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

Using fully nonlinear simulations of the two-dimensional Navier–Stokes equations at large Reynolds number (Re), we bracket a threshold amplitude above which a perturbed Gaussian monopole will relax to a quasi-steady, rotating tripole, and below which will relax to an axisymmetric monopole. The resulting quasi-steady structures are robust to small perturbations. We propose a means of measuring the decay rate of disturbances to asymptotic vortical structures wherein streamlines and lines of constant vorticity correspond in some rotating or translating frame. These experiments support the hypothesis that small or moderate deviations from asymptotic structures decay through inviscid and viscous mixing.

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Quasi-steady monopole and tripole attractors for relaxing vortices

This paper reviews some aspects of topography effects on the dynamics of barotropic monopolar, dipolar and tripolar vortices in a rotating fluid. It is shown that the modulated point-vortex model (essentially based on conservation of potential vorticity) is capable of describing the flow evolution correctly, as can be concluded from comparisons with numerical simulations and laboratory observations.

Topography effects on vortices in a rotating fluid

This paper addresses the stirring properties of interacting two-dimensional vortex structures. It is well known from previous studies that essential features of dipole-dipole and dipole-monopole interactions, like partner exchange, are captured by simple point-vortex models of these structures. In order to investigate the mixing induced by such interactions, the method of contour kinematics is adopted here, which allows us to trace fluid particles that are initially confined within some closed contour. For example, by tracing the deformations of the separatrix enclosing the ‘atmosphere’ of a point-vortex dipole during the interaction with another single point vortex or vortex pair, the contour kinematics method provides information about the processes of entrainment and detrainment that take place during the interaction. Comparison of the modelling results with both full numerical simulations and laboratory experiments on dye-visualised vortex interactions in a stratified fluid reveals a surprisingly good qualitative agreement.

Interacting two-dimensional vortex structures: Point vortices, contour kinematics and stirring properties

The evolution of a tripolar vortex under the influence of a parabolic topography – like the free surface of a rotating fluid – is studied experimentally and with a point-vortex model. Laboratory experiments reveal that tripoles generated off-axis become asymmetric and the whole structure travels towards the centre of the tank along an anticyclonic spiral. During this translation the structure rotates quasi-periodically with the core pairing alternately with one of the satellites. An asymmetric point-vortex tripole (with the central vortex located at a distance [varepsilon] from the middle point of the configuration) displays a periodic motion which is qualitatively similar to the motion of the laboratory tripoles. The exchange of fluid between the three vortices as a function of the perturbation parameter [varepsilon] is studied using the lobe-dynamics technique. A point-vortex tripole modulated on the basis of conservation of potential vorticity reproduces quantitatively the trajectories of the individual vortices measured in the laboratory. As in the experiments, the model shows that fluid is strongly stirred in the region surrounding the vortex cores and that the tripole carries a finite amount of fluid.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211209600002X

Unsteady behaviour of a topography-modulated tripole

This paper reports on an experimental study of vortices in a stratified fluid. The vortices were generated by two different stirring devices, viz. a rotatting sphere and a rotating bent rod. It was found that the vortices created with the rotating sphere are mostly axisymmetric and stable, whereas the vortices produced with the bent rod generally show instabilities, under certain conditions leading to the formation of a tripolar vortex. This report concentrates on this tripolar structure and presents quantitative information about the flow obtained through streak photography of tracer particles.

R. Van Sluis

Papers

Formation of a tripolar vortex in a stratified fluid