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Vortex Streets on a Sphere

DOI: 10.1155/2011/712704

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Abstract:

We consider flows on a spherical surface and use a transformation to transport some well-known periodic two-dimensional 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 two-dimensional 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 two-dimensional 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 large-scale vortices arising following the roll-up 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 two-dimensional vortex streets can be found in standard texts on hydrodynamics such as IN [6, 7]. In this paper, we are interested in transporting these well-known vortex streets from the plane to a curved two-dimensional 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 charge-on-a-sphere 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

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