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

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.

💡 Research Summary

The paper presents a comprehensive experimental investigation of global magnitude fluctuations in a confined von Kármán swirling flow driven by two counter‑rotating stirrers. By recording the stirrers’ rotation rates with high‑resolution encoders and measuring the torque exerted on the stirrers, the authors obtain long time series that capture both fast and slow dynamics. Two distinct global observables are examined: (1) the shear stress integrated over the stirrer surfaces, which represents the normal stresses imposed by the turbulent flow, and (2) the global angular displacement of the fluid, defined as the difference between the two stirrers’ rotation rates.

For the shear stress, the probability density function (PDF) exhibits heavy tails that are well described by a Lévy‑stable distribution, indicating that extreme‑value statistics dominate. Despite this non‑Gaussian PDF, the power spectral density (PSD) of the shear stress is essentially flat at high frequencies, resembling white noise. This suggests that the underlying process is a nearly Gaussian, uncorrelated fluctuation driven by rapid, independent turbulent collisions on the stirrer surfaces. The coexistence of a heavy‑tailed PDF with a white‑noise spectrum highlights the subtle distinction between the distribution of amplitudes and the temporal correlation structure.

In contrast, the global rotation angle θ(t) behaves as a one‑dimensional diffusion process with anomalous scaling. The variance of angular increments Δθ(t,Δt)=θ(t+Δt)−θ(t) scales as ⟨Δθ²⟩∝Δt^ν with ν≈1.36, which is larger than the ν=1 expected for ordinary Brownian motion. Simultaneously, the PSD of the rotation rate follows a 1/f^α law with α≈0.37, indicating long‑range temporal correlations. These two exponents satisfy the theoretical relationship expected for a Lévy‑flight or Lévy‑walk type superdiffusive process, confirming that the global rotation is a manifestation of 1‑D superdiffusion.

The key insight of the study is that the same experimental system can display both extreme‑value (non‑Gaussian) statistics and Gaussian statistics, depending on which global observable is considered. The shear stress reflects the small‑scale, essentially memoryless turbulent forcing, while the global rotation integrates the effect of large‑scale coherent structures and retains a memory of past fluctuations, leading to superdiffusive behavior. This duality underscores the multi‑scale nature of turbulence: different physical quantities probe different parts of the cascade and may belong to distinct statistical universality classes.

The authors discuss the implications for turbulence theory. Traditional Kolmogorov frameworks focus on second‑order statistics and assume Gaussianity at inertial scales, but the present results demonstrate that higher‑order statistics and extreme events are essential for a complete description. Moreover, the identification of a superdiffusive global rotation suggests that large‑scale flow organization can be modeled using fractional diffusion equations, opening a pathway to incorporate long‑range correlations into turbulence models.

Finally, the paper proposes that such global statistical signatures are robust and could be observed in a variety of turbulent configurations, from atmospheric and oceanic flows to industrial mixers. Future work may explore how geometry, boundary conditions, or external forcing modify the balance between Gaussian white‑noise forcing and superdiffusive dynamics, thereby enriching our understanding of the complex statistical landscape of turbulent systems.