Electromagnetic instabilities of low-beta alpha/proton beams in space plasmas


Relative drifts between different species or particle populations are characteristic to solar plasma outflows, e.g., in the fast streams of the solar winds, coronal mass ejections and interplanetary shocks. This paper characterizes the dispersion and stability of the low-beta alpha/proton drifts in the absence of any intrinsic thermal anisotropies, which are usually invoked in order to stimulate various instabilities. The dispersion relations derived here describe the full spectrum of instabilities and their variations with the angle of propagation and plasma parameters. The results unveil a potential competition between instabilities of the electromagnetic proton cyclotron and alpha cyclotron modes. For conditions specific to a low-beta solar wind, e.g., at low heliocentric distances in the outer corona, the instability operates on the alpha cyclotron branch. The growth rates of the alpha cyclotron mode are systematically stimulated by the (parallel) plasma beta and/or the alpha-proton temperature ratio. One can therefore expect that this instability develops even in the absence of temperature anisotropies, with potential to contribute to a self-consistent regulation of the observed drift of alpha particles.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. Asbridge, J.R., Bame, S.J., Feldman, W.C., Montgomery, M.D.: J. Geophys. Res. 81(16), 2719 (1976). https://doi.org/10.1029/JA081i016p02719

    ADS  Article  Google Scholar 

  2. Gary, S.P.: Theory of Space Plasma Microinstabilities. Cambridge University Press, Cambridge (1993). https://doi.org/10.1017/CBO9780511551512

    Google Scholar 

  3. Gary, S.P., Yin, L., Winske, D., Reisenfeld, D.B.: J. Geophys. Res. Space Phys. 105(A9), 20989 (2000a)

    ADS  Article  Google Scholar 

  4. Gary, S.P., Yin, L., Winske, D., Reisenfeld, D.B.: Geophys. Res. Lett. 27(9), 1355 (2000b)

    ADS  Article  Google Scholar 

  5. Gomberoff, L., Valdivia, J.A.: J. Geophys. Res. Space Phys. 108(A1), 1050 (2003)

    ADS  Article  Google Scholar 

  6. Kasper, J.C., Stevens, M.L., Lazarus, A.J., Steinberg, J.T., Ogilvie, K.W.: Astrophys. J. 660(1), 901 (2007)

    ADS  Article  Google Scholar 

  7. Kohl, J.L., Noci, G., Antonucci, E., Tondello, G., Huber, M.C.E., Cranmer, S.R., Strachan, L., Panasyuk, A.V., Gardner, L.D., Romoli, M., Fineschi, S., Dobrzycka, D., Raymond, J.C., Nicolosi, P., Siegmund, O.H.W., Spadaro, D., Benna, C., Ciaravella, A., Giordano, S., Habbal, S.R., Karovska, M., Li, X., Martin, R., Michels, J.G., Modigliani, A., Naletto, G., Neal, R.H.O., Pernechele, C., Poletto, G., Smith, P.L., Suleiman, R.M.: Astrophys. J. 501(1), 127 (1998). https://doi.org/10.1086/311434

    ADS  Article  Google Scholar 

  8. Li, X., Habbal, S.R.: Sol. Phys. 190(1–2), 485 (1999). https://doi.org/10.1023/A:1005288832535

    ADS  Article  Google Scholar 

  9. Maneva, Y.G., Araneda, J.A., Marsch, E.: Astrophys. J. 783(2), 139 (2014). https://doi.org/10.1088/0004-637x/783/2/139

    ADS  Article  Google Scholar 

  10. Marsch, E.: Kinetic Physics of the Solar Wind Plasma. Physics and Chemistry in Space, vol. 21, p. 45. Springer, Berlin (1991)

