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, e HD(k) and e KW(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 ¿=ln¿(l/a) (l is the mean intervortex separation and a is the vortex core radius) between 10 and 15 the total energy flux e=e HD(k)+e KW(k) and the total energy spectrum E(k)=EHD(k)+EKW(k) are dominated by the HD motions for k2/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