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

Abstract

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.

Appendices

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

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

(14)

Making use of linear wave equation,

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

(15)

it is possible to obtain

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

(16)

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

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

(17)

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

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

(18)

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

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

(19)

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

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

(20)