Large-amplitude whistler waves and electron acceleration



[1] A recent observation shows that large-amplitude whistler waves propagating obliquely with respect to the ambient magnetic field may be responsible for energizing the radiation belt electrons to relativistic energies (MeV) within a time scale as short as a fraction of a second. Test-particle simulations available in the literature invariably adopt simple model wave forms for the oblique whistlers, such that rigorous analysis of these waves have not been performed to this date. The present Letter solves fully nonlinear cold electron fluid equation for obliquely propagating large-amplitude whistlers. Relativistic test particle simulation is then performed over these exact wave solutions, and it is shown that a population of initially low energy electrons can be accelerated to equation image(10) MeV energies, within a few seconds time scale.

1. Introduction

[2] One of the outstanding problems in magnetospheric research involves understanding the source and loss mechanism of the Earth's radiation belt electrons. The motion of charged particles in the Earth's radiation belt is adequately described by three adiabatic invariants, the gyro-motion, bounce motion along the inhomogeneous field, and the longitudinal drift motion around the Earth [Roederer, 1970; Watt, 1994]. However, a number of transient effects can break these invariants, and thus lead to the acceleration and loss of the particles. Of these, various plasma waves in the inner magnetosphere are thought of as the primary cause of the acceleration and loss processes.

[3] Whistler waves have been suggested as being important for energizing relativistic electrons since the early days [Kennel and Petschek, 1966]. Most theoretical efforts rely on quasi-linear diffusion theory, which assumes that whistler mode chorus emission has low amplitudes, and that acceleration occurs over multiple random interactions between many waves and particles [Summers et al., 1998; Roth and Hudson, 1999; Summers and Ma, 2000; Meredith et al., 2001; Horne and Thorne, 2003; Horne et al., 2005; Albert and Young, 2005]. The quasi-linear diffusion process, however, is a slow mechanism with typical timescales for energization and scattering occurring on the order of hours to days.

[4] Recently STEREO spacecraft detected unusually large amplitude whistler waves. Cattell et al. [2008] report observation by STEREO of whistler waves with amplitudes higher than ∼240 mV/m, when the spacecraft passed through Earth's dawn-side outer radiation belt. The observed peak frequency is similar to the whistler chorus (∼0.2fce) although there are distinct differences. Local magnetic field was ∼300 nT–350 nT, while the wave magnetic field was δB ∼ 0.5–2 nT, meaning δB/Bequation image(0.01) or so. These coherent waves exhibited lack of drift in frequency and were characterized by highly oblique angles of propagation (∼45°–70°), and they had large longitudinal electric field component. With the observed background electron number density in the range of 2 ∼ 5 cm−3, the ratio of local plasma to electron gyrofrequency is approximately 3 or so.

[5] Cattell et al. [2008] also carried out test-particle simulation and stated that electrons can rapidly gain energy up to an order of MeV in less that a fraction of a second, and that such a finding accounts for the rapid enhancement in electron intensities observed between the STEREO-B and STEREO-A passage during this event. However, the authors did not discuss the details of their test particle simulation. Bortnik et al. [2008] and Tao and Bortnik [2010] performed test particle simulation of oblique whistler acceleration problem similar to that discussed by Cattell et al. [2008], but they used simple model waves rather than computing the waves from rigorous equations.

[6] Recently, Sauer and Sydora [2010] worked out the whistler oscilliton theory to account for the observation by Cattell et al. [2008]. They assumed that obliquely propagating whistlers are locally generated by field-aligned energetic electron beam, which then saturates into an oscilliton-like state in nonlinear regime. The authors obtained a reasonable comparison between the oscilliton wave form and the observation.

