Large negative velocity gradients in Burgers turbulence

Abstract

We consider 1D Burgers equation driven by large-scale white-in-time random force. The tails of the velocity gradients probability distribution function (PDF) are analyzed by saddle-point approximation in the path integral describing the velocity statistics. The structure of the saddle-point (instanton), that is velocity field configuration realizing the maximum of probability, is studied numerically in details. The numerical results allow us to find analytical solution for the long-time part of the instanton. Its careful analysis confirms the result of [Phys. Rev. Lett. 78 (8) 1452 (1997) [chao-dyn/9609005]] based on short-time estimations that the left tail of PDF has the form ln P(u_x) \propto -|u_x|^(3/2).