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Three-vortex quasi-geostrophic dynamics in a two-layer fluid. Part 1. Analysis of relative and absolute motions

Published online by Cambridge University Press:  01 February 2013

M. A. Sokolovskiy ,
K. V. Koshel  and
J. Verron
M. A. Sokolovskiy
Affiliation:
Water Problems Institute, RAS, 3, Gubkina Str., Moscow, 119991, Russia Southern Federal University, 8a Mil’chakova Str., Rostov-on-Don, 344090, Russia
K. V. Koshel*
Affiliation:
V. I. Il’ichev Pacific Oceanological Institute, FEB RAS, 43 Baltiyskaya Str., Vladivostok, 690041, Russia Far Eastern Federal University, 8 Sukhanova Str., Vladivostok, 690950, Russia
J. Verron
Affiliation:
Laboratoire des Ecoulements Géophysiques et Industriels, CNRS, BP 52, 38041, Grenoble, CEDEX 9, France
*
†Email address for correspondence: [email protected]
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Abstract

The results presented here examine the quasi-geostrophic dynamics of a point vortex structure with one upper-layer vortex and two identical bottom-layer vortices in a two-layer fluid. The problem of three vortices in a barotropic fluid is known to be integrable. This fundamental result is also valid in a stratified fluid, in particular a two-layer one. In this case, unlike the barotropic situation, vortices belonging to the same layer or to different layers interact according to different formulae. Previously, this occurrence has been poorly investigated. In the present work, the existence conditions for stable stationary (translational and rotational) collinear two-layer configurations of three vortices are obtained. Small disturbances of stationary configurations lead to periodic oscillations of the vortices about their undisturbed shapes. These oscillations occur along elliptical orbits up to the second order of the Hamiltonian expansion. Analytical expressions for the parameters of the corresponding ellipses and for oscillation frequencies are obtained. In the case of finite disturbances, vortex motion becomes more complicated. In this case we have made a classification of all possible movements, by analysing phase portraits in trilinear coordinates and by computing numerically the characteristic trajectories of the absolute and relative vortex motions.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Geophysical and Geological Flows: Quasi-geostrophic flows Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013 , pp. 232 - 254
Copyright
©2013 Cambridge University Press

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