# Vortex Turbulence in Linear Schroedinger Wave Mechanics

## TL;DR

This paper demonstrates quantum turbulence with vortex dynamics as exact solutions of the Schrödinger equation, revealing universal scaling laws and distinct vortex behaviors in 2D and 3D without classical cascade phenomena.

## Contribution

It introduces a novel exact solution framework for quantum turbulence in Schrödinger wave mechanics, highlighting universal scaling and unique vortex properties.

## Key findings

- Quantum turbulence exhibits vortex creation, annihilation, and interactions.
- Universal energy spectrum scaling as k^-1 in steady state.
- Quantum vortices behave as nonlinear waves, not classical vortices.

## Abstract

Quantum turbulence that exhibits vortex creation, annihilation and interactions is demonstrated as an exact solution of the time-dependent, free-particle Schr\"odinger 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 \Deltar^-2, where \Deltar 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.