VORTEX RECONNECTIONSAND STRETCHING

IN QUANTUM FLUIDS

Carlo F. Barenghi

School of Mathematics, Newcastle University,

Newcastle upon Tyne,

UK

VORTICES IN QUANTUM FLUIDS

density velocity

order parameter

quantisation of circulation

core radius a~ healing length ξ = ħ(mE0)-1/2

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

Example:Reconnection of vortex ring with vortex line

(NLSE)

Example:Reconnection of vortex ring with vortex line

(NLSE)

Substitute

classical Continuity and (quasi) Euler equations:

where

At scale r, quantum stress/pressure ~ ħ²/(mE0 r²) ~1 for r~ξIn 4He: ξ≈10-8 cm << vortex separation δ≈10-3 or 10-4 cm

and

into NLSE and get

and → reconnections

SUPERFLUID vs EULER FLUID

superfluid = reconnecting Euler fluid

Example of role played by reconnections:rotating counterflow in 4He

Tsubota, Araki & Barenghi, PRL 90, 205301, 2003; PRB 69, 134515, 2004

Ω=0

Ω=0.05 s-1

Ω=0.01 s-1

Tsubota, Araki & Barenghi, PRL 90, 205301, 2003; PRB 69, 134515, 2004

Example of role played by reconnections:rotating counterflow in 4He

Maurer & Tabeling, EPL 43, 29, 1998

Experiment

Araki, Tsubota & Nemirowskii,PRL 89, 145301, 2002Vortex filament model

Kobayashi & Tsubota,PRL 94, 665302, 2005

NLSE model

CLASSICAL TURBULENCE

Nore, Abid & Brachet,PRL 78, 3896, 1997

NLSE model

Kolmogorov energy spectrumE(k)≈ε2/3 k-5/3

wavenumber k~1/r, energy dissipation rate ε

CLASSICAL TURBULENCEVortex stretching drives the energy cascade

Intensification of vorticity (angular velocity) through conservation

of angular momentum

Vorticity Vorticity equation

Coherent structures

S. Goto, JFM 605, 355, 2008: Energy cascade can be caused by stretching of smaller-scalevortices in larger-scale strainsexisting between vortex pairs

CLASSICAL TURBULENCE

She, Jackson & Orszag, Nature 344, 226, 1990Vincent & Meneguzzi JFM 225, 1, 1991

Farge & et, PRL 87, 054501, 2001

Problem: there is no classical stretching for quantised vortices

Coherent structures

S. Goto, JFM 605, 355, 2008: Energy cascade can be caused by stretching of smaller-scalevortices in larger-scale strainsexisting between vortex pairs

CLASSICAL TURBULENCE

She, Jackson & Orszag, Nature 344, 226, 1990Vincent & 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

Evidence for bundles ?

Kivotides, PRL 96 175301, 2006 Morris, Koplik & Rouson, PRL 101, 015301, 2008

Alamri Youd & Barenghi, 2008

Reconnection of vortex bundlesAlamri, Youd & Barenghi,

2008

NLSEmodel

7 strands

Alamri Youd & Barenghi, 2008

Reconnection of vortex bundlesAlamri, Youd & Barenghi,

2008

NLSEmodel

5 strands

Alamri Youd & Barenghi, 2008

Reconnection of vortex bundlesAlamri, Youd & Barenghi,

2008

NLSEmodel

9 strands

Alamri Youd & Barenghi, 2008

Reconnection of vortex bundlesAlamri, Youd & Barenghi,

2008

vortexfilamentmodel

Reconnection of vortex bundlesAlamri, Youd & Barenghi,

2008

vortexfilamentmodel

Alamri, Youd & Barenghi,

2008

vortexfilamentmodel

Reconnection of vortex bundles

Length Curvature PDF of curvature

NLSEmodelReconnection of vortex bundles

Length

Alamri, Youd & Barenghi,

2008

Note that length increases by 30 % while energy is conserved within 0.1 %

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.