Scaling laws for localised states in a nonlocal amplitude equation

Abstract

It is well known that, although a uniform magnetic field inhibits the onset of small amplitude thermal convection in a layer of fluid heated from below, isolated convection cells may persist if the fluid motion within them is sufficiently vigorous to expel magnetic flux. Such fully nonlinear (“convecton”) solutions for magnetoconvection have been investigated by several authors. Here we explore a model amplitude equation describing this separation of a fluid layer into a vigorously convecting part and a magnetically-dominated part at rest. Our analysis elucidates the origin of the scaling laws observed numerically to form the boundaries in parameter space of the region of existence of these localised states, and importantly, for the lowest thermal forcing required to sustain them.

Keywords:

• Magnetoconvection
• Bifurcation
• Asymptotic
• Variational

Acknowledgements

J.H.P. Dawes gratefully acknowledges the support of the Royal Society through a University Research Fellowship, and thanks Douglas Gough, Mike Proctor and Nigel Weiss for useful discussions and comments. C.J. Penington was supported by the EPSRC. Both authors are grateful to the anonymous referees for their comments on an earlier version of this article.