Understanding and utilising relationships between coherent structures and almost invariant sets in function space

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Turbulent Superstructures have a crucial influence on the dynamical behaviour of fluids, for example the transport of mass, momentum and energy. They are a special case of so-called coherent sets, which are investigated in the theory of dynamical systems. In fluid dynamics, there are two fundamental points of view: a physical perspective, which describes the movement of points in three-dimensional space, and an abstract point of view, from which the evolution of the velocity field is described by a differential equation in a certain function space. So far, the development of turbulent superstructures and, more generally, of coherent sets in fluid flows has mainly been analysed from the first perspective. The basic idea of this project is to analyse fluid flows from the function space perspective and to develop in this way a novel approach to the analysis of numerous problems related to the development of turbulent superstructures. In the long run we expect that e.g. new insights into phenomena such as the El Niño southern oscillation are obtained. To this end, we develop numerical methods for the detection of invariant and almost invariant sets in the function space of fluid flow systems. For this we will adapt a set oriented approach, that was originally developed for finite-dimensional systems and was recently extended to infinite-dimensional systems, to the partial differential equations that describe fluid flows. This technique has so far successfully been applied to delay differential equations. For the application to partial differential equations additional questions arise, which are addressed in this project. For example, observables suitable for the reconstruction of the system behaviour have to be identified. In order to avoid infeasibly high computational effort, efficient algorithmic realisations of each step of the procedure have to be developed. Reduced models based on Galerkin projections and the Proper Orthogonal Decomposition as well as models derived from the Koopman operator allow to approximate the dynamics with significantly reduced numerical effort. In order to apply the new methods to the analysis of coherent structures and their origin it is necessary to investigate the relationships between (almost) invariant sets in function space and coherent sets in physical space. For this purpose, we consider simplified model systems as well as, together with partners from the priority programme, specific fluid flows that are also investigated experimentally. We expect that these techniques allow to obtain new insights into the mechanisms behind the origin of turbulent superstructures.