Collapse of three vortices on a sphere

Kidambi, R. and Newton, P. (1999) Collapse of three vortices on a sphere. Il nuovo cimento C, 22 C (6). pp. 779-792. ISSN 1826-9885

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The self-similar collapse of three point vortices moving on the surface of a sphere of radius R is analysed and compared with known results from the corresponding planar problem described in (AREF H., Motion of three vortices, Phys. Fluids, 22 (1979) 393-400; NOVIKOV E. A., Dynamics and statistics of a system of vortices, Sov. Phys. JETP, 41 (1975) 937-943; NOVIKOV E. A. and SEDOV Y., Vortex collapse, Sov. Phys. JETP, 50 (1979) 297-301; SYNGE J. L., On the motion of three vortices, Can. J. Math., 1 (1949) 257-270). An important conserved quantity is the center of vorticity vector c4(!i41 3 Gi xi )O!i41 3 Gi, which must have length R for collapse to occur. Collapse trajectories occur in pairs, called “partner states”, which have two distinct collapse times t2Et1. The collapse time that is achieved for a given configuration depends on the sign of the parallelpiped volume formed by the vortex position vectors, hence depends on whether the vortices (G1 , G2 , G3 ) are arranged in a right-handed or left-handed sense. From a given collapsing configuration, one can obtain the partner state by reversing the signs of the Gi’s, or, alternatively, by using a discrete symmetry associated with the initial configuration that leaves all relative distances unchanged, but reverses the sign of the parallelepiped volume. In the plane, there is only one collapse time associated with a given configuration—the partner state is one that expands self-similarly (AREF H., Motion of three vortices, Phys. Fluids, 22 (1979) 393-400). Formulas for the collapsing trajectories are derived and compared with the planar formulas. The collapse trajectories are then projected onto the stereographic plane where a new Hamiltonian system is derived governing the vortex motion. In this projected plane, the solutions are not self-similar. In the last section, the collapse process is studied using tri-linear coordinates, which reduces the system to a planar one.

Item Type: Article
Additional Information: Paper presented at the International Workshop on “Vortex Dynamics in Geophysical Flows”, Castro Marina (LE), Italy, 22-26 June 1998.
Uncontrolled Keywords: Turbulence and diffusion ; Rotational flow and vorticity ; Vortex dynamics ; Conference proceedings
Subjects: 500 Scienze naturali e Matematica > 550 Scienze della Terra > 551 Geologia, Idrologia, Meteorologia
Depositing User: Marina Spanti
Date Deposited: 11 Oct 2019 14:55
Last Modified: 11 Oct 2019 14:55

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