Note on Forced Burgers Turbulence

Abstract

A putative powerlaw range of the probability density of velocity gradient in high-Reynolds-number forced Burgers turbulence is studied. In the absence of information about shock locations, elementary conservation and stationarity relations imply that the exponent $-\alpha$ in this range satisfies $\alpha\ge3$, if dissipation within the powerlaw range is due to isolated shocks. A generalized model of shock birth and growth implies $\alpha=7/2$ if initial data and forcing are spatially homogeneous and obey Gaussian statistics. Arbitrary values $\alpha\ge3$ can be realized by suitably constructed homogeneous, non-Gaussian initial data and forcing.