Gradual Eddy-Wave Crossover in Superfluid Turbulence

We revise the theory of superfluid turbulence near the absolute zero of temperature and suggest a differential approximation model for the energy fluxes in the k-space, εHD(k) and εKW(k), carried, respectively, by the collective hydrodynamic (HD) motions of quantized vortex lines and by their individual uncorrelated motions known as Kelvin waves (KW). The model predicts energy spectra of the HD and the KW components of the system, ℰHD(k) and ℰKW(k), which experience a smooth crossover between different regimes of motion over a finite range of scales. For an experimentally relevant range of Λ≡ln (ℓ/a) (ℓ is the mean intervortex separation and a is the vortex core radius) between 10 and 15 the total energy flux ε=εHD(k)+εKW(k) and the total energy spectrum ℰ(k)=ℰHD(k)+ℰKW(k) are dominated by the HD motions for k<2/ℓ. In this region ℰ(k) follows the HD spectrum with constant energy flux ε≃εHD=const.: ℰ(k)∝k−5/3 for smaller k and tends to equipartition of the HD energy ℰ(k)∝k2 for larger k. This bottleneck accumulation of the energy spectrum is milder than the one predicted before in (L’vov et al. in Phys. Rev. B 76:024520, 2007) based on a model with sharp HD-KW transition. For Λ=15, it results in a prediction for the effective viscosity ν ′≃0.004κ (κ is the circulation quantum) which is in a reasonable agreement with its experimental value in 4He low-temperature experiment ≈0.003κ (Walmsley et al. in Phys. Rev. Lett. 99:265302, 2007). For k>2/ℓ, the energy spectrum is dominated by the KW component: almost flux-less KW component close to the thermodynamic equilibrium, ℰ≈ℰKW≈const at smaller k and the KW cascade spectrum ℰ(k)→ℰKW(k)∝k−7/5 at larger k.