Nonlinearly generated plasma waves as a model for enhanced ion acoustic lines in the ionosphere



[1] Observations from the EISCAT Svalbard Radar, for instance, demonstrate that the symmetry of the naturally occurring ion line can be broken by an enhanced, non-thermal, level of fluctuations, i.e., Naturally Enhanced Ion-Acoustic Lines (NEIALs). In a significant number of cases, the entire ion spectrum can be distorted, with the appearance of a third line, corresponding to a propagation velocity significantly below the ion acoustic sound speed. By numerical simulations, we consider one possible model accounting for the observations, suggesting that a primary process can be electron acoustic waves excited by a cold electron beam. Subsequently, an oscillating two-stream instability excites electron plasma waves which in turn decay to asymmetric ion lines. Our code solves the full Vlasov equation for electrons and ions, with the dynamics coupled through the electrostatic field derived from Poisson's equation.

1. Introduction

[2] Some of the most versatile and widely used tools for studying the Earth's ionosphere are incoherent scatter radars, giving both ion-acoustic and electron plasma fluctuation spectra. In many cases it is found that the ion-line signal is distorted [Buchert et al., 1999; Grydeland et al., 2003]. The two ion-lines can have different amplitudes and be corresponding to velocities which do not match the expected ion acoustic sound speed. Often, also an unshifted ion line can be observed, between the up- and down-shifted lines. Several models can account for some of these features: the symmetry of the natural ion-line is broken, for instance, if a current is flowing through the plasma [Sedgemore-Schulthess and St.-Maurice, 2001; Wahlund et al., 1992]. If an electron beam enhances electron plasma waves (Langmuir waves) significantly above the thermal level, then ion acoustic waves can be excited by parametric decay of these waves. Such models were invoked for instance by Forme [1999] and Forme et al. [2001]. Consistent with the basic features of this proposed model, observations of simultaneously enhanced levels of ion and electron plasma waves have been reported [Strømme et al., 2005]. Even earlier, it was pointed out that an external “pump-wave” could give effects similar to those observed [Dysthe et al., 1977; Fejer, 1977].

[3] The non-shifted (or weakly shifted) ion line can be explained by two basically different models. Since the propagation velocity differs significantly from the sound speed, the line is not a natural fluid mode. It is then either a feature which has to be continuously maintained by some external agency, the electron beam for instance, or, alternatively, it is a natural mode existing beyond a standard fluid model, a linear kinetic van Kampen-Case mode, or a nonlinear BGK-mode [Bernstein et al., 1957; Schamel, 1986]. Either of these modes can in principle have any velocity, even when the physical conditions of positive definite velocity distributions is imposed. In reality, the velocities will be restricted to the range of thermal velocities of the appropriate species. Significantly different propagation velocities require very “artificial” shapes of the velocity distributions. The parameterized models suggested by Schamel [1986] represent one way of imposing conditions on the distribution functions, in order to make them match physically realistic velocity distributions at large distances from the structures.

[4] It seems that a wave-decay model may be the most promising for explaining the observations, at least concerning the of features up- and down-shifted ion lines [Sedgemore-Schulthess and St.-Maurice, 2001; Grydeland et al., 2003; Guio and Forme, 2006]. In these models it is assumed, seemingly with no exceptions, that the basic dispersion relation for the Langmuir waves is the one obtained for a thermal plasma, i.e., ω2 = ωpe2 + 3uTe2k2, in terms of the plasma frequency ωpe and the electron thermal velocity uTe = equation image. The enhanced level of beam-excited plasma waves is assumed to follow this dispersion relation, with a spectrum determined solely by the velocity and dispersion of the electron beam.

[5] One problem concerning the model based on Langmuir wave decay seems to be that sometimes very short wavelength primary Langmuir waves are needed to account for the observations, below a few tens of Debye lengths, λDe. It might be possible to find a low velocity electron beam which generates unstable waves for u < 4uTe, but the decay Langmuir wave (“daughter wave”) obtained from these will be strongly Landau damped, implying that the growth rate of the decay instability becomes negligible. Also, the effects connected with plasma inhomogeneity have been largely ignored, although a consistent treatment of the ionospheric plasma density gradient can bring new understanding of the observational results [Kontar and Pécseli, 2005].

