Whistler wave scattering of energetic electrons past 90°

Whistler waves, characterized by their right-handed polarization and frequencies below the electron cyclotron frequency, are prevalent in astrophysical plasmas such as the corona, solar wind, magnetosphere, and intracluster medium (ICM).^6,8,9,29,53 Resonant whistler wave scattering represents one of the most important mechanisms that regulates the transport and loss of energetic particles.^{12,19–21,34} The suppression of particle transport in astrophysical systems, requires, of course, that particles undergo scattering over their full range of pitch angle with respect to the ambient magnetic field. An ongoing issue in the scientific literature is whether whistler waves can scatter particles beyond 90°. The issue is significant in various contexts, including Earth's radiation belts, the solar wind, and the ICM.^30,37,43,45 This issue is a variant of the “90° problem” commonly discussed in the cosmic-ray community.^10,15,50 In this paper, we present particle-in-cell (PIC) simulations that explore the consequences of a 90° scattering barrier. We show that the pileup of particles at 90° in the presence of such a barrier would lead to particle distributions with large reservoirs of free energy even in large astrophysical systems. Such distributions drive strong whistler waves that scatter energetic electrons past 90°, producing nearly isotropic pitch-angle distributions. Thus, we suggest that a 90° scattering barrier can not survive.

The degree of scattering is significantly influenced by the wave amplitude, propagation angle, and bandwidth, with larger amplitudes, oblique propagation, and finite bandwidth enhancing the scattering effect.^19,24,43,46 Strong scattering results in the appearance of perpendicular bumps at resonant velocities in the velocity distribution.^24,45,55 Particles at resonant velocities exhibit periodic oscillations within an interval of $v∥$⁠, known as the resonance band (or resonance width). As the wave amplitude increases, the bands broaden and can eventually overlap. Above the overlap threshold, the particle motion becomes diffusive, potentially allowing particles to cross the 90° pitch angle barrier.^{19,24,37,46,55}

Understanding whether energetic electrons can be scattered past 90° is crucial for calculating heat flux inhibition during solar flares. In the corona, particles can be accelerated to very high energies (sometimes above an MeV) during the energy release triggered by magnetic reconnection.^27,28,35 An accelerated, relativistic, free-streaming electron will take around 0.1 s to escape from the energy release region (⁠ $∼104$ km), which is about two orders of magnitude shorter than the observed lifetimes of energetic electrons.^11,27,28,39 This discrepancy, along with the prevalence of looptop hard x-ray sources^17,26,39,44 and the large difference between looptop and footpoint nonthermal electron flux,⁵¹ indicates that the transport of energetic electrons in the corona is strongly inhibited. Possible mechanisms responsible for such suppression and scattering of heat flux have been considered, including double layers,^31–33 magnetic traps,²² and magnetic turbulence.^4,5,23 PIC simulations have shown that strong whistlers can be self-generated by solar flare-generated heat flux and significantly scatter particles into the perpendicular (with respect to the ambient magnetic field) direction.^37,45 Whistler scattering of electrons is a significant process in other systems. In the Earth's bow shock transition layer, observations suggest that nonthermal electrons are locally scattered by whistler waves, resulting in pitch-angle broadening beyond 90°, a finding verified by test particle simulations.⁴³ Similarly, in the Earth's radiation belts, electromagnetic ion cyclotron (EMIC) waves can scatter relativistic particles via cyclotron resonance, causing a reduction in electron flux through atmospheric precipitation.^1,36,41,49 Test particle simulations show that electrons at 90° pitch angle can be directly scattered by large-amplitude EMIC waves, leading to significant pitch angle changes.³⁰ In the solar wind, the heat flux is carried by the “strahl” component, which forms a field-aligned beam and is scattered into the superthermal “halo” component.^16,38,52 The evolution of strahl electrons in the heliosphere is under active study.^18,40,54 Evidence from PIC simulations shows that the strahl excites oblique whistler waves, driving over-90° pitch-angle scattering and suppressing, but not eliminating the heat flux.⁴⁰ 

To summarize, in this study, we investigate the role of whistler waves in the scattering of energetic electrons, particularly addressing the challenge of pitch-angle scattering at 90°. Our PIC simulations reveal that strong oblique whistler waves, excited by an initial energetic heat flux distribution based on a presumed blockage of scattering past 90°, efficiently scatter electrons past the 90° barrier. The result is a nearly isotropic distribution. This mechanism has significant implications for understanding electron heat flux regulation in various astrophysical contexts, including solar flares, solar wind, and cosmic ray transport. The paper is organized as follows: In Sec. II, we discuss our numerical approach and the initial electron velocity distribution that would be present in the presence of a hypothetical transport barrier at a 90° pitch angle. Section III contains simulation results that show whistler wave excitation and strong electron scattering past 90°. Finally, we present our conclusions in Sec. IV.

