Level Crossing Analysis of Burgers Equation in 1+1 Dimensions

TL;DR

This paper analyzes the frequency of level crossings in the Burgers equation to measure turbulence fluctuations, especially before singularity formation, providing insights into the velocity field behavior.

Contribution

It introduces the level crossing frequency as a new measure for turbulence fluctuations in Burgers equation, especially in the inviscid limit.

Findings

Derived the average frequency of positive slope crossings in Burgers turbulence.

Showed that the crossing frequency effectively characterizes velocity fluctuations.

Analyzed the behavior of crossings before singularity formation.

Abstract

We investigate the average frequency of positive slope $\nu_{\alpha}^{+}$, crossing the velocity field $u(x)- \bar u = \alpha$ in the Burgers equation. The level crossing analysis in the inviscid limit and total number of positive crossing of velocity field before creation of singularities are given. The main goal of this paper is to show that this quantity, $\nu_{\alpha}^{+}$, is a good measure for the fluctuations of velocity fields in the Burgers turbulence.