An introduction to superfluidity and quantum turbulence

Transcription

1 An introduction to superfluidity and quantum turbulence Acknowledgments: Joy Allen, Andrew Baggaley, Nick Parker, Nick Proukakis, Yuri Sergeev, Nugzar Suramlishvili, Daniel Wacks, Angela White, Anthony Youd (

2 Summary Quantum fluids BEC Two-fluid model Vortex lines: reconnections, friction Quantum turbulence: energy spectrum, velocity statistics, comparison with classical turbulence

3 Quantum fluids Liquid 4 He (T c = 2.17K) Liquid 3 He (T c 10 3 K) atomic condensates: 87 Rb, 23 Na, 7 Li, etc (T c 10 7 K) Underlying physics: Bose-Einstein condensation Ideal Bose gas: at T < T c a finite fraction of the bosons falls into the lowest accessible state n 0 = condensate density ( ) n 0 T 3/2 n = 1 T c T c = critical temperature

4 Bose-Einstein condensation Atom with momentum p = mv has wavelength λ = h/p Average kinetic energy mv 2 /2 k B T Wavelength increases with decreasing T : λ h mkb T Compare λ against the average distance between atoms, d: BEC occurs when λ d

5 Two-fluid model Landau & Tisza: Quantum ground state superfluid component Thermal excitations normal fluid component (caveat) Accounts for unusual behaviour of 4 He for T < T c (thermal/mechanical effects, second sound, counterflow) Normal fluid Superfluid Density ρ n ρ s Velocity v n v s Entropy S 0 Viscosity η 0 ρ = ρ n + ρ s

6 Critical velocity Landau: shape of dispersion curve ɛ(p) explains superfluidity Energy/momentum conservation: E 1 = E 2 + ɛ P 1 = P 2 + p (McClintock 1977) Object of mass m moving at speed V 1 creates excitation of energy ɛ and momentum p if V 1 ɛ/p Critical velocity V c = min(ɛ/p)

7 Quantum of circulation Macroscopic Ψ = Ae iφ v s = ( /m) φ Quantum of circulation (Onsager 1948, Vinen 1961) v s dr = /m = κ C Vortex lattice in rotating helium Packard et al 1982 Bewley, Lathrop & Sreenivasan 2006 Ketterle et al 2001

8 Gross Pitaevskii Equation (GPE) Assume weak interactions (realistic for atomic gases, idealised for helium) V (x x ) = gδ(x x ) g = 4πa s 2 /m a s = scattering length i Ψ t = 2 2m 2 Ψ + gψ Ψ 2 + V trap Ψ Ψ(r) 2 d 3 r = N V trap = 1 2 ω2 x 2 trapping potential

9 Fluid dynamics interpretation of GPE Substitute Ψ = Ae iφ into GPE, define and get density ρ = ψ 2 velocity v = ( /m) φ ρ t + (ρv) = 0 (Continuity eq.) ( vj ρ t + v k ) v j x k = p x j + Σ jk x k (Quasi Euler eq.) Pressure p = g ( ) 2 2m 2 ρ2 Quantum stress Σ jk = ρ 2 ln ρ 2m x j x k At scales larger than ξ = / mµ Σ jk negligible Euler eq.

10 Vortex line and wave solutions of GPE Vortex is a hole of thickness ξ Phase Perturb uniform solution V c = c Perturb straight vortex Kelvin wave

11 Vortex reconnections Conjectured (Schwarz 1988) Proved (Koplik & Levine 1993) Observed (Paoletti et al 2008) using δ(t) = A κ(t 0 t) (Tebbs & CFB, JLTP 2011) (Kida & Takaoka 1988) Classical reconnections: rely on viscosity Moffatt s 1969 helicity theorem blow-up?

12 Vortex filament model Schwarz 1988: ξ << l model vortex line as reconnecting space curve s(ξ, t) Biot-Savart law: ds dt = κ (z s) dz 4π z s 3 Schwarz used LIA: ds dt βs s Biot-Savart is slow: CPU N 2 Tree algorithm is faster: CPU N log N (Baggaley & CFB 2011)

13 Friction At T > 0 thermal excitations interact with vortex fields/cores: mutual friction force between normal fluid and superfluid ds dt = v s + αs (v n v s ) α s [s (v n v s )] α, α difficult to calculate, but known from experiments Hot topic for atomic BECs (Jackson & CFB, PRA 2009): i ψ ) ( t = 2 2m 2 + V + gn c + 2gñ ir ψ f t + p m f U pf = C 22 + C 12 Condensate and thermal cloud densities: n c (r, t) = ψ(r, t) 2, ñ(r, t) = dp f (p, r, t) (2π ) 3

