THIS IS A CONTINUATION OF N00014-14-1-0633 Koopman Mode Decomposition and Mixing in Fluid Flows
Abstract
Executive Summary Two recent developments are gaining in popularity in analysis of complex uid ows of Navy in- terest. These developments are 1) the Koopman Mode Decomposition and 2) Mesohyperbolicity analysis. Here we propose to couple these developments into a theory that is capable of describ- ing mixing in complex, aperiodic ows using a strongly reduced representation - the Koopman Mode Decomposition - that enables application of both classical (e.g. lobe dynamics) and novel mesohyperbolicy-based tools to the problem of mixing in aperiodic ows. Firstly, we propose to utilize Koopman Decomposition methods to address the problem of mix- ing in ocean ows. Ocean ows commonly exhibit periodic components due to daily and seasonal forcing e ects. Those forcing e ects cause both super harmonic and subharmonic response due to nonlinearity of the dynamics. We hypothesize that the attractor in this case is of skew-periodic type. In order to verify this hypothesis, we will study well-resolved ocean models. The decom- position in this consists of time-mean, quasi periodic part and continuous-spectrum part. Notice that the continuous-spectrum part can be modeled as a stochastic process. In our work we will characterize the stochastic nature of the continuous spectrum part and consider it after we have thoroughly explored the mixing properties of the deterministic part, consisting of the quasi-periodic mean. Time-periodic examples of the deterministic part are well-studied in 1 1/2 degree of freedom dynamical systems. Some studies exist in three-dimensional time periodic case as well. However, the quasi-periodic case was studied much less in two dimensions and not at all in three dimensions. We propose to develop a theory of mixing in aperiodic ows by considering separate Koopman Modes as distinct contributions to overall mixing dynamics. From chaotic advection studies we know much about what this theory would look like in two-dimensional examples. For example, for ows with simple, periodic time dependence we know that lobe dynamics plays a large role in description of mixing in this case. We propose to explore the relationship between mesohyperbolicity and lobe dynamics in this context. In numerical simulation of periodically oscillating double gyre ows we have already observed positive and negative mesohyperbolicity concentration within lobes. We will explore this connection mathematically, using the formal apparatus of lobe dynamics. In addition, we have shown that in KAM regions, mesohyperbolicity concepts are related to Green s residue criterion. We will explore this connection further, pursuing Green s criterion ability to detect invariant structures. This will enable a deeper understanding of mixing dynamics in such ows and enable extensions to quasi-periodic ows that we will address next. Mesohyperbolicity theory for quasi-periodic ows will be developed. Its connection with lobe dynamics will be established. Invariant structures will be studied by pursuing extension of ideas from Greene s residue criterion via mesohyperbolicity connection. Finally, ows with fully aperiodic time dependence will be studied. First the splitting of the ow into its mean, quasi-periodic and continuous spectrum components will be established. Then, the developed knowledge of mixing in quasi-periodic ows will be supplemented by additive contribution from continuous spectrum dynamics. This will enable a more complete understanding of mixing in aperiodic ows via a staged approach where the analytical di culties induced by time dependence are added in the natural order of complexity. The team consists of the PI and a graduate student.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 10, 2016
- Source ID
- N000141612120
Entities
People
- Igor Mezić
Organizations
- Office of Naval Research
- United States Navy
- University of California, Santa Barbara