Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T14:40:07.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear fast sausage waves in homogeneous magnetic flux tubes

Published online by Cambridge University Press:  17 December 2015

Badma B. Mikhalyaev  and
Michael S. Ruderman
Badma B. Mikhalyaev
Affiliation:
Department of Theoretical Physics, Kalmyk State University, 11 Pushkin Str., Elista 358000, Russia
Michael S. Ruderman*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK Space Research Institute (IKI), Russian Academy of Sciences, 84/32 Profsoyuznaya Str., Moscow 117997, Russia
*
†Email address for correspondence: m.s.ruderman@sheffield.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider fast sausage waves in straight homogeneous magnetic tubes. The plasma motion is described by the ideal magnetohydrodynamic equations in the cold plasma approximation. We derive the nonlinear Schrödinger equation describing the nonlinear evolution of an envelope of a carrier wave. The coefficients of this equation are expressed in terms Bessel and modified Bessel functions. They are calculated numerically for various values of parameters. In particular, we show that the criterion for the onset of the modulational or Benjamin–Fair instability is satisfied. The implication of the obtained results for solar physics is discussed.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 6 , December 2015 , 905810611
Copyright
© Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th edn. Dover.Google Scholar
Dubard, P., Gaillard, P., Kleina, C. & Matveev, V. B. 2010 On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation. Eur. Phys. J. Special Topics 185, 247–258.CrossRefGoogle Scholar
Edwin, P. M. & Roberts, B. 1983 Wave propagation in a magnetic cylinder. Solar Phys. 88, 179–191.CrossRefGoogle Scholar
Harvey, J. V. 1977 Observations of small-scale photospheric magnetic fields. Highlights Astron. 4, 223–239.CrossRefGoogle Scholar
Huang, J., Tan, B., Zhang, Y., Karlický, M. & Mészárosová, H. 2014 Quasi-periodic pulsations with varying period in multi-wavelength observations of an X-class flare. Astrophys. J. 791, A44.CrossRefGoogle Scholar
Inglis, A. R., Nakariakov, V. M. & Melnikov, V. F. 2008 Multi-wavelength spatially resolved analysis of quasi-periodic pulsations in a solar flare. Astron. Astrophys. 487, 1147–1153.CrossRefGoogle Scholar
Kopylova, Yu. G., Melnikov, A. V., Stepanov, A. V., Tsap, Yu. T. & Goldvarg, T. B. 2007 Oscillations of coronal loops and second pulsations of solar radio emission. Astron. Lett. 33, 706–713.CrossRefGoogle Scholar
Kopylova, Yu. G., Stepanov, A. V. & Tsap, Yu. T. 2002 Radial oscillations of coronal loops and microwave radiation from solar flares. Astron. Lett. 28, 783–791.CrossRefGoogle Scholar
Merzlyakov, E. G. 1985 Nonlinear long wave modulation in symmetric wave of a plane magnetic layer. Fluid Dyn. 20, 305–309.CrossRefGoogle Scholar
Molotovshchikov, A. L. & Ruderman, M. S. 1987 Long nonlinear waves in a compressible magnetically structured atmosphere. IV: slow sausage waves in a magnetic tube. Solar Phys. 109, 247–263.CrossRefGoogle Scholar
Nakariakov, V. M. & Melnikov, A. V. 2009 Quasi-periodic pulsations in solar flares. Space Sci. Rev. 149, 119–151.CrossRefGoogle Scholar
Nakariakov, V. M. & Oraevsky, V. N. 1995 Resonant interactions of modes of coronal magnetic tubes. Solar Phys. 160, 289–302.CrossRefGoogle Scholar
Nakariakov, V. M. & Roberts, B. 1995 On fast magnetosonic coronal pulsations. Solar Phys. 159, 399–402.CrossRefGoogle Scholar
Nakariakov, V. M., Roberts, B. & Petrukhin, N. S. 1997 Nonlinear dynamics of fast magnetosonic waves ducted by a smooth plasma inhomogeneity. J. Plasma Phys. 58, 315–327.CrossRefGoogle Scholar
Nakariakov, V. M., Melnikov, A. V. & Reznikova, V. E. 2003 Global sausage modes of coronal loops. Astron Lett. 412, L7–L10.CrossRefGoogle Scholar
Oliver, R., Ruderman, M. S. & Terradas, J. 2015 Propagation and dispersion of sausage wave trains in magnetic flux tubes. Astrophys. J. 806, A56.CrossRefGoogle Scholar
Priest, E. R. 1978 The structure of coronal loops. Solar Phys. 58, 57–87.CrossRefGoogle Scholar
Prudnikov, A. P., Brychkov, Yu. A. & Marichev, O. I. 1998 Integrals and Series: Volume 2: Special Functions. Gordon and Breach Science Publishers.Google Scholar
Roberts, B. 1985 Solitary waves in magnetic flux tubes. Phys. Fluids 28, 3280–3286.CrossRefGoogle Scholar
Rosenberg, H. 1970 Evidence for MHD pulsations in the solar corona. Astron. Astrophys. 9, 159–162.Google Scholar
Ruderman, M. S. 2003 Nonlinear waves in the magnetically structured solar atmosphere. In Turbulence, Waves, and Instabilities in the Solar Plasma (ed. Erdélyi, R. et al. ), pp. 239–265. Kluwer.CrossRefGoogle Scholar
Ruderman, M. S. 2006 Nonlinear waves in the solar atmosphere. Phil. Trans. R. Soc. Lond. A 364, 485–504.Google ScholarPubMed
Su, J. T., Shen, Y. D., Liu, Y., Liu, Y. & Mao, X. J. Imaging observations of quasi-periodic pulsations in solar flare loops with SDO/AIA . Astrophys. J. 755, 113.Google Scholar
Vaiana, G. S. & Rosner, R. 1978 Recent advances in coronal physics. Annu. Rev. Astron. Astrophys. 16, 393–428.CrossRefGoogle Scholar
Van Doorsselaere, T., Brady, C. S., Verwichte, E. & Nakariakov, V. M. 2008 Seismological demonstration of perpendicular density structuring in the solar corona. Astron. Astrophys. 491, L9–L12.CrossRefGoogle Scholar
Zabolotskaya, E. A. & Shvartsburg, A. B. 1987 Nonlinear acoustic waveguide. Sov. Phys. Acoust. 33, 221–222.Google Scholar
Zabolotskaya, E. A. & Shvartsburg, A. B. 1988 Propagation of finite-amplitude waves a waveguide. Sov. Phys. Acoust. 34, 493–495.Google Scholar
Zakharov, V. E. & Ostrovsky, L. A. 2009 Modulation instability: the beginning. Physica D 338, 540–548.CrossRefGoogle Scholar
Zhelyazkov, I., Murawski, K., Goossens, M., Nenovski, P. & Roberts, B. 1994 Modulations of slow sausage surface waves travelling along a magnetized slab. J. Plasma Phys. 51, 291–308.CrossRefGoogle Scholar
Zwaan, C. 1978 On the appearance of magnetic flux in the solar photosphere. Solar Phys. 60, 213–240.CrossRefGoogle Scholar