Discovering multiscale and self-similar structure with data-driven wavelets.

Many materials, processes, and structures in science and engineering have important features at multiple scales of time and/or space; examples include biological tissues, active matter, oceans, networks, and images. Explicitly extracting, describing, and defining such features are difficult tasks, at least in part because each system has a unique set of features. Here, we introduce an analysis method that, given a set of observations, discovers an energetic hierarchy of structures localized in scale and space. We call the resulting basis vectors a "data-driven wavelet decomposition." We show that this decomposition reflects the inherent structure of the dataset it acts on, whether it has no structure, structure dominated by a single scale, or structure on a hierarchy of scales. In particular, when applied to turbulence-a high-dimensional, nonlinear, multiscale process-the method reveals self-similar structure over a wide range of spatial scales, providing direct, model-free evidence for a century-old phenomenological picture of turbulence. This approach is a starting point for the characterization of localized hierarchical structures in multiscale systems, which we may think of as the building blocks of these systems.

Swimmers' wake structures are not reliable indicators of swimming performance.

The structure of swimmers' wakes is often assumed to be an indicator of swimming performance, that is, how momentum is produced and energy is consumed. Here, we discuss three cases where this assumption fails. In general, great care should be taken in deriving any conclusions about swimming performance from the wake flow pattern.

Efficient cruising for swimming and flying animals is dictated by fluid drag.

Many swimming and flying animals are observed to cruise in a narrow range of Strouhal numbers, where the Strouhal number [Formula: see text] is a dimensionless parameter that relates stroke frequency f, amplitude A, and forward speed U. Dolphins, sharks, bony fish, birds, bats, and insects typically cruise in the range [Formula: see text], which coincides with the Strouhal number range for maximum efficiency as found by experiments on heaving and pitching airfoils. It has therefore been postulated that natural selection has tuned animals to use this range of Strouhal numbers because it confers high efficiency, but the reason why this is so is still unclear. Here, by using simple scaling arguments, we argue that the Strouhal number for peak efficiency is largely determined by fluid drag on the fins and wings.