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

湍流上层结构对流体的动力学行为有至关重要的影响，例如质量、动量和能量的传递。它们是在动力系统理论中研究的所谓相干集的一种特殊情况。在流体动力学中，有两种基本观点：一种是物理观点，它描述点在三维空间中的运动；另一种是抽象观点，它用一定函数空间中的微分方程来描述速度场的演变。到目前为止，湍流上层结构的发展，更一般地说，流体流动中的相干集主要是从第一个角度分析的。这个项目的基本思想是从功能空间的角度分析流体流动，并以这种方式发展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，我们期望获得对El Niño南方涛动等现象的新见解。为此，我们发展了在流体流动系统的函数空间中检测不变量集和几乎不变量集的数值方法。为此，我们将采用一种面向集合的方法，它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。该方法目前已成功地应用于求解时滞微分方程。对于偏微分方程的应用产生了额外的问题，这些问题将在本项目中解决。例如，必须确定适合于重建系统行为的可观察对象。为了避免不可行的高计算量，必须开发有效的算法实现程序的每一步。基于Galerkin投影和适当正交分解的简化模型以及由Koopman算子导出的模型允许以显着减少的数值努力来近似动力学。为了将新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中的（几乎）不变集与物理空间中的相干集之间的关系。为此，我们考虑简化的模型系统，以及与优先计划的合作伙伴一起，也对实验研究的特定流体流动进行了研究。我们期望这些技术能够对湍流上层建筑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.