest normal-fluid turbulence motions. This damping action is much weaker than viscous dissipation effects in a corresponding pure normal-fluid turbulence. The energy spectrum of superfluid turbulence presents the k−3 scaling that characterizes the spiral superfluid vorticity patterns of normal vortex tube–superfluid vortex interactions. The corresponding k−2 pressure spectrum signifies the singular nature of superfluid vorticity. These two scalings coincide in wavenumber space with the Kolmogorov regime in the normal-fluid turbulence. We compute a fractal dimension df ≈ 1.652 for superfluid vorticity. Due to simpler underlying superfluid vortex dynamics in relation to the strongly nonlinear classical vortex dynamics, this fractal dimension is smaller than the corresponding dimension of vortex tube centrelines in classical turbulence.

JFM classification

Complex Fluids: Quantum fluids Turbulent Flows: Turbulent Flows Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 668 , 10 February 2011 , pp. 58 - 75
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Andersson, N., Sidery, T. & Comer, G. L. 2007 Superfluid neutron star turbulence. Mon. Not. R. Astron. Soc. 381, 747–756.CrossRefGoogle Scholar
Barenghi, C. F., Donnelly, R. J. & Vinen, W. F. 1983 Friction on quantized vortices in Helium II. A review. J. Low Temp. Phys. 52, 189–247.CrossRefGoogle Scholar
Barenghi, C. F. & Niemela, J. J. (Ed.) 2009 Quantum turbulence, Special Issue: J. Low Temp. Phys. 156.Google Scholar
Barenghi, C. F. & Samuels, D. C. 2002 Evaporation of a packet of quantized vorticity. Phys. Rev. Lett. 89, 155302–155306.CrossRefGoogle ScholarPubMed
Barenghi, C. F., Samuels, D. C., Bauer, G. H. & Donnelly, R. J. 1997 Superfluid vortex lines in a model of turbulent flow. Phys. Fluids 9, 2631–2644.CrossRefGoogle Scholar
Bewley, G. P., Lathrop, D. P. & Sreenivasan, K. R. 2006 Superfluid helium: visualization of quantized vortices. Nature 44, 588.CrossRefGoogle Scholar
Chorin, A. 1994 Vorticity and Turbulence. Springer.CrossRefGoogle Scholar
Davidson, P. A. 2004 Turbulence. Oxford University Press.Google Scholar
Finne, A. P., Eltsov, V. B., Hanninen, R., Kopnin, N. B., Kopu, J., Krusius, M., Tsubota, M. & Volovik, G. E. 2006 Dynamics of vortices and interfaces in superfluid 3He. Rep. Prog. Phys. 69, 3157–3230.CrossRefGoogle Scholar
Fisher, S. N. & Pickett, G. R. 2008 Quantum turbulence in superfluid 3He at very low temperatures. Prog. Low Temp. Phys. 16, 147–194.CrossRefGoogle Scholar
Gardiner, C. 2004 Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Springer.CrossRefGoogle Scholar
Griffin, A., Kinuni, T. & Zaremba, E. 2009 Bose-Condensed Gases at Finite Temperatures. Cambridge University Press.CrossRefGoogle Scholar
Guo, W., Wright, J. D., Cahn, S. B., Nikkel, J. A. & McKinsey, D. N. 2009 Metastable helium molecules as tracers in superfluid 4He. Phys. Rev. Lett. 102, 235301–235305.CrossRefGoogle ScholarPubMed
Hall, H. E. & Vinen, W. F. 1956 The rotation of liquid Helium II. Part II. The theory of mutual friction in uniformly rotating Helium II. Proc. R. Soc. Lond. A 238, 215–234.Google Scholar
Henderson, K. L. & Barenghi, C. F. 2004 Superfluid Couette flow in an enclosed annulus. Theor. Comput. Fluid Dyn. 18, 183–196.CrossRefGoogle Scholar
Idowu, O. C., Kivotides, D., Barenghi, C. F. & Samuels, D. C. 2000 Equation for self-consistent superfluid vortex line dynamics. J. Low Temp. Phys. 120, 269–280.CrossRefGoogle Scholar
Jackson, B., Barenghi, C. F. & Proukakis, N. P. 2007 Matter wave solitons at finite temperatures. J. Low Temp. Phys. 148, 387–391.CrossRefGoogle Scholar
