Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Hall-magnetohydrodynamic waves in flowing ideal incompressible solar-wind plasmas: reconsidered

  • 8 Accesses


It is well established that the magnetically structured solar atmosphere supports the propagation of MHD waves along various kind of jets including also the solar wind. It is well-known as well that under some conditions, namely high enough jet speeds, the propagating MHD modes can become unstable against to the most common Kelvin–Helmholtz instability (KHI). In this article, we explore how the propagation and instability characteristics of running along a slow solar wind MHD modes are affected when they are investigated in the framework of the ideal Hall-magnetohydrodynamics. Hall-MHD is applicable if the jet width is shorter than or comparable to the so called Hall parameter \(l_{ \mathrm{Hall}} = c/\omega _{\mathrm{pi}}\) (where \(c\) is the speed of light and \(\omega _{\mathrm{pi}}\) is the ion plasma frequency). We model the solar wind as a moving with velocity \(\boldsymbol {v}_{0}\) cylindrical flux tube of radius \(a\), containing incompressible plasma with density \(\rho _{\mathrm{i}}\) permeated by a constant magnetic field \(\boldsymbol {B} _{\mathrm{i}}\). The surrounding plasma is characterized with its density \(\rho _{\mathrm{e}}\) and magnetic field \(\boldsymbol {B}_{\mathrm{e}}\). The dispersion relation of MHD waves is derived in the framework of both standard and Hall-MHD and is numerically solved with input parameters: the density contrast \(\eta = \rho _{\mathrm{e}}/\rho _{\mathrm{i}}\), the magnetic fields ratio \(b = {B}_{\mathrm{e}}/{B}_{\mathrm{i}}\), and the Hall scale parameter \(l_{\mathrm{Hall}}/a\). It is found that the Hall current, at moderate values of \(l_{\mathrm{Hall}}/a\), stimulates the emerging of KHI of the kink (\(m = 1\)) and high-mode (\(m \geqslant 2\)) MHD waves, while for the sausage wave (\(m = 0\)) the trend is just the opposite—the KHI is suppressed.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. Ballai, I., Thelen, J.C., Roberts, B.: Solitary waves in a Hall solar wind plasma. Astron. Astrophys. 404, 701 (2003). https://doi.org/10.1051/0004-6361:20030580

  2. Birn, J., Galsgaard, K., Hesse, M., et al.: Forced magnetic reconnection. Geophys. Res. Lett. 32, L06105 (2005). https://doi.org/10.1029/2004GL022058

  3. Cally, P.S., Khomenko, E.: Fast-to-Alfvén mode conversion mediated by the hall current. I. Cold plasma model. Astrophys. J. 814, 106 (2005). https://doi.org/10.1088/0004-637X/814/2/106

  4. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Clarendon, Oxford (1961), Chap. 11

  5. Edwin, P.M., Roberts, B.: Wave propagation in a magnetically structured atmosphere. III: The slab in a magnetic environment. Sol. Phys. 76, 239 (1982). https://doi.org/10.1007/BF00170986

  6. Edwin, P.M., Roberts, B.: Wave propagation in a magnetic cylinder. Sol. Phys. 88, 179 (1983). https://doi.org/10.1007/BF00196186

  7. Hagstrom, G.I., Hameiri, E.: Traveling waves in Hall-magnetohydrodynamics and the ion-acoustic shock structure. Phys. Plasmas 21, 022109 (2014). https://doi.org/10.1063/1.4862035

  8. Harris, E.G.: On a plasma sheath separating regions of oppositely directed fields. Nuovo Cimento 23, 115 (1962). https://doi.org/10.1007/BF02733547

  9. Huba, J.D.: Hall magnetohydrodynamics in space and laboratory plasmas. Phys. Plasmas 2, 2504 (1995). https://doi.org/10.1063/1.871212

  10. Khomenko, E.: On the effects of ion-neutral interactions in solar plasmas. Plasma Phys. Control. Fusion 59, 014038 (2016). https://doi.org/10.1088/0741-3335/59/1/014038

  11. Leroy, M.H.J., Keppens, R.: On the influence of environmental parameters on mixing and reconnection caused by the Kelvin-Helmholtz instability at the magnetopause. Phys. Plasmas 24, 012906 (2017). https://doi.org/10.1063/1.4974758

  12. Lighthill, M.J.: Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Philos. Trans. R. Soc. Lond. A 252, 397 (1960). https://doi.org/10.1098/rsta.1960.0010

  13. Martínez-Gómez, D., Soler, R., Terradas, J.: Onset of the Kelvin-Helmholtz instability in partially ionized magnetic flux tubes. Astron. Astrophys. 578, A104 (2015). https://doi.org/10.1051/0004-6361/201525785

