Ensemble of Fixed Points in Multi-branch Shell Models of Turbulent Cascades

TL;DR

This paper investigates fixed points in multi-branch shell models of turbulence, revealing a continuum of solutions including the Kolmogorov solution, and uses stochastic methods to analyze energy cascade properties.

Contribution

It introduces a recursive, geometrically governed shell model with a continuum of fixed points, including the Kolmogorov solution, and links stochastic sampling to turbulence intermittency.

Findings

Existence of a continuum of fixed points including Kolmogorov solution

Stochastic characterization of self-similarity and intermittency

Numerical simulations confirm theoretical predictions

Abstract

Stationary solutions of a shell model of turbulence defined on a dyadic tree topology are studied. Each node's amplitude is expressed as the product of amplitude multipliers associated with its ancestors, providing a recursive representation of the cascade process. A geometrical rule governs the tree growth, and we prove the existence of a continuum of fixed points, including the Kolmogorov solution, that sustain a strictly forward energy cascade. Sampling along randomly chosen branches defines a homogeneous Markov chain, enabling a stochastic characterization of extended self-similarity and intermittency through the spectral properties of the associated Feynman-Kac operators. Numerical simulations confirm the theoretical predictions, showing that multi-branch shell models offer a minimal yet physically rich framework for exploring the complexity of nonlinear energy transfer across scales.