Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers Equation

TL;DR

This paper explores how fractal Fourier mode decimation affects intermittency and energy transfer in the stochastic Burgers equation, revealing that minimal mode reduction can significantly alter turbulence characteristics.

Contribution

It provides the first detailed analysis of intermittency suppression in the Burgers equation due to fractal Fourier mode decimation, highlighting the role of mode correlations in shock formation.

Findings

Intermittency diminishes as fractal dimension D drops below 1.

A transition from intermittent to non-intermittent behavior occurs near D=1.

Strong localized shocks depend on complex correlations among Fourier modes.

Abstract

We present theoretical and numerical results for the one-dimensional stochastically forced Burgers equation decimated on a fractal Fourier set of dimension $D$. We investigate the robustness of the energy transfer mechanism and of the small-scale statistical fluctuations by changing $D$. We find that a very small percentage of mode-reduction ($D \lesssim 1$) is enough to destroy most of the characteristics of the original non-decimated equation. In particular, we observe a suppression of intermittent fluctuations for $D <1$ and a quasi-singular transition from the fully intermittent ($D=1$) to the non-intermittent case for $D \lesssim 1$. Our results indicate that the existence of strong localized structures (shocks) in the one-dimensional Burgers equation is the result of highly entangled correlations amongst all Fourier modes.