Solar Physics

, Volume 291, Issue 4, pp 1143–1157 | Cite as

On the Ratio of Periods of the Fundamental Harmonic and First Overtone of Magnetic Tube Kink Oscillations

Article

Abstract

We study kink oscillations of thin magnetic tubes. We assume that the density inside and outside the tube (and possibly also the cross-section radius) can vary along the tube. This variation is assumed to be of such a form that the kink speed is symmetric with respect to the tube centre and varies monotonically from the tube ends to the tube centre. Then we prove a theorem stating that the ratio of periods of the fundamental mode and first overtone is a monotonically increasing function of the ratio of the kink speed at the tube centre and the tube ends. In particular, it follows from this theorem that the period ratio is lower than two when the kink speed increases from the tube ends to its centre, while it is higher than two when the kink speed decreases from the tube ends to its centre. The first case is typical for non-expanding coronal magnetic loops, and the second for prominence threads. We apply the general results to particular problems. First we consider kink oscillations of coronal magnetic loops. We prove that, under reasonable assumptions, the ratio of the fundamental period to the first overtone is lower than two and decreases when the loop size increases. The second problem concerns kink oscillations of prominence threads. We consider three internal density profiles: generalised parabolic, Gaussian, and Lorentzian. Each of these profiles contain the parameter \(\alpha\) that is responsible for its sharpness. We calculate the dependence of the period ratio on the ratio of the mean to the maximum density. For all considered values of \(\alpha\) we find that a formula relating the period ratio and the ratio of the mean and maximum density suggested by Soler, Goossens, and Ballester (Astron. Astrophys.575, A123, 2015) gives a sufficiently good approximation to the exact dependence.

