Scaling and intermittency in Burgers turbulence

We use the mapping between the Burgers equation and the problem of a directed polymer in a random medium in order to study the fully developed turbulence in the N-dimensional forced Burgers equation. The stirring force corresponds to a quenched (spatiotemporal) 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, which is not the regime usually considered in the case of polymers. 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, which should become exact in the limit N→. 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 uq(r) scale as rζ(q) with ζ(q)1 for all q1 [in particular, the q=3 moment agrees with Kolmogorov's scaling ζ(3)=1]. This strong ''intermittency'' is related to the large-scale singularities of the velocity field, which is concentrated on an (N-1)-dimensional frothlike structure, which is in turn related to the one-step replica-symmetry-broken nature of the associated disordered problem. We also discuss the similarities and differences between Burgers turbulence and hydrodynamical turbulence and we comment on the anomalous tracer fluctuations in a Burgers turbulent field. Since this replica approach is rather unusual in turbulence problems, we provide all the necessary details of the method.