Unstable Equilibria and Invariant Manifolds in Quasi-Two-Dimensional Kolmogorov-like Flow

TL;DR

This study combines experiments and simulations to investigate how unstable equilibrium solutions and their invariant manifolds influence the dynamics of weakly turbulent, electromagnetically driven shallow fluid flows, revealing their role in turbulent behavior.

Contribution

It identifies and characterizes unstable equilibria and their invariant manifolds in a realistic two-dimensional flow model, demonstrating their significance in turbulent dynamics.

Findings

Unstable equilibria are closely visited by turbulent flow trajectories.

Invariant manifolds are shadowed by turbulent trajectories over large state space distances.

31 unstable equilibria were computed and analyzed.

Abstract

Recent studies suggest that unstable, non-chaotic solutions of the Navier-Stokes equation may provide deep insights into fluid turbulence. In this article, we present a combined experimental and numerical study exploring the dynamical role of unstable equilibrium solutions and their invariant manifolds in a weakly turbulent, electromagnetically driven, shallow fluid layer. Identifying instants when turbulent evolution slows down, we compute 31 unstable equilibria of a realistic two-dimensional model of the flow. We establish the dynamical relevance of these unstable equilibria by showing that they are closely visited by the turbulent flow. We also establish the dynamical relevance of unstable manifolds by verifying that they are shadowed by turbulent trajectories departing from the neighborhoods of unstable equilibria over large distances in state space.