Differential approximation for Kelvin-wave turbulence
نویسنده
چکیده
I present a nonlinear differential equation model 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. 1 Kelvulence: cascades and spectra. Kelvin waves propagating on a thin vortex filament were proposed by Svistunov to be a vehicle for the turbulent cascades in superfluids near zero temperature [1]. Presently it is a widely accepted view well supported by the theory and numerical simulations, see e.g. [2, 5–8]. I will reffer to the state characterised by random nonlinearly interacting Kelvin waves as “Kelvulence” (i.e. Kelvin turbulence). Recently, Kozik and Svistunov [6] 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 waveaction that corresponds to the constant Kolmogorov-like cascade of energy from small to large wavenumbers, nk ∼ k . (1) Because the number of waves in the leading resonant process is even (i.e. 6), KE conserves not only the total energy but also the total waveaction of the system. The systems with two positive conserved quantities are known in turbulence to possess a dual cascade behavior. For the Kelvin waves, besides the direct energy cascade there also exists an inverse cascade of waveaction, the spectrum for which was recently found by Lebedev [9], nk ∼ k . (2) Interestingly, such −3 spectrum was suggested before by Vinen based on a totally different argument involving the energy feeding via vortex-line self-crossings [2]. It was recently reported to be observed in numerics by Vinen, Tsubota and Mitani [7], where it was argued that the −3 exponent arises when
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