Statistics of decaying Burgers turbulence

Freely decaying Burgers turbulence at low and moderate Reynolds number R is studied by mapping closure and by numerical simulations. Good agreement is found for the single-point probability distribution function (pdf) of velocity gradient ξ, for wave number spectra, and for various integral statistical quantities. The pdf of ξ exhibits a nearly exponential tail at large negative ξ, and a cusp with sharp cutoff at a positive value of ξ. The pdf of fluid density has a power-law tail, so that high-order moments of density do not exist. Asymptotic analysis of the mapping closure as R→∞ yields an inertial-range wave number spectrum of kinetic energy with the form E(k)∝k-2. The pdf cusp becomes infinite as R→∞ and gives rise to a flatness <ξ4>/<ξ2>2∝R in the limit