Kivotides, D. 2006 Coherent structure formation in turbulent thermal superfluids. Phys. Rev. Lett. 96, 17530–17534.CrossRefGoogle ScholarPubMed
Kivotides, D. 2007 a Relaxation of superfluid vortex bundles via energy transfer to the normal fluid. Phys. Rev. B76, 054503–054515.CrossRefGoogle Scholar
Kivotides, D. 2007 b Superfluid contributions to general turbulence theory. J. Low Temp. Phys. 148, 287–291.CrossRefGoogle Scholar
Kivotides, D. 2008 a Normal-fluid velocity measurement and superfluid vortex detection in thermal counterflow turbulence. Phys. Rev. B 78, 224501–224505.CrossRefGoogle Scholar
Kivotides, D. 2008 b Motion of a spherical solid particle in thermal counterflow turbulence. Phys. Rev. B 77, 174508–174513.CrossRefGoogle Scholar
Kivotides, D., Barenghi, C. F. & Samuels, D. C. 2001 a Superfluid vortex reconnections at finite temperature. Europhys. Lett. 54, 774–778.CrossRefGoogle Scholar
Kivotides, D., Barenghi, C. F. & Samuels, D. C. 2001 b Fractal dimension of superfluid turbulence. Phys. Rev. Lett. 87, 155301–155305.CrossRefGoogle ScholarPubMed
Kivotides, D., Vassilicos, J. C., Barenghi, C. F., Khan, M. A. I & Samuels, D. C. 2001 c Quantum signature of superfluid turbulence. Phys. Rev. Lett. 87, 275302–275306.CrossRefGoogle ScholarPubMed
Kivotides, D., Barenghi, C. F. & Sergeev, Y. A. 2008 a Interactions between particles and quantized vortices in superfluid helium. Phys. Rev. B 77, 014527–014540.CrossRefGoogle Scholar
Kivotides, D., Sergeev, Y. A. & Barenghi, C. F. 2008 b Dynamics of solid particles in a tangle of superfluid vortices at low temperatures. Phys. Fluids 20, 055105–055115.CrossRefGoogle Scholar
Kivotides, D. & Leonard, A. 2003 Quantized turbulence physics. Phys. Rev. Lett. 90, 234503–234507.CrossRefGoogle ScholarPubMed
Kivotides, D. & Leonard, A. 2004 Geometrical physics of the many vortex filament problem. Europhys. Lett. 66, 69–75.CrossRefGoogle Scholar
Kivotides, D. & Wilkin, S. L. 2008 Collisions of solid particles with vortex rings in superfluid helium. J. Fluid Mech. 605, 367–387.CrossRefGoogle Scholar
Kivotides, D. & Wilkin, S. L. 2009 Elementary vortex processes in thermal superfluid turbulence. J. Low Temp. Phys. 156, 163–181.CrossRefGoogle Scholar
Koplik, J. & Levine, H. 1993 Vortex reconnection in superfluid helium. Phys. Rev. Lett. 71, 1375–1379.CrossRefGoogle ScholarPubMed
Leggett, A. J. 2006 Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems. Oxford University Press.CrossRefGoogle Scholar
Lundgren, T. S. 2003 Linearly forced isotropic turbulence. Center for Turbulence Research, Annual Research Briefs, p. 461.Google Scholar
Maurer, J. & Tabeling, P. 1998 Local investigation of superfluid turbulence. Europhys. Lett. 43, 29–34.CrossRefGoogle Scholar
Melatos, A. & Peralta, C. 2010 Gravitational radiation from hydrodynamic turbulence in a differentially rotating neutron star. Astrophys. J. 709, 77–87.CrossRefGoogle Scholar
Mine, M., Okumura, M., Sunaga, T. & Yamanaka, Y. 2007 Quantum field theoretical description of dynamical instability of trapped Bose–Einstein condensates. J. Low Temp. Phys. 148, 331–336.CrossRefGoogle Scholar
Morris, K., Koplik, J. & Rouson, D. W. I. 2008 Vortex locking in direct numerical simulations of quantum turbulence. Phys. Rev. Lett. 101, 015301–015305.CrossRefGoogle ScholarPubMed
Nemirovskii, S. K. 2010 Diffusion of inhomogeneous vortex tangle and decay of superfluid turbulence. Phys. Rev. B 81, 064512–064522.CrossRefGoogle Scholar
