▪ Abstract This article reviews the transport properties of coherent vortices in rotating barotropic flows. It is shown that vortices induce regular Lagrangian motion inside their cores and are highly impermeable to inward and outward particle fluxes. Passive tracers can be trapped inside vortex cores for long times and are transported by the vortex motion over large distances. Absolute dispersion in vortex-dominated flows is discussed by studying particle dynamics in 2D turbulence, point-vortex systems, and subsurface float trajectories in the ocean. Finally, it is shown that anticyclonic coherent vortices in cyclonically rotating reference frames can concentrate heavy impurities (e.g. dust grains) in their cores. This process may play an important role in the formation of planetesimals in the early solar nebula.

http://www.o3d.org/abracco/anto_review.pdf

Transport by coherent barotropic vortices

In Sect. 1.1 of this chapter we explain the subject of Geophysical Fluid Dynamics, and give the description of main vortex structures that have become objects of the present book: (a) Heton – a two-layer vortex with opposite rotations in different layers, and (b) Intrathermocline lens, which is studied in this work as a vortex patch in the intermediate layer of a three-layer ocean model. In this section, we propose also a short review of works in the related topics. Section 1.2 is a mathematical introduction. It contains all main formulae which are used further in Chaps. 2 and 3.

The Introductory Chapter

The dynamics of passively advected particles in either integrable or chaotic point vortex systems and in two‐dimensional (2‐D) turbulence is studied. For point vortices, it is shown that the regular or chaotic nature of the particle trajectories is not determined by the Eulerian chaoticity of the vortex motion, but rather by pure Lagrangian quantities, such as the distance of an advected particle from the vortex centers. In fact, each point vortex turns out to be surrounded by a regular island, where the advected particles are trapped and their Lagrangian Lyapunov exponent is zero, even though the vortex itself may perform a chaotic trajectory. In the field between the vortices, passive particles undergo chaotic advection with an associated positive Lyapunov exponent. For well‐separated vortices, even at large times, the advected particles do not cross the boundary between the chaotic sea and the regular islands surrounding the vortices. A similar situation holds in the case of forced‐dissipative 2‐D turbulence, where particles trapped in the interior of the coherent structures have a null Lagrangian Lyapunov exponent, while those in the background turbulent sea move chaotically. This gives clear evidence of the important role played by chaotic advection, even in complex Eulerian flows.

http://personalpages.to.infn.it/~boffetta/Papers/bbpv94.pdf

Chaotic advection in point vortex models and two-dimensional turbulence

Invariant manifolds can serve as a geometric template for the study of chaotic advection. In particular, global stable and unstable manifolds of invariant hyperbolic or normally hyperbolic sets form the boundaries of chaotic tangles in physical space, and these manifolds criss-cross one another to form an intricate network of lobes in the tangles. Manifold invariance implies that these lobes of fluid evolve from one to another in a well-defined manner, and it is in the context of lobe dynamics in the tangles that one exploits the invariant manifold template as a skeletal backbone for the study of transport, stretching, and mixing of fluid under chaotic advection, all of which are intimately connected. More concretely, in the study of fluid transport, one can use segments of the invariant manifolds to partition the irregular flow into regions of qualitatively different fluid motion, such as open flow versus closed flow, or different rolls, and then perform a lobe-dynamic study of fluid transport into, and out of, these regions (e.g. entrainment and detrainment) in a geometrically exact context, specified in terms of images and pre-images of a set of turnstile lobes. Mixing can be thought of as a consequence of barrier destruction, and transport across partially destroyed barriers can be studied in a lobe-dynamic context, providing a basic measure of mixing. A practical example of a transport calculation is the use of Melnikov theory to obtain analytical expressions for lobe areas, and hence a measure of flux in the mixing region, which allows for efficient searches through parameter space in order to enhance or diminish a mixing process. The stretching of fluid elements can be studied in the context of the lobes repeatedly stretching, folding, and wrapping around one another in an approximately self-similar manner. Such a picture can be viewed as a generalization of the horseshoe map paradigm for stretching in 2D chaotic tangles, which restricts its interest to a Cantor set near hyperbolic fixed points. A symbolic description of the evolution of lobe boundaries provides a framework for studying the global topology of, and mechanisms for, enhanced stretching in chaotic tangles. In particular, it is found that, though stretching under chaotic flows can be viewed in terms of products of weakly correlated events, there is a range of stretch, spatial, and temporal scales associated with these products, leading to a range of stretch processes. Invariant manifold geometry plays a central role in these stretch processes. For example, the invariant hyperbolic or normally hyperbolic sets have special relevance as engines of good stretching, the turnstile lobes play a fundamental role of re-orienting line elements between each successive pass by the hyperbolic sets, and intersections of stable and unstable manifolds act as partitions between segments of lobe boundaries that experience qualitatively different stretch histories. A notable consequence of the range of scales in the stretch processes is the range of statistics, and hence multifractal properties, associated with the high-stretch tails of finite-time Lyapunov exponent distributions, which has significant impact on interfacial evolution and striation width in the small-scale limit. Regions in physical space of vanishingly small initial measure associated with these tails are thus shown to be able to play a significant role under the flow. The mixing of passive scalars or vectors as a result of such stochastic effects as molecular diffusion or chemical reactions tends to wash out the structure associated with invariant manifolds. However, one can study this interplay between advection and mixing in the context of lobe evolution. In particular, mixing efficiency is seen to depend not only on the stretch experienced by lobe boundaries, but also on the thickness of the lobes, and their separation from other lobes. Though essentially all chaotic advection studies have been in the context of 2D time-periodic velocity fields, we show how an invariant manifold template can apply to quite general circumstances, such as quasiperiodic and indeed aperiodic time dependences, and 3D velocity fields. In all cases, the invariant manifold templates are seen to have physical reality, and the unstable manifolds act as a dominant structure in the regions of irregular flow.

Invariant manifold templates for chaotic advection

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

Dynamics of vortex structures in a stratified rotating fluid

Two‐dimensional inviscid flows governed by a vortex pair in the presence of another point vortex or vortex pair on an unbounded plane are considered analytically and via numerical simulations. For some integrable cases of vortex motion, the stirring process of a fixed closed volume of surrounding fluid (‘‘atmosphere’’) initially trapped by vortex pair is investigated. Using the full classification of vortex movement types, it is shown that for all cases of vortex pair direct and exchange scattering the stirring process is regular. Some internal atmosphere regions conserve their existence and form after vortex interaction, resulting a ‘‘solitonlike’’ behavior. For general cases of vortex pair mutual trapping, the stirring process is chaotic. For limiting cases of vortex motions, the fluid particles reveal a regular behavior. A simple model for the qualitative description of recent experiments on vortex dipoles interaction is discussed. Although clearly an extreme idealization, the model appears to shed some light on what to expect in laboratory experiments.

T. P. Konovaljuk

Papers

Advection of a vortex pair atmosphere in a velocity field of point vortices