Solar Physics

, Volume 289, Issue 11, pp 4105–4115 | Cite as

Resonant Damping of Propagating Kink Waves in Time-Dependent Magnetic Flux Tube

I. Zero Plasma-\(\pmb{\upbeta}\)


We explore the notion of resonant absorption in a dynamic time-dependent magnetised plasma background. Very many works have investigated resonance in the Alfvén and slow MHD continua under both ideal and dissipative MHD regimes. Jump conditions in static and steady systems have been found in previous works, connecting solutions at both sides of the resonant layer. Here, we derive the jump conditions in a temporally dependent, magnetised, inhomogeneous plasma background to leading order in the Wentzel–Kramers–Billouin (WKB) approximation. Next, we exploit the results found in Williamson and Erdélyi (Solar Phys. 289, 899, 2014) to describe the evolution of the jump condition in the dynamic model considered. The jump across the resonant point is shown to increase exponentially in time. We determined the damping as a result of the resonance over the same time period and investigated the temporal evolution of the damping itself. We found that the damping coefficient, as a result of the evolution of the resonance, decreases as the density gradient across the transitional layer decreases. This has the consequence that in such time-dependent systems resonant absorption may not be as efficient as time progresses.


MHD waves Resonant damping Time-dependent medium 


  1. Andries, J., Arregui, I., Goossens, M.: 2005, Determination of the coronal density stratification from the observation of harmonic coronal loop oscillations. Astrophys. J. Lett. 624, L57. ADSCrossRefGoogle Scholar
  2. Andries, J., van Doorsselaere, T., Roberts, B., Verth, G., Verwichte, E., Erdélyi, R.: 2009, Coronal seismology by means of kink oscillation overtones. Space Sci. Rev. 149, 3. ADSCrossRefGoogle Scholar
  3. Aschwanden, M.J.: 2004, Physics of the Solar Corona. An Introduction, Praxis, Chichester, 23. Google Scholar
  4. Banerjee, D., Erdélyi, R., Oliver, R., O’Shea, E.: 2007, Present and future observing trends in atmospheric magnetoseismology. Solar Phys. 246, 3. ADSCrossRefGoogle Scholar
  5. Bender, C.M., Orszag, S.A.: 1978, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 484. MATHGoogle Scholar
  6. Cally, P.S.: 1991, Phase-mixing and surface waves: A new interpretation. J. Plasma Phys. 45, 453. ADSCrossRefGoogle Scholar
  7. De Moortel, I.: 2009, Longitudinal waves in coronal loops. Space Sci. Rev. 149, 65. ADSCrossRefGoogle Scholar
  8. De Moortel, I., Hood, A.W.: 2004, The damping of slow MHD waves in solar coronal magnetic fields. II. The effect of gravitational stratification and field line divergence. Astron. Astrophys. 415, 705. ADSCrossRefGoogle Scholar
  9. Díaz, A.J., Oliver, R., Ballester, J.L.: 2006, Fast magnetohydrodynamic oscillations in coronal loops with heating profiles. Astrophys. J. 645, 766. ADSCrossRefGoogle Scholar
  10. Edwin, P.M., Roberts, B.: 1982, Wave propagation in a magnetically structured atmosphere III. Solar Phys. 76, 239. ADSCrossRefGoogle Scholar
  11. Erdélyi, R., Fedun, V.: 2010, Magneto-acoustic waves in compressible magnetically twisted flux tubes. Solar Phys. 263, 63. ADSCrossRefGoogle Scholar
  12. Erdélyi, R., Verth, G.: 2007, The effect of density stratification on the amplitude profile of transversal coronal loop oscillations. Astron. Astrophys. 462, 743. ADSCrossRefGoogle Scholar
  13. Goossens, M., Erdélyi, R., Ruderman, M.S.: 2011, Resonant MHD waves in the solar atmosphere. Space Sci. Rev. 158, 289. ADSCrossRefGoogle Scholar
  14. Goossens, M., Hollweg, J.V., Sakurai, T.: 1992, Resonant behavior of MHD waves on magnetic flux tubes. III – effect of equilibrium flow. Solar Phys. 138, 233. ADSCrossRefGoogle Scholar
