A vortex ring on a sphere: the case of total vorticity equal to zero

The stability of a ring of vortices has attracted the interest of researchers for over a century. Recent beautiful observations of polygonal configurations of vortices present in the atmospheres of Jupiter and Saturn, and of polygonal jets in the Earth's atmosphere, have revived the interest in the subject. In the observed cases, the vortex ring is in the presence of a central vortex. We present analytical and numerical results about the linear, spectral and Lyapunov stability of a ring in the presence of polar vortices. Motivated by both atmospheric observations we considered the special case of total vorticity equal to zero. Such a case has also the very nice property of beinguniversal , i.e. not depending on a choice of gauge. We considered the two cases of fixed and non-fixed polar vortices. A ring in the northern (respectively, southern) hemisphere is stabilized by the presence of a northern (respectively, southern) polar vortex of suitable strength, in agreement with what is observed numerically and atmospherically.This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.