Study of the formation of the quiet-time electron slot, which divides the radiation belt electrons into an inner and an outer zone. The pitch-angle diffusion of radiation belt electrons resulting from resonant interactions with the observed plasmaspheric whistler-mode wave band is quantitatively investigated. The effects of wave propagation obliquely to the geomagnetic field direction with the resulting diffusion at all cyclotron-harmonic resonances and the Landau resonance are evaluated along with the effects of interactions occuring at all geomagnetic latitudes. The results obtained account for the long-term stability of the inner radiation zone, the location of its outer edge as a function of electron energy, and the removal of electrons to levels near zero throughout the slot. Computed pitch-angle distributions and precipitation decay rates are in good agreement with slot-region observations.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA077i019p03455%4010.1002/%28ISSN%292169-9402.RADBELTS

Pitch-angle diffusion of radiation belt electrons within the plasmasphere.

Review of modeling of losses and sources of relativistic electrons in the outer radiation belt II: Local acceleration and loss

[1] Much current research in magnetospheric physics is directed toward understanding the dramatic variations in relativistic (>1 MeV) electron fluxes that can take place in the Earth's outer zone during magnetic storms. The behavior of outer-zone energetic electrons is partly controlled by the competing mechanisms of acceleration and loss that result from wave-particle interactions, in particular electron gyroresonance with ELF, VLF, and electromagnetic ion cyclotron (EMIC) waves. Powerful techniques for treating gyroresonant wave-particle interactions are provided by quasi-linear diffusion theory. In this paper, we derive formulae for the quasi-linear (momentum, mixed, and pitch-angle) diffusion coefficients for cyclotron resonance with field-aligned electromagnetic waves of any mode and general spectral density. The formulae are fully exact, expressed in closed analytical form, and easily computable. Our results can therefore be readily used to determine accurate diffusion rates for many forms of wave-particle interaction in the magnetosphere and other space plasmas. We find that momentum diffusion rates for MeV electrons in gyroresonance with VLF chorus can be less than a day in the lower-density regions outside the plasmasphere. The mechanism of stochastic acceleration by VLF chorus could therefore be instrumental in generating relativistic electrons during the recovery phase of a magnetic storm. Pitch-angle diffusion rates of MeV electrons scattered by EMIC waves along the plasmapause can approach the limit of strong diffusion. EMIC wave scattering could hence contribute significantly to electron precipitation loss over the course of a storm. Codes designed to model electron dynamics in the radiation belts need to incorporate, in addition to radial (cross-L) diffusion, resonant diffusion due to electron gyroresonance with ELF, VLF, and EMIC waves. In order to determine the quasi-linear diffusion coefficients for such codes, observational data are required on the power spectral density, spatial distribution, and temporal variation of these wave modes. Currently, only limited wave data sets are available. In addition, problems of numerical instability associated with diffusion codes need to be solved. The development of fully comprehensive models of radiation belt electron dynamics remains a considerable scientific challenge.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2005JA011159

Quasi‐linear diffusion coefficients for field‐aligned electromagnetic waves with applications to the magnetosphere

