Interacting dipole pairs on a rotating sphere

Paul K Newton, Houman Shokraneh

Abstract

The evolution, interaction and scattering of 2N point vortices grouped into equal and opposite pairs (N-dipoles) on a rotating unit sphere are studied. A new coordinate system made up of centres of vorticity and centroids associated with each dipole is introduced. With these coordinates, the nonlinear equations for an isolated dipole diagonalize and one directly obtains the equation for geodesic motion on the sphere for the dipole centroid. When two or more dipoles interact, the equations are viewed as an interacting billiard system on the sphere—charged billiards—with long-range interactions causing the centroid trajectories to deviate from their geodesic paths. Canonical interactions are studied both with and without rotation. For two dipoles, the four basic interactions are described as exchange-scattering, non-exchange-scattering, loop-scattering (head on) and loop-scattering (chasing) interactions. For three or more dipoles, one obtains a richer variety of interactions, although the interactions identified in the two-dipole case remain fundamental.

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Footnotes

    • Received September 7, 2007.
    • Accepted February 6, 2008.
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