[6] The simple Langmuir wave decay can seemingly give a good account for the unsymmetrically enhanced ion lines, but can not directly explain the weakly shifted component. In our first attempt to explain this feature we considered the possibility of excitation of ion phase-space vortices, or ion holes. These BGK-type structures are well known from laboratory experiments and numerical simulations [Pécseli et al., 1984]. It has been found, however, that excitation of ion holes is ineffective when the electron/ion temperature ratio is below two [Pécseli et al., 1984], and this is after all the most common parameter range for many ionospheric conditions. Analytical and numerical studies by Shukla and Eliasson [2003] have demonstrated that ion phase space holes can be maintained by an enhanced level of Langmuir waves even for moderate ratios Te/Ti. One purpose of the present study was to search for self-consistently and spontaneously generated ion holes with a trapped electron wave component. In order to make the conditions for ion hole formation demanding, we choose a temperature ratio of Te/Ti = 1.

[7] Also electron phase space vortices, or electron holes [Schamel, 1986], can be excited and they will have interest in the present context also because such structures can have, in principle, any velocity also one below the ion sound speed. Electron holes are well known from laboratory experiments [Saeki et al., 1979], but their possible role in the generation of NEIALs and the intermediate ion acoustic signature is unknown.

2. Beam-Driven Electron Plasma Waves

[8] When the electron beam velocity is much larger than any of the relevant thermal velocities we can have the case where the most unstable phase velocity is somewhere between the beam and background plasma velocities, at a region where the electron velocity distribution is close to vanishing. For this case we might as well assume both the beam and background distributions are adequately represented by δ-functions. The instability can then be modeled by a simple two-electron beam model. In the other limit, the phase velocity uph = ω/k of the most unstable mode will be close to the beam velocity ub so that ubuTbuph < ub, with uTb = equation image being the thermal velocity of the electron beam. For this fully kinetic limit, the instability is of the standard Landau-type, where the growth rate is determined by the slope of the electron velocity distribution at the phase velocity. The full time evolution can involve both models: in an initial phase we can have the standard cold-beam model to apply, until the waves have reached an amplitude where they effectively scatter the beam so that it disperses in velocity. The resulting “plateau” in the electron velocity distribution gives rise to an electron acoustic branch [Henry and Treguier, 1972]. When a significant number of particles are scattered into the velocity range of the linearly most unstable waves, the process will continue at a rate determined by the fully kinetic theory.

[9] The linear dispersion relation can be solved analytically in the cold electron limit (albeit with a lengthy general result), in which case the beam velocity enters via the normalizations. The result for a reference case is shown in Figure 1. The important part is here the unstable branch, appearing as a weakly dispersive electron acoustic mode. The most unstable part of this mode appears here at frequencies below the electron plasma frequency.

Figure 1.

Dispersion relation for the reference case with nb/n0 = 0.1, with ℜ{ω} given by a full line, and equation image{ω} ≥ 0 given by a dashed line, while a thin dashed line gives the beam velocity for reference.

[10] For a wide range of parameters with αnb/n0 ≥ 0.1, the most unstable wave is found for kub/ωpe ≈ 1.2. For smaller α → 0 we have kub/ωpe → 1. For α = 1/2 we find the most unstable wave for ub/2, as expected by symmetry reasons. For small α we find that the most unstable phase velocity increases, ultimately to reach ub. As an approximation, we have the phase velocity for the most unstable wave as uphub(1 − α). As an approximate criterion for the beaming instability to be relevant, rather than the Landau instability, we have ub(1 − α) < ubuTb, or α > uTb/ub for uTbub.

[11] The electron beam is exciting electron plasma waves by the beaming instability. The dominant wave component will be the one with the largest temporal growth rate, which here corresponds to a frequency below ωpe. When these waves reach a sufficient intensity, they can excite new waves by the oscillating two-stream instability (see Chen [1984] for an excellent summary). The low frequency component of these wave-modes need not be represented by a dispersion relation, and can in principle also be stationary, while the high frequency wave-component will have a frequency ωωpe.

3. Numerical Results

[12] The basic equations considered here are here the Vlasov equations for the electrons and ions, with the dynamics coupled through Poisson's equation. Details of the code have been presented elsewhere [Eliasson, 2001; Shukla and Eliasson, 2003]. We use here a spatial simulation interval of 620 λDe. The basic set of equations is solved for given initial conditions, which is standard for these types of problems. This gives a simplified alternative to the full problem with conditions imposed at a boundary, while at the same time retaining the important physics. Our analysis is restricted to one spatial dimension for practical computational reasons. In particular for the case of Langmuir wave decay we do not expect this to pose any serious limitation, also because the growth rate for the decay instability is known to have a maximum for aligned wave-vectors. Also, for the observations relevant to the present study [Grydeland et al., 2003] the radar beam is basically directed along the magnetic field lines, inviting a comparison with models referring to that direction and ignoring the magnetic field.

