J. Phys. IV France 08 (1998) Pr6-27-Pr6-38
Disordered systems and Burger's turbulenceM. Mézard
Laboratoire de Physique Théorique de l'École Normale Supérieure, 24 rue Lhomond, 75231 Paris cedex 05, France
In order to celebrate the memory of my friend Giovanni, I could not imagine a better topic than one which lies at the interface between the studies of turbulence and tlie statistical mechanics of disordered systems, his two favorite fields in physics. I have recently written a paper on this subject, in the Proceedings of the International Conference of Mathematical Physics 1997, and what I have written here is basically the content of this recent paper. The basic properties of scaling and intermittency in fully developped turbulence can be studied in a lot of details in the case of stirred Burgers turbulence, using the mapping between Burgers' equation and the problem of a directed polymer in a random medium. In the limit of large dimenisions, 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.
© EDP Sciences 1998