Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Flavio Tuteri, Sergio Chibbaro, Alexandros Alexakis

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.

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.
Paper Structure (11 sections, 1 theorem, 63 equations, 6 figures)

This paper contains 11 sections, 1 theorem, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Foundations
  3. System definition
  4. Problem statement
  5. Geometrical interpretation
  6. Sampling
  7. Multiplicative random walk
  8. Ergodicity and intermittency
  9. Simulations
  10. Conclusions
  11. Inductive construction

Key Result

Proposition 2.3.1

Given $\lambda>1$, for any $\eta\in(0,1)$ and any $\xi>\xi_0(\lambda,\eta)$, where there exists a continuum of admissible iterations of the recurrence relation (recurrence) that satisfy both the boundedness condition in $I$ and the flux-positivity constraints (half_spaces).

Figures (6)

  • Figure 1: Schematic representation of the $2$-adic tree with one descent path highlighted.
  • Figure 2: Schematic of the nonlinear interactions affecting the red-highlighted node, showing contributions from first and second neighbors in the GOY-like model.
  • Figure 3: Simulation of $3000$ steps of the Markov chain. The top-left panel shows the evolution of the amplitude multiplier, demonstrating its confinement within the prescribed region. The top-right panel displays the trajectory of the flux multipliers, bounded between $\epsilon$ and $1-\epsilon$. The bottom panels illustrate the remaining geometric functions relevant to the construction. In particular, the right inset shows a zoomed portion for $\lvert\phi\rvert$, highlighting the absence of patterns associated with extended self-similarity.
  • Figure 4: Probability distributions of the multipliers and geometric functions defined in the sampling process. Results are obtained from $10^7$ samples, showing collapse for sufficiently large $l$. In the top-right panel, a uniform distribution is observed for the flux multipliers; the dashed lines denote a $10\%$ deviation from uniform probability over the support.
  • Figure 5: Probability distributions of velocity and energy flux, rescaled by their maxima for the sake of visualization. The scaling factors are shown below.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 2.3.1
  • proof