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

TL;DR

This paper analyzes the decaying Burgers turbulence in multiple dimensions, developing a saddle-point method to derive the asymptotic probability distributions of density and velocity increments, including rare-event tails and shock formation effects.

Contribution

It introduces a saddle-point approach based on spherical instantons to compute probability distributions and their tails in decaying Burgers turbulence with self-similar evolution.

Findings

Derived asymptotic probability distributions for density and velocity increments.

Extended the method to include shock formation and rare-event tail analysis.

Provided high-mass tail estimates for shock singularities.

Abstract

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).