Estimée Höldérienne pour le problème à 3 points vortex avec des modèles α (original) (raw)
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1988
Abstract: By generalizing the method of contour dynamics to the quasigeostrophic two layer model, we have proposed and solved a number of fundamental problems in the dynamics of rotating and stratified vorticity fields. A variety of rotating and translating potential vorticity equilibria (V-states) in one and two layers have been obtained, shedding new light on potential vorticity dynamics in the geostrophic context.
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Journal of Fluid Mechanics, 1999
An existence theorem for localised stationary vortex solutions in an external shear ow i s proved for three-dimensional quasigeostrophic ow i n a n u n bounded domain. The external ow is a linear shear ow whose strength varies linearly with height. The ow conserves an in nite family of Casimir integrals. Flows that have the same value of all Casimir integrals are called isovortical ows, and the potential vorticity-PV-elds of isovortical ows are strati ed rearrangements of one another. The theorem guarantees the existence of a maximum energy ow in any family of isovortical ows that statis es the following conditions: the PV-anomaly must have compact support, it must have the same sign everywhere, and this sign must be the same as the sign of the external shear over the vertical interval to which the support of the PV-anomaly is con ned. This ow represents a stationary and localised vortex, and the maximum-energy property implies that it is formally stable. Newton Institute preprint no. NI 96 019.
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Journal of Nonlinear Science, 2019
A two-layer quasigeostrophic model is considered. The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a circle of radius R in one of the layers is presented. The vortices have identical intensity and length scale is γ −1 > 0. The problem has three parameters: N , γ R and β, where β is the ratio of the fluid layer thicknesses. The stability of the stationary rotation is interpreted as orbital stability. The instability of the stationary rotation is instability of system reduced equilibrium. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The parameter space (N , γ R, β) is divided on three parts: A is the domain of stability in an exact nonlinear setting, B is the linear stability domain, where the stability problem requires the nonlinear analysis, and C is the instability domain. The case A takes place for N = 2, 3, 4 for all possible values of parameters γ R and β. In the case of N = 5, we have two domains: A and B. In the case N = 6, part B is curve, which divides the space of parameters (γ R, β) into the domains: A and C. In the case of N = 7, there are all three domains: A, B and C. The instability domain C takes place always if N = 2n 8. In the case of N = 2 + 1 9, there are two domains: B and C. The results of research are presented in two versions: for parameter β and parameter α, where α is the difference between layer thicknesses. A number of statements about the stability of the Thomson N -gon is obtained for the systems of interacting particles with the general Hamiltonian depending only on distances between the particles. The results of theoretical analysis are confirmed by numerical calculations of the vortex trajectories.