Non-unique self-similar blowups in shell models: insights from dynamical systems and machine-learning

TL;DR

This paper investigates the self-similar blowup phenomena in the Sabra shell model of turbulence, combining bifurcation analysis and machine learning to reveal multiple non-universal blowup profiles and deepen understanding of turbulence scaling behaviors.

Contribution

It introduces two novel strategies—bifurcation analysis and machine learning—to analyze and characterize non-unique self-similar blowups in shell models, advancing the understanding of turbulence singularities.

Findings

Identification of homoclinic bifurcations converging to Sabra solutions

Discovery of a continuous family of non-universal blowup profiles

Machine learning reveals multiple blowup exponents and pulse structures

Abstract

Strong numerical hints exist in favor of a universal blowup scenario in the Sabra shell model, a popular cascade model of 3D turbulence, which features complex velocity variables on a geometric progression of scales $\ell_n \propto \lambda ^{-n}$. The blowup is thought to be of self-similar type and characterized by the finite-time convergence towards a universal profile with non-Kolmogorov (anomalous) small-scale scaling $\propto \ell_n^{x}$. Solving the underlying nonlinear eigenvalue problem has however proven challenging, and prior insights mainly used the Dombre-Gilson renormalization scheme, transforming self-similar solutions into solitons propagating over infinite rescaled time horizon. Here, we further characterize Sabra blowups by implementing two strategies targeting the eigenvalue problem. The first involves formal expansion in terms of the bookkeeping parameter $\delta = (1-x)\log \lambda$, and interpretes the self-similar solution as a (degenerate) homoclinic bifurcation. Using standard bifurcation toolkits, we show that the homoclinic bifurcations identified under finite-truncation of the series converge to the observed Sabra solution. The second strategy uses machine-learning optimization to solve directly for the Sabra eigenvalue. It reveals an intricate phase space, with the presence of a continuous family of non-universal blowup profiles, characterized by various number $N$ of pulses and exponents $x_N\ge x$.