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Formation of singularities in Hall magnetohydrodynamics

Published online by Cambridge University Press:  26 August 2009

MANUEL NÚÑEZ
MANUEL NÚÑEZ*
Affiliation:
Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain
*
†Email address for correspondence: mnjmhd@am.uva.es
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Abstract

Hall magnetohydrodynamics has proved a useful model in several physical phenomena, in particular, in fast magnetic reconnection. The induction equation for this model involves the nonlinear Hall term which has been suspected to imply loss of regularity and in particular formation of singularities of the current density. Numerical simulations strongly suggest that this is the case, but so far a rigorous proof was lacking. We show that for a particular axisymmetric geometry, certain integrals of the magnetic field satisfy a differential inequality that leads to a finite-time blow-up of the field gradient, and therefore of the current density. We may interpret this as a breakdown of the field regularity and the formation of discontinuous solutions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 634 , 10 September 2009 , pp. 499 - 507
Copyright
Copyright © Cambridge University Press 2009

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