Global picture of self-similar and non-self-similar decay in Burgers turbulence

Abstract

This paper continues earlier investigations of the decay of Burgers turbulence in one dimension from Gaussian random initial conditions of the power-law spectral type E0(k)an. Depending on the power n, different characteristic regions are distinguished. The main focus of this paper is to delineate the regions in wave number k and time t in which self-similarity can (and cannot) be observed, taking into account small-k and large-k cutoffs. The evolution of the spectrum can be inferred using physical arguments describing the competition between the initial spectrum and the new frequencies generated by the dynamics. For large wave numbers, we always have a ka2 region, associated with the shocks. When n is less than 1, the large-scale part of the spectrum is preserved in time and the global evolution is self-similar, so that scaling arguments perfectly predict the behavior in time of the energy and integral scale. If n is larger than 2, the spectrum tends for long times to a universal scaling form independent of the initial conditions, with universal behavior k2 at small wave numbers. In the interval 2