Modeling shock-accelerated solar energetic particles coupled to interplanetary Alfven waves

We present an idealized model simulating the coupled evolution of the distributions of multispecies shock-accelerated energetic ions and interplanetary Alfven waves in gradual solar energetic particle (SEP) events. Particle pitch-angle diffusion coefficients are expressed in terms of wave intensities, and wave growth rates in terms of momentum gradients of SEP distributions, by the same quasilinear theory augmented with resonance broadening. The model takes into consideration various physical processes: for SEPs, particle motion, magnetic focusing, scattering by Alfven waves, solar wind convection, and adiabatic deceleration; for the waves, WKB transport and amplification by streaming SEPs. Shock acceleration is heuristically represented by continuous injection of prescribed spectra of SEPs at a moving shock front. We show the model predictions for two contrasting sets of SEP source spectra, fast weakening and softening in one case and long lasting and hard in the other. The results presented include concurrent time histories of multispecies SEP intensities and elemental abundance ratios, as well as sequential snapshots of the following: SEP intensity energy spectra, Alfven wave spectra, particle mean free paths as functions of rigidity, and spatial profiles of SEP intensities and mean free paths. Wave growth plays a key role in both cases, although the magnitude of the wave growth differs greatly, and quite different SEP abundance variations are obtained. In these simulations, the maximum wave growth rate is large, but small relative to the wave frequency, and everywhere the total wave magnetic energy density remains small relative to that of the background magnetic field. The simulations show that, as the energetic protons stream outward, they rapidly amplify the ambient Alfven waves, by several orders of magnitude in the inner heliosphere. Energetic minor ions find themselves traveling through resonant Alfven waves previously amplified by higher velocity protons. The nonuniformly growing wave spectra alter the rigidity dependence of particle scattering, resulting in complex time variations of SEP abundances at large distances from the Sun. The greatly amplified waves travel outward in an expanding and weakening "shell," creating an expanding and falling "reservoir" of SEPs with. at spatial intensity profiles behind, while in and beyond the shell the intensities drop steeply. The wave-particle resonance relation dynamically links the evolving characteristics of the SEP and Alfven wave distributions in this new mode of SEP transport. We conclude that wave amplification, the counterpart to the scattering of streaming particles required by energy conservation, plays an essential role in the transport of SEPs in gradual SEP events. The steep proton-amplified wave spectra just upstream of the shock suggest that they may also be important in determining the elemental abundances of shock-accelerated SEP sources.