Radiation belt electrons are strongly affected by resonant interactions with cyclotron‐resonant waves. For broad band, small amplitude waves the interactions are well described by quasi‐linear diffusion in pitch angle and energy, but coherent, large amplitude waves such as strong whistler mode chorus call for a different treatment. The standard nonlinear framework reduces the problem to that of a classical pendulum. This picture has generally been confirmed by many numerical simulations, but recent studies have uncovered additional, complex behavior, not captured by the pendulum model, for particles with low pitch angle. We show that avoiding a commonly made approximation leads to a more general but still tractable “second fundamental model” Hamiltonian, which involves not one but two regions of phase trapping. We analyze its phase portraits in detail, and perform representative test particle simulations with slowly changing parameters. We find that the trajectories encompass traditional phase bunching and phase trapping as well as additional behavior best understood using the new model.
Analytical results for phase bunching in the pendulum model of wave-particle interactions
Radiation belt electrons are strongly affected by resonant interactions with cyclotron-resonant waves. In the case of a particle passing through resonance with a single, coherent wave, a Hamiltonian formulation is advantageous. With certain approximations, the Hamiltonian has the same form as that for a plane pendulum, leading to estimates of the change at resonance of the first adiabatic invariant I , energy, and pitch angle. In the case of large wave amplitude (relative to the spatial variation of the background magnetic field), the resonant change in I and its conjugate phase angle ξ are not diffusive but determined by nonlinear dynamics. A general analytical treatment of slow separatrix crossing has long been available and can be used to give the changes in I associated with “phase bunching,” including the detailed dependence on ξ , in the nonlinear regime. Here we review this treatment, evaluate it numerically, and relate it to previous analytical results for nonlinear wave-particle interactions. “Positive phase bunching” can occur for some particles even in the pendulum Hamiltonian approximation, though the fraction of such particles may be exponentially small.
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- Award ID(s):
- 1847818
- NSF-PAR ID:
- 10399054
- Date Published:
- Journal Name:
- Frontiers in Astronomy and Space Sciences
- Volume:
- 9
- ISSN:
- 2296-987X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract -
This work compares several versions of the equations of motion for a test particle encountering cyclotron resonance with a single, field-aligned whistler mode wave. The gyro-averaged Lorentz equation produces both widespread phase trapping (PT) and “positive phase bunching” of low pitch angle electrons by large amplitude waves. Approximations allow a Hamiltonian description to be reduced to a single pair of conjugate variables, which can account for PT as well as phase bunching at moderate pitch angle, and has recently been used to investigate this unexpected bahavior at low pitch angle. Here, numerical simulations using the Lorentz equation and several versions of Hamiltonian-based equations of motion are compared. Similar behavior at low pitch angle is found in each case.more » « less
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This study analyzes the effects of electromagnetic ion cyclotron (EMIC) waves on relativistic electron scattering and losses in the Earth’s outer radiation belt. EMIC emissions are commonly observed in the inner magnetosphere and are known to reach high amplitudes, causing significant pitch angle changes in primarily > 1 MeV electrons via cyclotron resonance interactions. We run test-particle simulations of electrons streaming through helium band waves with different amplitudes and wave normal angles and assess the sensitivity of advective and diffusive scattering behaviors to these two parameters, including the possibility of very oblique propagation. The numerical analysis confirms the importance of harmonic resonances for oblique waves, and the very oblique waves are observed to efficiently scatter both co-streaming and counter-streaming electrons. However, strong finite Larmor radius effects limit the scattering efficiency at high pitch angles. Recently discussed force-bunching effects and associated strong positive advection at low pitch angles are, surprisingly, shown to cause no decrease in the phase space density of precipitating electrons, and it is demonstrated that the transport of electrons into the loss cone balances out the scattering out of the loss cone. In the case of high-amplitude obliquely propagating waves, weak but non-negligible losses are detected well below the minimum resonance energy, and we identify them as the result of non-linear fractional resonances. Simulations and theoretical analysis suggest that these resonances might contribute to subrelativistic electron precipitation but are likely to be overshadowed by non-resonant effects.more » « less
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In this study we consider the Hamiltonian approach for the construction of a map for a system with nonlinear resonant interaction, including phase trapping and phase bunching effects. We derive basic equations for a single resonant trajectory analysis and then generalize them into a map in the energy/pitch-angle space. The main advances of this approach are the possibility of considering effects of many resonances and to simulate the evolution of the resonant particle ensemble on long time ranges. For illustrative purposes we consider the system with resonant relativistic electrons and field-aligned whistler-mode waves. The simulation results show that the electron phase space density within the resonant region is flattened with reduction of gradients. This evolution is much faster than the predictions of quasi-linear theory. We discuss further applications of the proposed approach and possible ways for its generalization.more » « less
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Abstract Relativistic electron losses in Earth's radiation belts are usually attributed to electron resonant scattering by electromagnetic waves. One of the most important wave modes for such scattering is the electromagnetic ion cyclotron (EMIC) mode. Within the quasi‐linear diffusion framework, the cyclotron resonance of relativistic electrons with EMIC waves results in very fast electron precipitation to the atmosphere. However, wave intensities often exceed the threshold for nonlinear resonant interaction, and such intense EMIC waves have been shown to transport electrons away from the loss cone due to the
force bunching effect. In this study we investigate if this transport can block electron precipitation. We combine test particle simulations, low‐altitude observations of EMIC‐driven electron precipitation by the Electron Losses and Fields Investigations mission, and ground‐based EMIC observations. Comparing simulations and observations, we show that, despite the low pitch‐angle electrons being transported away from the loss cone, the scattering at higher pitch angles results in the loss cone filling and electron precipitation.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3389/fspas.2022.971358
