Seeking earthly measures: Algorithms for the detection, tracking and investigation of coherent structures in non-autonomous dynamical systems

Abstract

We develop and test algorithms for the detection, tracking and investigation of coherent structures in non-autonomous dynamical settings. Coherent structures are spatially varying regions that disperse minimally over time and organise motion in non-autonomous systems. Understanding and characterising the dynamical behaviour of these structures, as well as maximising the information about them that can be extracted from data and models of the underlying flows, is important for understanding how transport and mixing properties develop as dynamical systems evolve. The so-called transfer operator point of view can be interpreted as tracking the evolution of an initial ensemble of trajectories, or density, through time. Transfer-operator based methods were first found useful in the identification of invariant measures for dynamical systems, and later in the detection of almost-invariant sets. Coherent structures are the time-dependent analogue of such sets. In the form of oceanic eddies and atmospheric vortices, coherent structures play important roles in biogeophysical phenomena and influence the weather of our planet. Recent developments in multiplicative ergodic theory yield the foundations for our algorithms. The acclaimed theorem of Oseledets provides for a spectral type decomposition for time-dependent dynamical systems. From this perspective, Oseledets spaces play the role of time-varying (generalised) eigenspaces, each of which is associated to a characteristic value, or Lyapunov exponent, that dictates the rate at which the associated objects expand or decay. These spaces and their Lyapunov exponents can be approximated utilising singular value decomposition algorithms.The algorithms we develop employ Ulam type discretisations of transfer operators induced by dynamical systems. As in the case of Oseledets decompositions, the objects approximated by our algorithms provide multilayered descriptions of time-dependent systems. These algorithms help locate coherent structures and are useful for detecting time windows within which coherent structures undergo fundamental structural changes, such as merging and splitting events. They also allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. Furthermore, we develop and analyse algorithms that utilise a localised transfer operator in the detection and characterisation of coherent structures and their associated lifespans. This permits an exploration of the core structures influencing transport, despite having incomplete knowledge of the dynamics. Moreover, the ability to characterise lifespans of coherent structures allows one to narrow in on time windows of interest, particularly those in which fundamental changes are expected to occur.The algorithms we develop are evaluated using vector fields generated by periodically, and quasi-periodically, driven double well potentials, the Boussinesq equations, and also a geophysical dataset corresponding to the splitting of the Southern Polar Vortex. In each case, our algorithms provide valuable information regarding the accurate detection of coherent structures, the paths they track through time, and time windows within which they experience fundamental changes.