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On Algebraic and Geometric Aspects of Fluid Dynamics On algebraic and geometric aspects of fluid dynamics: New perspectives based on Nambu mechanics and its applications to atmospheric dynamics Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) am Fachbereich Geowissenschaften der Freien Universität Berlin vorgelegt von Annette Müller Berlin, 20. März 2018 1. Gutachter PD Dr. Peter Névir Freie Universität Berlin Fachbereich Geowissenschaften Institut für Meteorologie Carl-Heinrich-Becker-Weg 6-10 12165 Berlin, Deutschland 2. Gutachter Prof. Dr. Henning Rust Freie Universität Berlin Fachbereich Geowissenschaften Institut für Meteorologie Carl-Heinrich-Becker-Weg 6-10 12165 Berlin, Deutschland 3. Gutachter Prof. Dr. Rupert Klein Freie Universität Berlin Fachbereich Mathematik and Informatik Institut für Mathematik Arnimallee 2-6 14195 Berlin, Deutschland Tag der Disputation 26. Juni 2018 i Abstract Vortices play a crucial role in atmospheric dynamics across all scales. An- ticyclonic and cyclonic rotating vortices, also known as high-and low pres- sure systems, determine our weather on the larger, synoptic scale. Such larger-scale vortices have horizontal radii of about 1000 km and their ver- tical 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 lo- cally. 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-dimen- sional vortex dynamics. Nambu mechanics can be seen as a generaliza- tion 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 vor- tex 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 incom- pressible, inviscid fluids. We will explore this algebraic approach in the second part of this thesis. First, we will introduce a novel matrix represen- tations for the Lie algebra for two- and three-dimensional vortex dynamics. From these Lie algebra representations we will derive novel Lie group rep- resentations 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 dif- ferent perspective leading to a better understanding of various problems. As an example for the applicability of the here derived algebraic ap- proach, we will show how splitting storms can be explained by analyz- ing 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 ii 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 investi- gate new perspectives on fluid dynamics and show several applications to atmospheric dynamics. iii Zusammenfassung Auf allen Skalen in unserer Atmosphäre haben Wirbel einen Einfluss auf unser Wettersystem. Auf den größeren Skalen bestimmen die Antizyklo- nen und Zyklonen, welche auch unter dem Namen Hoch- und Tiefdruckge- biet bekannt sind, unser Wetter in den mittleren Breiten. Sie entscheiden, ob es stürmisch, sonnig oder regnerisch ist. Weil die horizontalen Radii solch großskaliger, synoptischer Wirbel ca. 1000 km misst, kann deren Dynamik mathematisch mit dem Gleichungssystem der quasi-zweidimensionalen Hy- drodynamik ausgedrückt werden. Superzellen hingegen sind Beispiele von Wirbeln auf kleinerer Skala, die unser Wetter lokal beeinflussen. Die hor- izontale Ausdehnung einer Superzelle beträgt ca. 10 km. Somit haben die horizontalen und vertikalen Skalen solcher kleiner skaligen Wirbel die gleiche Größenordnung und es müssen dreidimensionale Modelle verwen- det werden, um die Dynamik von Wirbeln auf der kleineren, konvektiven Skala bestmöglichst zu erfassen. Um Wirbel auf den verschiedenen Skalen zu beschreiben, werden wir in dieser Arbeit die Nambu-Darstellungen der zwei-und dreidimensionalen Wirbeldynamik betrachten. Nambu-Mechanik kann als Verallgemeinerung der Hamilton’schen Sichtweise aufgefasst werden, wobei als konstituieren- de Größen neben der Energie auch Wirbelerhaltungsgrößen fungieren. Im ersten Teil der Arbeit werden wir zeigen, wie die zusätzliche Wirbelerhal- tunsgröße in der diskreten Nambu-Darstellung eine neue, geometrische Klassifikation von Punktwirbelbewegungen ermöglicht. Zudem werden wir das idealisierte Konzept