Lagrangian scale decomposition via the graph Fourier transform

Scale decomposition is ubiquitous in the analysis of complex fluid flows. Often with an eye towards reduced order modelling, considerable effort has been invested in the development of novel sets of basis functions for such decompositions. The most successful sets of basis functions have been developed in the Eulerian perspective, where traditional tools of calculus and newer methods from data science can be most easily leveraged. Here, we take advantage of recent interest in graph-based approaches to Lagrangian coherence as well as new methods for the scale analysis of complex networks to introduce a new scale decomposition that is instead fully Lagrangian and based on transport. To do so, we adapt a technique from network science know as the graph Fourier transform and develop a novel graph correlation function that allows us to quantitatively describe our Lagrangian decomposition as a function of scale. This method allows better interpretability of the dynamic consequences of kinematic coherence as well as the ability to perform traditional Eulerian tasks (such as decomposition, filtering, and compression) on Lagrangian quantities. We illustrate our techniques on examples drawn from coherent-structure analysis, ocean mixing, and cloud physics.