Nonlinear Dynamics

The past decade has seen substantial developments in the understanding of Lagrangian transport by fluid flows, using tools inspired by dynamical systems theory.

Lagrangian Coherent Structures (LCSs): The variational method of LCSs is a geometrical approach that seeks to determine the key material lines that organize flow transport. The focus of the research is to develop these methods to be reliable, accessible and applicable to real-world ocean problems. For example, although these methods have been applied to ocean surface transport, there has been no accounting for the impact of surface wind drag (windage) on floating objects, which is known to be a key factor in oil spill prediction. We have incorporated windage into LCSs analysis and shown how important this consideration is, by performing a case study of the Australian West Coast, where there is huge o_shore development and a fragile coral reef environment [24]. We have also demonstrated how LCSs can be used to uncover the connectivity of coral reef systems, with a view to assisting the design of marine sanctuaries [25], and investigated practical LCS classi_cation schemes.

Braid Theory: Another approach we are pursuing is the topological method of Braid Theory. This approach encodes complex trajectories of drifters in uid ows as concise algebraic data, enabling e_cient identi_cation of transport barriers. In contrast to the variational LCSs approach, which _nds dominant repelling and attracting material lines, Braid Theory identi_es closed loops that contain coherent regions. And whereas variational LCSs analysis requires extensive velocity _eld data, the Braid Theory approach relies on the trajectories of a modest number of drifters, which is a form of ocean data that is widely available. We have obtained funding from the NSF Dynamical Systems program to test this method in laboratory experiments and on _eld data, for which we participated in a 2014 field study.