Differential approximation for Kelvin wave turbulence

I present a nonlinear differential equation model (DAM) for the spectrum of Kelvin waves on a thin vortex filament. This model preserves the original scaling of the six-wave kinetic equation, its direct and inverse cascade solutions, as well as the thermodynamic equilibrium spectra. Further, I extend DAM to include the effect of sound radiation by Kelvin waves. I show that, because of the phonon radiation, the turbulence spectrum ends at a maximum frequency of ω* ∼ (ε3 c s 20 /κ16)1/13, where ε is the total energy injection rate, c s is the speed of sound, and κ is the quantum of circulation.