Anomalous resonance between low-energy particles and electromagnetic plasma waves

Wave–particle interactions are critical for energy transfer and dissipation in collisionless space and astrophysical plasmas. Classical cyclotron resonance assumes the particle’s angular velocity is set solely by the background magnetic field, yielding the linear resonance condition ω − κv∥ − Ω = 0 and resonant velocity V∥ = (ω − Ω)/κ. This framework underpins explanations of ion and electron dynamics driven by EMIC and chorus waves and has observational support from MMS-era measurements of phase bunching. However, the derivation neglects wave-associated forces (qv × B⊥ and qE∥). For large-amplitude waves and low-energy particles, these forces can substantially modify particle angular velocity, invalidating the classical condition. The study tests the hypothesis that low-energy ions can undergo anomalous resonance—resonant behavior at energies and pitch angles inconsistent with the classical condition—producing phase relationships (in-phase and antiphase) between wave magnetic fields and ion fluxes. The aim is to demonstrate and quantify this anomalous resonance using MMS observations and test-particle simulations and to assess its implications for wave–particle energy exchange and wave evolution.

Prior work has established cyclotron resonance as a key process in near-Earth and planetary environments, with EMIC waves (below ion gyrofrequency) affecting ions and whistler-mode chorus (below electron gyrofrequency) accelerating electrons, contributing to radiation belt dynamics and atmospheric precipitation. MMS observations enabled direct identification of cyclotron resonance via gyro-phase–bunched ion flux stripes matching classical resonance predictions and phase-space proton hills associated with EMIC chirping. Nonetheless, classical theory relies on linear approximations that omit wave-associated forces. Simulations have indicated that small pitch-angle particles can experience anomalous resonance due to qv × B⊥, and theory has shown that qE∥ can also alter angular velocity, enabling trapped trajectories for particles far from classical resonance. Hamiltonian treatments predict two resonance islands (at ζ = 0° and 180°), but clear observational evidence has been lacking. This study addresses that gap.

Observations: MMS four-spacecraft measurements on 2020-01-14 (19:23:30–19:23:50 UTC) in the dawnside magnetosheath were analyzed. Instruments: Fast Plasma Investigation (3D ion distributions, ~150 ms cadence), Fluxgate Magnetometer (magnetic fields), Electric field Double Probes (electric fields), and Hot Plasma Composition Analyzer (ion species; ~99.5% protons). Spacecraft had small separations (10–40 km); data were averaged over the four spacecraft. Field-aligned coordinates (FAC) were defined using a low-pass filtered background field B0; wave fields were bandpass filtered (0.25–0.65 Hz). Wave characteristics were derived: left-hand circular polarization, negligible parallel components, Poynting flux predominantly antiparallel, phase speed estimated from Faraday’s law using E⊥ and B⊥. Doppler shift and bulk flow were used to estimate phase speed in plasma and spacecraft frames and wavelength. Gyro-phase spectra were constructed from high-cadence ion distributions; pitch-angle bins (e.g., 90° ± 11.25°, 0°–50°) and energy channels (e.g., 53, 441, 974, 1269 eV) were examined for phase-bunching. Simulations: Test-particle simulations applied Liouville’s theorem to compute ion phase-space density (PSD) along backward-traced trajectories. Initial ion distributions (plasma rest frame) were gyrotropic Maxwellians fitted to MMS pre-growth observations (19:23:23–19:23:28 UTC) over 50–1000 eV: proton temperature T = 130 eV, density n0 = 8.5 cm⁻3. In the spacecraft frame, distributions were shifted to match observed bulk velocity (−20, 10, 35) km s⁻1 (FAC). Electromagnetic fields (plasma rest frame) described a Gaussian-profiled, monochromatic EMIC wave packet propagating opposite to B0 with equal phase and group velocities: ωp = −kVph = 3.28 s⁻1; wavelength λ ≈ 383 km; packet center at z = 5λ; width L = 3λ. Maximum fields: Emax = 0.8 mV m⁻1 and Bmax = Emax/Vgp = 4 nT; uniform background B0 = 43.5 nT. A virtual spacecraft moving at constant speed in the plasma rest frame sampled distributions every 0.3 s. To reflect instrumental binning, 81 particles were traced per cell of finite widths in gyro-phase (30°), pitch angle (22.5°), and energy channel, totaling 60,264 particles. PSDs were averaged per bin. Additional analyses removed the Gaussian envelope to study constant-amplitude wave behavior (B1 = 4 nT) and to map phase-space trajectories (ζ, dζ/dt; v⊥–v∥; energy–ζ; v∥–ζ). Theoretical development: Starting from the full equation of motion in wave fields, the study derives a modified phase evolution equation retaining the wave-associated term, leading to a full resonance condition with an additional amplitude- and v⊥-dependent term. The criterion for validity of classical theory is obtained by comparing the nonlinear term to trapping frequencies, yielding a practical inequality involving B1/B0, |Vw − v∥|, and VA.

