SCALING AND INTERMITTENCY IN BURGERS' TURBULENCE

TL;DR

This paper investigates turbulence in the forced Burgers' equation across multiple dimensions by mapping it to directed polymers, deriving an exact velocity difference distribution, and revealing strong intermittency with universal scaling exponents.

Contribution

It introduces a novel Ansatz for the velocity field in high-dimensional Burgers' turbulence, enabling exact calculation of velocity difference distributions and revealing universal scaling laws.

Findings

Moments of velocity differences scale as r^{ta(q)} with ta(q)=1 for q  1.

Velocity field exhibits strong intermittency linked to large-scale singularities.

The velocity field concentrates on a froth-like structure of dimension N-1.

Abstract

We use the mapping between Burgers' equation and the problem of a directed polymer in a random medium in order to study the fully developped turbulence in the $N$ dimensional forced Burgers' equation. The stirring force corresponds to a quenched (spatio temporal) random potential for the polymer. The properties of the inertial regime are deduced from a study of the directed polymer on length scales smaller than the correlation length of the potential. From this study we propose an Ansatz for the velocity field in the large Reynolds number limit of the forced Burgers' equation in $N$ dimensions. This Ansatz allows us to compute exactly the full probability distribution of the velocity difference $u(r)$ between points separated by a distance $r$ much smaller than the correlation length of the forcing. We find that the moments $<u^q(r)>$ scale as $r^{\zeta(q)}$ with $\zeta(q) \equiv 1$ for all $q \geq 1$. This strong `intermittency' is related to the large scale singularities of the velocity field, which is concentrated on a $N-1$ dimensional froth-like structure.