Quasilinear regime and rare-event tails of decaying Burgers turbulence.
We study the decaying Burgers dynamics in d dimensions for random Gaussian initial conditions. We focus on power-law initial energy spectra, such that the system shows a self-similar evolution. This is the case of interest for the "adhesion model" in cosmology and a standard framework for "decaying Burgers turbulence." We briefly describe how the system can be studied through perturbative expansions at early time or large scale (quasilinear regime). Next, we develop a saddle-point method, based on spherical instantons, that allows to obtain the asymptotic probability distributions P(eta_{r}) and P(Theta_{r}) , of the density and velocity increment over spherical cells, reached in the quasilinear regime. Finally, we show how this approach can be extended to take into account the formation of shocks and we derive the rare-event tails of these probability distributions, at any finite time and scale. This also gives the high-mass tail of the mass function of pointlike singularities (shocks in the one dimensional case).
Valageas
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Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, France.