[13] We have analyzed a large parameter range, ub/uTe = 2–10, nb/n0 = 0.01–0.1, and Teb/Te0 = 0.2–0.6. We attempted to use representative plasma parameters, so that the numerical results can be used for qualitative and to some extent also quantitative comparisons with observations. Practical limitations are however imposed by the numerical code, which means that the effects have to be observable within at most a few thousand electron plasma periods. This condition restricts our mass ratio and also imposes limits on the relative beam density nb/n0. In order to have a well defined electrostatic wave problem, we choose the initial condition so that there is no DC-current in the system, i.e. let the background electron population drift slightly in the opposite direction of the beam.

[14] In Figure 2 we show the space-time evolution of the ion density. In particular we note the evolution of narrow, localized spikes and depletions having distributed propagation velocities. Figure 3 shows the (ω, kx)-distribution of the electron plasma oscillations in the beam-plasma system. Similarly, Figure 4 shows the corresponding variation for the ion density. Figures 3 and 4 possess an (ω, kx) → (−ω, −kx)-symmetry, but the properties of the low frequency parts are best seen in a representation retaining this redundancy.

Figure 2.

Space-time evolution of the ion density. Axes are in normalized units for M/m = 400, ub = 10uTe, nb = n0/10 and Teb = Te/5.

Figure 3.

Temporal and spatial Fourier transforms of the electron density, shown in an (ω, kx)-plane. A dashed line gives the electron beam velocity.

Figure 4.

Temporal and spatial Fourier transforms of the ion density, shown in an (ω, kx)-plane, see also Figure 3. The dashed line with a negative slope gives the sound speed.

[15] We have analyzed the electron density variations of the initial ∼100 electron plasma periods of the simulations, and find here only activity on the low-frequency electron acoustic-like branch of the dispersion relation, see Figure 1. The large line-width reflects the growth-rate of the linear instability. Simultaneously, we find a strong scattering in velocity space of the beam electrons. In the same time-interval, we find negligible activity in the ion density. As the linear instability saturates, we find that most of the high frequency part of the dispersion relation becomes populated, see Figure 3. Investigations of the ion density at later times show first the evolution of a long wavelength spatially varying density, which is almost stationary, and does not follow any linear dispersion relation. Later we find a slow evolution of a backward traveling ion acoustic wave and also a smaller amplitude forward propagating component, which is barely noticeable in Figure 4. The latter component originates from the decay from the beam modes to the ion sound mode and a backward traveling electron plasma wave. The thin backward traveling electron branch in Figure 3, has a curvature due to thermal effects ignored in Figure 1. All our simulations, with parameters listed before, contained the essential features described here.

[16] We have studied both electron and ion phase-spaces. Electron holes can be observed at late stages of the saturated instability. By comparison with Figure 2 we note that the largest electron hole is formed close to a density enhancement. Although we note a strong activity also in ion phase space, we find it interesting that no ion holes are formed. The numerical results of Pécseli et al. [1984] would indeed indicate that no such ion holes should be present, but as already mentioned, these observations refer to a case without electron plasma wave activity.

4. Discussions and Conclusions

[17] We have presented results from full numerical solutions of the coupled electron-ion Vlasov equations, using parameters that are relevant also for ionospheric conditions, but nonetheless differing from those that can be modeled by standard studies based on for instance Zakharov-type equations with energy sources and sinks [Guio and Forme, 2006].

[18] Our numerical results show the following features relevant for the observations of NEIALs: (1) The oscillating two stream instability [Chen, 1984] driven by the electron beam generated electron acoustic waves excites high frequency as well as low frequency oscillations. The latter does not necessarily follow the ion acoustic dispersion relation and can appear as a weakly shifted line. (2) The oscillating two stream instability simultaneously excites electron plasma waves with ωωpe, that can decay to another electron wave and an ion sound wave. An important point is that the electron acoustic mode can participate in the decay [Hanssen et al., 1994; Skjæraasen et al., 1996]. A part of the decay product is an unsymmetric population of sound waves propagating in opposite directions. (3) Late in the evolution of the waveforms, we find that electron phase-space vortices (but no ion vortices) are formed. These saturate in nonlinear ion-acoustic pulses, propagating at a speed larger than the linear ion sound speed [Schamel, 1986]. Many of these electron holes can be quite faint. Of those macroscopically noticeable (with density variations exceeding ∼5%), we have approximately a density of 1 electron hole per 100–300 λDe, with an individual lifetime of ∼200 electron plasma periods, τpe. These values refer to α = 0.1. The hole density decreases for decreasing α. The electron holes are transient phenomena, but they can be formed by electron trapping at any late time of the saturated stage of the instability. Very large electron holes are formed during the first phase of the linear instability, where the initial electron beam is dispersed, but these holes disrupt within the first ∼50τpe.


[19] Valuable discussions with Bengt Eliasson are gratefully acknowledged.