Stability and evolution of uniform-vorticity dipoles

Makarov, Viatcheslav Georgievich; Kizner, Ziv

doi:10.1017/s0022112010006026

Cited by 22 publications

(22 citation statements)

References 22 publications

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“…Given the recent results of Makarov & Kizner (2011) it is unsurprising that, in the case of the Pierrehumbert family, we found no evidence of detrainment of rotational fluid into a trailing vortex tail, even when thick-cored members of the family were subjected to the largest of the perturbations considered (δ = −0.05). Figure 10 depicts the evolution of a Pierrehumbert vortex pair with α = 1.2 under a perturbation of δ = −0.05.…”

Section: Response Of the Pierrehumbert Family Of Vortex Pairsmentioning

confidence: 57%

“…Dritschel (1995) examined the linear stability of the family of dipoles, and used contour dynamics to find the nonlinear stability bounds for asymmetric perturbations. Recently, Makarov & Kizner (2011) used contour dynamics methods to show that all members of the Pierrehumbert family are stable with respect to symmetric perturbations. However, the nonlinear response of this family to prolate perturbations of the type described in § 2 has not been previously reported.…”

Section: Response Of the Pierrehumbert Family Of Vortex Pairsmentioning

confidence: 99%

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Perturbation response and pinch-off of vortex rings and dipoles

O’Farrell

Dabiri

2012

J. Fluid Mech.

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The nonlinear perturbation response of two families of vortices, the Norbury family of axisymmetric vortex rings and the Pierrehumbert family of two-dimensional vortex pairs, is considered. Members of both families are subjected to prolate shape perturbations similar to those previously introduced to Hill's spherical vortex, and their response is computed using contour dynamics algorithms. The response of the entire Norbury family to this class of perturbations is considered, in order to bridge the gap between past observations of the behaviour of thin-cored members of the family and that of Hill's spherical vortex. The behaviour of the Norbury family is contrasted with the response of the analogous two-dimensional family of Pierrehumbert vortex pairs. It is found that the Norbury family exhibits a change in perturbation response as members of the family with progressively thicker cores are considered. Thin-cored vortices are found to undergo quasi-periodic deformations of the core shape, but detrain no circulation into their wake. In contrast, thicker-cored Norbury vortices are found to detrain excess rotational fluid into a trailing vortex tail. This behaviour is found to be in agreement with previous results for Hill's spherical vortex, as well as with observations of pinch-off of experimentally generated vortex rings at long formation times. In contrast, the detrainment of circulation that is characteristic of pinch-off is not observed for Pierrehumbert vortex pairs of any core size. These observations are in agreement with recent studies that contrast the formation of vortices in two and three dimensions. We hypothesize that transitions in vortex formation, such as those occurring between wake shedding modes and in vortex pinch-off more generally, might be understood and possibly predicted based on the observed perturbation responses of forming vortex rings or dipoles.

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Section: Response Of the Pierrehumbert Family Of Vortex Pairsmentioning

confidence: 57%

Section: Response Of the Pierrehumbert Family Of Vortex Pairsmentioning

confidence: 99%

Perturbation response and pinch-off of vortex rings and dipoles

O’Farrell

Dabiri

2012

J. Fluid Mech.

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“…However, all of these states are linearly unstable (Dritschel et al 2018). Opposite-signed vortex patch equilibria having both unequal area and vorticity were obtained by Makarov and Kizner (2011). These states generally rotate about a common point, e.g.…”

Section: Introductionmentioning

confidence: 99%

The interaction of two asymmetric quasi-geostrophic vortex patches

Jalali

Dritschel

2018

Geophysical & Astrophysical Fluid Dynamics

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Herein we study the general interaction of two vortex patches in a single-layer quasi-geostrophic shallowwater flow. Steadily-rotating equilibrium states are found over a wide parameter space spanning the Rossby deformation length, vortex area ratio, potential vorticity ratio, and gap between their innermost edges. A linear stability analysis is then used to identify the critical gap separating stable and unstable solutions, over the entire range of area and potential vorticity ratios, and for selected values of the Rossby deformation length. A representative set of marginally unstable equilibrium states are then slightly perturbed and evolved by an accurate contour dynamics numerical algorithm to understand the long-term fate of the instabilities. Not all instabilities lead to vortex merger; many in fact are characterised by weak filamentation and a small adjustment of the vortex shapes, without merger. Stronger instabilities lead to material being torn from one vortex and either wrapped around the other or reduced to ever thinning filamentary debris. A portion of the vortex may survive, or it may be completely strained out by the other.

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“…In numerical simulations, finite-core uniform-vorticity tripoles exhibit stable behavior in response to moderate perturbations that do not preserve the total circulation. 31 However, the analytical approach presented above does not apply to point-vortex tripoles perturbed in such a way.…”

Section: Perturbed Circulationsmentioning

confidence: 99%

Stability of point-vortex multipoles revisited

Kizner

2011

Physics of Fluids

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The point-vortex tripoles and pentapoles with zero total circulation are considered in the rigid-lid barotropic, equivalent-barotropic, and quasigeostrophic two-layer models. A tripole is assembled by three symmetrically arranged collinear vortices, while a pentapole by five vortices, of which one is located at the center of a square and four in the vertices of the square. The vortices on the sides, termed satellite vortices, are equal in strength and opposite in sign to the central vortex. To fulfill the zero-total-circulation condition, the central vortex is taken to be twice as strong as each of the satellite vortices in a tripole and four times as strong in a pentapole. In the two-layer model, two cases are distinguished, namely, the flat multipoles whose vortices are all located in the same layer and the carousel multipoles whose central vortex and satellite vortices reside in different layers. In all the models, the tripoles are shown to be nonlinearly stable and pentapoles, generally, unstable. Carousel pentapoles comparable in their size with the Rossby radius, and smaller, are exceptional in that they are stable to centrally symmetric perturbations (and, presumably, to arbitrary perturbations). The simple proof of the tripole stability is based on the fact that among the possible three-vortex configurations with zero total circulation characterized by the same (fixed) value of the Hamiltonian, there exists only one tripole, and, within the iso-Hamiltonian sheet, the squared linear momentum vanishes at this unique tripole only. This approach, being in essence universal for all models, works only with tripoles. For instance, a quadrupole cannot be treated in such a way, because there is a continuum of configurations of four vortices with zero total circulation on which the squared impulse vanishes. Dealing with pentapoles, we consider the perturbations that do not violate the central symmetry of the vortex configuration, fix the angular momentum, and examine the second derivatives of the Hamiltonian on the iso-momentum sheet.

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Stability and evolution of uniform-vorticity dipoles

Cited by 22 publications

References 22 publications

Perturbation response and pinch-off of vortex rings and dipoles

Perturbation response and pinch-off of vortex rings and dipoles

The interaction of two asymmetric quasi-geostrophic vortex patches

Stability of point-vortex multipoles revisited

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