3D MHD Coronal Oscillations about a Magnetic Null Point: Application of WKB Theory

Abstract

This paper is a demonstration of how the WKB approximation can be used to help solve the linearised 3D MHD equations. Using Charpit’s method and a Runge – Kutta numerical scheme, we have demonstrated this technique for a potential 3D magnetic null point, B=[x,ε y,−(ε+1)z]. Under our cold-plasma assumption, we have considered two types of wave propagation: fast magnetoacoustic and Alfvén waves. We find that the fast magnetoacoustic wave experiences refraction towards the magnetic null point and that the effect of this refraction depends upon the Alfvén speed profile. The wave and thus the wave energy accumulate at the null point. We have found that current buildup is exponential and the exponent is dependent upon ε. Thus, for the fast wave there is preferential heating at the null point. For the Alfvén wave, we find that the wave propagates along the field lines. For an Alfvén wave generated along the fan plane, the wave accumulates along the spine. For an Alfvén wave generated across the spine, the value of ε determines where the wave accumulation will occur: fan plane (ε=1), along the x-axis (0<ε<1) or along the y-axis (ε>1). We have shown analytically that currents build up exponentially, leading to preferential heating in these areas. The work described here highlights the importance of understanding the magnetic topology of the coronal magnetic field for the location of wave heating.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bender, C.M., Orszag, S.A.: 1978, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, Singapore.

    MATH  Google Scholar 

  2. Beveridge, C., Priest, E.R., Brown, D.S.: 2002, Solar Phys. 209, 333 – 347.

    Article  ADS  Google Scholar 

  3. Brown, D.S., Priest, E.R.: 2001, Astron. Astrophys. 367, 339 – 346.

    Article  ADS  Google Scholar 

  4. Cairns, R.A., Lashmore-Davies, C.N.: 1983, Phys. Fluids 26, 1268 – 1274.

    MATH  Article  ADS  Google Scholar 

  5. Craig, I.J., McClymont, A.N.: 1993, Astrophys. J. 405, 207 – 215.

    Article  ADS  Google Scholar 

  6. Craig, I.J., Watson, P.G.: 1992, Astrophys. J. 393, 385 – 395.

    Article  ADS  Google Scholar 

  7. De Moortel, I.: 2005, Philos. Trans. Roy. Soc. A 363, 2743 – 2760.

    Article  ADS  Google Scholar 

  8. De Moortel, I.: 2006, Philos. Trans. Roy. Soc. A 364, 461 – 472.

    Article  ADS  Google Scholar 

  9. De Moortel, I., Hood, A.W., Ireland, J., Arber, T.D.: 1999, Astron. Astrophys. 346, 641 – 651.

    ADS  Google Scholar 

  10. Evans, G., Blackledge, J., Yardley, P.: 1999, Analytical Methods for Partial Differential Equations, Springer, London.

    Google Scholar 

  11. Galsgaard, K., Priest, E.R., Titov, V.S.: 2003, J. Geophys. Res. 108, 1 – 12.

    Article  Google Scholar 

  12. Heyvaerts, J., Priest, E.R.: 1983, Astron. Astrophys. 117, 220 – 234.

    MATH  ADS  Google Scholar 

  13. Hood, A.W., Brooks, S.J., Wright, A.N.: 2002, Proc. Roy. Soc. A 458, 2307 – 2325.

    MATH  Article  ADS  MathSciNet  Google Scholar 

  14. Khomenko, E.V., Collados, M.: 2006, Astrophys. J. 653, 739 – 755.

    Article  ADS  Google Scholar 

  15. McDougall, A.M.D., Hood, A.W.: 2007, Solar Phys. 246, 259 – 271.

    Article  ADS  Google Scholar 

  16. McLaughlin, J.A., Hood, A.W.: 2004, Astron. Astrophys. 420, 1129 – 1140.

    Article  ADS  Google Scholar 

  17. McLaughlin, J.A., Hood, A.W.: 2005, Astron. Astrophys. 435, 313 – 325.

    Article  ADS  Google Scholar 

  18. McLaughlin, J.A., Hood, A.W.: 2006a, Astron. Astrophys. 452, 603 – 613.

    MATH  Article  ADS  Google Scholar 

  19. McLaughlin, J.A., Hood, A.W.: 2006b, Astron. Astrophys. 459, 641 – 649.

    Article  ADS  Google Scholar 

  20. Nakariakov, V.M., Roberts, B.: 1995, Solar Phys. 159, 399 – 402.

    Article  ADS  Google Scholar 

  21. Nakariakov, V.M., Roberts, B., Murawski, K.: 1997, Solar Phys. 75, 93 – 105.

    Article  ADS  Google Scholar 

  22. Nakariakov, V.M., Verwichte, E.: 2005, Living Reviews in Solar Physics 2, http://www.livingreviews.org/lrsp-2005-3 (cited August 2007).

  23. Parnell, C.E., Smith, J.M., Neukirch, T., Priest, E.R.: 1996, Phys. Plasmas 3, 759 – 770.

    Article  ADS  Google Scholar 

  24. Pontin, D.I., Galsgaard, K.: 2007, J. Geophys. Res. 112, 3103 – 3116.

    Article  Google Scholar 

  25. Pontin, D.I., Bhattacharjee, A., Galsgaard, K.: 2007, Phys. Plasmas 14, 2106 – 2119.

    Google Scholar 

  26. Priest, E.R., Titov, V.S.: 1996, Philos. Trans. Roy. Soc. 354, 2951 – 2992.

    MATH  Article  ADS  MathSciNet  Google Scholar 

  27. Titov, V.S., Hornig, G.: 2000, Phys. Plasmas 7, 3350 – 3542.

    Article  MathSciNet  Google Scholar 

  28. Weinberg, S.: 1962, Phys. Rev. 6, 1899 – 1909.

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to J. A. McLaughlin.

Additional information

Guest Editors: Laurent Gizon and Paul Cally

Electronic Supplementary Material

Video file

Video file

Video file

Video file

Video file

Video file

Video file

Rights and permissions

Reprints and Permissions

About this article

Cite this article

McLaughlin, J.A., Ferguson, J.S.L. & Hood, A.W. 3D MHD Coronal Oscillations about a Magnetic Null Point: Application of WKB Theory. Sol Phys 251, 563–587 (2008). https://doi.org/10.1007/s11207-007-9107-2

Download citation

Keywords

  • Magnetohydrodynamics: waves, propagation
  • Magnetic fields: models
  • Heating: coronal