Quasi-linear regime and rare-event tails of decaying Burgers turbulence

Details Quasi-linear regime and rare-event tails of decaying Burgers turbulence Quasi-linear regime and rare-event tails of decaying Burgers turbulence

2009-05-12

arxiv.org/abs/0905.1910v1

Phys. Rev. E 80, 016305 (2009)

Physics

Condensed Matter

Statistical Mechanics

32 pages

Scientific paper

10.1103/PhysRevE.80.016305

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 (quasi-linear regime). Next, we develop a saddle-point method, based on spherical instantons, that allows to obtain the asymptotic probability distributions $\cP(\eta_r)$ and $\cP(\ctheta_r)$, of the density and velocity increment over spherical cells, reached in the quasi-linear 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 point-like singularities (shocks in the one dimensional case).

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