A one dimensional analysis of turbulence and its intermittence for the d-dimensional stochastic Burgers equation

TL;DR

This paper analyzes the stochastic Burgers equation in one dimension, revealing how the geometry of certain classical mechanical structures causes rapid, stochastic turbulence intermittence.

Contribution

It introduces a reduced one-dimensional action function to study turbulence and its intermittence in the stochastic Burgers equation, linking geometric structures to turbulent behavior.

Findings

Geometry of caustic and Maxwell set can change rapidly, causing turbulence

Turbulence intermittence is demonstrated through recurrence of two processes

Analysis connects classical mechanics structures to stochastic turbulence

Abstract

The inviscid limit of the stochastic Burgers equation is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one dimensional) action function. We demonstrate that the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature. The intermittence of this turbulence is demonstrated in terms of the recurrence of two processes.