Bottleneck crossover between classical and quantum superfluid turbulence

Transcription

1 between classical and quantum superfluid turbulence Mathematics Institute University of Warwick Joint work with V.Lvov and O.Rudenko (Weizmann). Inertial-Range Dynamics and Mixing Workshop, INI-Cambridge, Sept

2 Outline 1 Turbulence at T = 0 Hydro and wave turbulence Pre-bottleneck picture 2 3

3 Superfluid Turbulence at T = 0. Turbulence at T = 0 Hydro and wave turbulence Pre-bottleneck picture Rotating superfluid turbulence. De Graaf et al. (2007). Tangle of vortex lines with quantized circulation κ = h/m and cores of atomic size. Biot-Savart model: ṙ = κ ds (r s) 4π. r s 3 Cutoff at r s = a (core radius). Implicitly, such a formulation introduces a particular shape of the vortex core profile.

4 Processes at different scales. Turbulence at T = 0 Hydro and wave turbulence Pre-bottleneck picture Rotating superfluid turbulence. De Graaf et al. (2007). Classical Turbulence at scales > intervortex separation l, Kelvin waves at scales < l. Dissipation by phonon radiation at very small scales.

5 Pre-bottleneck picture. Turbulence at T = 0 Hydro and wave turbulence Pre-bottleneck picture At scales > l : E k = C hd ǫ 2/3 k 5/3 At l : reconnections. At scales < l : C kw Λ ( κ 7 ǫ/l 8) 1/5 k 7/5 (Kozik&Svistunov 04). (K41 spectrum). Effective viscosity measured by turbulence decay rate (Stalp et al 00). Assume that K41 extends down to l. Vicinity of l contains most vorticity (hence vortex line density). ν ǫl4 κ 2 κ

6 Bottleneck scenario Energy density k -5/3 k 2 k -7/5 k Lvov, Nazarenko & Rudenko (2007). Sharp transition from HD to KW at kl 1. HD part consists of K41 and thermalized ranges. kl Impossible to match K41 and KS04 at l. Wave turbulence is less efficient than strong hydro turbulence and cannot cope with Kolmogorov cascade, bottleneck. Leith 67 model: ǫ(k) = 1 8 k 11 d(e E k /k 2 ) k. dk "Warm cascade" (Connaughton&SN 04) E k = k 2 [(24/11)ǫk 11/2 + T 3/2 ] 2/3. Effective viscosity ν = Λ 5 κ.

7 / Turbulence decay measurements (T): Quasi-classical regime: spin-down ion jet towed grid fit to KS2008 Quantum regime: CVRs T (K) Manchester group. Walmsley et al 07

8 Gradual crossover range. Self-induced velocity ṽelocity induced by neighbor lines at scale l v Λ 1/2 l. Thus, KW can propagate already at this scale. Assuming sharp crossover at l v we get weaker bottleneck: ν = Λκ. In reality, the eddy-wave transition is gradual in the range from l v to l. We can expect Λκ < ν < Λ 5 κ. Differential Approximation Model for coupled HD and KW turbulence to describe the gradual crossover Leith-like for HD, Hasselmann-like for KW (Lvov, SN & Rudenko 08).

9 Gradual bottleneck scenario. Smooth transition from HD to KW (from Lvov, SN &Rudenko 08). E hd /E kw, ǫ hd /ǫ and ǫ kw /ǫ. Note range where E hd E kw but ǫ hd ǫ kw (from Lvov, SN &Rudenko 08).

10 Waves Bottleneck scenario. Waves Lvov, Nazarenko & Rudenko (2007). Flow Flow Kozik &Svistunov proposal or (3 sharp-transition) reconnection dominated ranges between the HD and KW cascades. Even more radical, - no KW at all. Reconnection/fragmentation cascade down to the phonon radiation scales (may be for non-polarized turbulence, e.g. counterflow?) K41 eddies implies strong polarization of vortex lines and their clustering into bundles. Reconnection inside the bundles of co-oriented lines is suppressed. Reconnection occurs in-between of the bundles: mechanism for HD KW transfers at fixed scale, but not for the cross-scale flux.

11 Conclusions The bottleneck theory predicts reduction of ν by a factor in the range from Λ to λ 5. There is a simple argument for Λ 11/3. The gradual crossover theory predicts existence of a range where the spectrum is wave dominated but the flux is eddy dominated. Passive advection of vortex lines at these scales like suggested by Roche and Barenghi (interpreting Lancaster data)? The bottleneck picture relies on assumption of large K41 range, - may be correct for turbulence produced by classical means (rotation, towed grid, but not counterflow). Crucial next step: accurate numerical modeling of turbulence with a wide scale range including those > and < than l.