Locality and stability of the cascades of two-dimensional turbulence

TL;DR

This paper analyzes the locality and stability of cascades in two-dimensional turbulence using a mathematical framework, confirming the local nature of both enstrophy and inverse energy cascades and detailing their stability conditions.

Contribution

It provides a rigorous analysis of the locality and stability of two-dimensional turbulence cascades, extending previous hypotheses and clarifying conditions for their stability.

Findings

Both enstrophy and inverse energy cascades are non-perturbatively local.

Inverse energy cascade is unconditionally statistically stable.

Enstrophy cascade stability requires large-scale dissipation and zero downscale energy dissipation.

Abstract

We investigate and clarify the notion of locality as it pertains to the cascades of two-dimensional turbulence. The mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions, first introduced by L'vov and Procaccia. As a point of departure we use a revised version of the system of hypotheses that was proposed by Frisch for three-dimensional turbulence. We show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non-perturbative statistical locality. We also investigate the stability conditions for both cascades. We have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade. For the enstrophy cascade, statistical stability requires large-scale dissipation and a vanishing downscale energy dissipation. A careful discussion of the subtle notion of locality is given at the end of the paper.