| Summary: | We present a decay scenario of superfluid turbulence (ST)—structured or non-structured tangle of quantized vortex lines—in the fundamental limit of very low temperatures, when the friction with normal component is negligible. At length scales smaller than the typical interline separation, Kelvin waves (kelvons)—the distortion waves on quantized vortex lines—play a key part in relaxation. We develop a kinetic theory of kelvons, which allows us to prove the existence of a cascade of Kelvin waves and find its spectrum. Being the only dissipative channel of vortex dynamics in this regime, radiation of sound by kelvons is responsible for the short-wavelength cutoff of the cascade. We derive a general Hamiltonian of vortex-phonon interaction starting with the hydrodynamic action. On the basis of this formalism, we calculate the rate of sound radiation by ST and estimate the value of the short-wavelength cutoff. Incidentally, the vortex-phonon interaction theory allows us to resolve apparent paradoxes of the Kosterlitz-Thouless theory of U(1) phase transitions in two dimensions. As long as vorticity quantization remains irrelevant for the long-wave physics, superfluid turbulence supports a regime macroscopically identical to the Kolmogorov cascade of a normal liquid. We analyze the transformation of the Kolmogorov cascade into the Kelvin-wave cascade, revealing a chain of three distinct intermediate cascades, supported by local-induction motion of the vortex lines, and distinguished by specific reconnection mechanisms. The most prominent qualitative feature predicted is unavoidable production of vortex rings of the size of the order of inter-vortex distance.
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