The Tale of Two Tangles: Dynamics of "Kolmogorov" and "Vinen" turbulences in 4 He near T =0

1. Euromech 491, Exeter 2007 The Tale of Two Tangles:Dynamics of "Kolmogorov" and "Vinen" turbulences in 4He near T=0 Paul Walmsley, Steve May, Alexander Levchenko, Andrei Golov (thanks: Henry Hall, JOE VINEN) • Different types of tangles and their dissipation at T=0 • Production of random and structured tangles • Detection of turbulence by ballistic vortex rings • Results for both types of tangles • Conclusions

2. Introduction In classical turbulence dissipation is via vorticity w and viscosity n: In superfluid, turbulence is made of quantized vortices and their tangles of density L Superfluid 4He has zero molecular viscosity, n = 0 Conversion of flow energy into heat is mediated by quantized vortices: (n’ – “effective kinematic viscosity”) dE/dt = -nw2 dE/dt = -n’(kL)2 Circulation quantum, k = h/m = 10-3 cm2s-1 Core a0 ~ 0.1 nm L = 10 – 105 cm-2 l = L-1/2 = 0.03 – 3 mm

3. Random (“Vinen”) vs. structured (“Kolmogorov”) state No correlations between vortices, hence only one length scale, l = L-1/2 All energy in vortex line tension Vinen’s equation: dL/dt = -zkL2 Free decay: E(t) ~ t -1 L(t) = 1.2 n’-1t-1 Expectations: n’ ~ k Eddies of different sizes >> l Most of energy in the largest eddy If the largest eddy saturates at d and decays within turnover time: Free decay: E(t) ~ t -2 L(t) = (1.5/k)d n’-1/2 t -3/2 Expectations: T > 1K, n’ ~ k T = 0, n’ - ? Dissipation: -dE/dt = n’(kL)2

4. T = 0 T = 1.6 K phonon emission Quantum d Quasi-classical l= L-1/2 k 4.5 cm l ~ 40 nm 0.03 mm - 3 mm Kolmogorov Kelvin waves (Svistunov PRB 1995) T = 1.6 K T = 0 Bottleneck? (L’vov, Nazarenko, Rudenko PRB 2007) Simulations by Tsubota, Araki, Nemirovskii (PRB 2000)

5. Possible scenarios in 4He at T=0 Kolmogorov cascade Kelvin-wave cascade n’(Kolmogorov) - ? n’(Vinen) ~ k • The nature of the transfer of energy from Kolmogorov to Kelvin cascade is debated: • - Accumulation of energy/vorticity at scale ~ l(L’vov, Nazarenko, Rudenko, PRB2007): • n’(Vinen) / n’(Kolmogorov) ~ (ln(l/a))5 ~ 106 • Reconnections should ease the problem (Kozik-Svistunov, cond-mat 2007): • n’(Vinen) / n’(Kolmogorov) ~ ln(l/a) ~ 15

6. reconnections of vortex bundles reconnections between neighbors in the bundle self – reconnections (vortex ring generation) purely non-linear cascade of Kelvin waves (no reconnections) From Kolmogorov to Kelvin-wave cascade (Kozik & Svistunov, 2007) length scale crossover to QT phonon radiation

7. Random (“Vinen”) vs. structured (“Kolmogorov”) state No correlations between vortices, hence only one length scale, l = L-1/2 All energy in vortex line tension Vinen’s equation: dL/dt = -zkL2 Free decay: E(t) ~ t -1 L(t) = 1.2 n’-1t-1 Expectations: n’ ~ k Eddies of different sizes >> l Most of energy in the largest eddy If the largest eddy saturates at d and decays within turnover time: Free decay: E(t) ~ t -2 L(t) = (1.5/k)d n’-1/2 t -3/2 Expectations: T > 1K, n’ ~ k T = 0, n’ - ? Dissipation: -dE/dt = n’(kL)2

