Asymmetric vortex merger: mechanism and criterion

The merging of two unequal co-rotating vortices in a viscous fluid is investigated. Two-dimensional numerical simulations of initially equal sized Lamb-Oseen vortices with differing relative strengths are performed. Results show how the disparity in deformation rates between the vortices alters the interaction. Key physical mechanisms associated with vortex merging are identified. A merging criterion is formulated in terms of the relative timing of core detrainment and destruction. A critical strain parameter is defined to characterize the establishment of core detrainment. This parameter is shown to be directly related to the critical aspect ratio in the case of symmetric merger

Offsprings of a point vortex

The distribution engendered by successive splitting of one point vortex are considered. The process of splitting a vortex in three using a reverse three-point vortex collapse course is analysed in great details and shown to be dissipative. A simple process of successive splitting is then defined and the resulting vorticity distribution and vortex populations are analysed

Collision of dipolar vortices on a beta-plane

The interaction of two dipoles moving perpendicularly to the gradient of background vorticity is studied both numerically and experimentally. In the numerical computations the vorticity distribution is represented either by four point vortices (point-vortex model) or by thousands of them (vortex-in-cell method). The simplest model is used to study the dynamics and the advection of fluid particles in two kinds of interaction: coaxial couples of equal strength but different size, and equal parallel couples with a nonzero impact parameter (the distance between the dipoles’ axis). As a result of the interaction fluid masses are exchanged between the two dipoles and between each dipole and the ambient fluid. In the case of equal coaxial couples the amount of fluid exchanged depends on the gradient of ambient vorticity, with the largest mass exchange occurring always between the eastward traveling dipole and the ambient fluid. The collision of parallel couples with nonzero impact parameter leads to a large mass exchange, either because several interactions may occur or because when two independent couples arise, they have a nonuniform motion. Laboratory experiments in a rotating fluid (with a flat sloping bottom providing the ß effect), confirm that an elastic interaction is a rare event. The unstable trajectory of the westward traveling dipole, as well as small perturbations unavoidable in the laboratory, invariably lead to collisions of nonaligned dipoles. The gross features of the vortex motion, as well as of the mass exchange, are well modeled using the point-vortex model, whereas the vortex-in-cell method reproduces many details of the vortex motion, the evolution of the vorticity field, and the exchange of mass

Collision of dipolar vortices on a beta-plane

The interaction of two dipoles moving perpendicularly to the gradient of background vorticity is studied both numerically and experimentally. In the numerical computations the vorticity distribution is represented either by four point vortices (point-vortex model) or by thousands of them (vortex-in-cell method). The simplest model is used to study the dynamics and the advection of fluid particles in two kinds of interaction: coaxial couples of equal strength but different size, and equal parallel couples with a nonzero impact parameter (the distance between the dipoles’ axis). As a result of the interaction fluid masses are exchanged between the two dipoles and between each dipole and the ambient fluid. In the case of equal coaxial couples the amount of fluid exchanged depends on the gradient of ambient vorticity, with the largest mass exchange occurring always between the eastward traveling dipole and the ambient fluid. The collision of parallel couples with nonzero impact parameter leads to a large mass exchange, either because several interactions may occur or because when two independent couples arise, they have a nonuniform motion. Laboratory experiments in a rotating fluid (with a flat sloping bottom providing the ß effect), confirm that an elastic interaction is a rare event. The unstable trajectory of the westward traveling dipole, as well as small perturbations unavoidable in the laboratory, invariably lead to collisions of nonaligned dipoles. The gross features of the vortex motion, as well as of the mass exchange, are well modeled using the point-vortex model, whereas the vortex-in-cell method reproduces many details of the vortex motion, the evolution of the vorticity field, and the exchange of mass

Experimental study of dipolar vortices on a topographic β-plane

The behaviour of dipolar vortices in a rotating fluid with a sloping bottom (simulating the variation of the Coriolis parameter on the Earth, with the direction of steepest bottom slope corresponding with the northern direction) has been investigated in the laboratory. Dipoles were generated by moving a vertical cylinder through the fluid. Dye photographs provided qualitative information, whereas quantitative information about the evolving flow field was obtained by streak photography. Dipoles initially directed under a certain angle relative to the west-east axis showed meandering or cycloid-like trajectories. Some asymmetries between east-travelling dipoles (ETD’s) and west-travelling dipoles (WTD’s) were observed. ETD’s are stable in the trajectory sense: a small deviation from zonal motion results in small oscillations around the equilibrium latitude. WTD’s are unstable: small initial deviations produce large displacements in northern or southern directions. This asymmetry arises because the vorticity of a dipole moving westward is anticorrelated with the ambient vorticity, while the vorticities are correlated when the dipole moves eastward. ETD’s increase in size and eventually split into two independent monopoles, the rate of growth depending on the gradient of planetary vorticity. WTD’s are initially more compact but owing to the large displacements in the meridional direction strong asymmetries in the circulation of the two halves are produced, resulting in a large deformation of the weaker part. The experimental observations show good qualitative agreement with analytical and numerical results obtained using a modulated point-vortex model

Chaotic transport bij dipolar vortices on a B-plane.

During the meandering motion of a dipolar vortex on a ß-plane mass is exchanged both between the dipole and the ambient fluid and between the two dipole halves. The mass exchange (as well as the meandering motion) is caused by variations of the relative vorticity of the vortices due to conservation of potential vorticity. Previous studies have shown that a modulated point-vortex model captures the essential features in the dipole evolution. For this model we write the equations of motion of passive tracers in the form of a periodically perturbed integrable Hamiltonian system and subsequently study transport using a ‘dynamical-systems theory’ approach. The amount of mass exchanged between different regions of the flow is evaluated as a function of two parameters: the gradient of ambient vorticity, ß, and the initial direction of propagation of the dipole, a0. Mass exchange between the dipole and the surroundings increases with increasing both ß and a0. The exchange rate (amount of mass exchanged per unit time) increases with ß and has a maximum for a particular value of a0 ([approximate] 0.62p). Dipolar vortices in a rotating fluid (with a sloping bottom providing the ‘topographic’ ß-effect) show, in addition to the relative vorticity variations, a second perturbation that leads to exchange of mass. The points where vorticity is extreme approach each other as the dipole moves to shallower parts of the fluid and separate as the couple moves to deeper parts. This mechanism is studied independently and it is shown to lead to a stronger exchange between the dipole halves and the ambient fluid but no exchange between the two dipole halves

Unsteady behaviour of a topography-modulated tripole

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