Scattering and energization by large-amplitude whistler-mode waves in the evolution of solar wind electron distributions and Hamiltonian analysis of resonant interactions

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electrons. Simulations are run to record the response of a wide initial phase space volume with uniform waves and wave packets. Using a Hamiltonian analysis, resonant responses at different harmonics of the cyclotron frequency are included in the simulation. A numerical integration scheme that combines the Hamiltonian analysis and the relativistic 3d particle tracing deployed on a high performance cluster enables accurate mapping and large-scale statistical studies of phase space responses. Observations of electron distributions from WIND at 1 AU are used for normalization. This enables extrapolations for core, halo, and strahl electrons evolution with the numerical Green's function method. Results provide evidence for pitch angle broadening of the strahl and energization of core and halo electrons. This model can also provide results that are applicable to a number of different wave-particle interactions in the heliosphere for comparison to in-situ measurements.

The solar wind
Solar wind electrons include the cold core, hot halo, and magnetic field-aligned strahl populations. Figure 1 shows a model distribution based on Wilson et al. (2019), showing from left to right, the core, halo, strahl and total distributions. Strahl electrons are expected to remain narrow in pitch angle and field-aligned as they propagate outward from the Sun due to the conservation of the magnetic moment. However, in-situ data shows strahl pitch angle widths from 5 to 90 degrees (Anderson et al., 2012). The radial evolution of electron distributions has been examined using data from satellites at radial distances from the Sun ranging from Parker Solar Probe (PSP) inside ~ 0.  Whistler-mode waves have long been proposed as a potential mechanism for scattering of solar wind electrons. Theoretical arguments indicated that the waves must propagate sunward to match the resonance condition if the waves propagate parallel to the interplanetary magnetic field. Most theoretical and simulation studies have assumed parallel propagation. Electric field waveform captures from STEREO at 1 AU provided evidence that large amplitude narrowband whistlers were frequently obliquely propagating, enabling resonant interactions with electrons without the need for sunward propagation (Breneman et al., 2010;Cattell et al., 2020a). Our study is designed to assess the role of whistler-mode waves in the scattering and energization of solar wind electrons, using a particle tracing simulation.

Whistler-mode waves
Whistlers are right-hand polarized electromagnetic waves with frequency between the ion and electron cyclotron frequencies. The dispersion relation is given by

Resonant harmonics are orders of the isolated Bessel decomposition term in the Hamiltonian. The resonant condition is
In our range of interested energy, we observe three harmonics: the cyclotron resonance ( ), the Landau resonance ( ), and the anomalous cyclotron resonance ( ).
We aim to repeat these techniques for simulations of broader spectra of particle distribution and study its evolution in the solar wind conditions.

Simulation parameters based on 1 AU STEREO observations
Constant background magnetic field.
Waves propagate in the (xz) plane with various wave normal angles from 5 to 65 degrees. Wave amplitude is large (~10 mV/m) corresponding with .
Particles injected at the origin with kinetic energy and pitch angle .
Each and result is weighted using the distribution shown in Figure 1.

Equations of motion
We use the Lorentz force to derive our equations of motion.
Boris' algorithm, a volume-preserving scheme widely used in Particle-In-Cell simulations, is used to integrate for the numerical solutions.

Lyapunov exponents
Lyapunov exponents are a measure for "chaos". They determine the extent to which numerical solutions that are originally arbitrarily close diverge from each other. If the Lyapunov exponents are positive, any small perturbation in initial condition results in drastically different solutions, in other words, exponentially increasing distance. Numerical errors are also exponentially increased. Figure 4. Separation of two phase space trajectories (Sandri, 1996).
We implement a variational approach to estimate the Lyapunov exponents in our dynamical system (Benettin et al., 1980). A unit volume is attached to the initial position in 6D phase space and its contraction/expansion is calculated from the Jacobian of the Lorentz force. Figure 4 shows an instance of the expansion in 1 dimension. We use Runge-Kutta of the 4th order to estimate this numerically for local precision. For each dimension in phase space, the ith Lyapunov exponent is where is the principal axis of the parallelpiped in the jth dimension. Our simulation is then essentially an initial value problem Figure 5. Time series solution for an initially stationary particle at the origin being carried with the inbound whistler (Landau resonance).
From top to bottom, the panels show (1) the resonant harmonic, (2) the kinetic energy, (3) the pitch angle, (4) the Lyapunov exponents. Figure 5 shows a time series of the system's characteristic quantities and the convergence of the Lyapunov exponents in the last panel. Each of the dimension is unimportant since the volume is free to rotate in phase space and the algorithm does not take account for that. However, it is significant whether the total exponent, the extent to which the original unit volume contracts or expands, is reasonably close to zero. This determines whether our solutions are convergent since Boris' algorithm preserves volume in phase space (Qin et al., 2013). Also, it is interesting to note that there is at least one chaotic dimension. This reflects the resonant interactions when the particle is in the vicinity of a resonant harmonic, where the adiabatic invariant is violated and the system goes through irreversible change.

