Extreme statistics, Gaussian statistics, and superdiffusion in global magnitude fluctuations in turbulence

Abstract: Extreme value statistics, or extreme statistics for short, refers to the statistics that characterizes rare events of either unusually high or low intensity: climate disasters like floods following extremely intense rains are among the principal examples. Extreme statistics is also found in fluctuations of global magnitudes in systems in thermal equilibrium, as well as in systems far from equilibrium. A remarkable example in this last class is fluctuations of injected power in confined turbulence. Here we report results in a confined von K\'arm\'an swirling flow, produced by two counter-rotating stirrers, in which quantities derived from the same global magnitude ---the rotation rate of the stirrers--- can display both, extreme and Gaussian statistics. On the one hand, we find that underlying the extreme statistics displayed by the global shear of the flow, there is a nearly Gaussian process resembling a white noise, corresponding to the action of the normal stresses exerted by the turbulent flow, integrated on the flow-driving surfaces of the stirrers. On the other hand, the magnitude displaying Gaussian statistics is the global rotation rate of the fluid, which happens to be a realization of a 1D diffusion where the variance of the angular increments $\theta(t+\Delta t) - \theta(t)$ scales as $\Delta t^{\nu}$, while the power spectral density of the rotation rate follows a $1/f^{\alpha}$ scaling law. These scaling exponents are found to be $\alpha \approx 0.37$ and $\nu \approx 1.36$, which implies that this process can be described as a 1D superdiffusion.