We performed the PIC simulations with the code p3d,⁵⁸ which calculates particle trajectories using the relativistic Newton–Lorentz equations and advances the electromagnetic fields with Maxwell's equations. The system is initialized with a uniform background magnetic field $By=B0$ so the y direction is taken to be the parallel direction in the discussion that follows.

The context of the present simulations of electron scattering is the transport of energetic electrons in solar flares. Some models of electron energization in flares suggest that during magnetic reconnection, a combination of parallel electric fields and Fermi reflection in contracting and merging flux ropes drives energy gain, with particles gaining most energy parallel to the ambient magnetic field.^2,7 Particle distributions are well described by kappa distributions that extend many decades in energy.² Thus, energized electrons in flares should be well-described by kappa distributions with strong anisotropy, with parallel velocities being much greater than perpendicular velocities. Such distributions are also motivated by flare observations that exhibit power-law distributions of nonthermal electrons.^{3,14,27,42,56} It has already been established that anisotropic kappa distributions drive oblique whistlers that efficiently scatter electrons at least up to 90° in pitch angle.^37,45 The question to be addressed is the efficiency of scattering beyond 90°.

Figure 1 shows the initial electron velocity distribution function, characterized by an energetic semi-circle kappa distribution in the positive $v∥$ domain and a cold beam distribution in the negative $v∥$ domain. The heat flux is scattered into a uniform distribution with pitch angles $0<θ<90°$⁠, but the scattering is stopped by the 90° barrier. Again, the form of the cold return current beam is supported by a prior study³¹ where the cold return current was self-generated by the injection of hot plasma into a cold background. This setup, which is similar to previous studies featuring a strong heat flux and a return beam,^37,45 can represent energetic particle outflow from magnetic reconnection-driven solar flares. Simulations² have shown that reconnection can drive the energetic outflow to a marginal firehose state where $β∥/β⊥∼2$⁠. The parameter $vd/vAe$⁠, a key factor in our system for wave instability and scattering efficiency, falls within a reasonable range of observational data. Our value for β is too low to properly describe electron transport in the intracluster medium (ICM).^{9,13,46,48,57} However, a drift speed above the electron Alfvén speed is the regime of interest for the ICM. Further discussion of how the simulations might apply to the ICM is presented in Sec. IV.

The dimensions of the simulation domain are $Ly=2Lx=81.92 de$⁠, and the time step is $0.008Ωe−1$⁠. Ions are initialized with a cold Maxwellian distribution with temperature T_c and do not play an important role due to the large assumed mass ratio, $mi/me$ = 1600. The simulation uses 560 particles per species per cell and has a grid of 1024 × 2048 cells. We adopt periodic boundary conditions in both directions.

The initial distribution (see Fig. 1) drives the system unstable and amplifies magnetic fluctuations. Figure 2 shows the time evolution of the box-averaged out-of-plane magnetic field variance $⟨Bz2/B02⟩$⁠, which serves as an overall indicator of wave activity in the system. The fluctuations grow exponentially at the beginning and reach a short-lived plateau at $tΩe∼10$⁠. Two-dimensional (2D) images of B_z at this moment (not shown here) reveal an oblique mode with a very short wavelength (⁠ $∼0.1de$⁠). Because this mode is smaller in amplitude and transient in time, its impact on particle scattering can be neglected. The magnetic fluctuations reach their first peak at $tΩe∼30$⁠, decrease slightly to form another plateau, and then grow again to reach a second peak at $tΩe∼80$⁠. Corresponding 2D images at these times are shown in Figs. 3(a) and 3(b). After the second peak, the fluctuations decay slowly. A similar two-peak pattern in magnetic fluctuation has been observed in previous studies.^45,47