der Punktwirbel auf Hoch-und Tiefdruckgebi- ete übertragen und blockierende Wetterlagen erklären. Im zweiten Teil dieser Arbeit werden wir die kontinuierliche Nambu- Mechanik und die daraus abgeleitete algebraische Struktur der Wirbeldy- namik erörtern und auf sich teilende Superzellen anwenden. Aus der Helm- holtz’schen Wirbelgleichung kann eine Nambu-Klammer abgeleitet wer- den, die eine Lie algebra erzeugt. Wir werden eine Matrixdarstellung der Lie Algebra einführen und daraus verschiedene Gruppendarstellungen für zwei- und dreidimensionale Wirbelbewegungen ableiten. Dies ermöglicht eine neue, algebraische Sichtweise, mit der die Mechanismen verschiedener atmosphärische Prozesse neu verstanden werden können. Als Beispiel wer- den wir den Zerfall von Superzellen mithilfe der neuen Gruppe für inkom- pressible, dreidimensionale Wirbeldynamik erklären. Hierbei werden wir Helizitätsfelder hinsichtlich deren Vorzeichenstrukturen analysieren und iv aus den verschiedenen Vorzeichen der Helizität Rückschlüsse auf den Zer- fallsmechanismus von Wirbeln ziehen. Damit könnte der algebraische An- satz auch für ein besseres Verständnis in Bereichen der Turbulenztheorie herangezogen werden. Im letzten Kapitel erläutern wir, wie die im Rahmen dieser Arbeit herge- leitete Gruppe angewendet werden kann, um Wirbelgeodedäten abzuleiten. Dabei werden wir zeigen, dass in zwei Dimensionen die Geodäten mit aus- gezeichneten Punktwirbelsystemen verbunden sind, die wir bereits im er- sten Teil dieser Arbeit diskutiert haben werden. Zusammengefasst betrachten wir in dieser Arbeit sowohl algebraische als auch geometrische Aspekte der Wirbeldynamik, die durch Anwendung der Nambu-Mechanik sichtbar werden. Dabei versuchen wir mithilfe math- ematischer Konzepte physikalische Mechanismen atmosphärischer Phänomene zu erklären. Contents Abstracti Zusammenfassung iii 1 INTRODUCTION1 I Geometric aspects of discrete fluid dynamics9 2 Geometric aspects of vortex dynamics 11 2.1 Nambu formulation of the Lorenz equations . 13 2.1.1 A Hamilton view on the Lorenz equations . 14 2.1.2 Nambu formulation of the Lorenz equations . 15 2.2 Discrete Nambu mechanics . 17 2.2.1 Transition to canonical Hamiltonian dynamics . 18 2.2.2 Nambu mechanics for n = 3 degrees of freedom . 19 3 Nambu mechanics of point vortex theory 23 3.1 Introduction to point vortex dynamics . 23 3.2 Geometric classification . 31 3.3 Periodic motion . 34 3.4 Relative equilibrium . 35 3.5 Collapse and expanding state . 36 3.6 Nambu formulation for n point vortices . 40 3.7 Summary . 41 4 Atmospheric blockings 43 4.1 Atmospheric scales of circulation . 44 4.2 Relative point vortex equilibria . 46 4.3 Methods and data . 48 4.4 Atmospheric blockings . 52 v CONTENTS vi 4.4.1 Example 1: Omega-block over Russia 2010 . 55 4.4.2 Example 2: Omega-block over North Pacific 2011 . 57 4.5 Modes of disturbed equilibria . 57 4.6 Statistical corroboration . 58 4.7 Summary . 60 II Algebraic aspects of continuous fluid dynamics 63 5 Introduction part II 65 6 Group theory of vortex dynamics 71 6.1 Group theory of vortex dynamics . 71 6.2 Introduction to group theory . 75 6.2.1 Lie group . 87 6.2.2 Lie algebra . 90 6.3 Summary . 96 7 Continuous Nambu mechanics 99 7.1 Nambu formulation of 2D vortex dynamics . 99 7.1.1 Casimir functionals . 101 7.2 Nambu formulation of 3D vortex dynamics . 104 7.2.1 Equations of motion . 104 7.2.2 Formulating the Energy and the Helicity in terms of the vorticity vector . 105 7.2.3 The Nambu bracket of 3D vortex dynamics . 107 7.3 A Lie algebra for 3D vortex dynamics . 109 7.3.1 The law of Biot-Savart . 111 7.4 Summary . 114 8 The Vortex groups 115 8.1 Introduction . 115 8.1.1 The Heisenberg Algebra and Heisenberg group of mass point mechanics . 117 8.1.2 The concept of Quaternions . 121 8.2 A vortex algebra and group for 2D flows . 123 8.2.1 Introducing a matrix representation of vh(2) . 124 8.2.2 Derivation of the
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