Parker, N. G., Proukakis, N. P., Barenghi, C. F. & Adams, C. S. 2004 Controlled vortex sound interaction in atomic Bose–Einstein condensate. Phys. Rev. Lett. 92, 160403–160407.CrossRefGoogle Scholar
Peralta, C., Melatos, A., Giacobello, M. & Ooi, A. 2005 Global three-dimensional flow of a neutron superfluid in a spherical shell in a neutron star. Astrophys. J. 635, 1224–1232.CrossRefGoogle Scholar
Peralta, C., Melatos, A., Giacobello, M. & Ooi, A. 2006 Transitions between turbulent and laminar superfluid vorticity states in the outer core of a neutron star. Astrophys. J. 651, 1079–1091.CrossRefGoogle Scholar
Peralta, C., Melatos, A., Giacobello, M. & Ooi, A. 2008 Superfluid spherical Couette flow. J. Fluid Mech. 609, 221–274.CrossRefGoogle Scholar
Pitaevskii, L. & Stringari, S. 2003 Bose–Einstein Condensation. Oxford University Press.Google Scholar
Samuels, D. C. & Kivotides, D. 1999 A damping length scale for superfluid turbulence. Phys. Rev. Lett. 83, 5306–5310.CrossRefGoogle Scholar
Schwarz, K. W. 1995 Three-dimensional vortex dynamics in superfluid 4He: line–line and line-boundary interactions. Phys. Rev. B 31, 5782–5804.CrossRefGoogle Scholar
Sidery, T., Andersson, N. & Comer, G. L. 2008 Waves and instabilities in dissipative rotating superfluid neutron stars. Mon. Not. R. Astron. Soc. 385, 335–348.CrossRefGoogle Scholar
Skrbek, L. & Vinen, W. F. 2008 The use of vibrating structures in the study of quantum turbulence. Prog. Low Temp. Phys. 16, 195–246.CrossRefGoogle Scholar
Sonin, E. B. 1997 Magnus force in superfluids and superconductors. Phys. Rev. B 55, 485–501.CrossRefGoogle Scholar
Stalp, S. R., Skrbek, L. & Donnelly, R. J. 1999 Decay of grid turbulence in a finite channel. Phys. Rev. Lett. 82, 4831–4835.CrossRefGoogle Scholar
Stone, M. 2000 Iordanskii force and the gravitational Aharonov–Bohm effect for a moving vortex. Phys. Rev. B 61, 11780–11786.CrossRefGoogle Scholar
Thouless, D. J., Ao, P. & Niu, Q. 1996 Transverse force on a quantized vortex in a superfluid. Phys. Rev. Lett. 76, 3758–3762.CrossRefGoogle Scholar
Tsubota, M., Araki, T. & Vinen, W. F. 2003 Diffusion of an inhomogeneous vortex tangle. Physica B 329, 224–225.CrossRefGoogle Scholar
Tsubota, M. & Kobayashi, M. 2008 Energy spectra of quantum turbulence. Prog. Low Temp. Phys. 16, 1–43.Google Scholar
Tulin, M. P. & Wu, J. 1977 Additive effects on free turbulent flows. Phys. Fluids 20, S109–S113.CrossRefGoogle Scholar
Vassilicos, J. C. & Brasseur, J. G. 1996 Self-similar spiral flow structure in low-Reynolds-number isotropic and decaying turbulence. Phys. Rev. E 54, 467–485.CrossRefGoogle ScholarPubMed
Vinen, W. F. 2000 Classical character of turbulence in a quantum liquid. Phys. Rev. B 61, 1410–1420.CrossRefGoogle Scholar
Vinen, W. F. & Niemela, J. J. 2002 Quantum turbulence. J. Low Temp. Phys. 128, 167–231.CrossRefGoogle Scholar
Volovik, G. E. 1998 Vortex versus spinning string: Iordanskii force and gravitational Aharonov–Bohm effect. J. Exp. Theor. Phys. Lett. 67, 881–886.CrossRefGoogle Scholar
Wexler, C. 1997 Magnus and Iordanskii forces in superfluids. Phys. Rev. Lett. 79, 1321–1325.CrossRefGoogle Scholar
Wilkin, S. L., Barenghi, C. F. & Shukurov, A. 2007 Magnetic structures produced by the small-scale dynamo. Phys. Rev. Lett. 99, 134501–134505.CrossRefGoogle ScholarPubMed
Wilson, J. R. & Mathews, G. J. 2007 Relativistic Numerical Hydrodynamics. Cambridge University Press.Google Scholar
Zhang, T. & Van Sciver, S. W. 2005 The motion of micron-sized particles in He II counterflow as observed by the PIV technique. J. Low Temp. Phys. 138, 865–870.CrossRefGoogle Scholar