Self Similar Properties of Avalanche Statistics in a Simple Turbulent Model

TL;DR

This paper investigates the statistical properties of energy avalanches in a simplified turbulence model, revealing scaling laws and suggesting broader applicability to other avalanche phenomena.

Contribution

It introduces a simplified turbulence model exhibiting avalanche-like energy drops and analyzes their scaling properties, linking durations and sizes through a novel scaling argument.

Findings

Probability distributions of waiting times and avalanche sizes show clear scaling behavior.

Waiting times and avalanche sizes are uncorrelated but their scaling properties are related.

The approach may apply to other systems with avalanche dynamics like seismic events.

Abstract

In this paper, we consider a simplified model of turbulence for large Reynolds numbers driven by a constant power energy input on large scales. In the statistical stationary regime, the behaviour of the kinetic energy is characterised by two well defined phases: a laminar phase where the kinetic energy grows linearly for a (random) time $t_w$ followed by abrupt avalanche-like energy drops of sizes $S$ due to strong intermittent fluctuations of energy dissipation. We study the probability distribution $P[t_w]$ and $P[S]$ which both exhibit a quite well defined scaling behaviour. Although $t_w$ and $S$ are not statistically correlated, we suggest and numerically checked that their scaling properties are related based on a simple, but non trivial, scaling argument. We propose that the same approach can be used for other systems showing avalanche-like behaviour such as amorphous solids and seismic events.