Disordered systems and Burgers' turbulence

TL;DR

This paper explores the scaling and intermittency in forced Burgers turbulence, using a mapping to directed polymers in random media and replica symmetry breaking to analyze velocity differences and shock wave regions.

Contribution

It introduces a novel approach linking Burgers turbulence to directed polymer models, providing a detailed probability distribution of velocity differences and insights into intermittency and shock phenomena.

Findings

Strong intermittency linked to shock waves and metastable states.

Replica symmetry breaking yields the full distribution of velocity differences.

Connections to renormalization group flow in disordered systems.

Abstract

Talk presented at the International Conference on Mathematical Physics (Brisbane 1997). This is an introduction to recent work on the scaling and intermittency in forced Burgers turbulence. The mapping between Burgers' equation and the problem of a directed polymer in a random medium is used in order to study the fully developped turbulence in the limit of large dimensions. The stirring force corresponds to a quenched (spatio temporal) random potential for the polymer, correlated on large distances. A replica symmetry breaking solution of the polymer problem provides 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. This exhibits a very strong intermittency which is related to regions of shock waves, in the fluid, and to the existence of metastable states in the directed polymer problem. We also mention some recent computations on the finite dimensional problem, based on various analytical approaches (instantons, operator product expansion, mapping to directed polymers), as well as a conjecture on the relevance of Burgers equation (with the length scale playing the role of time) for the description of the functional renormalisation group flow for the effective pinning potential of a manifold pinned by impurities.