Figures 3(a) and 3(b) are 2D images of the out-of-plane magnetic fluctuations, $Bz/B0$⁠, at $tΩe=30$ and 80, respectively. Figure 3(a) corresponds to the first peak in the amplitudes of the magnetic fluctuations. The fluctuations correspond to oblique waves with a vector pitch angle of approximately 60°. The B_z component has a peak value around 0.75B₀, while the magnitude of the B_x component (not shown here) is smaller. Averaging $δB2=Bz2+Bx2$ over the entire domain, we find the mean fluctuation amplitude is $δB/B0∼0.1$⁠. Figure 3(b) is taken during the second peak of wave activity and shows a similar oblique mode, except that the angle of propagation with respect to the ambient field is reduced to approximately 40°. Fast Fourier transforms (FFT) of the fluctuations shown in panels (a) and (b) (not shown) indicate the wave patterns correspond to a dominant oblique mode with a finite bandwidth. The amplitudes of the parallel modes are small.

Figure 3(c) is a space-time diagram taken from the vertical dashed line in Fig. 3(a). Two major phases of wave activity are apparent: a downward-traveling wave in the time interval $20≲tΩe≲60$ with a phase speed of approximately $0.3vAe$ and an upward-moving wave in the interval $70≲tΩe≲160$ with a smaller phase speed of around $0.1vAe$⁠. Whistlers traveling in different directions suggest that they are driven by different electron components. The initial downward whistler is resonant with and driven primarily by the downward-traveling beam (Eq. 8), while the later upward whistler is driven by the energetic electrons moving in the upward (positive $v∥$⁠) direction (Eq. 6). Further evidence for this interpretation comes from the time dependence of the electron distribution functions.

Figure 4 shows hodograms of the magnetic field components, B_z vs B_x, from a fixed location in the simulation domain. Figure 4(a) is from the time interval $37≤tΩe≤60$⁠, which approximately coincides with the period of the downward-traveling whistlers. Figure 4(b) is in the interval $70≤tΩe≤160$⁠, which is the upward-moving phase. Figure 4(a) exhibits a left-handed elliptical polarization, while Fig. 4(b) shows a right-handed polarization. Considering the wave direction, Fig. 4(a) is from the viewpoint of the receiver (the wave goes to the viewer), and Fig. 4(b) is from the viewpoint of the source (the wave goes away from the viewer). Both perspectives confirm that these magnetic fluctuations exhibit right-handed polarization and hence are consistent with being whistlers. Another perspective on the rotation of the whistler at early time is that the return current electrons are dominating the wave dynamics. In the frame of the return current, whose negative velocity (⁠ $2.12vAe$⁠) exceeds that of the whistler (⁠ $0.1vAe$⁠), the whistler has a right-hand rotation direction.

Shown in Fig. 5 are the electron velocity-space distributions at several times during the simulation. In Fig. 5(a), the initial gap around $v∥=0$ is filled in. A significant portion of the return current, which initially had $v∥<0$⁠, has moved toward $v∥=0$ and into the region with $v∥>0$⁠. Thus, the negative velocity whistler is scattering and extracting energy from the return current particles as in the normal heat flux whistler instability.⁴⁵ At the same time, some $v∥>0$ electrons have moved across the $v∥=0$ boundary into the region $v∥<0$⁠. Such particles are also potentially contributing to the growth of the downward propagating whistler, although whether they contribute net energy to the downward waves is unclear. In Fig. 5(b), at the second peak of wave activity (⁠ $tΩe=80$⁠), the distribution function in the negative $v∥$ plane is significantly broadened, indicating a substantial number of the energetic electrons in the positive $v∥$ plane scattered past the 90° barrier. This demonstrates that the pitch angle diffusion coefficient at 90° is nonzero. The diffusion of $v∥>0$ electrons toward $v∥=0$ contributes energy to the positive propagating whistler that has reached its peak amplitude at this time. Despite the strong whistler-driven scattering, there are no obvious horn-like structures in velocity space as was seen in previous simulations.⁴⁵ This is likely due to the overlap of resonance bands.^19,55 It is noteworthy that, although the evolution of the distribution function seems to indicate a one-direction net flow from the upper plane to the lower plane, previous studies suggest, based on single particle trajectories, that many particles may have crossed the $v∥=0$ threshold multiple times.^24,37,47

At the end of the simulation, shown in Fig. 5(c), the final distribution appears nearly isotropic, indicating complete scattering. A significant number of electrons passing the 90° barrier implies a substantial reduction in the parallel heat flux. This process occurred on a rapid timescale of hundreds of electron cyclotron periods, demonstrating that whistler scattering is an efficient mechanism for inhibiting heat flux and confining escaping high-energy electrons in solar flares.

