Equilibria and stability of four point vortices on a sphere

David Gerard Dritschel

doi:10.1098/rspa.2020.0344

Equilibria and stability of four point vortices on a sphere

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Abstract

This paper discusses the problem of ﬁnding the equilibrium positions of four point vortices, of generally unequal circulations, on the surface of asphere. A random search method is developed which utilises a modiﬁcation of the linearised equations to converge on distinct equilibria. Many equilibria (47 and possibly more) may exist for prescribed circulations and angular impulse. A linear stability analysis indicates that they are generally unstable,though stable equilibria do exist. Overall, there is a surprising diversity of equilibria, including those which rotate about an axis opposite to the angular impulse vector.

Original language	English
Article number	20200344
Number of pages	26
Journal	Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences
Volume	476
Issue number	2241
Early online date	30 Sept 2020
DOIs	https://doi.org/10.1098/rspa.2020.0344
Publication status	Published - 30 Sept 2020

Keywords

Vortex dynamics
Surfaces
Point vortices

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10.1098/rspa.2020.0344

Dritschel-2020-Equilibria-and-stability-PTRSA-AAM
Copyright © 2020 the Author(s). Published by the Royal Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1098/rspa.2020.0344.
Accepted author manuscript, 20.5 MBLicence: CC BY

Geophysical Vortices: The Structure, stability and interaction of geophysical vortices
Reinaud, J. N. (PI), Dritschel, D. G. (CoI) & Scott, R. K. (CoI)
EPSRC
5/01/10 → 1/11/13
Project: Standard

Cite this

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abstract = "This paper discusses the problem of ﬁnding the equilibrium positions of four point vortices, of generally unequal circulations, on the surface of asphere. A random search method is developed which utilises a modiﬁcation of the linearised equations to converge on distinct equilibria. Many equilibria (47 and possibly more) may exist for prescribed circulations and angular impulse. A linear stability analysis indicates that they are generally unstable,though stable equilibria do exist. Overall, there is a surprising diversity of equilibria, including those which rotate about an axis opposite to the angular impulse vector.",

keywords = "Vortex dynamics, Surfaces, Point vortices",

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note = "Funding: UK Engineering and Physical Sciences Research Council (grant EP/H001794/1).",

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N1 - Funding: UK Engineering and Physical Sciences Research Council (grant EP/H001794/1).

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Y1 - 2020/9/30

N2 - This paper discusses the problem of ﬁnding the equilibrium positions of four point vortices, of generally unequal circulations, on the surface of asphere. A random search method is developed which utilises a modiﬁcation of the linearised equations to converge on distinct equilibria. Many equilibria (47 and possibly more) may exist for prescribed circulations and angular impulse. A linear stability analysis indicates that they are generally unstable,though stable equilibria do exist. Overall, there is a surprising diversity of equilibria, including those which rotate about an axis opposite to the angular impulse vector.

AB - This paper discusses the problem of ﬁnding the equilibrium positions of four point vortices, of generally unequal circulations, on the surface of asphere. A random search method is developed which utilises a modiﬁcation of the linearised equations to converge on distinct equilibria. Many equilibria (47 and possibly more) may exist for prescribed circulations and angular impulse. A linear stability analysis indicates that they are generally unstable,though stable equilibria do exist. Overall, there is a surprising diversity of equilibria, including those which rotate about an axis opposite to the angular impulse vector.

KW - Vortex dynamics

KW - Surfaces

KW - Point vortices

U2 - 10.1098/rspa.2020.0344

DO - 10.1098/rspa.2020.0344

M3 - Article

SN - 1364-5021

VL - 476

JO - Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences

JF - Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences

IS - 2241

M1 - 20200344

ER -

Equilibria and stability of four point vortices on a sphere

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Geophysical Vortices: The Structure, stability and interaction of geophysical vortices

Cite this