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VORTEX RECONNECTIONS AND STRETCHING IN QUANTUM FLUIDS Carlo F. Barenghi School of Mathematics, Newcastle University, Newcastle upon Tyne, UK

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VORTICES IN QUANTUM FLUIDS densityvelocity order parameter quantisation of circulation core radius a~ healing length ξ = ħ(mE 0 )-1/2

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QUANTUM TURBULENCE Reconnections Postulated by Schwarz 1985 (vortex filament model) Confirmed by Koplik & Levine 1993 (NLSE model) CFB Hanninen, Eltsov, Krusius et al isotropic vortex tangle twisted vortex state

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Example: Reconnection of vortex ring with vortex line (NLSE)

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Example: Reconnection of vortex ring with vortex line (NLSE)

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Substitute classical Continuity and (quasi) Euler equations: where At scale r, quantum stress/pressure ~ ħ²/(mE 0 r²) ~1 for r~ξ In 4 He: ξ≈10 -8 cm << vortex separation δ≈10 -3 or 10 -4 cm and into NLSEand get and→ reconnections SUPERFLUID vs EULER FLUID superfluid = reconnecting Euler fluid

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Example of role played by reconnections: rotating counterflow in 4 He Tsubota, Araki & Barenghi, PRL 90, 205301, 2003; PRB 69, 134515, 2004 Ω=0 Ω=0.05 s -1 Ω=0.01 s -1

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Tsubota, Araki & Barenghi, PRL 90, 205301, 2003; PRB 69, 134515, 2004 Example of role played by reconnections: rotating counterflow in 4 He

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Maurer & Tabeling, EPL 43, 29, 1998 Experiment Araki, Tsubota & Nemirowskii, PRL 89, 145301, 2002 Vortex filament model Kobayashi & Tsubota, PRL 94, 665302, 2005 NLSE model CLASSICAL TURBULENCE Nore, Abid & Brachet, PRL 78, 3896, 1997 NLSE model Kolmogorov energy spectrum E(k)≈ε 2/3 k -5/3 wavenumber k~1/r, energy dissipation rate ε

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CLASSICAL TURBULENCE Vortex stretching drives the energy cascade Intensification of vorticity (angular velocity) through conservation of angular momentum VorticityVorticity equation

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Coherent structures S. Goto, JFM 605, 355, 2008: Energy cascade can be caused by stretching of smaller-scale vortices in larger-scale strains existing between vortex pairs CLASSICAL TURBULENCE She, Jackson & Orszag, Nature 344, 226, 1990 Vincent & Meneguzzi JFM 225, 1, 1991 Farge & et, PRL 87, 054501, 2001 Problem: there is no classical stretching for quantised vortices

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Coherent structures S. Goto, JFM 605, 355, 2008: Energy cascade can be caused by stretching of smaller-scale vortices in larger-scale strains existing between vortex pairs CLASSICAL TURBULENCE She, Jackson & Orszag, Nature 344, 226, 1990 Vincent & Meneguzzi JFM 225, 1, 1991 Farge & et, PRL 87, 054501, 2001 Problem: there is no classical stretching for quantised vortices Solution: think of quantised vortex bundles

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Evidence for bundles ? Kivotides, PRL 96 175301, 2006Morris, Koplik & Rouson, PRL 101, 015301, 2008

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Alamri Youd & Barenghi, 2008 Reconnection of vortex bundles Alamri, Youd & Barenghi, 2008 NLSE model 7 strands

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Alamri Youd & Barenghi, 2008 Reconnection of vortex bundles Alamri, Youd & Barenghi, 2008 NLSE model 5 strands

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Alamri Youd & Barenghi, 2008 Reconnection of vortex bundles Alamri, Youd & Barenghi, 2008 NLSE model 9 strands

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Alamri Youd & Barenghi, 2008 Reconnection of vortex bundles Alamri, Youd & Barenghi, 2008 vortex filament model

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Reconnection of vortex bundles Alamri, Youd & Barenghi, 2008 vortex filament model

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Alamri, Youd & Barenghi, 2008 vortex filament model Reconnection of vortex bundles Length CurvaturePDF of curvature

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NLSE model Reconnection of vortex bundles Length Alamri, Youd & Barenghi, 2008 Note that length increases by 30 % while energy is conserved within 0.1 %

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Conclusions 1. Concept of quantised vortex bundle strengthens the analogy between quantum turbulence and classical turbulence. 2. Quantised vortex bundles are so robust that they can undergo reconnections. 3. Large amount of coiling of vortex strands confirms Kerr (Nonlinearity 9, 271, 1996) and the conjecture by Holm and Kerr (PRL 88, 244501, 2002) on the generation of helicity in nearly singular events of the Euler equation.

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