Abstract:
We determine and characterise relative equilibria for arrays of point vortices in a
three-dimensional quasi-geostrophic flow. The vortices are equally spaced along two horizontal
rings whose centre lies on the same vertical axis. An additional vortex may be placed along
this vertical axis. Depending on the parameters defining the array, the vortices on the two rings
are of equal or opposite sign. We address the linear stability of the point vortex arrays. We
find both stable equilibria and unstable equilibria, depending on the geometry of the array. For
unstable arrays, the instability may lead to the quasi-regular or to the chaotic motion of the
point vortices. The linear stability of the vortex arrays depends on the number of vortices in
the array, on the radius ratio between the two rings, on the vertical offset between the rings
and on the vertical offset between the rings and the central vortex, when the latter is present.
In this case the linear stability also depends on the strength of the central vortex. The nonlinear
evolution of a selection of unstable cases is presented exhibiting examples of quasi-regular
motion and of chaotic motion.
Keywords:quasi-geostrophy, point vortex dynamics, equilibria, vortex arrays.
References