Dynamics of Vortex Structures in a Stratified Rotating Fluid

1. The Introductory Chapter

1.1 Introduction

1.2 The mathematical introduction

1.2.1 The derivation of potential vortex conservation equations

1.2.2 Formal solution. Integral invariants

1.2.3 Contour dynamics method

1.2.4 Stationary axisymmetric solution

1.2.5 An approach to studying the stability of a axisymmetric two-layer vortex

1.2.6 The structure of simplest types of external field

1.2.7 A limiting case of discrete vortices

1.2.8 Phase portraits. Choreographies

1.2.9 Three-layer model equations

2. Dynamics of Discrete Vortices

2.1 Two vortices in a two-layer fluid

2.2 2A vortices in a two-layer fluid

2.2.1 The case of arbitrary A

2.2.2 Case A = 2

2.2.2.1 Two hetons with zero total linear momentum and nonzero angular momentum

2.2.2.2 Two hetons with nonzero total linear momentum and zero angular momentum

2.2.2.3 Two hetons with zero total linear and angular momenta

2.2.2.4 Vortex structures: warm heton-cold heton, two antihetons, two "horizontal" pairs

2.3 A + 1 vortices in a two-layer fluid

2.3.1 Vortex structures with zero total momentum at A 2 (free motion)

2.3.2 Vortex structures with zero total momentum at A 2 (motion in an external field)

2.3.2.1 Analysis of static states

2.3.2.2 Stationary solutions at A = 2

2.3.2.3 Analysis of optimal perturbation frequencies at A = 2

2.3.2.4 Origination of chaos at A = 2

2.3.3 The case of nonzero total momentum at A = 2

2.3.3.1 Phase portraits in trilinear coordinates

2.3.3.2 Analysis of steady states

2.3.3.3 Classification of motions of triangular vortex structures: trajectories of absolute motion, choreographies

2.3.3.4 Analysis of weakly perturbed tripolar collinear states

2.3.3.5 Regular advection near stationary configurations

2.3.3.6 Chaotic advection near stationary configurations

2.4 Heton structures in a three-layer fluid

3 Dynamics of Finite-core Vortices

3.1 Studying the linear stability of a two-layer vortex

3.1.1 A vortex with a vertical axis: two circular vortex patches

3.1.1.1 Heton with vertical axis

3.1.1.2 "Ballistic" propagation ; law of vortex domain boundary: application to deep convection in the ocean

3.1.1.3 Analogy with A-symmetric structure of discrete hetons

3.1.1.4 Noncompensated two-layer vortex

3.1.2 Annular two-layer vortex: four circular vortex patches

3.1.2.1 Studying the stability of rings

3.1.2.2 Modeling the transformations of an oceanic ring into

smaller vortex structures

3.2 The impact of finite perturbations

3.2.1 Heton with a tilted axis: two initially circular patches

3.2.2 Stationary translation hetonic V-states

3.2.3 Heton with a vertical axis: two initially elliptic vortex patches

3.3 Interaction between two hetons

3.3.1 Two hetons with vertical axes

3.3.2 Heton with a vertical axis and heton with a tilted axis

3.3.3 Two hetons with tilted axes, the case of zero total momentum

3.3.4 Two hetons with tilted axes, the case of nonzero total momentum

3.3.5 Interaction between a warm and a cold hetons

3.4 The effect of external field on heton motion

3.5 Vortex patch dynamics in a three-layer model

3.5.1 Stability study of a three-layer vortex

3.5.2 Modeling the motion of meddies

3.5.2.1 Merging of two initially circular vortex patches

3.5.2.2 Evolution of elliptic vortex patch

3.5.2.3 On detecting lenses on oceanic surface

3.5.2.4 On the effect of bottom topography on the motion of lenses

3.5.2.5 Dynamics of medies in the flow over submerged hills

3.5.3 Examples of interaction between three-layer vortices

4 The Concluding Chapter

4.1 Concluding remarks

4.2 Outlook to heton problems

4.3 Discussion

Appendix A. E.J. Hopfinger. Experimental study of hetons

Appendix B. M.A. Sokolovskiy. In memory of my Teacher

Index