1. The Tale of Two Tangles: Dynamics of "Kolmogorov" and "Vinen" turbulences in 4 He near T=0 Paul Walmsley, Steve May, Alexander Levchenko, Andrei Golov (thanks: Henry Hall, JOE VINEN) Euromech 491, Exeter 2007 1.Different types of tangles and their dissipation at T=0 2.Production of random and structured tangles 3.Detection of turbulence by ballistic vortex rings 4.Results for both types of tangles 5.Conclusions

2. In classical turbulence dissipation is via vorticity  and viscosity : In superfluid, turbulence is made of quantized vortices and their tangles of density L Superfluid 4 He has zero molecular viscosity, = 0 Conversion of flow energy into heat is mediated by quantized vortices: ( ’ – “effective kinematic viscosity”) dE/dt = - ’(  L) 2 dE/dt = -  2 Introduction Circulation quantum,  = h/m = 10 -3 cm 2 s -1 Core a 0 ~ 0.1 nm L = 10 – 10 5 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 = -   L 2 Free decay: E(t) ~ t -1 L(t) = 1.2 ’ -1 t -1 Expectations: ’ ~  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/  )  d ’ -1/2 t -3/2 Expectations: T > 1K, ’ ~  T = 0, ’ - ? Dissipation: -dE/dt = ’(  L) 2

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

5. Possible scenarios in 4 He at T=0 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): ’(Vinen) / ’(Kolmogorov) ~ (ln( l/a )) 5 ~ 10 6 - Reconnections should ease the problem (Kozik-Svistunov, cond-mat 2007): ’(Vinen) / ’(Kolmogorov) ~ ln( l/a ) ~ 15 Kolmogorov cascadeKelvin-wave cascade ’(Vinen) ~  ’(Kolmogorov) - ?

6. From Kolmogorov to Kelvin-wave cascade (Kozik & Svistunov, 2007) crossover to QT reconnections of vortex bundles reconnections between neighbors in the bundle self – reconnections (vortex ring generation) purely non-linear cascade of Kelvin waves (no reconnections) length scale 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 = -   L 2 Free decay: E(t) ~ t -1 L(t) = 1.2 ’ -1 t -1 Expectations: ’ ~  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/  )  d ’ -1/2 t -3/2 Expectations: T > 1K, ’ ~  T = 0, ’ - ? Dissipation: -dE/dt = ’(  L) 2

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

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

10. 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/R Rings as injected: E 0 = 30 eV, R 0 = 0.8  m, v = 11 cm/s In liquid helium, an injected electron creates a bubble of radius ~ 20 A Charged rings have large capture diameter ~ 1  m (c.f. typical inter-vortex distance of ~ 30 - 3000  m) E Ions in helium

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  m 2. Charged quantized vortex rings (T < 0.8 K), trapping diameter ~1  m Rotating cryostat was used to calibrate trapping diameter vs. electric field and temperature

12. 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) d phonons k l = L -1/2 Quasi-classical Quantum KolmogorovKelvin waves 0.03 - 3 mm4.5 cm ~ 40 nm

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

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

15. ’ ver = 0.17  ’ hor = 0.13 

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: 50 s initial injection t Probe pulse

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

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. 1 s injection 0.3 s probe pulse a time, t, after injection from left 10 V/cm field Maximum line density occurs at about 4 seconds

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

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

21. Horizontal vs. vertical direction HorizontalVertical

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

27. Summary We have used charged vortex rings to probe turbulence in superfluid 4 He 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 . Structured tangles decay as L ~ t -3/2 which is consistent with a developed Kolmogorov cascade saturated at cell size. The effective kinematic viscosity is 0.003 . ‘(random) / ’(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.