Solar Physics

, Volume 289, Issue 3, pp 899–909 | Cite as

Linear MHD Wave Propagation in Time-Dependent Flux Tube

I. Zero Plasma-β


MHD waves and oscillations in sharply structured magnetic plasmas have been studied for static and steady systems in the thin tube approximation over many years. This work will generalize these studies by introducing a slowly varying background density in time, in order to determine the changes to the wave parameters introduced by this temporally varying equilibrium, i.e. to investigate the amplitude, frequency, and wavenumber for the kink and higher order propagating fast magnetohydrodynamic wave in the leading order approximation to the WKB approach in a zero-β plasma representing the upper solar atmosphere. To progress, the thin tube and over-dense loop approximations are used, restricting the results found here to the duration of a number of multiples of the characteristic density change timescale. Using such approximations it is shown that the amplitude of the kink wave is enhanced in a manner proportional to the square of the Alfvén speed, \(V_{\mathrm{A}}^{2}\). The frequency of the wave solution tends to the driving frequency of the system as time progresses; however, the wavenumber approaches zero after a large multiple of the characteristic density change timescale, indicating an ever increasing wavelength. For the higher order fluting modes the changes in amplitude are dependent upon the wave mode; for the m=2 mode the wave is amplified to a constant level; however, for all m≥3 the fast MHD wave is damped within a relatively small multiple of the characteristic density change timescale. Understanding MHD wave behavior in time-dependent plasmas is an important step towards a more complete model of the solar atmosphere and has a key role to play in solar magneto-seismological applications.


MHD Time-dependent density Waves 



The authors thank M.S. Ruderman and R. Morton for a number of useful discussions. RE acknowledges M. Kéray for patient encouragement and is also grateful to NSF, Hungary (OTKA, Ref. No. K83133) for support received.