- MMS observed large-amplitude hydrogen-band EMIC waves (0.25–0.65 Hz), left-hand circularly polarized, with negligible parallel components. Derived parameters: spacecraft-frame phase speed ≈ 165 km s⁻1; plasma-frame phase speed ≈ 200 km s⁻1; wavelength ≈ 383–400 km; background B0 ≈ 43.5 nT; proton gyrofrequency Ω ≈ 4.17 s⁻1. - Perpendicular-moving ions (pitch ≈ 90°) from ~50 eV to ~1 keV showed clear gyro-phase–bunched PSD stripes in phase with the wave magnetic field, indicating strong acceleration/deceleration (tens to hundreds of eV), with enhanced PSD when ions move along B1 (ζ = 0°). - Quasi-parallel-moving ions (pitch 0°–50°) at ~1 keV (e.g., 974 and 1269 eV) showed phase-bunching antiphase to the wave magnetic field, with strongest enhancement near ζ = 180°. - These phase-bunched signatures occur at energies and pitch angles incompatible with the classical cyclotron resonance prediction. Using ω = 2.7 s⁻1, Ω = 4.17 s⁻1, and Vws = 165 km s⁻1 gives V∥,res ≈ 90 km s⁻1 (spacecraft frame; 55 km s⁻1 plasma frame), implying pitch angles (e.g., 27° at 53 eV; 78° at 974 eV) that do not match the observed phase-bunched populations. - Test-particle simulations reproduce the observed gyro-phase spectra and energy spectra, including both in-phase (ζ ≈ 0°) and antiphase (ζ ≈ 180°) bunching, validating the anomalous resonance scenario. - Phase-space analysis reveals two resonance islands centered at ζ = 0° and ζ = 180°. Trapped ions within these islands exhibit significant pitch-angle, energy, and v∥ variations (more pronounced for ζ = 0° islands for 90° pitch ions), while traversing ions pass through without trapping. - The full resonance condition includes an additional term proportional to the wave amplitude and dependent on v⊥, showing that wave-associated forces (qE∥ and qv × B1) can reduce or increase the effective angular velocity, shifting resonant velocities. In extreme cases, the resonant velocity can approach the wave phase velocity and even reverse sign relative to the classical prediction. - A practical criterion delineating when classical theory fails is given by B1/B0 ≪ |Vw − v∥|/VA. For representative trapped ions in this event, left- and right-hand sides are comparable (e.g., ~0.3 vs ~1.2 for a 441 eV, 90° proton; ~0.3 vs ~0.06 for a 974 eV, low-pitch proton), indicating violation of the classical criterion and occurrence of anomalous resonance. - The threshold for v∥ separating classical from anomalous behavior is ~500 km s⁻1 (≈1.5 keV proton) in this event, consistent with observations that phase-bunching appears below ~1.5 keV. - Time-varying wave amplitude (Gaussian envelope) displaces island centers from ζ = 0°/180° with deviations governed by the island expansion rate (∝ dB1/B1dt), explaining initial phase offsets of PSD peaks; alignment improves as amplitude variations subside.

The observations and simulations demonstrate that low-energy ions can resonate with large-amplitude EMIC waves at energies and pitch angles inconsistent with the classical cyclotron resonance condition. Including wave-associated forces in the phase dynamics yields a full resonance condition that predicts two concurrent resonance islands (ζ = 0° and 180°), consistent with the observed in-phase and antiphase gyro-phase bunching. This resolves the discrepancy between classical theory and MMS measurements by showing that the wave’s electric field and the Lorentz force from B1 nonlinearly modify particle angular velocity. The result implies substantially altered wave–particle energy exchange pathways: perpendicular ions (near 90° pitch) primarily experience qE∥-driven reduction of angular velocity and strong energy modulation near ζ = 0°, whereas low-pitch ions experience qv × B1-enhanced angular velocity and trapping near ζ = 180°. The derived criterion B1/B0 ≪ |Vw − v∥|/VA delineates the regime where classical theory applies; its violation at sub-1.5 keV energies explains the prevalence of anomalous resonance in the observed parameter space. Temporal modulation of wave amplitude further affects trapping by shifting island centers and widths, enabling energy transfer through phase-lagged currents. These findings suggest that standard models of diffusion, acceleration, and precipitation based on classical resonance may underestimate or mischaracterize energy exchange in regions with strong wave amplitudes and low background fields.

This work presents the first direct observational evidence, corroborated by test-particle simulations, of anomalous resonance between low-energy ions and large-amplitude EMIC waves. The key contributions are: (1) identification of in-phase and antiphase gyro-phase–bunched ion fluxes at energies and pitch angles incompatible with classical cyclotron resonance; (2) validation of a revised resonance framework that incorporates wave-associated forces, predicting and explaining two coexisting resonance islands; and (3) a practical criterion delineating when classical theory breaks down. Given the high phase-space densities of low-energy particles, anomalous resonance likely exerts strong control on wave–particle energy exchange and wave evolution, particularly in regions of weak magnetic field. Future research could extend this analysis across broader plasma environments, include magnetic field inhomogeneity and self-consistent wave–particle feedback, and assess impacts on global radiation belt dynamics and ion precipitation.

- The analysis focuses on a single EMIC wave event in the magnetosheath; broader statistical validation across events and regions is not presented. - Test-particle simulations assume a uniform background magnetic field and neglect field curvature/mirror forces (consistent with observations here but limiting applicability to inhomogeneous regions). - The prescribed wave model is a Gaussian-profiled, monochromatic packet with equal group and phase velocities; self-consistent wave growth/damping and multi-mode spectra are not included. - The criterion and thresholds are evaluated for event-specific parameters; generalization to other plasma conditions (e.g., different compositions, densities) requires further study.