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

湍流超结构对流体的动力学行为有着至关重要的影响，例如质量、动量和能量的传输。它们是动力系统理论中研究的所谓相干集的特例。在流体力学中，有两种基本的观点：一种是物理观点，它描述了点在三维空间中的运动；另一种是抽象的观点，从这种观点出发，速度场的演化可以用某个函数空间中的微分方程来描述。到目前为止，湍流超结构的发展，以及更广泛地说，流体流动中的相干集的发展主要是从第一个角度来分析的。这个项目的基本思想是从功能空间的角度分析流体流动，并以这种方式开发一种新的方法来分析与湍流上层建筑发展有关的许多问题。从长远来看，我们期望对厄尔尼诺南方振荡等现象有新的认识。为此，我们发展了检测流体流动系统函数空间中不变集和几乎不变集的数值方法。为此，我们将采用一种面向集合的方法，这种方法最初是为有限维系统开发的，最近被扩展到无限维系统，用于描述流体流动的偏微分方程组。到目前为止，这一技术已经成功地应用于时滞微分方程。对于偏微分方程式的应用，出现了额外的问题，这些问题将在本项目中解决。例如，必须确定适合重建系统行为的观测值。为了避免不可能的高计算量，必须开发程序每一步的高效算法实现。基于Galerkin投影和适当的正交分解的简化模型以及从Koopman算子导出的模型允许以显著减少的数值工作量来逼近动力学。为了将新的方法应用于相干结构及其起源的分析，有必要研究函数空间的(几乎)不变集与物理空间的相干集之间的关系。为此，我们考虑简化的模型系统，以及与优先方案的合作伙伴一起，对也进行实验研究的特定流体流动进行研究。我们期望这些技术能够对湍流超结构的起源背后的机制获得新的见解。

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.