One-dimensional hydrodynamic model generating a turbulent cascade

TL;DR

This paper introduces a one-dimensional PDE model that mimics turbulent cascades, conserving enstrophy and exhibiting key turbulence features like energy spectra and intermittency, providing insights into turbulence dynamics.

Contribution

It presents a minimal mathematical model that reproduces turbulence-like cascades and properties, advancing understanding of turbulence phenomena through a simplified framework.

Findings

The model exhibits enstrophy cascade under forcing and viscosity.

It shows a broad energy spectrum with deviations from simple dimensional analysis.

The system displays intermittency and self-similarity characteristic of turbulence.

Abstract

As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analogue (enstrophy) in the inviscid case. With a large-scale forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency and self-similarity in the dynamical system structure.