Stability of streets of vortices on surfaces of revolution with a reflection symmetry

Abstract

Helmholtz’s theory of ideal vortex motion in two dimensions is generalized to flows on curved surfaces. The existence of a generalized vortex stream function is proved and used to generate conservation laws. In particular, the angular moment of circulation is related to invariance under scale transformations. The theory is used to derive criteria for stability of vortex streets on surfaces of revolution having symmetry under reflection in a plane whose normal is the axis of revolution. For the special case of the sphere it is found that only those vortex streets having six or fewer vortices per ring can be stable and that, in contradistinction to the results of von Karman, both symmetric and staggered vortex streets can be stable.