8. Available information Towed grid in 4He (Oregon): Vibrating grid in 3He-B (Lancaster):

9. Experimental challenges:- How to produce turbulence at T < 1K?- How to detect it?

10. E Ions in helium In liquid helium, an injected electron creates a bubble of radius ~ 20 A Vortex rings are nucleated by such ions at T < 1 K; electron stays trapped by vortex (binding energy ~ 50 K) Ring dynamics: E ~ R , v ~ 1/RRings as injected: E0 = 30 eV,R0 = 0.8 mm, v = 11 cm/s Charged rings have large capture diameter ~ 1mm(c.f. typical inter-vortex distance of ~ 30 - 3000mm)

11. Turbulence detection: We developed techniques to measure L by scattering a beam of probe particles: 1. Free ions (T > 0.8 K), trapping diameter ~ 0.1 mm 2. Charged quantized vortex rings (T < 0.8 K), trapping diameter ~1 mm Rotating cryostat was used to calibrate trapping diameter vs. electric field and temperature

12. Quantum phonons d Quasi-classical l= L-1/2 k 4.5 cm 0.03 - 3 mm l ~ 40 nm Kolmogorov Kelvin waves Turbulence production: • We developed techniques to produce either structured or random tangles: • 1. Impulsive spin-down to rest (works at any temperatures) • Energy injected at the largest scale (structured tangle) • 2. Jet of free ions in stationary helium (T > 0.8 K) • Energy injected at the largest scale (structured tangle) • 3. Beam of small vortex rings in stationary helium (T< 0.8 K) • Energy mainly injected on scale << l(random tangle)

13. We can inject rings from the side We can also inject rings from the bottom Experimental Cell The experiment is a cube with sides of length 4.5 cm containing 4He (P = 0.1 bar). 4.5 cm We can create an array of vortices by rotating the cryostat

14. 4.5 cm 1. Random Tangles Produced by Charged Vortex Rings

15. n’ver = 0.17 k n’hor = 0.13 k

16. Tangle decay We probe the decay after a long injection by sending a short pulse a time, t, after stopping injection. Signal applied to injector: Probe pulse 50 s initial injection t

17. Tangle decay t-1 Decay of tangle generated by long pulse Decay of tangle generated by short pulses

18. Tangle Growth & Decay in Centre of Cell We can probe the growth of the tangle by first sending a pulse from the left tip and then use a pulse from the bottom tip to probe the vortex line density in the centre of the cell. The tangle grows and fills the whole cell.L~1/t, agrees well with our other measurements. Maximum line density occurs at about 4 seconds 10 V/cm field 1 s injection 0.3 s probe pulse a time, t, after injection from left

19. Tangle decay: varying temperature For T = 0.08 K – 0.5 K, n’ = (0.15 ± 0.03) k

20. Stopping rotation 2. Structured tangles Impulsive stopping rotation: (from a vortex array to L=0 through 3D turbulence) W ~ 1 rad/s W = 0

21. Horizontal vs. vertical direction Horizontal Vertical

22. Scaling with Angular Velocity one initial revolution

23. Low vs. High Temperature: horizontal

24. High Temperatures: spin-down vs. ion-injection

25. Low vs. High Temperatures

26. “Vinen” (random) tangle Low vs. High Temperature “Kolomogorov” (structured) tangle

27. Summary • We have used charged vortex rings to probe turbulence in superfluid 4He in the T=0 limit. • The decay of a tangle produced by either injected current or impulsive spin-down have been studied. • Random tangles decay as L = t-1. This is consistent with Vinen’s equation with the effective kinematic viscosity of 0.15 k. • Structured tangles decay as L ~ t-3/2which is consistent with a developed Kolmogorov cascade saturated at cell size. The effective kinematic viscosity is 0.003 k. • n‘(random) / n’(Kolmogorov) ~ 50. Bottleneck between the two cascades? However, not as huge an effect as if reconnections were suppressed. • Techniques of great potential. More detailed studies to follow.