    Google Scholar 

  11. Marsch, E.: Living Rev. Sol. Phys. 3(1), 1 (2006). https://doi.org/10.12942/lrsp-2006-1

    ADS  Article  Google Scholar 

  12. Marsch, E., Livi, S.: J. Geophys. Res. 92(A7), 7263 (1987). https://doi.org/10.1029/JA092iA07p07263

    ADS  Article  Google Scholar 

  13. Marsch, E., Mühlhäuser, K.-H., Rosenbauer, H., Schwenn, R., Neubauer, F.: J. Geophys. Res. 87(A1), 35 (1982)

    ADS  Article  Google Scholar 

  14. Maruca, B.A., Kasper, J.C., Gary, S.P.: Astrophys. J. 748(2), 137 (2012)

    ADS  Article  Google Scholar 

  15. Matteini, L., Hellinger, P., Goldstein, B.E., Landi, S., Velli, M., Neugebauer, M.: J. Geophys. Res. 118(6), 2771 (2013). https://doi.org/10.1002/jgra.50320

    Article  Google Scholar 

  16. Melrose, D.B.: Instabilities in Space and Laboratory Plasmas. Cambridge University Press, Cambridge (1986). https://doi.org/10.1017/CBO9780511564123

    Google Scholar 

  17. Neugebauer, M.: Fundam. Cosm. Phys. 7, 131 (1981)

    ADS  Google Scholar 

  18. Reisenfeld, D.B., Gary, S.P., Gosling, J.T., Steinberg, J.T., McComas, D.J., Goldstein, B.E., Neugebauer, M.: J. Geophys. Res. 106(A4), 5693 (2001)

    ADS  Article  Google Scholar 

  19. Revathy, P.: J. Geophys. Res. 83(A12), 5750 (1978)

    ADS  Article  Google Scholar 

  20. Robbins, D.E., Hundhausen, A.J., Bame, S.J.: J. Geophys. Res. 75(7), 1178 (1970)

    ADS  Article  Google Scholar 

  21. Stansby, D., Perrone, D., Matteini, L., Horbury, T.S., Salem, C.S.: Astron. Astrophys. 623, 2 (2019). https://doi.org/10.1051/0004-6361/201834900

    ADS  Article  Google Scholar 

  22. Verscharen, D., Bourouaine, S., Chandran, B.D.G., Maruca, B.A.: Astrophys. J. 773(1), 8 (2013)

    ADS  Article  Google Scholar 

  23. von Steiger, R., Geiss, J., Gloeckler, G., Galvin, A.B.: Space Sci. Rev. 72(1), 71 (1995). https://doi.org/10.1007/BF00768756

    ADS  Article  Google Scholar 

Download references


The authors acknowledge support from the Katholieke Universiteit Leuven (Grant No. SF/17/007, 2018), Ruhr-University Bochum, and Alexander von Humboldt Foundation. These results were obtained in the framework of the projects SCHL 201/35-1 (DFG–German Research Foundation), C14/19/089 (KU Leuven), G0A2316N (FWO-Vlaanderen), and C 90347 (ESA Prodex 9). M.A.R. acknowledges Punjab Higher Education Commission (PHEC) Pakistan for granted Postdoctoral Fellowship FY 2017-18. S.M. Shaaban gratefully acknowledges support by a Postdoctoral Fellowship (Grant No. 12Z6218N) of the Research Foundation Flanders (FWO-Belgium). P.H.Y. acknowledges NASA Grant NNH18ZDA001N-HSR and NSF Grant 1842643 to the University of Maryland, and the BK21 plus program from the National Research Foundation (NRF), Korea, to Kyung Hee University.

Author information



Corresponding author

Correspondence to S. M. Shaaban.

Additional information

Publisher’s Note

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


Appendix A: Polarization vector

For an ambient magnetic field vector directed along \(z\) axis, \(\hat{\mathbf{b}}={\mathbf{B}}_{0}/|B_{0}|=\hat{\mathbf{z}}\) and the wave vector lying in \(xz\) plane, \({\mathbf{k}}=k_{\perp }\hat{\mathbf{x}}+k_{\parallel }\hat{\mathbf{z}} =\hat{\mathbf{x}}k \sin \theta +\hat{\mathbf{z}}k\cos \theta \), we define three orthogonal unit vectors, following (Melrose 1986), \(\hat{\boldsymbol{\kappa }}=\hat{\mathbf{x}}\sin \theta +\hat{\mathbf{z}}\cos \theta \), \(\hat{\mathbf{a}}=\hat{\mathbf{y}}\), and \(\hat{\mathbf{t}}=\hat{\mathbf{x}}\cos \theta -\hat{\mathbf{z}}\sin \theta \). Then the unit wave electric field vector is given by

$$ \hat{\mathbf{e}}({\mathbf{k}})=\frac{\delta {\mathbf{E}}}{|\delta E|} = \frac{K\,\hat{\boldsymbol{\kappa }}+T\,\hat{\mathbf{t}} +i\,\hat{\mathbf{a}}}{(K^{2}+T^{2}+1)^{1/2}}. $$

