Rotating multipoles on the f- and γ-planes

Kizner, Ziv; Khvoles, R.; McWilliams, James C.

doi:10.1063/1.2432915

Cited by 27 publications

(30 citation statements)

References 43 publications

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“…6, 8-10 and the references therein). In conformity with the experiments, our most recent numerical results (that will be published elsewhere) indicate that the tripole, quadrupole, and higher semi-analytical multipolar solutions with zero total circulation [8][9][10] are stable, feebly stable, and unstable, respectively. For all that, in studying the stability of coherent vortical structures, any progress achieved analytically (even if owing to serious simplifications) is always appreciated.…”

Section: Introductionsupporting

confidence: 81%

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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Section: Introductionsupporting

confidence: 81%

Stability of point-vortex multipoles revisited

Kizner

2011

Physics of Fluids

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“…It was shown that these semianalytical solutions reproduce the main features of experimental tripoles 13,14 and numerical quadrupoles 29 concerning their shape, flow pattern, and the form of the vorticitystreamfunction relationship. This work was later extended by Kizner et al 30 in order to construct semianalytical solutions for multipolar vortices with more satellite vortices and to incorporate the quasigeostrophic equivalent-barotropic ͑1 1 2 -layer͒ and ␥-plane approximations.…”

Section: Introductionmentioning

confidence: 99%

“…Also, evidence of the onset of hexapolar and heptapolar vortices was found. The experimentally produced multipoles will be compared with the semianalytical solutions for multipolar vortices obtained by Kizner et al, 30 with the emphasis put on the form of the vorticity versus streamfunction relation and the crosssectional distribution of vorticity.…”

Section: Introductionmentioning

confidence: 99%

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Laboratory experiments on multipolar vortices in a rotating fluid

2010

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The instability properties of isolated monopolar vortices have been investigated experimentally and the corresponding multipolar quasisteady states have been compared with semianalytical vorticity-distributed solutions to the Euler equations in two dimensions. A novel experimental technique was introduced to generate unstable monopolar vortices whose nonlinear evolution resulted in the formation of multipolar vortices. Dye-visualization and particle imaging techniques revealed the existence of tripolar, quadrupolar, and pentapolar vortices. Also evidence was found of the onset of hexapolar and heptapolar vortices. The observed multipolar vortices were found to be unstable and generally broke up into multipolar vortices of lesser complexity. The characteristic flow properties of the quadrupolar vortex were in close agreement with the semianalytical model solutions. Higher-order multipolar vortices were observed to be susceptible to strong inertial oscillations.

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“…15,16 More recently, numerical and laboratory experiments [17][18][19][20][21][22][23][24][25][26] indicate the existence of coherent rotating tripolar vortex structures, and higher connectivity analogs have also been found numerically. 27,28 Further analytical studies of the vortex equilibrium problem can be found in other studies.…”

Section: Introductionmentioning

confidence: 99%

New families of vortex patch equilibria for the two-dimensional Euler equations

2017

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Various modified forms of contour dynamics are used to compute multipolar vortex equilibria, i.e., configurations of constant vorticity patches which are invariant in a steady rotating frame. There are two distinct solution families for “N + 1” point vortex-vortex patch equilibria in which a finite-area central patch is surrounded by N identical point vortices: one with the central patch having opposite-signed vorticity and the other having same-signed vorticity to the satellite vortices. Each solution family exhibits limiting states beyond which no equilibria can be found. At the limiting state, the central patch of a same-signed equilibrium acquires N corners on its boundary. The limiting states of the opposite-signed equilibria have cusp-like behaviour on the boundary of the central patch. Linear stability analysis reveals that the central patch is most linearly unstable as it approaches the limiting states. For equilibria comprising a central patch surrounded by N identical finite-area satellite patches, again two distinct families of solutions exist: one with the central patch and satellite patches having the same-signed vorticity and the other in which they are opposite-signed. In each family, there are two limiting behaviours in which either the central patch or the satellite patches develop corners or cusps. Streamline plots and time-dependent simulations indicate that opposite-signed multipolar equilibria are robust structures and same-signed equilibria are generally less stable. Streamlines also reveal stable and unstable (saddle point) stagnation points, indicating the existence of new equilibria in which additional patches of vorticity are “grown” at the stagnation points. Examples of such equilibria are computed, and a general numerical routine is briefly described for finding even more complex finite-area equilibria. Finally, new nested polygonal vortex equilibria consisting of two sets of polygonally arranged vortex patches (named “N + N” equilibria here) are computed for two distinct cases: one with the corners of the polygons aligned with each other and the other when they are staggered. Various limiting states are computed for these equilibria. Time-dependent simulations reveal that the aligned equilibria are susceptible to instability, while the staggered equilibria survive a relatively long time. In some parameter regimes, following instability, these structures evolve into known structures such as “N + 1” multipolar vortex equilibria and N-polygon co-rotating equilibria.

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Rotating multipoles on the f- and γ-planes

Cited by 27 publications

References 43 publications

Stability of point-vortex multipoles revisited

Stability of point-vortex multipoles revisited

Laboratory experiments on multipolar vortices in a rotating fluid

New families of vortex patch equilibria for the two-dimensional Euler equations

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