[7] In the present Letter, we approach the problem differently from that of Sauer and Sydora [2010]. The oscilliton solution depends on a number of assumptions including quasi-neutrality assumption. In general, oblique whistlers have a compressional component. It is generally known that compressional waves always steepen eventually, whether in plasmas or neutral fluids. Since obliquely propagating whistler have a compressional component, under certain circumstances, oblique whistlers could also exhibit steepening behavior. For this reason, in this Letter we investigate the wave property associated with oblique whistlers on the basis of rigorous analysis. We also perform test-particle simulation over the exact wave solution and demonstrate that indeed the electrons can be energized to relativistic level within a time scale of seconds.

2. Large-Amplitude Whistlers

[8] For the present purpose we start from cold fluid equations without any linearization. Treating the ions as stationary neutralizing background with density n0 (this is different from the whistler oscilliton theory in which ion dynamics play a key role), in one-dimensional approximation the exact cold electron fluid equations are given by

equation image

where ∇ = equation image(∂/∂x) and d/dt = ∂/∂t + v · ∇. From the above it is clear that Bx = const. Suppose that the ambient magnetic field is given by

equation image

and that waves propagate along x axis. For exactly parallel propagation, θ = 0, it is straightforward to show that (1) supports an exact nonlinear wave solution given by

equation image

where Ωe = eB0/mec is the electron gyrofrequency defined with respect to Bx = B0, and Δk represents the wave amplitude (especially, the amplitude associated with the wave electric field). Here, e and me stand for unit electric charge and electron mass, respectively. Other notations are standard, E and B being electric and magnetic field vectors, respectively, n and v being electron fluid density and velocity, respectively, and c representing the speed of light in vacuo.

[9] Note that solution (3) represent an arbitrarily large-amplitude whistler wave. This is because it does not involve density perturbation, n = n0 = const. In general, when nonlinear waves involve density perturbation (i.e., compressional wave), the amplitude cannot be arbitrarily high, since beyond a certain critical amplitude the density may become negative, which is unphysical. However, since the exact parallel propagation is characterized by purely transverse wave, in principle, the amplitude can be arbitrarily high. Of course, once we include thermal and kinetic effects, then high-amplitude waves will trap particles, which will in turn lead to other nonlinear effects. However, within the context of cold plasma theory, solution (3) is not only exact but it also represents arbitrarily large amplitude wave.

[10] It is also straightforward to show that the dispersion relation ω = ωk associated with solution (3) is given by

equation image

which has the exact mathematical form as the cold plasma result. Here, ωpe2 = 4πn0e2/me is the square of the plasma frequency defined with respect to the density n0. It is imperative to emphasize that the above dispersion relation was derived within the context of the fully nonlinear equation without any linearization involved. For ωk < Ω, the above dispersion relation supports whistler wave solution. It should be noted that the above exact solution (3) and (4) was not reported in the present form in the literature, although it is a fairly straightforward exercise to obtain such a solution.

[11] Of course, the observation by Cattell et al. [2008] corresponds to whistler waves that are propagating in oblique directions. Specifically, the propagation angle is anywhere between θ = 45° to 70°. The exact solution (3) and (4), on the other hand, is valid only for exactly parallel propagation. We thus need to extend the solution to oblique angles of propagation (θ ≠ 0), for which the set of nonlinear equations (1) do not enjoy exact analytical solution. Consequently, we solve the equation by numerical means.

[12] Unlike the parallel propagation, oblique whistler wave solution is compressional. As a result, such waves will eventually steepen. In the case of Cattell et al.'s [2008] observation, the ratio of wave to ambient magnetic field is quite low, δB/Bequation image(0.01). Consequently, the wave steepening is expected to occur over a relatively long time scale. In order to demonstrate the steepening behavior, let us consider an artificially high wave amplitude for the moment. Specifically, we consider an initial wave amplitude that is ten times the observed value of δB/B ∼ 0.01. Thus, with artificially high δB/B ∼ 0.1, we may accelerate the wave steepening process. Other than the artificial wave amplitude we choose all other physical parameters as in the observation by Cattell et al. [2008], to wit, the wave propagation angle is θ = 70°, the ratio of plasma to gyrofrequency is ωpee = 3, and the normalized wave frequency is ωe = 0.2. The result is shown in Figure 1. We have solved all the physical quantities on the basis of equation (1), but here we show only the density perturbation and longitudinal electric field Ex, which are sufficient to prove that the oblique whistler wave is compressional.

