Coherence, Transport, and Inference in Turbulent Dynamical Systems
Abstract
Analyzing and understanding features in chaotic and turbulent systems is central to developing modern systems that can thrive in fluid media. From ships in oceans to airplanes in the air, to chemical mixing and biological processes, and so forth, both the local scale of the vessels as well as the macro scale of weather and oceanic currents, it is clear that the Naval relevance of these issues call for ever better technology to analyze the underlying processes. This project concerns analysis of chaotic and turbulent systems, from a Lagrangian perspective, specifically toward a modern perspective of coherent structures for understanding transport as well as persistence of underlying structures whether they be blooms of plankton in the oceans, or cascade of energy in turbulence. We emphasize two major thrusts in this work, both working toward this major theme of understanding aspects of simplicity embedded in nonlinear systems. • Transport from a theory of Shape Coherence: Coherence has clearly become a central concept of interest in nonautonomous dynamical systems, particularly in the study of turbulent flows, with many recent papers designed toward describing, quantifying and constructing such sets, [16, 18, 24, 32, 33, 40, 50]. There have been a wide range of notions of coherence, from spectral, [27], to set oriented, [11] and through transfer operators [4], as well as variational principles. A general perspective of set oriented analysis of coherence seems to emphasize a discussion of transport. A number of theories are developed to model and analyze the dynamics in the Lagrangian perspective, such as the geodesic transport barriers [24] and transfer operators method [18]. Whatever the perspectives taken, generally it may be summarized that coherent structures can be taken as a region of simplicity, within the observed time scale and over a stated spatial scale, perhaps embedded within an otherwise possibly turbulent flow, [16, 18, 24, 33]. Here we will review our recent theory of Shape Coherence and describe significant new computational and theoretical directions we plan to develop. These will include relating coherence to a theory of transport for nonautonomous systems, extending to a three-dimensionality and coherence, deepening connections between geometry, di?erential geometry, measurable dynamics, and dynamical systems. Application of these concepts to oceanographic and atmospheric flow problems and including understanding advection, di?usion-reaction-problems such as plankton bloom growth in the oceans and estuaries. • Inference in Spatiotemporal Dynamical Systems: Inverse Problems and System Inference: We have been developing convex optimization methods for systems inference, designed for data sets from remote sensing platforms, as summarized in our review article, [70] and the recent PhD thesis, [69] and also [76]. This includes methods for inferring both advective information (time varying vector fields) and parametric information regarding system identification, especially for problems from ecological oceanographic systems as information remotely by satellite imagery. This continues to be a promising thread with aspects to develop further that we describe here, to continue to build a practical tool for modeling spatiotemporal systems remotely and therefore to inform the analytic coherence and transport discussion of the first part of this project described above. We will adapt recent Bayesian-based data fusion methods [73] to the problem of improving our current estimates to mixed data forms, suitable for improving already developed estimates from remotely observed scalar fields to the possibility of mixed data including from sparsely placed floaters (buoys). Such improvements should significantly help with missing data from a remote sensor based method alone which is subject to occlusion.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 08, 2016
- Source ID
- N000141512093
Entities
People
- Eric Bollt
Organizations
- Clarkson University
- Office of Naval Research
- United States Navy