The interscale behaviour of uncertainty in three-dimensional Navier-Stokes turbulence

TL;DR

This paper investigates the scale-dependent behavior of uncertainty in 3D turbulence, revealing a self-similar inverse cascade mechanism that influences the growth and distribution of uncertainty across scales.

Contribution

It introduces a scale-by-scale uncertainty energy budget equation and identifies a self-similar equilibrium cascade of decorrelation in turbulence.

Findings

Identification of a self-similar inverse cascade of uncertainty

Demonstration of power law growth of uncertainty scales and energy

Revealing the role of compressions in stochasticity and chaoticity

Abstract

We derive the scale-by-scale uncertainty energy budget equation and demonstrate theoretically and computationally the presence of a self-similar equilibrium cascade of decorrelation in an inertial range of scales during the time range of power law growth of uncertainty in statistically stationary homogeneous turbulence. This cascade is predominantly inverse and driven by compressions of the reference field's relative deformation tensor and their aligments with the uncertainty velocity field. Three other subdominant cascade mechanisms are also present, two of which are forward and also dominated by compressions and one of which, the weakest and the only non-linear one of the four, is inverse. The uncertainty production and dissipation scalings which may follow from the self-similar equilibrium cascade of decorrelation lead to power law growths of the uncertainty integral scale and the average uncertainty energy which are also investigated. Compressions are not only key to chaoticity, as previously shown, but also to stochasticity.