Gas-dynamical Description of the Propagation of a Cloud of Hot Electrons Through a Plasma with Decreasing Density

S. Sobhanian¤ and G. R. Foroutan†
¤Faculty of Physics, Tabriz University, Tabriz 51664, Iran.
†Physics Department, Faculty of Science, Sahand University of Technology, 51335-1996, Tabriz, Iran
Abstract. The dynamic of a cloud of hot electrons propagating through a cold background plasma is investigated in the
regime of fast relaxation, when the relaxation time of the beam distribution function is much shorter than its propagation time.
Gas-dynamic theory is extended to include the effect of decreasing background number density on the evolution of the beam
and wave distribution functions. Numerical simulations of the quasilinear equations show that the evolution of the system is
self-similar and satisfies the condition of gas-dynamic theory. In spite of decrease in the number of hot electrons the beam
relaxation is always fast enough to generate a high level of Langmuir waves which is related to decrease in the number density
of background plasma. Contrary to the case of uniform plasma, the velocity of propagation of the beam is no longer constant
and decreases during the course of propagation. The results are especially appropriate for the propagation of hot electron
beams through solar corona plasma and generation of Langmuir waves and the associated type III radio bursts.
Keywords: solar flares, quasilinear relaxation, gas-dynamic theory, Langmuir waves
PACS: 52.35.Ra, 52.35.Qz. 52.35.Fp, 52.35.-y52.35-g
It is widely accepted that type III radio bursts are generated by the hot electron beams ejected during solar flares and
traveling fromthe Sun toward the Earth [1, 2, 3]. While streaming through solar corona plasma, the hot electrons appear
as a bump on the tail of background distribution function and can excite Langmuir plasma oscillations. The Langmuir
waves are then partially transformed into radio emission at fundamental and harmonic of plasma frequency when their
intensity reaches the threshold of three wave interactions [4]. The Langmuir waves also influence the propagation of
the electron beam and result in relaxation of the beam in velocity space. If the amplitude of the Langmuir waves is
sufficiently small such that their energy is much less than the thermal energy of surrounding plasma, then the evolution
of beam-plasma system can be described in the frame of quasilinear theory.
The dynamic spectra of type III radio bursts presents radio emission with a fast drift toward smaller frequencies.
The frequency drift is associated with the decrease of plasma density and consequently the local plasma frequency
declines as the beam propagates away from the Sun. The presence of plasma inhomogeneity has two main effects.
Firstly, the relaxation time of the beam and generation of Langmuir waves depend on the background number density.
Secondly, the Langmuir waves experience a shift in their wavenumber due to variation of local refractive index [5].
The quasilinear equations describing the evolution of the beam distribution function and energy density of Langmuir
waves are nonlinear and the analytical solutions can not be obtained without simplifying assumptions. Furthermore, the
recent work on the propagation of a cloud of hot electrons through plasma [6] revealed that the beam and the associated
Langmuir waves evolve in a self-similar way. The aim of this work is to extend and investigate the applicability of the
gas-dynamic theory to situations where the background number density decreases with distance.
The resonant one-dimensional quasilinear equations for evolution of beam.

where f (v,x, t) and W(k,x, t) are, respectively, the electron and Langmuir-wave distribution functions, x and v denote
the position and velocity coordinates of the electron at time t, k is the plasma wave number along the electron beam
velocity direction (the x-direction).
The wave frequency w(k) is given by the Langmuir dispersion relation w2 = w2
p +3k2ve
2, where w2
p = ne2/me0,
ve = (kBTe/m)1/2 and m, e, ve, Te, and n are the electron mass, charge, thermal velocity, temperature, and number
density, respectively. The Debye wave number is kD =wp/ve and w = kv describes the Cerenkov resonance condition.
In Eqs. (1) and (2) f is a sum of beam electrons fs and thermal background electrons fT and the first and second terms
on the right hand sides represent induced and spontaneous emission, respectively.
Using the assumption of fast relaxation, Ryutov and Sagdeev [7] developed a gas-dynamic description of the quasilinear
equations and obtained their time asymptotic solutions which is called mathematically the weak solutions. They
assumed that, the beam distribution function is always in a plateau state, with the plateau specified by its height and
upper boundary in velocity space. We first review the essential equations neglecting the effect of decreasing number
density. Then the system of quasilinear equations (1) and (2) can be written.

Figures 2(a)–2(c) display log contours of the electron distribution at a given time as a function of coordinate and
velocity on a color scale for three different scales of plasma inhomogeneity as in Fig. 1. The high level vertical strip at
the left side of each figure corresponds to the background electron distribution. Figures 2(d)–2(f) display log contours
of the corresponding Langmuir-wave distribution in coordinate and phase velocity space. It is seen that the system

FIGURE 2. Contours of the electron distribution and the corresponding spectral density of waves for three differnt scales of
plasma inhomogeneity as in Fig. 1. The scale of inhomogeneity decreases from (a) to (c).
keeps its self similarity in the presence of plasma inhomogeneity.
Figures 3(a)–3(c) represent log contours of the total wave energy, for three different levels of inhomogeneity as in
Figs. 1 and 2. There are two main effects related to the increase of the strength of inhomogeneity. First, as mentioned
above the level of total wave energy increases with increasing inhomogeneity. Second, the Langmuir waves experience
a shift in their wavenumber as propagate through an inhomogeneous plasma. Consequently, these waves become out of
resonance with the beam and cannot be reabsorbed and participate in subsequent evolution of the system. Therefore,
due to loss of energy, the velocity of propagation of the beam decreases and the trajectory of the maximum wave
energy in (x, t) plane deviates from straight line.

FIGURE 3. Contours of the total wave energy density U(x, t) for three different levels of plasma inhomogeneity as in Figs. 1 and
2. Like Fig. 2 the scale of inhomogeneity decreases from (a) to (c).
1. T. Takakura, and H. Shibahashi, Sol. Phys., 46, 323–346 (1976).
2. G. R. Megelssen, and D. F. Smith, Sol. Phys., 55, 211–240 (1977).
3. L. Muschietti, Sol. Phys., 130, 201–228 (1990).
4. P. A. Robinson and I. H. Cairns, Sol. Phys., 181, 363–394 (1998).
5. E. P. Kontar, Plasma Phys. Control. Fusion,43, 589–601 (2001).
6. G. R. Foroutan, B. Li, P. A. Robinson, I. H. Cairns and M. Moslehi-Fard, Phys. Plasmas, 12, 042905 (2005).
7. D. D. Ryutov and R. Z. Sagdeev, Soviet Phys. JETP, 31, 396–399 (1970).

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