On algebraic and geometric aspects of fluid dynamics: New perspectives based on Nambu mechanics and its applications to atmospheric dynamics

Abstract

Vortices play a crucial role in atmospheric dynamics across all scales. Anticyclonic and cyclonic rotating vortices, also known as high-and low pressure systems, determine our weather on the larger, synoptic scale. Such larger-scale vortices have horizontal radii of about 1000 km and their vertical extend is much smaller. Therefore, their motion can be described by quasi-two-dimensional fluid dynamics. Rotating supercells are examples of vortices on smaller scale that occur and influence our weather more locally. The radius of a supercell is about 10 km. Therefore, the horizontal and vertical length scales have about the same order of magnitude and three- dimensional fluid dynamical models are used to describe vortex motions on the smaller, convective scale.

In order to gain a comprised description of vortices on different scales, this thesis concerns the Nambu representation for two- and three-dimensional vortex dynamics. Nambu mechanics can be seen as a generalization of Hamilton's formulation. The equations of motion are represented in terms of two constitutive conserved quantity: the energy and a vortex-related conserved quantity. In the first part of this thesis, we will show how this concept allows for a novel, geometric classification of planar point vortex motions. Furthermore, we will use the idealized point vortex model to explain atmospheric blockings.

Regarding further conserved quantities, the Nambu representation of the Helmholtz vorticity equation provides an algebraic structure for incompressible, inviscid fluids. We will explore this algebraic approach in the second part of this thesis. First, we will introduce a novel matrix representations for the Lie algebra for two- and three-dimensional vortex dynamics. From these Lie algebra representations we will derive novel Lie group representations for two- and three-dimensional vortex flows. This approach can be seen as a structural integration of the vorticity equation, because the vortex group is directly derived from the Helmholtz vorticity equation. Now, we can regard incompressible, inviscid vortex dynamics from a different perspective leading to a better understanding of various problems.

As an example for the applicability of the here derived algebraic approach, we will show how splitting storms can be explained by analyzing helicity density fields with respect to their sign structure. Moreover, the explanation of splitting storms and the associated breakup of vortices might lead to a better understanding of turbulent structures. Finally, we will show how the vortex algebra allows for the investigation of shortest paths of point vortices. Such vortex geodesics can be compared to special point vortex constellations that we will have discussed in the first part of the thesis. We will also outline a concept for the derivation of 3D vortex geodesics.

In summary, this thesis will concern algebraic and geometric studies of fluid dynamics combining the different disciplines of mathematics, physics and meteorology. In this way, based on Nambu mechanics, we will investigate new perspectives on fluid dynamics and show several applications to atmospheric dynamics.