# Renormalized analytic solution for the enstrophy cascade in two-dimensional quantum turbulence

## TL;DR

This paper develops a renormalized analytical model for the enstrophy cascade in two-dimensional quantum turbulence, revealing a $k^{-3}$ energy spectrum and connecting turbulence dynamics with phase-ordering in superfluids.

## Contribution

It introduces a Fokker-Planck based renormalization approach to describe the enstrophy cascade in 2D quantum turbulence, providing new insights into vortex dynamics and energy spectra.

## Key findings

- The energy spectrum follows a $k^{-3}$ power law.
- Cascade dynamics reach equilibrium after about four eddy turnover times.
- The turbulence cascade is linked to phase-ordering in superfluids.

## Abstract

The forward enstrophy cascade in two-dimensional quantum turbulence in a superfluid film connected to a thermal bath is investigated using a Fokker-Planck equation based on Kosterlitz-Thouless renormalization. The steady-state cascade is formed by injecting vortex pairs of large initial separation at a constant rate. They diffuse with a constant flux to smaller scales, finally annihilating when reaching the core size. The energy spectrum varies as $k^{-3}$, similar to the spectrum known for 2D classical-fluid enstrophy cascades. The dynamics of the cascade can also be studied, and for the case of a sharply peaked initial vortex-pair distribution, it takes about four eddy turnover times for the system to evolve to the decaying $k^{-3}$ cascade, in agreement with recent computer simulations. These insights into the nature of the cascade also allow a better understanding of the phase-ordering process of temperature-quenched 2D superfluids, where the decay of the vorticity is found to proceed via the turbulent cascade. This connection with turbulence may be a fundamental characteristic of phase-ordering in general.