Making use of linear wave equation,

$$\begin{aligned} \left [\epsilon _{ij}-N^{2}\left (\delta _{ij}-k_{i}k_{j}/k^{2} \right )\right ] \delta E_{j}=0, \end{aligned}$$

it is possible to obtain

$$\begin{aligned} \delta E_{x} &= \frac{\epsilon _{xx}-N^{2}}{\epsilon _{xy}}\,\delta E_{y}, \\ \delta E_{z} &= - \frac{N^{2}\sin \theta \cos \theta }{\epsilon _{zz}-N^{2}\sin ^{2}\theta } \frac{\epsilon _{xx}-N^{2}}{\epsilon _{xy}}\,\delta E_{y}. \end{aligned}$$

Upon direct comparison with Eq. (14) one may identify

$$\begin{aligned} K &= - \frac{i\sin \theta \,(\epsilon _{zz}-N^{2})\,\epsilon _{xy}}{\epsilon _{xx}\,\epsilon _{zz}-N^{2}\,A}, \\ T &= - \frac{i\cos \theta \,\epsilon _{zz}\,\epsilon _{xy}}{\epsilon _{xx}\,\epsilon _{zz}-N^{2}\,A}, \end{aligned}$$

where \(A = \epsilon _{xx}\sin ^{2}\theta +\epsilon _{zz}\cos ^{2}\theta \). Upon making use of Eq. (5), we further obtain

$$\begin{aligned} K &= -M\sin \theta ,\qquad T=-M\cos \theta , \\ M &= \frac{x^{2}Q_{+}Q_{-}+\delta P_{+}P_{-}(x^{2}+\frac{1}{4})}{x^{2}D+q^{2}P_{+}P_{-}Q_{+}Q_{-}\mu ^{2}}, \end{aligned}$$

where various quantities, \(P_{\pm }\), \(Q_{\pm }\), and \(D\), as well as normalized wave number and frequency, \(q=ck/\omega _{pp}\) and \(x=\omega /\Omega _{p}\), are defined in Eq. (8).

Appendix B: Parameter \(R\)

In the growth rate expression (9) appears a quantity \(\partial (\omega ^{2}N^{2})/\partial \omega \), which in normalized form, is defined by

$$ R=\frac{\Omega _{p}}{\omega _{pp}^{2}} \frac{\partial (\omega ^{2}N^{2})}{\partial \omega } = \frac{\partial q^{2}}{\partial x}. $$

Making use of Eqs. (6) and (8), the desired quantity \(R\) can readily be computed as

$$\begin{aligned} R =& \frac{x}{2P_{+}P_{-}Q_{+}Q_{-}q^{2}\mu ^{2} +D(1+\mu ^{2})x^{2}} \\ & \times \left [\left (\frac{2Q_{+}Q_{-}}{P_{+}P_{-}} + \frac{\delta }{2}\frac{P_{+}P_{-}}{Q_{+}Q_{-}}\right )q^{2}(1+\mu ^{2}) \right . \\ & +2\delta ^{2}P_{+}P_{-}Q_{+}Q_{-} + \frac{2x^{2}(x^{2}-2)Q_{+}Q_{-}}{P_{+}P_{-}} \\ & \left .+\frac{\delta x^{2}}{4} \frac{8x^{4}(2x^{2}-5)+3(9x^{2}-2)}{P_{+}P_{-}Q_{+}Q_{-}}\right ]. \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rehman, M.A., Shaaban, S.M., Yoon, P.H. et al. Electromagnetic instabilities of low-beta alpha/proton beams in space plasmas. Astrophys Space Sci 365, 107 (2020). https://doi.org/10.1007/s10509-020-03823-4

Download citation


  • Plasma
  • Instability
  • Waves
  • Solar wind