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

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)∼∣k∣ⁿ. 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 k⁻² 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<n the leading behavior is self-similar, independent of n and with universal behavior k² at small wave number. When 1<n<2, the spectrum has three scaling regions: first, a ∣k∣ⁿ region at very small k’s with a time-independent constant; second, a k² region at intermediate wave numbers; finally, the usual k⁻² region. In the remaining interval n<−3 the small-k cutoff dominates and n also plays no role. We find also (numerically) the subleading term ∼k² in the evolution of the spectrum in the interval −3<n<1. High-resolution numerical simulations have been performed confirming both scaling predictions and analytical asymptotic theory.