Inferring Multi-Scale Cascade Structure from a Single Lagrangian Trajectory via Hidden Markov Switching

Abstract

We show that a single Lagrangian trajectory encodes not only temporal but also spatial cascade structure--recoverable without access to a spatially resolved Eulerian velocity field. We report a Markov-Switching Multifractal (MSM) framework that infers hidden multi-scale cascade states directly from single-particle trajectories. Kolmogorov scaling fixes the model's switching rates, leaving a single calibrated intermittency parameter; exact Bayesian filtering then returns the posterior of the layer-resolved cascade state along the trajectory. When we apply the MSM framework independently to each trajectory of a particle pair, we find that the inferred states carry genuine spatial information: consistent with Richardson's locality picture, the Kolmogorov $r^{1/3}$ scaling is preserved when fine-scale layers are randomized, but collapses once the randomization crosses the layer set by the pair separation. Single trajectories thus encode the spatial organization of the cascade, opening a route to cascade diagnostics for turbulent flows accessible primarily through sparse or individual Lagrangian trajectories--from particle-tracking velocimetry to atmospheric and oceanographic tracking--where the instantaneous Eulerian field is out of reach.