Kelvin-wave turbulence generated by vortex reconnections

Reconnections of quantum vortex filaments create sharp bends which degenerate into propagating Kelvin waves. These waves cascade their energy down-scale and their waveaction up-scale via weakly nonlinear interactions, and this is the main mechanism of turbulence at the scales less than the inter-vortex distance. In case of an idealised forcing concentrated around a single scale k0, the turbulence spectrum exponent has a pure direct cascade form −17/5 at scales k > k0 [2] and a pure inverse cascade form −3 at k < k0 [3]. However, forcing produced by the reconnections contains a broad range of Fourier modes. What scaling should one expect in this case? In this Letter I obtain an answer to this question using the differential model for the Kelvin wave turbulence introduced in [4]. The main result is that the direct cascade scaling dominates, i.e. the reconnection forcing is more or less equivalent to a low-frequency forcing. 1 Differential equation model for Kelvin wave turbulence. Superfluid turbulence, when excited at scales much greater than the mean separation between quantum vortices, behaves similarly to turbulence in classical fluids at such large scales in that it develops a Richardson-like cascade characterised by Kolmogorov spectrum [1]. However, quantum turbulence starts feeling discreteness when the energy cascade reaches down to the length-scales comparable to the mean inter-vortex separation distance. In superfluids near zero temperature, there is no normal component and, therefore, there is no viscid of frictional dissipation in the system. Even though part of the turbulent energy is lost to sound radiation during the vortex reconnection processes, the major part of it is believed to be continuing to cascade to the scales below the inter-vortex separation scale via nonlinear interactions of Kelvin waves [2, 5–9]. Following [4] I will refer to this state characterised by random nonlinearly interacting Kelvin waves as “kelvulence” (i.e. Kelvin turbulence). Kozik and Svistunov [7] used the weak turbulence approach to kelvulence and derived a six-wave kinetic equation (KE) for the spectrum of weakly nonlinear Kelvin waves. Based on KE, they derived a spectrum of