Bounds on the Global Attractor of 2D Incompressible Turbulence in the Palinstrophy-Enstrophy-Energy Space

TL;DR

This paper investigates bounds on the global attractor of 2D incompressible turbulence in the palinstrophy-enstrophy-energy space, revealing that previous analytic bounds significantly overestimate actual values due to non-sharp estimates of certain inner products.

Contribution

The study identifies the lack of sharp estimates for key inner products in the analysis of turbulence attractors and discusses implications for more accurate bounds.

Findings

Analytic bounds overestimate attractor projections compared to simulations.

Expected value of the advection term inner product is zero for isotropic turbulence.

Sobolev inequalities like Ladyzhenskaya and Agmon's are not sharp for these bounds.

Abstract

Analytic bounds on the projection of the global attractor of 2D incompressible turbulence in the palinstrophy--enstrophy plane [Dascaliuc, Foias, and Jolly 2005, 2010] are observed to vastly overestimate the values obtained from numerical simulations. This is due to the lack of a good estimate for the inner product $(\cal{B}(u,u),A^2u)$ of the advection term and the biLaplacian. Sobolev inequalities like Ladyzhenskaya or Agmon's inequalities yield an upper bound that we show is not sharp. In fact, for statistically isotropic turbulence, the expected value of $(\cal{B}(u,u),A^2u)$ is zero. The implications for estimates on the behaviour of the global attractor are discussed.