5-degree wave
Results from a simulation with a wave normal angle of 5 degrees. The video shows how the system changes in time (normalized to the wave period). Simulated particles are categorized by initial kinetic energy (horizontal axis) and initial pitch angle (vertical axis). The  Strahl angle width is broadened after 52 wave periods. Counter-streaming particles are deflected and energized across the co-streaming region, then get scattered back and de-energized by the hard boundary defined by the anomalous cyclotron resonance. Overall less negative Lyapunov exponents. There is no hard scattering boundary.

30-degree wave
Instead of only sunward particles, all particles in the initial distribution interact with all three harmonics. The interaction is isotropic.
The upper bound in the Lyapunov exponents ensures that this statistical structure is numerically valid (except for the counter-streaming region around 600 eV).
For obliquely propgating waves (as are seen at 1 AU by STEREO), pitch angle scattering is more efficient.
Strahl population becomes isotropic. The total population gains a larger drift in the parallel direction.

5-degree wave
Particle trajectory in kinetic energy vs. pitch angle domain under interaction with 5-degree wave. The particles are colored by initial kinetic energy (left) and initial pitch angle (right).
Clearly isolated interaction with each harmonic.
Anomalous cyclotron resonance deflects and scatters particle but does not energize.
Landau resonance scatters and energizes particles.
Counter-streaming particles reach steady state slower than particles already in Landau resonance.

30-degree wave
Particle trajectory in kinetic energy vs. pitch angle domain under interaction with 30-degree wave. The particles are colored by initial kinetic energy (left) and initial pitch angle (right).
[VIDEO] https://www.youtube.com/embed/Y1JoJi8kb5Q?rel=0&fs=1&modestbranding=1&rel=0&showinfo=0 [VIDEO] https://www.youtube.com/embed/vPdqgFhKzl8?rel=0&fs=1&modestbranding=1&rel=0&showinfo=0 ABSTRACT Whistler-mode waves have often been proposed as a plausible mechanism for pitch angle scattering and energization of electron populations in the solar wind. Theoretical work suggested that whistler waves with wave vectors parallel to the interplanetary magnetic field must counter-propagate (sunward) to the electrons for resonant interactions to occur. However, recent studies reveal the existence of obliquely propagating, high amplitude, and coherent waves consistent with the whistler-mode. Initial results from a particle tracing simulation demonstrated that these waves were able to scatter and energize electrons. That simulation was limited and did not examine a broad range of electron distributions. We have adapted the original particle tracing code for the solar wind with wave parameters observed by the STEREO satellites and to model core, halo and strahl electrons.
Simulations are run to record the response of a wide initial phase space volume with uniform waves and wave packets. Using a Hamiltonian analysis, resonant responses at different harmonics of the cyclotron frequency are included in the simulation. A numerical integration scheme that combines the Hamiltonian analysis and the relativistic 3d particle tracing deployed on a high performance cluster enables accurate mapping and large-scale statistical studies of phase space responses. Observations of electron distributions from WIND at 1 AU are used for normalization. This enables extrapolations for core, halo, and strahl electrons evolution with the numerical Green's function method. Results provide evidence for pitch angle broadening of the strahl and energization of core and halo electrons. This model can also provide results that are applicable to a number of different wave-particle interactions in the heliosphere for comparison to in-situ measurements.