  14. Miteva, R., Mann, G.: On nonlinear waves in Hall-MHD plasma. J. Plasma Phys. 74, 607 (2008). https://doi.org/10.1017/S0022377808007058

  15. Nakariakov, V.M.: MHD oscillations in solar and stellar coronae: current results and perspectives. Adv. Space Res. 39, 1804 (2007). https://doi.org/10.1016/j.asr.2007.01.044

  16. Nariyuki, Y., Hada, T.: Magnetohydrodynamic parametric instabilities of parallel propagating incoherent Alfvén waves. Earth Planets Space 59, e13 (2007). https://doi.org/10.1186/BF03353093

  17. Nykyri, K., Otto, A.: Influence of the Hall term on KH instability and reconnection inside KH vortices. Ann. Geophys. 22, 935 (2004). https://doi.org/10.5194/angeo-22-935-2004

  18. Panday, B.P.: Surface waves in the partially ionized solar plasma slab. Mon. Not. R. Astron. Soc. 436, 1659 (2013). https://doi.org/10.1093/mnras/stt1682

  19. Panday, B.P., Wardle, M.: Hall magnetohydrodynamics of partially ionized plasmas. Mon. Not. R. Astron. Soc. 385, 2269 (2008). https://doi.org/10.1111/j.1365-2966.2008.12998.x

  20. Panday, B.P., Wardle, M.: Hall instability of solar flux tubes in the presence of shear flows. Mon. Not. R. Astron. Soc. 426, 1436 (2012). https://doi.org/10.1111/j.1365-2966.2012.21718.x

  21. Panday, B.P., Wardle, M.: Magnetic-diffusion-driven shear instability of solar flux tubes. Mon. Not. R. Astron. Soc. 431, 570 (2013). https://doi.org/10.1093/mnras/stt184

  22. Ruderman, M.S., Caillol, P.: Parametric instabilities of circularly polarized small-amplitude Alfvén waves in Hall plasmas. J. Plasma Phys. 74, 119 (2008). https://doi.org/10.1017/s0022377807006691

  23. Spitzer, L.: Physics of Fully Ionized Gases, 2nd edn. p. 28. Interscience, New York (1962)

  24. Terra-Homem, M., Erdélyi, R., Ballai, I.: Linear and non-linear MHD wave propagation in steady-state magnetic cylinders. Sol. Phys. 217, 199 (2003). https://doi.org/10.1023/B:SOLA.0000006901.22169.59

  25. Zaqarashvili, T.V., Vörös, Z., Zhelyazkov, I.: Kelvin–Helmholtz instability of twisted magnetic flux tubes in the solar wind. Astron. Astrophys. 561, A64 (2014). https://doi.org/10.1051/0004-6361/201322808

  26. Zhelyazkov, I.: MHD waves and instabilities in flowing solar flux-tube plasmas in the framework of Hall magnetohydrodynamics. Eur. Phys. J. D 55, 127 (2009). https://doi.org/10.1140/epjd/e2009-00217-3

  27. Zhelyazkov, I.: Hall-magnetohydrodynamic waves in flowing ideal incompressible solar-wind plasmas. Plasma Phys. Control. Fusion 52, 065008 (2010). https://doi.org/10.1088/0741-3335/52/6/065008

  28. Zhelyazkov, I.: Magnetohydrodynamic waves and their stability status in solar spicules. Astron. Astrophys. 537, A124 (2012). https://doi.org/10.1051/0004-6361/201117780

  29. Zhelyazkov, I., Debosscher, A., Goossens, M.: Fast surface waves in an ideal Hall-magnetohydrodynamic plasma slab. Phys. Plasmas 3, 4346 (1996). https://doi.org/10.1063/1.872050

  30. Zhelyazkov, I., Dimitrov, Z.: Kelvin–Helmholtz instability in a cool solar jet in the framework of Hall magnetohydrodynamics: a case study. Sol. Phys. 293, 11 (2018). https://doi.org/10.1007/s11207-017-1230-0

Download references


The work was supported by the Bulgarian Science Fund under project DNTS/INDIA 01/7. We are deeply indebted to the reviewer for his very helpful and constructive comments. The authors are also thankful to Dr. Snezhana Yordanova for plotting one figure.

Author information

Correspondence to I. Zhelyazkov.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhelyazkov, I., Dimitrov, Z. & Bogdanova, M. Hall-magnetohydrodynamic waves in flowing ideal incompressible solar-wind plasmas: reconsidered. Astrophys Space Sci 365, 29 (2020). https://doi.org/10.1007/s10509-020-3741-7

Download citation


  • Sun: solar wind
  • MHD waves: dispersion relation
  • Kelvin–Helmholtz instability