Origin of Non-Gaussian Statistics in Hydrodynamic Turbulence

Abstract

Turbulent flows are notoriously difficult to describe and understand based on first principles. One reason is that turbulence contains highly intermittent bursts of vorticity and strain rate with highly non-Gaussian statistics. Quantitatively, intermittency is manifested in highly elongated tails in the probability density functions of the velocity increments between pairs of points. A long-standing open issue has been to predict the origins of intermittency and non-Gaussian statistics from the Navier-Stokes equations. Here we derive, from the Navier-Stokes equations, a simple nonlinear dynamical system for the Lagrangian evolution of longitudinal and transverse velocity increments. From this system we are able to show that the ubiquitous non-Gaussian tails in turbulence have their origin in the inherent self-amplification of longitudinal velocity increments, and cross amplification of the transverse velocity increments.