Figure 1.

Numerical solution of oblique whistler wave for propagation angle α = 70°, and with wave frequency ωe = 0.2 and ratio of plasma to gyrofrequency ωpee = 3, that are typical of Cattell et al.'s [2008] observation, but we have imposed an artificially high wave amplitude δB/B = 0.1 (or more precisely, Δ = 0.1), which is an order of magnitude higher than the observed value. We chose such an arbitrarily high amplitude for the purpose of demonstrating the wave steepening process associated with compressional oblique whistler waves. Note how the density undergoes steepening while the longitudinal electric field Ex also becomes distorted from the initial sinusoidal form.

[13] In solving equation (1), we have adopted the following dimensionless quantities:

equation image

In dimensionless form the initial condition is chosen as follows:

equation image

The choice of Δ = 0.1 roughly corresponds to δB/B ∼ 0.1. The above choice of initial condition reduces to the exact nonlinear solution (3) and (4) in the limit of θ = 0. Although equation (6) is not an exact solution of nonlinear equation (1) for nonzero θ, one can show that it exactly satisfies linearized equation. We solved equation (1) with the above initial condition. In Figure 1 magenta lines represent the sinusoidal initial wave. As the initially monochromatic wave propagates at an angle θ = 70° with respect to the ambient field, it deviates from the initial monochromaticity and begins to get distorted as high harmonic components get generated. At normalized time t = 10 the density wave can be seen to steepen considerably. Of course, once we include thermal effects, collective dissipation will prevent the compressional waves to steepen indefinitely, but within the context of the present cold fluid electron theory, the compressional oblique whistlers undergo steepening and distortion over long time scale. The importance of Figure 1 is that when studying oblique whistlers one must beware that for sufficiently high wave amplitude and for sufficiently long wave propagation period, the use of sinusoidal model waves is not justified, and that one must exercise caution when modeling oblique whistlers.

[14] We now turn to wave amplitude that is closer to the actual observation, that is, δB/B ∼ 0.01. Since the parameter Δ can be considered as a proxy for δB/B, we choose Δ = 0.01 next. Other input parameters are exactly the same as in Figure 1. Shown in Figure 2 is the realistic oblique whistler wave solution. Even though dimensionless physical quantities, n, v, E, and B are solved from equation (1), we choose to show only the density, n, longitudinal electric field Ex, and one transverse electric field component, Ez. Figure 2 shows that for moderate wave amplitude the sinusoidal waveform is still retained at the end of computational time, t = 10. It is important to note that wave steepening will eventually occur for much longer wave propagation period. The salient point is that the oblique whistler solution thus obtained has a compressional component judging from finite density perturbation and longitudinal electric field Ex. Note also that the longitudinal electric field is much higher than the transverse electric field, which is consistent with observation [Cattell et al., 2008].

Figure 2.

Oblique whistler wave for propagation angle 70°, wave frequency ωe = 0.2, plasma to gyrofrequency ratio ωpee = 3, and wave amplitude δB/B ∼ 0.01 (or more precisely Δ = 0.01), as in STEREO observation [Cattell et al., 2008]. Note that for realistically moderate wave amplitude, oblique whistler wave still retains sinusoidal form within the time scale of interest. Even so, the wave is compressional judging from the fact that density perturbation is finite. Note also that the longitudinal electric field Ex dominates over transverse field Ez, just as in observation.

