The N -vortex problem on a rotating sphere. I Multi-frequency configurations

Author:

Newton Paul K1,Shokraneh Houman1

Affiliation:

1. Department of Aerospace & Mechanical Engineering and Department of Mathematics, University of Southern CaliforniaLos Angeles, CA 90089-1191, USA

Abstract

The problem of N -point vortices moving on a rotating unit sphere is considered. Through a sequence of linear coordinate transformations, which takes into account the orientation of the centre of vorticity vector with respect to the axis of rotation, we show how to reduce the problem to that on a non-rotating sphere, where the centre of vorticity vector is aligned with the z -axis. As a consequence, we prove that integrability on the rotating sphere is the same as on the non-rotating sphere, namely, the three-vortex problem on the rotating sphere is integrable for all vortex strengths, while the four-vortex problem is integrable in the special case where the centre of vorticity is zero. Rigid multi-frequency configurations that retain their shape while rotating about two independent axes with two independent frequencies are obtained, and necessary conditions for one- and two-frequency motions are derived. Examples including dipoles which exhibit global ‘wobbling’ and ‘tumbling’ dynamics, rings, and Platonic solid configurations are shown to undergo either periodic or quasi-periodic evolution on the rotating sphere depending on the ratio of the solid-body rotational frequency Ω to the rotational frequency ω associated with the rigid structure.

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

Reference38 articles.

1. Vortex Crystals

2. Nonintegrability of two problems in vortex dynamics

3. On the motion of a vortex on a rotating sphere;Bogomolov V.A;Izv. Atmos. Ocean. Phys,1985

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