Vortex turbulence in linear Schrödinger wave mechanics

Abstract

Quantum turbulence that exhibits vortex creation, annihilation and interactions is demonstrated as an exact solution of the time-dependent, free-particle Schrödinger equation evolved from a smooth random-phased initial condition. Relaxed quantum turbulence in 2D and 3D exhibits universal scaling in the steady-state energy spectrum as k−1 in small scales. Due to the lack of dissipation, no evidence of the Kolmogorov-type forward energy cascade in 3D or the inverse energy cascade in 2D is found, but the rotational and potential flow components do approach equi-partition in the scaling regime. In addition, the 3D vortex line–line correlation exhibits universal behaviour, scaled as Δr−2, where Δr is the separation between any two vortex line elements, in fully developed turbulence. We also show that the quantum vortex is not frozen to the matter, nor is the vortex motion induced by other vortices via Biot–Savart's law. Thus, the quantum vortex is actually a nonlinear wave, propagating at a speed very different from a classical vortex.