[15] To reiterate, oblique whistlers are generally compressional, and will undergo steepening (eventually), such that sinusoidal waveform observed by Cattell et al. [2008] is unusual. Moreover, the oscilliton solution obtained by Sauer and Sydora [2010], if it exists, must also be restricted to special conditions. We should caution that this does not necessarily mean that nonlinear oscillitons discussed by Sauer and Sydora [2010] do not exist. In their model, ion dynamics play a crucial role, whereas in the present paper, the ions are treated as a neutralizing background.

3. Acceleration of Relativistic Electrons

[16] We now consider the acceleration of relativistic electrons by oblique whistlers. We perform fully relativistic test-particle simulation. The underlying assumption is that in addition to the bulk cold plasma that supports the waves, there exists a small component of warm electrons. We solve the equation of motion for these electrons,

equation image

where the electric and magnetic field vectors E and B are computed from nonlinear cold electron fluid equation (1). The initial condition for the electrons must reflect the pre-existence of large-amplitude whistler waves, otherwise, immediately after the initial time step, the electrons will jostle around in response to the waves. These jostling motion may be mistaken for heating and acceleration. In order to prevent such an artifact, we initiate all electron test-particle runs with finite initial velocity that is prescribed by the same velocity perturbation as defined in equation (6). Let us call this velocity vw(0), where the subscript w stands for “wave” since this velocity component reflects direct particle response to the pre-existence of finite amplitude waves. We then added a random velocity perturbation on top. Consequently, the initial electron velocities are v(0) = vw(0) + vrandom. We choose vrandom according to Gaussian deviate scheme.

[17] Shown in Figure 3 is the result of the test-particle run. The initial ensemble of electrons have a Gaussian distribution with relatively low thermal energy. Since the system is periodic in x and one-dimensional we simply launched all test particles (4 × 104 in number) in one spatial location, x = 0. We also performed the same test particle runs with evenly distributed test electrons, but the result was identical. As we solve for the particle equation of motion (7) as well as the wave equation (1), we find that the initially warm electrons quickly gain energy within the timescale of seconds, to an energy level up to an order 10 MeV. This result unambiguously shows that the overall impact of large amplitude whistlers propagating in oblique directions is to interact with an entire ensemble, not just a fraction, of the electrons and efficiently energize them to relativistic energy levels within timescales of seconds.

Figure 3.

The result of test-particle simulation with oblique whistler wave. The wave solution is the same as that shown in Figure 2. An initial Gaussian distribution of low energy electrons can be see to be energized by interaction with oblique whistler waves to order 10 MeV energy levels within a few seconds. The final distribution is in a power-law form.

4. Conclusions and Discussion

[18] In the present Letter, we presented the test particle simulation of relativistic electrons for oblique whistler waves rigorously computed from nonlinear cold electron fluid equations. The result shows that oblique whistlers of the type detected by STEREO [Cattell et al., 2008] indeed can exist in plasmas, and that the acceleration of relativistic electrons by these waves is extremely efficient.

[19] Before we close, it is appropriate to mention several caveats. In the present Letter finite temperature associated with the background plasma is ignored, i.e., ∇P term in the fluid momentum equation is ignored. We also ignored the possible role of dynamic ions. It is pointed out in the literature that the ion dynamics is important for whistler oscilliton formation [Sauer and Sydora, 2010]. We do not address how these large-amplitude and almost coherent nonlinear whistler waves are generated in the first place. Sauer and Sydora [2010] invoke field-aligned beam as the free energy source for these waves, but the existence of such beams has not been confirmed observationally. We simply assume that the generation mechanism must be similar to that of chorus generation, i.e., driven by anisotropic electrons at the night-side magnetic equator. We do not consider the feedback effects of energetic electrons, or for that matter, the background plasma, on the large-amplitude waves. To address such an issue requires full particle-in-cell simulation. These tasks must belong to the future.


[20] This research was supported by WCU grant R31-10016 from the Korean Ministry of Education, Science and Technology.

[21] The Editor thanks Anthony Lui and an anonymous reviewer for their assistance in evaluating this paper.