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Evidence for singularity formation in a class of stretched solutions of the equations for ideal MHD

  • J. D. Gibbon
  • K. Ohkitani
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 71)

Abstract

A class of stretched solutions of the equations for incompressible, ideal 3D-MHD are studied using Elsasser variables V ± =U±B. This class takes the form V±=(v±, v 3 ± ) where v ±=v±(x,y,t) and v 3 ± (x,y,z,t)= ±(x,y,t)+β±(x,y,t). The chosen domain is of a tubular form which is infinite in the z-direction with periodic cross-section. This follows a previous study by the authors on this same class of solutions for the 3D Euler equations. In both cases the systems are of infinite energy. Strong numerical evidence for a finite time singularity in the Euler case was subsequently confirmed by a rigorous analytical proof by Constantin. In the MHD case, pseudo-spectral computations of the 2D partial differential equations for & γ ± , v ± and β ± valid on the cross-sectional domain provide evidence for a finite time blow-up in both the fluid and magnetic variables although an analytical proof for the existence of this singularity remains elusive

Keywords

Singularity Formation Vortex Tube Incompressible Euler Equation Magnetic Variable Infinite Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • J. D. Gibbon
    • 1
  • K. Ohkitani
    • 2
  1. 1.Department of Mathematics, Imperial College of ScienceTechnology and MedicineLondonUK
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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