Some properties of motion of A+1 vortices in a two-layer rotating fluid

Sokolovskiy, Mikhail; Verron, Jacques

doi:10.20537/nd0601002

Cited by 4 publications

(5 citation statements)

References 24 publications

(28 reference statements)

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“…The other two elliptic points correspond to nontrivial collinear states representing a uniformly moving vortex triplet. Such a structure was called a triton in [188], [189], [190], [191]. The existence condition for a triton is specified by the dispersion equation 20) and the velocity of its motion by the equality…”

Section: Nonzero Total Momentummentioning

confidence: 99%

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Gryanik¹,

Sokolovskiy²,

Verron³

2006

REG CHAOT DYN

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Studies of the properties of vortex motions in a stably stratified and fast rotating fluid that can be described by the equation for the evolution of a potential vortex in the quasi-geostrophic approximation are reviewed. Special attention is paid to the vortices with zero total intensity (the so-called hetons). The problems considered include self-motion of discrete hetons, the stability of a solitary distributed heton, and the interaction between two finite-core hetons. New solutions to the problems of three or more discrete vortices with a heton structure are proposed. The existence of chaotic regimes is revealed. The range of applications of the heton theory and the prospects for its future application, particularly in respect, to the analysis of the dynamic stage in the development of deep ocean convection, are discussed.

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Section: Nonzero Total Momentummentioning

confidence: 99%

Untitled

Gryanik¹,

Sokolovskiy²,

Verron³

2006

REG CHAOT DYN

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“…Note that, among carousel pentapoles, there is a static one for which l ¼ l ðPÞ S % 1:221; the pentapoles with l > l ðPÞ S rotate counterclockwise, and those with l < l ðPÞ S , clockwise. 24 So we can state that, in the carousel case, all counterclockwise pentapoles and the clockwise ones at l 0 < l < l ðPÞ S are unstable, whereas clockwise pentapoles with l < l 0 are stable to symmetric perturbations.…”

Section: Instability Of the Pentapolementioning

confidence: 93%

“…Ensembles of the first kind will be termed flat. Symmetric constantly rotating ensembles of the second kind were named carousels 24 or roundabouts 25 (in a later, concise English version of the same article); we will use the term "carousel. "…”

Section: B Ensembles Of Point Vorticesmentioning

confidence: 99%

“…On the uneven tripoles, the Hamiltonian H ¼ Hðr AB ; r AC ; r BC Þ ¼ 2ð1 þ eÞ ln r AB þ 2ð1 À eÞ ln r AC À ð1 À e 2 Þ ln r BC (24) becomes a monotonic function of the half-distance l between the satellite vortices B and C FðlÞ ¼ H½ð1 À eÞl; ð1 þ eÞl;2l ¼ ð3 þ e 2 Þ lnl þ const: (25) Thus, an even tripole is stable according to the generalized definition of stability. In numerical simulations, finite-core uniform-vorticity tripoles exhibit stable behavior in response to moderate perturbations that do not preserve the total circulation.…”

Section: Perturbed Circulationsmentioning

confidence: 99%

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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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“…0 ¼ s 0 ¼ 0) of the stationary problem was considered in the study by Sokolovskiy and Verron (2006); they showed that for any A, there exists a critical value cr 50, such that an appropriate solution exists whatever cr . The steady motion of the multipolar structure is a cyclonic rotation for 5 0 (a motion called inverse roundabout for which intralayer interaction is dominant); for ¼ 0 , steady states are found.…”

Section: Mathematical and Physical Frameworkmentioning

confidence: 99%

Baroclinic multipole evolution in shear and strain

Sokolovskiy

Koshel

Carton³

2010

Geophysical & Astrophysical Fluid Dynamics

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In a two-layer quasi-geostrophic model, the evolution of a symmetric baroclinic multipole, composed of a central vortex with strength in the upper layer, and A satellites with strength in the lower layer, is studied. This multipole is imbedded in a center-symmetric shear/strain field, either steady or time-periodic. Special attention is given to the case of the tripole (A ¼ 2). The stability of this tripole is assessed and its oscillations in the external field are analyzed. Conditions for resonance of these oscillations are derived and transition to chaos is described.

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Some properties of motion of A+1 vortices in a two-layer rotating fluid

Cited by 4 publications

References 24 publications

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Stability of point-vortex multipoles revisited

Baroclinic multipole evolution in shear and strain

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