Keywords

Corona Coronal magnetic loops Prominences Waves Oscillations 

References

  1. Andries, J., Arregui, I., Goossens, M.: 2005, Astrophys. J. Lett. 624, L57. ADS. DOI. ADSCrossRefGoogle Scholar
  2. Andries, J., Van Doorsselaere, T., Roberts, B., Verth, G., Verwichte, E., Erdélyi, R.: 2009, Space Sci. Rev. 149, 3. ADS. DOI. ADSCrossRefGoogle Scholar
  3. Arregui, I., Oliver, R., Ballester, J.L.: 2012, Living Rev. Solar Phys. 9, A2. ADS. DOI. ADSCrossRefGoogle Scholar
  4. Arregui, I., Andries, J., Van Doorsselaere, T., Goossens, M., Poedts, S.: 2007, Astron. Astrophys. 463, 333. ADS. DOI. ADSCrossRefGoogle Scholar
  5. Arregui, I., Soler, R., Ballester, J.L., Wright, A.N.: 2011, Astron. Astrophys. 533, A60. ADS. DOI. ADSCrossRefGoogle Scholar
  6. Aschwanden, M.J.: 2006, Phil. Trans. Roy. Soc. A 364, 417. ADS. DOI. ADSCrossRefGoogle Scholar
  7. Aschwanden, M.J., Schrijver, C.J.: 2011, Astrophys. J. 736, A102. ADS. DOI. ADSCrossRefGoogle Scholar
  8. Aschwanden, M.J., Terradas, J.: 2008, Astrophys. J. Lett. 686, L127. ADS. DOI. ADSCrossRefGoogle Scholar
  9. Aschwanden, M.J., Fletcher, L., Schrijver, C.J., Alexander, D.: 1999, Astrophys. J. 520, 880. ADS. DOI. ADSCrossRefGoogle Scholar
  10. Berger, T.E., Shine, R.A., Slater, G.L., Tarbell, T.D., Title, A.M., Okamoto, T.J., et al.: 2008, Astrophys. J. Lett. 676, L89. ADS. DOI. ADSCrossRefGoogle Scholar
  11. Coddington, E.A., Levinson, N.: 1955, Theory of Ordinary Differential Equations, McGraw–Hill, New York. MATHGoogle Scholar
  12. Díaz, A.J., Donelly, G.R., Roberts, B.: 2007, Astron. Astrophys. 476, 359. ADS. DOI. ADSCrossRefGoogle Scholar
  13. Díaz, A.J., Oliver, R., Ballester, J.L.: 2010, Astrophys. J. 725, 1742. ADS. DOI. ADSCrossRefGoogle Scholar
  14. Dymova, M., Ruderman, M.S.: 2005, Solar Phys. 229, 79. ADS. DOI. ADSCrossRefGoogle Scholar
  15. Dymova, M., Ruderman, M.S.: 2006a, Astron. Astrophys. 457, 1059. ADS. DOI. ADSCrossRefGoogle Scholar
  16. Dymova, M., Ruderman, M.S.: 2006b, Astron. Astrophys. 459, 241. ADS. DOI. ADSCrossRefGoogle Scholar
  17. Lin, Y., Engvold, O., Rouppe van der Voort, L.H.M., van Noort, M.: 2007, Solar Phys. 246, 65. ADS. DOI. ADSCrossRefGoogle Scholar
  18. Lin, Y., Soler, R., Engvold, O., Ballester, J.L., Langangen, O., Oliver, R., van der Voort, L.H.M.R.: 2009, Astrophys. J. 704, 870. ADS. DOI. ADSCrossRefGoogle Scholar
  19. Nakariakov, V., Ofman, L.: 2001, Astron. Astrophys. 372, L53. ADS. DOI. ADSCrossRefGoogle Scholar
  20. Nakariakov, V., Ofman, L., DeLuca, E.E., Roberts, B., Davila, J.M.: 1999, Science 285, 862. ADS. DOI. ADSCrossRefGoogle Scholar
  21. Ofman, L., Aschwanden, M.J.: 2002, Astrophys. J. 576, L153. ADS. DOI. ADSCrossRefGoogle Scholar
  22. Okamoto, T.J., Tsuneta, S., Berger, T.E., Ichimoto, K., Katsukawa, Y., Lites, B.W., et al.: 2007, Science 318, 1577. ADS. DOI. ADSCrossRefGoogle Scholar
  23. Orozco Surez, D., Díaz, A.J., Asensio Ramos, A., Trujillo Bueno, J.: 2014, Astrophys. J. Lett. 785, L10. ADS. DOI. ADSCrossRefGoogle Scholar
  24. Ruderman, M.S.: 2015, Solar Phys. 290, 423. ADS. DOI. ADSCrossRefGoogle Scholar
  25. Ruderman, M.S., Verth, G., Erdélyi, R.: 2008, Astrophys. J. 686, 694. ADS. DOI. ADSCrossRefGoogle Scholar
  26. Safari, H., Nasiri, S., Sobouti, Y.: 2007, Astron. Astrophys. 470, 1111. ADS. DOI. ADSCrossRefGoogle Scholar
  27. Soler, R., Goossens, M.: 2011, Astron. Astrophys. 531, A167. ADS. DOI. ADSCrossRefGoogle Scholar
  28. Soler, R., Goossens, M., Ballester, J.L.: 2015, Astron. Astrophys. 575, A123. ADS. DOI. ADSCrossRefGoogle Scholar
  29. Soler, R., Ruderman, M.S., Goossens, M.: 2012, Astron. Astrophys. 546, A82. ADS. DOI. ADSCrossRefGoogle Scholar
  30. Soler, R., Arregui, I., Oliver, R., Ballester, J.L.: 2010, Astrophys. J. 722, 1778. ADS. DOI. ADSCrossRefGoogle Scholar
  31. Terradas, J., Arregui, I., Oliver, R., Ballester, J.L.: 2008, Astrophys. J. Lett. 678, L153. ADS. DOI. ADSCrossRefGoogle Scholar
  32. Van Doorsselaere, T., Nakariakov, V.M., Verwichte, E.: 2007, Astron. Astrophys. 473, 959. ADS. DOI. ADSCrossRefGoogle Scholar
  33. Verth, G., Erdélyi, R.: 2007, Astron. Astrophys. 486, 1015. ADS. DOI. ADSCrossRefGoogle Scholar
  34. Verth, G., Erdélyi, R., Jess, D.B.: 2008, Astrophys. J. Lett. 687, L45. ADS. DOI. ADSCrossRefGoogle Scholar
  35. Verwichte, E., Nakariakov, V.M., Ofman, L., Deluca, E.E.: 2004, Solar Phys. 223, 77. ADS. DOI. ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • M. S. Ruderman
    • 1
    • 2
  • N. S. Petrukhin
    • 3
  • E. Pelinovsky
    • 4
    • 5
  1. 1.Solar Physics and Space Plasma Research Centre (SP²RC), School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK
  2. 2.Space Research Institute (IKI)Russian Academy of SciencesMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.Department of Nonlinear Geophysical ProcessesInstitute of Applied PhysicsNizhny NovgorodRussia
  5. 5.Nizhny Novgorod State Technical University n.a. R.E. AlekseevNizhny NovgorodRussia

Personalised recommendations