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

Pedram Emami and John C. Bowman Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

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 $(\mathcal{B}({\bm{u}},{\bm{u}}),A^{2}{\bm{u}})$ 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 $(\mathcal{B}({\bm{u}},{\bm{u}}),A^{2}{\bm{u}})$ is zero. The implications for estimates on the behaviour of the global attractor are discussed.

Key words and phrases:

palinstrophy, enstrophy, energy, global attractor, two-dimensional isotropic incompressible turbulence, hypoviscosity

1. Introduction

One of the most recent approaches in studying turbulence incorporates functional analysis tools to study turbulence on a solid mathematical basis. The Navier–Stokes equation is the generally accepted governing partial differential equation of turbulence. Unlike empirical or heuristic formulations, it provides a solid foundation for studying important intrinsic characteristics of turbulence.

Building on the work of