Probability distribution functions of derivatives and increments for decaying Burgers turbulence

A Lagrangian method is used to show that the power law with a - 7/2 exponent in the negative tail of the probability distribution function (PDF) of the velocity gradient and of velocity increments, predicted by E et al. [Phys. Rev. Lett. 78, 1903 (1997)] for forced Burgers turbulence, is also present in the unforced case. The theory is extended to the second-order space derivative whose PDF has power-law tails with exponent -2 at both large positive and negative values and to the time derivatives. PDF's of space and time derivatives have the same (asymptotic) functional forms. This is interpreted in terms of a random Taylor hypothesis.