Chaos, coherence, and the 'dark turbulence background'*

It has been about 50 years since the first papers on deterministic chaos appeared, and also since coherent structures were first reported in turbulent flows, and it is time to examine what have we learned that would not have been possible had these twin observations not happened. We now know that turbulence is not just a stochastic sea, but has a coherent component that typically covers a small part of the flow volume but explains a much larger fraction of the 'things that matter', such as energy or momentum transfer. We also know that chaos in a low-dimensional system is typically limited to a fraction of the state space, while the rest of the evolution is smooth. But it not always clear whether these facts are related. Is coherent turbulence the same as the invariant manifolds of a dynamical system (fixed points, etc.)? If so, why is it related to the 'things that matter'? Is this always the case? Most coherent structures in turbulence arise from the transient amplification of particular initial conditions. This is even true in modally unstable systems such as free shear flows. How are those initial conditions generated? Is there a coherent scaffold in which structures regenerate themselves? Or do the required initial conditions spontaneously appear from the 'dark matter' of incoherent turbulence? I will illustrate these questions with examples, and give a personal opinion of which of them require most attention.