Figure 6 shows the electron distribution function integrated over a constant-velocity phase space from the $v∥>0$ region (blue lines) and $v∥<0$ region (red lines). The dashed lines correspond to the initial distribution (Fig. 1), and the solid lines correspond to the final distribution [Fig. 5(c)]. The initial beam (red dashed line) decays and drifts to a lower velocity. The tail of the red dashed line starts at zero, as there are no energetic particles in the $v∥<0$ region initially. Due to scattering, the initial particles from the $v∥>0$ region (blue dashed line) decrease. As a result, shown by the solid blue and red lines, the high-energy tails of the final distributions are nearly equal, implying that around half of the particles from the $v∥>0$ region have been scattered to the $v∥<0$ region.

Using particle-in-cell simulations, we explored electron pitch-angle scattering past 90° by oblique whistler waves in a system with an initial heat flux of energetic electrons with pitch angles less than 90°. This distribution might result from a hypothetical barrier to electron scattering at 90°. We demonstrated that this heat flux drives strong oblique whistler waves (with magnitude $δB/B0∼0.1$ and a wave vector normal angle of approximately 60°). These waves scatter a large fraction (around 50%) past 90° into the negative parallel velocity domain, making the late-time distribution nearly isotropic. This result confirms that large-amplitude, oblique whistlers can grow during flares and efficiently inhibit heat flux transport in the corona.^37,45

The results of earlier simulations suggested that the strong heat flux that is expected from a reconnection-driven solar flare² would drive oblique whistlers of sufficiently large amplitude to drive electrons past pitch angles of 90°.^37,45 In other plasma environments with weaker heat fluxes and lower amplitude whistler waves, the turbulence might be unable to diffuse electrons past 90°. The result might be an effective 90° scattering barrier. However, our simulations suggest that electron pileup near 90° yields a system with additional free energy that drives large-amplitude, oblique waves that efficiently scatter electrons past 90°. Thus, we conclude that a scattering barrier at 90° cannot survive in a real system. We have shown that both oblique propagation and large amplitude of whistler waves are crucial for enabling the scattering of electrons past 90°. Oblique propagation allows for a Landau resonance for electrons with $|v∥/vthe|≪1$⁠, which requires strong wave power at $kde≪1$ [see Eq. (5)]. This condition cannot be achieved by parallel whistlers since their interaction with $|v∥/vthe|≪1$ electrons is associated with waves with $kde≫1$⁠, where the wave power is weak.

Our initial distribution was designed to model the energetic electron outflow from flares in the solar corona where the mean energetic electron flux is expected to exceed the electron Alfvén speed.² The electron Alfvén speed is the drift threshold for the onset of strong whistler-driven electron pitch angle scattering.³⁷ In the ICM, where the plasma β is around 100, typical thermal speeds are well above the electron Alfvén speed. The heat flux in the ICM is expected to develop as a result of the magnetic-field-aligned electron temperature gradient. Suppose that there is a 90° scattering barrier in such a system. The hot plasma would penetrate into the cold region while maintaining a thermal distribution with pitch angles in the range $0<θ<90°$⁠. At the same location in the absence of scattering across 90°, there would be a separate colder plasma with pitch angles 90° < θ < 180°. The consequence would be a distribution similar to that shown in Fig. 1. The differing temperatures of the positive and negative velocity electrons would again produce discontinuities across the $v∥=0$ boundary. With increasing penetration of the hot plasma into the cold region, the resulting distribution will drive oblique whistlers that will relax the gradient across the 90° boundary. Further simulations with a focus on how interpenetrating Maxwellian distributions rather than kappa distributions can drive whistlers should be carried out to clarify whether 90° scattering barriers can survive in the high β environment of the ICM.

The authors were supported by NASA (Grant Nos. 80NSSC20K0627, 80NSSC20K1813, and 80NSSC20K1277) and NSF (Grant No. PHY2109083). The simulations were performed in the National Energy Research Scientific Computing Center (NERSC). J.F.D. acknowledges informative discussions with attendees of the Princeton Center for Theoretical Science (PCTS) meeting on synergistic approaches to particle transport in magnetized turbulence: from the laboratory to astrophysics.

The authors have no conflicts to disclose.

Hanqing Ma: Data curation (equal); Formal analysis (equal); Resources (equal); Visualization (equal); Writing – original draft (lead). J. F. Drake: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). M. Swisdak: Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.