  1. Al-Ghafri, K.S., Ruderman, M.S., Erdélyi, R.: 2012, Longitudinal MHD waves in dissipative time-dependent plasma. Astrophys. J., submitted. Google Scholar
  2. Andries, J., Goossens, M., Hollweg, J.V., Arregui, I., Van Doorsselaere, T.: 2005, Coronal loop oscillations. Calculation of resonantly damped MHD quasi-mode kink oscillations of longitudinally stratified loops. Astron. Astrophys. 430, 1109 – 1118. ADSCrossRefGoogle Scholar
  3. Andries, J., Van Doorsselaere, T., Roberts, B., Verth, G., Verwichte, R., Erdélyi, E.: 2009, Coronal seismology by means of kink oscillation overtones. Space Sci. Rev. 149, 3 – 29. ADSCrossRefGoogle Scholar
  4. Aschwanden, M.J.: 2004, Physics of the Solar Corona. An Introduction, Praxis Publishing Ltd, Chichester, 28 – 30. Google Scholar
  5. Aschwanden, M.J., Terradas, J.: 2008, The effect of radiative cooling on coronal loop oscillations. Astrophys. J. 686, 127 – 130. ADSCrossRefGoogle Scholar
  6. Bender, C.M., Orszag, S.A.: 1978, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 484. MATHGoogle Scholar
  7. Berghmans, D., Clette, F.: 1999, Active region EUV transient brightenings. Solar Phys. 186, 207 – 229. ADSCrossRefGoogle Scholar
  8. De Moortel, I., Ireland, J., Walsh, R.W.: 2000, Observation of oscillations in coronal loops. Astron. Astrophys. 457, L23 – L26. Google Scholar
  9. Defouw, R.J.: 1976, Wave propagation along a magnetic tube. Astrophys. J. 209, 266 – 269. ADSCrossRefGoogle Scholar
  10. Dymova, M.V., Ruderman, M.S.: 2006, Resonantly damping oscillations of longitudinally stratified coronal loops. Astron. Astrophys. 457, 1059 – 1070. ADSCrossRefMATHGoogle Scholar
  11. Edwin, P.M., Roberts, B.: 1982, Wave propagation in a magnetically structured atmosphere III. Solar Phys. 76, 239 – 259. ADSGoogle Scholar
  12. Edwin, P.M., Roberts, B.: 1983, Wave propagation in a magnetic cylinder. Solar Phys. 88, 179 – 191. ADSGoogle Scholar
  13. Erdélyi, R., Al-Ghafri, K.S., Morton, R.J.: 2011, Damping of longitudinal magneto-acoustic oscillations in slowly varying coronal plasma. Solar Phys. 272, 73 – 89. ADSCrossRefGoogle Scholar
  14. Erdélyi, R., Fedun, V.: 2010, Magneto-acoustic waves in compressible magnetically twisted flux tubes. Solar Phys. 263, 63 – 85. ADSCrossRefGoogle Scholar
  15. Erdélyi, R., Verth, G.: 2007, The effect of density stratification on the amplitude profile of transversal coronal loop oscillations. Astron. Astrophys. 462, 743 – 751. ADSCrossRefGoogle Scholar
  16. Goossens, M., Erdélyi, R., Ruderman, M.: 2011, Resonant MHD waves in the solar atmosphere. Space Sci. Rev. 158, 289 – 338. ADSCrossRefGoogle Scholar
  17. Goossens, M., Hollweg, V.J., Sakurai, T.: 1992, Resonant behavior of MHD waves on magnetic flux tubes. III – Effect of equilibrium flow. Solar Phys. 138, 233 – 255. ADSCrossRefGoogle Scholar
  18. Goossens, M., Ruderman, M., Hollweg, J.: 1995, Dissipative MHD solutions for resonant Alfvén waves in 1-dimensional magnetic flux tubes. Solar Phys. 157, 75 – 102. ADSCrossRefGoogle Scholar
  19. Grossmann, W., Tataronis, J.: 1973, Decay of MHD waves by phase mixing. Z. Phys. 261, 217 – 236. ADSCrossRefGoogle Scholar
  20. Hasan, S.S.: 2008, Chromospheric dynamics. Adv. Space Res. 42, 86 – 95. ADSCrossRefGoogle Scholar
  21. Heyvaerts, J., Priest, E.R.: 1983, Coronal heating by phase-mixed shear Alfvén waves. Astron. Astrophys. 117, 220 – 234. ADSMATHGoogle Scholar
  22. Hollweg, J.V.: 1978, Alfvén waves in the solar atmosphere. Solar Phys. 56, 305 – 333. ADSGoogle Scholar
  23. Ionson, A.J.: 1978, Resonant absorption of Alfvénic surface waves and the heating of solar coronal loops. Astrophys. J. 226, 650 – 673. ADSCrossRefGoogle Scholar
  24. Morton, R.J., Erdélyi, R.: 2009, Transverse oscillations of a cooling coronal loop. Astrophys. J. 707, 750. ADSCrossRefGoogle Scholar
  25. Morton, R.J., Hood, A.W., Erdélyi, R.: 2010, Propagating magneto-hydrodynamic waves in a cooling homogeneous coronal plasma. Astron. Astrophys. 512, A43. CrossRefGoogle Scholar
  26. Narain, U., Ulmschneider, P.: 1996, Chromospheric and coronal heating mechanisms II. Space Sci. Rev. 75, 453 – 509. ADSCrossRefGoogle Scholar
  27. Narayanan, A.S.: 1991, Hydromagnetic surface waves in compressible moving cylindrical flux tubes. Plasma Phys. Control. Fusion 33, 333 – 338. ADSCrossRefGoogle Scholar
  28. O’Shea, E., Banerjee, D., Doyle, J.G., Fleck, B., Murtagh, F.: 2001, Active region oscillations. Astron. Astrophys. 368, 1095 – 1107. ADSCrossRefGoogle Scholar
  29. Roberts, B.: 1981a, Wave propagation in a magnetically structured atmosphere I. Solar Phys. 69, 27 – 38. ADSCrossRefGoogle Scholar
  30. Roberts, B.: 1981b, Wave propagation in a magnetically structured atmosphere II. Solar Phys. 69, 39 – 56. ADSCrossRefGoogle Scholar
  31. Roberts, B.: 2000, Wave and oscillations in the corona. Solar Phys. 193, 139 – 152. ADSCrossRefGoogle Scholar
  32. Ruderman, M., Erdélyi, R.: 2009, Transverse oscillations of coronal loops. Space Sci. Rev. 149, 199 – 228. ADSCrossRefGoogle Scholar
  33. Ruderman, M.S.: 2005, Damping mechanisms of coronal loop oscillations. In: Danesy, D., Poedts, S., De Groof, A., Andries, J. (eds.) The Dynamic Sun: Challenges for Theory and Observations, Proc. 11th European Solar Physics Meeting, ESA SP-600, 96.1 (on CDROM). Google Scholar
  34. Ruderman, M.S.: 2010, The effects of flows on transverse oscillations of coronal loops. Solar Phys. 267, 377 – 391. ADSCrossRefGoogle Scholar
  35. Ruderman, M.S., Roberts, B.: 2002, Resonantly damping oscillations of longitudinally stratified coronal loops. Astrophys. J. 577, 475 – 486. ADSCrossRefGoogle Scholar
  36. Spruit, H.C.: 1982, Propagation speeds and acoustic damping of waves in magnetic flux tubes. Solar Phys. 75, 3 – 17. ADSCrossRefGoogle Scholar
  37. Terra-Homem, M., Erdélyi, R., Ballai, I.: 2003, Linear and non-linear MHD wave propagation in steady-state magnetic cylinders. Solar Phys. 217, 199 – 223. ADSCrossRefGoogle Scholar
  38. Verth, G., Erdélyi, R.: 2008, Effect of longitudinal magnetic and density inhomogeneity on transversal coronal loop oscillations. Astron. Astrophys. 486, 1015 – 1022. ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

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

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