14 Quantum turbulence Experimentally, there are many ways to create a tangle of vortices: Heat current (Vinen 1957): Prague, Florida, Maryland Ultra sound (Schwarz & Smith 1981) Towed grid (Donnelly, Vinen & 1993) Ions: Manchester Vibrating sphere, grid, fork, wire: Regensburg, Osaka, Lancaster, Prague Propellers (Tabeling 1998): Grenoble Instabilities following rotation: Helsinki Laser spoon, shaking the trap (BEC): Sao Paulo

15 Quantum turbulence ξ = vortex core, l = average vortex spacing, D = system size Homogeneous: ( 3 He, 4 He) constant density, ξ << l << D parameters fixed by nature Non-Homogeneous: (BECs) non-uniform density, ξ < l < D control geometry, dimensions, strength and type of interactions

16 Kolmogorov energy spectrum Kolmogorov spectrum E(k) k 5/3 observed by Tabeling 1998 Computed by: -Nore & Brachet Kivotides & CFB Tsubota & al 2002, Kerr Baggaley & CFB 2011, etc

17 Turbulence at high temperature Superfluid vortices align with normal vorticity (Samuels, 1993; CFB, Bauer & al, PoF 1997) θ(t) = 2tan 1 (e αωt ) φ(t) = α Ωt (CFB & Hulton, PRL 2002) (Morris & Koplik PRL 2008) Polarization of vortices at large scale (Vinen & Niemela 2002)

18 Turbulence at high temperature v s and v n are both turbulent: self-consistent approach is needed Done for vortex ring (Kivotides & CFB, Science 2000) Simpler attempts: Two-fluid DNS model Two-fluid shell model (Roche, Leveque & CFB, EPL 2009) (Wacks & CFB 2011) Energy fluxes: NF: inertial viscous; SF: inertial friction

19 Turbulence near absolute zero Viscosity=0 but turbulence decays Sound = sink of kinetic energy (Vinen 2001) Moving vortices sound waves Reconnections sound pulses (Leadbeater, CFB, & al PRL 2001, PRA 2002, JLTP 2005)

20 Kelvin waves cascade Cascade to large k: energy radiated away (Svistunov 1995, Kivotides & CFB 2001, Vinen & Tsubota 2003) Two cascades: k << 1/l: Kolmogorov cascade k >> 1/l: Kelvin waves cascade Bottleneck between the two cascades? L vov & Nazarenko PRL 2010, PRB 2010 Kozik & Svistunov JLTP 2009, PRL 2008 (Baggaley, CFB PRB 2011)

21 Velocity statistics In 4 He: PDF(v x ) vx 3 (Paoletti & al 2008) In ordinary fluids: PDF(v x ) = Gaussian Experiment: Noullez & al (JFM 1997) Theory: Vincent & Meneguzzi (JFM 1991)

22 Velocity statistics Non-Gaussian statistics arise from singular nature of vortex (White, CFB & al PRL 2010)

23 Velocity statistics (Baggaley, CFB 2011) (Adachi, Tsubota 2011)

24 Final remarks Some open problems: Turbulence of normal fluid (Melotte & Barenghi 1998, Guo, McKinsey & al 2010) Laminar and turbulent profiles Equations of motion

25 Which equations? In classical turbulence we solve the Navier-Stokes equation. Which equation governs quantum turbulence? GPE: resolves vortex cores, T = 0 only, compressible. Vortex filament model: incompressible (no acoustic emission). Two-fluid equations: what is ω s? (micro) (meso) (macro)

26 What is ω s? Two-fluid equations: Dv n Dt = 1 ρ P ρ s ρ n S T +ν n 2 v n + F ρ n, The Hall Vinen friction Dv s Dt = 1 ρ P+S T F ρ s F = Bρ sρ n ρ ˆω s [ω s (v n v s ν s ω s ( ˆω s )] + works well for rotating cylinder (Hall & Vinen 1956) and for Taylor-Couette flow (CFB 1992). Its use for turbulence, or the use of the simpler F ω s (v n v s ), would assume that all vortex length L is polarised and contributes to ω s How to relate L to v s = ω s?