SCALING AND INTERMITTENCY IN BURGERS' TURBULENCE

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.