Investigating the connection between complexity of isolated trajectories and Lagrangian coherent structures

Abstract

Abstract. It is argued that the complexity of fluid particle trajectories provides the basis for a new method, referred to as the Complexity Method (CM), for estimation of Lagrangian coherent structures in aperiodic flows that are measured over finite time intervals. The basic principles of the CM are explained and the CM is tested in a variety of examples, both idealized and realistic, and in different reference frames. Two measures of complexity are explored in detail: the correlation dimension of trajectory, and a new measure -the ergodicity defect. Both measures yield structures that strongly resemble Lagrangian coherent structures in all of the examples considered. Since the CM uses properties of individual trajectories, and not separation rates between closely spaced trajectories, it may have advantages for the analysis of ocean float and drifter data sets in which trajectories are typically widely and non-uniformly spaced.

New Lagrangian diagnostics for characterizing fluid flow mixing

“…The MET and many other Lagrangian quantifiers (such as RD, FTLE, the hypergraph map and other averaged quantifiers [2,3,17,26,27,33]) have a common feature: asymptotically these converge to constants on ergodic sets and hence, in principle, may be utilized to divide the phase space to separate ergodic components. In many applications, the transient properties of these and other Lagrangian quantifiers were studied, showing that in some cases ridges of finite time realizations of these fields provide good predictors for dividing surfaces.…”

“…More generally, the study of the transient behaviour of the MET in τ, t 1 may reveal the structure (e.g. local dimension [33,44]) of the ergodic component. Possibly, it may reveal other transient transport processes, such as dividing surfaces (LCS) and the lobe structure [40].…”

“…Since asymptotically this value in each ergodic component converges to a common extreme value (similarly to other Lagrangian averages along trajectories [3,17,33]), such extremal fields provide the sought division into distinct ergodic components. Moreover, the extreme values of the selected observable may by themselves have significant physical meaning.…”

“…Previous works on the extreme values of an observable of dynamical systems have focused on the temporal dependence of a single chaotic trajectory for maps (mainly for chaotic dissipative maps), connecting it to the universal distributions appearing in the field of Extreme Value Statistics (EVS) on one hand and to Poincaré recurrences and local dimensionality of the attractors on the other (the connection to [33,44] may thus be intriguing). Here, we focus instead on utilizing the extreme value functionals as convenient spatial characteristics of dynamical systems with mixed phase space.…”

“…Some of the Lagrangian characteristics present the resulting observable after a certain integration time, with no information regarding the intermediate time dynamics (e.g. the AD and RD fields depend only on the initial and final location of the particles), whereas some of the other Lagrangian characteristics use averaging or integration along the trajectories [3,17,26,[33][34][35]. These Lagrangian fields are commonly used to identify regions of small and enhanced stretching and, in particular, are used to identify the spatial position of dividing surfaces between different regions, the Lagrangian Coherent Structures (LCS) [2,26,27].…”