Pdf's of Derivatives and Increments for Decaying Burgers Turbulence

TL;DR

This paper demonstrates that the power-law tails in the probability density functions of velocity gradients and derivatives in Burgers turbulence are present in both forced and unforced cases, extending the theory to second derivatives and time derivatives.

Contribution

The study extends the understanding of turbulence statistics by showing the universality of power-law tails in derivatives' PDFs for unforced Burgers turbulence using a Lagrangian approach.

Findings

Power-law tail with exponent -7/2 in velocity gradient PDFs.

Power-law tail with exponent -2 in second derivatives.

Similar asymptotic forms for space and time derivatives' PDFs.

Abstract

A Lagrangian method is used to show that the power-law with a -7/2 exponent in the negative tail of the pdf of the velocity gradient and of velocity increments, predicted by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) 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".