On the decay of Burgers turbulence

This work is devoted to the decay of random solutions of the unforced Burgers equation in one dimension in the limit of vanishing viscosity. The initial velocity is homogeneous and Gaussian with a spectrum proportional to k(n) at small wavenumbers k and falling off quickly at large wavenumbers. In physical space, at sufficiently large distances, there is an 'outer region', where the velocity correlation function preserves exactly its initial form (a power law) when n is not an even integer. When 1 < n < 2 the spectrum, at long times, has three scaling regions: first, a \k\(n) region at very small k with a time-independent constant, stemming from this outer region, in which the initial conditions are essentially frozen; second, a k? region at intermediate wavenumbers, related to a self-similarly evolving 'inner region' in physical space and, finally, the usual k(-2) region, associated with the shocks. The switching from the \k\(2) to the k(2) region occurs around a wavenumber k(s)(t) proportional to t-(1/[2(2-n)]), while the switching from k(2) to k(-2) occurs around k(L)(t) proportional to t(-1/2) (ignoring logarithmic corrections in both instances). When -1 < n < 1 there is no inner region and the long-time evolution of the spectrum is self-similar. When n = 2,4,6,... the outer region disappears altogether and the long-time evolution is again self-similar. For other values of n > 2, the outer region gives only subdominant contributions to the small-k spectrum and the leading-order long-time evolution is also self-similar. The key element in the derivation of the results is an extension of the Kida (1979) log-corrected lit law for the energy decay when n = 2 to the case of arbitrary integer or non-integer n > 1. A systematic derivation is given in which both the leading term and estimates of higher-order corrections can be obtained. It makes use of the Balian-Schaeffer (1989) formula for the distribution in space of correlated particles, which gives the probability of having a given domain free of particles in terms of a cumulant expansion. The particles are, here, the intersections of the initial velocity potential with suitable parabolas. The leading term is the Poisson approximation used in previous work, which ignores correlations. High-resolution numerical simulations are presented which support our findings.