Ensemble of Fixed Points in Multi-branch Shell Models of Turbulent Cascades
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.
Keywords
Cite
@article{arxiv.2601.04788,
title = {Ensemble of Fixed Points in Multi-branch Shell Models of Turbulent Cascades},
author = {Flavio Tuteri and Sergio Chibbaro and Alexandros Alexakis},
journal= {arXiv preprint arXiv:2601.04788},
year = {2026}
}