Inertial Manifolds for Certain Sub-Grid Scale $\alpha$-Models of Turbulence

TL;DR

This paper proves the existence of inertial manifolds for two sub-grid scale turbulence models in 2D, showing that their long-term behavior can be captured by finite-dimensional systems, simplifying analysis and computation.

Contribution

It establishes the existence of inertial manifolds for the simplified Bardina and modified Leray-$lpha$ turbulence models in two dimensions, a novel theoretical result.

Findings

Existence of inertial manifolds for the models

Finite-dimensional representation of long-term dynamics

Implication for simplified turbulence modeling

Abstract

In this note we prove the existence of an inertial manifold, i.e., a global invariant, exponentially attracting, finite-dimensional smooth manifold, for two different sub-grid scale $\alpha$-models of turbulence: the simplified Bardina model and the modified Leray-$\alpha$ model, in two-dimensional space. That is, we show the existence of an exact rule that parameterizes the dynamics of small spatial scales in terms of the dynamics of the large ones. In particular, this implies that the long-time dynamics of these turbulence models is equivalent to that of a finite-dimensional system of ordinary differential equations.