  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.
