Relevance of instantons in Burgers turbulence

¹ Department of Physics of Complex Systems, Weizmann Institute of Science - Rehovot 76100, Israel
² Theoretische Physik I, Ruhr-Universität Bochum - Universitätsstr. 150, D-44780 Bochum, Germany
³ Department of Mathematics, College of Staten Island, CUNY - Staten Island, NY, USA
⁴ Courant Institute, New York University - 251 Mercer Street, New York, USA

^(a) tobias.grafke@weizmann.ac.il

Received: 15 December 2014
Accepted: 29 January 2015

Abstract

Instanton calculations are performed in the context of stationary Burgers turbulence to estimate the tails of the probability density function (PDF) of velocity gradients. These results are then compared to those obtained from massive direct numerical simulations (DNS) of the randomly forced Burgers equation. The instanton predictions are shown to agree with the DNS in a wide range of regimes, including those that are far from the limiting cases previously considered in the literature. These results settle the controversy of the relevance of the instanton approach for the prediction of the velocity gradient PDF tail exponents. They also demonstrate the usefulness of the instanton formalism in Burgers turbulence, and suggest that this approach may be applicable in other contexts, such as 2D and 3D turbulence in compressible and incompressible flows.