Disordered systems and Burgers’ turbulence

The problem of fully developped turbulence is to characterize the statistical properties of the velocity field of a stirred fluid described by Navier stokes equations. The simplest scaling approach, due to Kolmogorov in 1941, gives a reasonable starting point, but it must be corrected due to the failure of naive scaling giving ‘intermittency’ corrections which are presumably associated with the existence of large scale structures. These scaling and intermittency properties can be studied analytically for the case of stirred Burgers turbulence, a kind of simplified version of Navier Stokes equations. 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 d dimensional forced Burgers’ equation, in the limit of large dimensions. The stirring force corresponds to a quenched (spatio temporal) random potential for the polymer. 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. Preprint LPTENS 97/66.