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 prop ortional to k(n) at small wavenumbers k and falling off quickly at lar ge wavenumbers. In physical space, at sufficiently large distances, th ere is an 'outer region', where the velocity correlation function pres erves exactly its initial form (a power law) when n is not an even int eger. When 1 < n < 2 the spectrum, at long times, has three scaling re gions: first, a \k\(n) region at very small k with a time-independent constant, stemming from this outer region, in which the initial condit ions are essentially frozen; second, a k? region at intermediate waven umbers, related to a self-similarly evolving 'inner region' in physica l space and, finally, the usual k(-2) region, associated with the shoc ks. The switching from the \k\(2) to the k(2) region occurs around a w avenumber 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) (igno ring logarithmic corrections in both instances). When -1 < n < 1 there is no inner region and the long-time evolution of the spectrum is sel f-similar. When n = 2,4,6,... the outer region disappears altogether a nd the long-time evolution is again self-similar. For other values of n > 2, the outer region gives only subdominant contributions to the sm all-k spectrum and the leading-order long-time evolution is also self- similar. The key element in the derivation of the results is an extens ion 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 system atic derivation is given in which both the leading term and estimates of higher-order corrections can be obtained. It makes use of the Balia n-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 pa rabolas. The leading term is the Poisson approximation used in previou s work, which ignores correlations. High-resolution numerical simulati ons are presented which support our findings.