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Leapfrogging vortex rings in the Landau–Lifshitz equation

Niemi, A.J.; Sutcliffe, P.M.

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A.J. Niemi


Vortex rings are ubiquitous in fluids, with smoke rings being a familiar example. The interaction of multiple vortex rings produces complex dynamical behaviour, such as the leapfrogging motion first analysed by Helmholtz more than a century and a half ago. Here we report on numerical investigations of vortex ring dynamics in a different setting from fluids, namely, as solutions of the Landau-Lifshitz equation that models the evolution of the local magnetization in a ferromagnetic medium. We present the results of the first study on the dynamics of interacting magnetic vortex rings and provide a novel link between fluids and magnetism, by showing that a range of phenomena familiar in fluids are reproduced in ferromagnets. This includes the leapfrogging motion of a pair of vortex rings and evidence for the chaotic dynamics of a trio of rings.


Niemi, A., & Sutcliffe, P. (2014). Leapfrogging vortex rings in the Landau–Lifshitz equation. Nonlinearity, 27(9),

Journal Article Type Article
Publication Date Sep 1, 2014
Deposit Date May 19, 2014
Publicly Available Date Jul 23, 2014
Journal Nonlinearity
Print ISSN 0951-7715
Electronic ISSN 1361-6544
Publisher IOP Publishing
Peer Reviewed Peer Reviewed
Volume 27
Issue 9


Accepted Journal Article (693 Kb)

Copyright Statement
© 2014 IOP Publishing Ltd. This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at

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