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Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence

Published online by Cambridge University Press:  02 May 2007

ANDRÉ THESS  and
OLEG ZIKANOV
ANDRÉ THESS
Affiliation:
Department of Mechanical Engineering, Ilmenau University of Technology, P. O. Box 100565, 98684 Ilmenau, Germany
OLEG ZIKANOV
Affiliation:
Department of Mechanical Engineering, University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, MI 48128-1491, USA
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Abstract

We report a theoretical investigation of the robustness of two-dimensional inviscid magnetohydrodynamic (MHD) flows at low magnetic Reynolds numbers with respect to three-dimensional perturbations. We use a combination of linear stability analysis and direct numerical simulations to analyse three problems, namely the flow in the interior of a triaxial ellipsoid, and two unbounded flows: a vortex with elliptical streamlines and a vortex sheet parallel to the magnetic field. The flow in a triaxial ellipsoid is found to present an exact analytical model which demonstrates both the existence of inviscid unstable three-dimensional modes and the stabilizing role of the magnetic field. The nonlinear evolution of the flow is characterized by intermittency typical of other MHD flows with long periods of nearly two-dimensional behaviour interrupted by violent three-dimensional transients triggered by the instability. We demonstrate, using the second model, that motion with elliptical streamlines perpendicular to the magnetic field becomes unstable with respect to the elliptical instability once the magnetic interaction parameter falls below a critical magnitude whose value tends to infinity as the eccentricity of the streamlines increases. Furthermore, the third model indicates that vortex sheets parallel to the magnetic field, which are unstable for any velocity and any magnetic field, emit eddies with vorticity perpendicular to the magnetic field. Whether the investigated instabilities persist in the presence of small but finite viscosity, in which case two-dimensional turbulence would represent a singular state of MHD flows, remains an open question.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 579 , 25 May 2007 , pp. 383 - 412
Copyright
Copyright © Cambridge University Press 2007

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