During the recovery phase of a magnetic storm, fluxes of relativistic (> 1 MeV) electrons in the inner magnetosphere (3 ≤ L ≤ 6) increase to beyond prestorm levels, reaching a peak ∼4 days after the initiation of the storm. In order to account for the generation of these "killer electrons" a model is presented primarily on the basis of the stochastic acceleration of electrons by enhanced whistler mode chorus. In terms of a quasi-linear formulation a kinetic (Fokker-Planck) equation for the electron energy distribution is derived comprising an energy diffusion coefficient based on gyroresonant electron-whistler mode wave interaction and parallel wave propagation, a source term representing substorm-produced (lower-energy) seed electrons, and a loss term representing electron precipitation due to pitch angle scattering by whistler mode waves and electromagnetic ion cyclotron (EMIC) waves. Steady state solutions for the electron energy distribution are constructed and fitted to an empirically derived relativistic Maxwellian distribution for the high-energy "hard" electron population at geosynchronous orbit. If the average whistler amplitude is sufficiently large, for instance, 75-400 pT, dependent on the values of the other model parameters, and assuming a background plasma density of N 0 = 10 cm -3 outside the plasmasphere, then a good fit to the empirical distribution is obtained and corresponds to a timescale for the formation of the high-energy steady state distribution of 3-5 days. For a lower representative value of the background plasma density, N 0 = 1 cm -3, smaller whistler amplitudes, in the range 13-72 pT, can produce the high-energy distribution in the required time frame of several days. It is concluded from the model calculations that the process of stochastic acceleration by gyroresonant electron-whistler mode wave interaction in conjunction with pitch angle scattering by EMIC waves constitutes a viable mechanism for generating killer electrons during geomagnetic storms. The mechanism is expected to be particularly effective for the class of small and moderate storms possessing a long-lasting recovery phase during which many substorms occur.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999JA900444

A model for generating relativistic electrons in the Earth's inner magnetosphere based on gyroresonant wave-particle interactions

We calculate quasi-linear transport and acceleration parameters for cosmic ray particles interacting resonantly with undamped fast-mode waves propagating in a low-β plasma. For super-Alfvenic particles and a vanishing cross-helicity state of the fast-mode waves, we demonstrate that the rate of adiabatic deceleration vanishes, and that the momentum and spatial diffusion coefficients can be calculated from the Fokker-Planck coefficient Dμμ. Adopting isotropic fast-mode turbulence with a Kolmogorov-like turbulence spectrum, we demonstrate that Dμμ is the sum of contributions from transit-time damping and gyroresonant interactions. Gyroresonance refers to | n | ≠ 0 resonant particle-wave interactions. Transit-time damping refers to the n = 0 interaction of particles with the compressive magnetic field component of the fast-mode waves. We show that transit-time damping provides the dominant contribution to pitch-angle scattering in the interval ≤ | μ | ≤ 1, where is the ratio of Alfven to particle speed. In the interval | μ | < , transit-time damping does not occur, and gyroresonance provides a small but finite contribution to particle scattering. As a consequence, the momentum diffusion coefficient is mainly determined by the transit-time damping contribution. On the other hand, since the spatial diffusion coefficient and the related mean free path are given by the average over μ of the inverse of Dμμ, these spatial transport parameters are determined by the contribution from the interval | μ | < . We also calculate the cosmic ray transport parameters for plasma turbulence consisting of a mixture of isotropic fast-mode waves and slab Alfven waves. Here, the momentum diffusion coefficient is determined by the transit-time damping of the fast-mode waves, and is a factor ln -1 larger than in the case of pure slab Alfven wave turbulence. The mean free path and the spatial diffusion coefficient are modified significantly from the pure fast-mode case, since the crucial scattering at | μ | < is now provided by gyroresonances with slab Alfven waves. The mean free path is a constant at nonrelativistic energies, and may account for the legendary λfit-λQLT discrepancy of solar energetic particles.

Quasi-linear Theory of Cosmic Ray Transport and Acceleration: The Role of Oblique Magnetohydrodynamic Waves and Transit-Time Damping

Equations are set up to describe the quasilinear behavior of a collisionless, relativistic plasma immersed in a uniform, homogeneous magnetic field. Use of the adiabatic assumption, together with an approximation to resonant denominators, enables an estimate to be given of the change of the distribution function due to the turbulent waves. In the particular cases of pure electrostatic waves, magneto‐sonic waves, and cyclotron waves, estimates are made of the rate of energy change of resonant particles. It is shown that some modes exhibit resonant diffusion only when relativistic particles are present and for which there is no nonrelativistic counterpart. It is believed that this calculation may be of some interest in situations where relativistic particles play, or are suspected of playing, a major role; for example, the galactic cosmic ray gas and nonthermal radio sources.

I. Lerche

Papers

Quasilinear theory of resonant diffusion in a magneto-active, relativistic plasma.