Instanton theory of Burgers shocks and intermittency

Onset of intermittency in stochastic Burgers hydrodynamics

We study the onset of intermittency in stochastic Burgers hydrodynamics, as characterized by the statistical behavior of negative velocity gradient fluctuations. The analysis is based on the response functional formalism, where specific velocity configurations-the viscous instantons-are assumed to play a dominant role in modeling the left tails of velocity gradient probability distribution functions. We find, as expected on general grounds, that the field-theoretical approach becomes meaningful in practice only if the effects of fluctuations around instantons are taken into account. Working with a systematic cumulant expansion, it turns out that the integration of fluctuations yields, in leading perturbative order, to an effective description of the Burgers stochastic dynamics given by the renormalization of its associated heat kernel propagator and the external force-force correlation function.

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).