Transition to Chaos in a Shell Model of Turbulence

TL;DR

This paper investigates how a shell model of turbulence transitions from stable fixed points to chaos as a control parameter varies, revealing bifurcations, limit cycles, and strange attractors consistent with turbulence phenomena.

Contribution

It demonstrates the transition to chaos in a turbulence shell model while preserving Navier-Stokes symmetries, using bifurcation analysis and bi-orthogonal decomposition.

Findings

Stable fixed point at low epsilon with Kolmogorov scaling

Hopf bifurcation leading to limit cycle at epsilon=0.3843

Onset of chaos with intermittent dynamics and strange attractors for epsilon > 0.3953

Abstract

We study a shell model for the energy cascade in three dimensional turbulence at varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. When a control parameter $\epsilon$ related to the strength of backward energy transfer is enough small, the dynamical system has a stable fixed point corresponding to the Kolmogorov scaling. This point becomes unstable at $\epsilon=0.3843...$ where a stable limit cycle appears via a Hopf bifurcation. By using the bi-orthogonal decomposition, the transition to chaos is shown to follow the Ruelle-Takens scenario. For $\epsilon > 0.3953..$ the dynamical evolution is intermittent with a positive Lyapunov exponent. In this regime, there exists a strange attractor which remains close to the Kolmogorov (now unstable) fixed point, and a local scaling invariance which can be described via a intermittent one-dimensional map.