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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 9, issue 2
Nonlin. Processes Geophys., 9, 79-86, 2002
https://doi.org/10.5194/npg-9-79-2002
© Author(s) 2002. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

Special issue: Theory and simulation of Solar System Plasmas, No. 3

Nonlin. Processes Geophys., 9, 79-86, 2002
https://doi.org/10.5194/npg-9-79-2002
© Author(s) 2002. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

  30 Apr 2002

30 Apr 2002

Dynamics of nonlinear resonant slow MHD waves in twisted flux tubes

R. Erdélyi1 and I. Ballai2 R. Erdélyi and I. Ballai
  • 1Space & Atmosphere Research Center (SPARC), Dept. of Applied Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UK
  • 2School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, UK

Abstract. Nonlinear resonant magnetohydrodynamic (MHD) waves are studied in weakly dissipative isotropic plasmas in cylindrical geometry. This geometry is suitable and is needed when one intends to study resonant MHD waves in magnetic flux tubes (e.g. for sunspots, coronal loops, solar plumes, solar wind, the magnetosphere, etc.) The resonant behaviour of slow MHD waves is confined in a narrow dissipative layer. Using the method of simplified matched asymptotic expansions inside and outside of the narrow dissipative layer, we generalise the so-called connection formulae obtained in linear MHD for the Eulerian perturbation of the total pressure and for the normal component of the velocity. These connection formulae for resonant MHD waves across the dissipative layer play a similar role as the well-known Rankine-Hugoniot relations connecting solutions at both sides of MHD shock waves. The key results are the nonlinear connection formulae found in dissipative cylindrical MHD which are an important extension of their counterparts obtained in linear ideal MHD (Sakurai et al., 1991), linear dissipative MHD (Goossens et al., 1995; Erdélyi, 1997) and in nonlinear dissipative MHD derived in slab geometry (Ruderman et al., 1997). These generalised connection formulae enable us to connect solutions obtained at both sides of the dissipative layer without solving the MHD equations in the dissipative layer possibly saving a considerable amount of CPU-time when solving the full nonlinear resonant MHD problem.

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