Vortex Dynamics on a Cylinder

Spatially periodic rows of point vortices in a 2-dimensional ideal fluid have long attracted the attention of fluid dynamicists, one of the earliest and the most popular instances being Kármán’s vortex street [6], [15, photos 94–98]. The general problem is as follows: analyse the motion of an infinite configuration consisting of vortices z1, . . . , zN ∈ C with vorticities Γ1, . . . ,ΓN ∈ R together with their translates {zk + 2πrm | k = 1, . . . , N,m ∈ Z}, where 2πr > 0 is the spatial period of translation. Traditionally the problem is analysed on the plane C, but in this paper we place the vortices on a cylinder C/2πrZ (fig. 1). Though the two pictures—periodic planar