The global picture of self-similar and not self-similar decay in Burgers Turbulence

Abstract

This paper continue earlier investigations on the decay of Burgers turbulence in one dimension from Gaussian random initial conditions of the power-law spectral type $E_0(k)\sim|k|^n$. 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 wavenumbers, we always have $k^{-2}$ region, associated to the shocks. When $n$ is less than one, 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 of the integral scale. If $n$ is larger than two, the spectrum tends for long times to a universal scaling form independent of the initial conditions, with universal behavior $k^2$ at small wavenumbers. In the interval $2<n$ the leading behaviour is self-similar, independent of $n$ and with universal behavior $k^2$ at small wavenumber. When $1<n<2$, the spectrum has three scaling regions : first, a $|k|^n$ region at very small $k$\ms1 with a time-independent constant, second, a $k^2$ region at intermediate wavenumbers, 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 $\sim k^2$ 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.