Type-III Electron Beams: 3D Quasilinear Effects

Abstract

A conventional model for the generation of Langmuir waves in Type-III radio bursts is based on a one-dimensional (1D) version of the quasilinear equations. In this model a wave with phase velocity \(v_{\phi }\) resonates with an electron with velocity \(v=v_{\phi }\), causing the waves to grow at a rate \(\propto {\mathrm{d}}F(v)/{\mathrm{d}}v>0\), where \(F(v)\) is the 1D-distribution function. The backreaction on the electrons drives the electrons towards a plateau distribution: \(\mathrm{d}F(v)/{\mathrm{d}}v\to 0\). In the 3D-generalization, none of these features apply: waves with phase speed \(v_{\phi }\) can resonate with electrons with speed \(v< v_{\phi }\), depending on the angle between the wave normal and the electron velocity, wave growth occurs only if the distribution function is both an increasing function of \(v\) and also has an anisotropic pitch-angle distribution, and the backreaction involves diffusion in both speed \(v\) and in pitch-angle \(\alpha \). In this article we discuss implications of the generalization from 1D to 3D on models for Type-III bursts. An effect that is absent in 1D, but may be important in 3D, is scattering of Langmuir waves by turbulence in the ambient plasma. Pitch-angle scattering by the scattered Langmuir waves may play an important role in the evolution of the Type-III beam.

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Appendices

Appendix A: Semiclassical Formalism

We start from the kinetic equation for the waves in semiclassical form based on the probability of spontaneous emission (e.g. Tsytovich, 1966; Melrose, 1986, p. 80) for Langmuir waves,

w L ( p , k ) = π e 2 ω p 2 ε e ħ ω L k 2 δ ( ω L k v cos χ ) , cos χ = cos α cos θ + sin α sin θ cos ( ϕ ψ ) ,
(23)

where \(\chi \) is the angle between \({\boldsymbol{k}}\) and \({\boldsymbol{v}}\), and \(\omega _{\mathrm{L}}\) is the frequency of the Langmuir waves, which we approximate by the plasma frequency [\(\omega _{\mathrm{p}}\)] assuming \(k\lambda _{\mathrm{De}}\ll 1\). In the axisymmetric case, the probability may be replaced by its average over azimuthal angle, which involves averaging the \(\delta \)-function in Equation 23 over azimuthal angle. Denoting this average by angular brackets and writing \(\cos \chi =\omega _{\mathrm{L}}/kv=v_{\phi }/v\), the average is zero except for \(\cos \alpha _{-}\le \cos \alpha \le \cos \alpha _{+}\) or \(\cos \theta _{-}\le \cos \theta \le \cos \theta _{+}\), where it is given by

$$ \langle \delta (\cos \chi -\cos \chi )\rangle = \frac{1}{\pi F(\alpha ,\theta ,\chi )}, $$
(24)

with

$$\begin{aligned} F(\alpha ,\theta ,\chi ) =&(1+2\cos \alpha \cos \theta \cos \chi - \cos ^{2}\alpha -\cos ^{2}\theta -\cos ^{2}\chi )^{1/2}, \\ \cos \alpha _{\pm } =&\cos (\theta \mp \chi ), \qquad \cos \theta _{\pm }=\cos (\alpha \mp \chi ). \end{aligned}$$
(25)

The identities

$$ \frac{\partial F}{\partial \cos \theta }=- \frac{\cos \theta -\cos \alpha \cos \chi }{F}, \qquad \frac{\partial ^{2}F}{\partial \cos \theta \partial \cos \alpha }= \sin ^{2}\chi \frac{\partial }{\partial \cos \chi }\frac{1}{F}, $$
(26)

with \(F=F(\alpha ,\theta ,\chi )\), and the integral

$$ \int _{\cos \alpha _{-}}^{\cos \alpha _{+}} \frac{\mathrm{d}\cos \alpha }{F(\alpha ,\theta ,\chi )}=\pi $$
(27)

is used below.

