Dynamic multiscaling in stochastically forced Burgers turbulence

TL;DR

This paper investigates dynamic multiscaling in stochastically forced Burgers turbulence, revealing multiple characteristic time scales, non-Gaussian distributions, and power-law tails through theoretical analysis and numerical simulations.

Contribution

It introduces the concept of interval collapse times and analytically derives multiscaling exponents, advancing understanding of turbulence with shocks in one dimension.

Findings

Multiple characteristic time scales exist in Burgers turbulence.

The distribution of collapse times has a power-law tail.

Analytical and numerical results are in good agreement.

Abstract

We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of $\textit{interval collapse times}$ $\tau_{\rm col}$, the time taken for an interval of length $\ell$, demarcated by a pair of Lagrangian tracers, to collapse at a shock. By calculating the dynamic scaling exponent of the order-$p$ moment of $\tau_{\rm col}$, we show that (a) there is $\textit{not one but an infinity of characteristic time scales}$ and (b) the probability distribution function of $\tau_{\rm col}$ is non-Gaussian and has a power-law tail. Our study is based on (a) a theoretical framework that allows us to obtain dynamic-multiscaling exponents analytically, (b) extensive direct numerical simulations, and (c) a careful comparison of the results of (a) and (b). We discuss possible generalizations of our work to dimensions $d >1 $, for the stochastically forced Burgers equation, and to other compressible flows that exhibit turbulence with shocks.