  The existence and dynamical role of particular unstable Navier-Stokes
solutions (exact coherent structures) is revealed in laboratory studies of weak
turbulence in a thin, electromagnetically-driven fluid layer. We find that the
dynamics exhibit clear signatures of numerous unstable equilibrium solutions,
which are computed using a combination of flow measurements from the experiment
and fully-resolved numerical simulations. We demonstrate the dynamical
importance of these solutions by showing that turbulent flows visit their state
space neighborhoods repeatedly. Furthermore, we find that the unstable manifold
associated with one such unstable equilibrium predicts the evolution of
turbulent flow in both experiment and simulation for a considerable period of
time.