Appendix B: Balancing Focusing and Diffusion

The Fokker–Planck equation for the evolution of the angular dependence of a distribution function \(f\) due to systematic changes in pitch angle, described by a Fokker–Planck coefficient [\(\langle {\mathrm{d}}\cos \alpha /{\mathrm{d}}t\rangle \)] and a diffusion coefficient [\(D_{\alpha \alpha }\)], is

$$ \frac{\partial f}{\partial t}=\frac{\partial }{\partial \cos \alpha } \left [-\left \langle \frac{\mathrm{d}\cos \alpha }{\mathrm{d}t}\right \rangle f +D_{\alpha \alpha }\frac{\partial f}{\partial \cos \alpha } \right ]. $$
(28)

For an electron beam propagating towards decreasing \(B(s)\), conservation of the magnetic moment implies \(\sin ^{2}\alpha /B=\text{constant}\), and hence

$$ \left \langle \frac{\mathrm{d}\cos \alpha }{\mathrm{d}t}\right \rangle = \frac{1}{2\cos \alpha }\frac{v_{\mathrm{b}}}{L_{\mathrm{b}}}, \qquad L_{\mathrm{b}}=- \left |\frac{\mathrm{d}}{\mathrm{d}s}\ln B(s)\right |^{-1}. $$
(29)

Assuming isotropic scattering, such that \(D_{\alpha \alpha }\) does not depend on \(\alpha \), the Fokker–Planck Equation 28 with Equation 29 may be written as

$$ \frac{\partial f}{\partial t}=\frac{\partial }{\partial \cos \alpha } \left [ D_{\alpha \alpha }(\cos \alpha )^{\tau } \frac{\partial }{\partial \cos \alpha }(\cos \alpha )^{-\tau } f\right ], \qquad \tau =\frac{v_{\mathrm{b}}}{2L_{\mathrm{b}}D_{\alpha \alpha }}. $$
(30)

The steady-state solution of Equation 30 is

$$ f\propto (\cos \alpha )^{\tau }. $$
(31)

B.1 Absorption Coefficient for Distribution Equation 6

The absorption coefficient for the distribution in Equation 6 may be evaluated by expanding \(\phi (\alpha )\) in Legendre polynomials and using results derived by Melrose and Stenhouse (1977). For \(N=0,1,2\) one uses

$$ P_{1}(x)=x,\quad P_{3}(x)=\frac{1}{2}(5x^{3}-3x),\quad P_{5}(x)= \frac{1}{8}(63x^{5}-70x^{3}+15x), $$
(32)

to write \(\cos ^{2N+1}\alpha \) in terms of Legendre polynomials. The integrals involved in evaluating Equation 13 are then of the form

$$ \frac{1}{\pi }\int _{\cos \alpha _{-}}^{\cos \alpha _{+}} \frac{P_{l}(\alpha )\, \mathrm{d}\cos \alpha }{[(\cos \alpha _{+}-\cos \alpha )(\cos \alpha -\cos \alpha _{-})]^{1/2}} =P_{l}(\cos \theta )P_{l}(\cos \chi ). $$
(33)

One finds

$$ \gamma _{\mathrm{A}}(k,\theta )= \frac{-2\pi ^{2}\omega _{\mathrm{p}}^{4}}{n_{\mathrm{e}}k^{3}}\frac{1}{v_{\phi }} \int _{v_{\phi }}^{\infty }{\mathrm{d}}vf_{0}(v)\sum _{n=0}^{N}a^{N}_{2n+1} P_{2n+1}( \cos \theta )P'_{2n+1}(\cos \chi ), $$
(34)

with

$$ a^{0}_{1}=1,\qquad a^{1}_{1}=\frac{3}{5}, \quad a^{1}_{3}=\frac{2}{5}, \qquad a^{2}_{1}=\frac{3}{7}, \quad a^{2}_{3}=\frac{4}{9}, \quad a^{2}_{5}= \frac{8}{63}. $$
(35)

The result given by Equation 14 for \(\cos \theta =1\) follows from Equations 34 and 35 with \(P_{l}(1)=1\).

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Melrose, D.B., Harding, J. & Cairns, I.H. Type-III Electron Beams: 3D Quasilinear Effects. Sol Phys 296, 42 (2021). https://doi.org/10.1007/s11207-021-01783-8

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Keywords

  • Solar flares
  • Radio bursts
  • Plasma instabilities