  15. Goossens, M., Ruderman, M., Hollweg, J.: 1995, Dissipative MHD solutions for resonant Alfvén waves in 1-dimensional magnetic flux tubes. Solar Phys. 157, 75. ADSCrossRefGoogle Scholar
  16. Grossmann, W., Tataronis, J.: 1973, Decay of MHD waves by phase mixing. Z. Phys. 261, 217. ADSCrossRefGoogle Scholar
  17. Heyvaerts, J., Priest, E.R.: 1983, Coronal heating by phase-mixed shear Alfvén waves. Astron. Astrophys. 117, 220. ADSMATHGoogle Scholar
  18. Ionson, J.A.: 1978, Resonant absorption of Alfvénic surface waves and the heating of solar coronal loops. Astrophys. J. 226, 650. ADSCrossRefGoogle Scholar
  19. Krall, N.A., Trivelpiece, A.W.: 1986, Principles of Plasma Physics, San Francisco Press, San Francisco. Section 8.6.2. Google Scholar
  20. Mathioudakis, M., Jess, D.B., Erdélyi, R.: 2013, Alfvén waves in the solar atmosphere. From theory to observations. Space Sci. Rev. 175, 1. ADSCrossRefGoogle Scholar
  21. Morton, R.J., Erdélyi, R.: 2010, Application of the theory of damping of kink oscillations by radiative cooling of coronal loop plasma. Astron. Astrophys. 519, A43. CrossRefGoogle Scholar
  22. Morton, R.J., Hood, A.W., Erdélyi, R.: 2010, Propagating magneto-hydrodynamic waves in a cooling homogenous coronal plasma. Astron. Astrophys. 512, A23. ADSCrossRefGoogle Scholar
  23. Morton, R.J., Verth, G., Jess, D.B., Kuridze, D., Ruderman, M.S., Mathioudakis, M., Erdélyi, R.: 2012, Observations of ubiquitous compressive waves in the Sun’s chromosphere. Nature Commun. 3, 1315. ADSCrossRefGoogle Scholar
  24. Porter, L.J., Klimchuk, J.A., Sturrock, P.A.: 1994, The possible role of MHD waves in heating the solar corona. Astrophys. J. 435, 482. ADSCrossRefGoogle Scholar
  25. Priest, E.R.: 2003, Solar Magnetohydrodynamics, Reidel, Dordrecht, 38. Google Scholar
  26. Roberts, B.: 2000, Waves and oscillations in the corona – (invited review). Solar Phys. 193, 139. ADSCrossRefGoogle Scholar
  27. Ruderman, M.S.: 2011, Resonant damping of kink oscillations of cooling coronal magnetic loops. Astron. Astrophys. 534, A78. ADSCrossRefGoogle Scholar
  28. Ruderman, M., Erdélyi, R.: 2009, Transverse oscillations of coronal loops. Space Sci. Rev. 149, 199. ADSCrossRefGoogle Scholar
  29. Ruderman, M., Verth, G., Erdélyi, R.: 2008, Transverse oscillations of longitudinally stratified coronal loops with variable cross section. Astrophys. J. 686, 694. ADSCrossRefGoogle Scholar
  30. Sakurai, T., Goossens, M., Hollweg, J.: 1991, Resonant behavior of MHD waves on magnetic flux tubes 1. Solar Phys. 133, 227. ADSCrossRefGoogle Scholar
  31. Soler, R., Ruderman, M.S., Goossens, M.: 2012, Damped kink oscillations of flowing prominence threads. Astron. Astrophys. 546, A82. ADSCrossRefGoogle Scholar
  32. Van Doorsselaere, T., Andries, J., Poedts, S., Goossens, M.: 2004, Damping of coronal loop oscillations: calculation of resonantly damped kink oscillations of one-dimensional nonuniform loops. Astrophys. J. 606, 1223. ADSCrossRefGoogle Scholar
  33. Verth, G., Erdélyi, R.: 2008, Effect of longitudinal magnetic and density inhomogeneity on transversal coronal loop oscillations. Astron. Astrophys. 486, 1015. ADSCrossRefMATHGoogle Scholar
  34. Voitenko, Y., Goossens, M.: 2000, Competition of damping mechanisms for the phase-mixed Alfvén waves in the solar corona. Astron. Astrophys. 357, 1086. ADSGoogle Scholar
  35. Wang, T.: 2011, Standing slow-mode waves in hot coronal loops: observations, modeling, and coronal seismology. Space Sci. Rev. 158, 397. ADSCrossRefGoogle Scholar
  36. Williamson, A., Erdélyi, R.: 2014, Linear MHD wave propagation in time-dependent flux tube. I. Zero plasma-β. Solar Phys. 289, 899. ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Solar Physics and Space Plasma Research Centre (SP2RC), School of MathematicsUniversity of SheffieldSheffieldUK

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