
Vortex Streets on a SphereDOI: 10.1155/2011/712704 Abstract: We consider flows on a spherical surface and use a transformation to transport some wellknown periodic twodimensional vortex streets to that spherical surface to arrive at some new expressions for vortex streets on a sphere. 1. Introduction For fluid flow on a twodimensional plane, the vorticity at a point is twice the angular rotation. A point vortex is a model of a flow in which the vorticity is zero except at the point itself where the vorticity is infinite, so that there is a nonzero circulation around the point. The study of point vortices on the plane, and other twodimensional manifolds such as the cylinder, sphere, and torus, has a long history, dating back to the 19th century with Helmholtz [1] initiating the point vortex model and Kirchhoff [2] and Lin [3] formulating it as a Hamiltonian dynamical system. In this paper, we are concerned primarily with vortex streets, which consist of one or more periodic rows of point vortices, the simplest of which is a single infinite row of identical vortices [4]. These have important applications in engineering and geophysics, with a single row having been used to model the quasisteady largescale vortices arising following the rollup of a shear layer, and double rows, or von Kármán vortex streets [5], having been used to model the shedding of eddies behind a bluff body. An overview of twodimensional vortex streets can be found in standard texts on hydrodynamics such as IN [6, 7]. In this paper, we are interested in transporting these wellknown vortex streets from the plane to a curved twodimensional manifold, the surface of a sphere. Flows on a sphere are important because of applications to planetary atmospheres. In his classic monograph, Lamb [6] briefly outlines a method of determining the motion of vortices on a curved manifold and discusses how some of the 19th century work on electrical conduction, such as chargeonasphere problems, by Boltzmann, Kirchhoff, T？pler, and others could be applied to the problem of point vortices on the sphere although Gromeka [8] appears to have been the first to study vortices on a sphere specifically. More recently, the formulation of the motion of vortex streets on curved manifolds has been examined in more detail by Hally [9], with several subsequent studies [10–12] delving more deeply into the formulation of vortex motion on a sphere. A review of some of the work on point vortices on vortices on a sphere can be found in [13]. One interesting thread of research [14–16] has involved using numerical methods, such as contour surgery, to study the motion of
