Coupled Hydromagnetic Wave Excitation and Ion Acceleration at an Evolving Coronal/Interplanetary Shock

An analytical quasilinear theory is presented for the evolution of a "gradual" event consisting of solar energetic particles (SEPs) accelerated at an evolving coronal/interplanetary shock. The upstream ion transport is described by the two-stream moments of the focused transport equation, which accommodate the large streaming anisotropies observed near event onset. The proton transport equations and a wave kinetic equation are solved together for the coupled behavior of the hydromagnetic waves and the energetic protons. The theory includes diffusive shock acceleration, ion advection with the solar wind, spatial diffusion upstream of the shock, magnetic focusing, wave excitation by the energetic protons, and minor ions as test particles. A number of approximations are made for analytical tractability. The predictions reproduce the observed phases of most gradual SEP events: onset, a "plateau" with large streaming anisotropy, an "energetic storm particle" (ESP) enhancement prior to shock passage, and the decaying "invariant spectra" after shock passage. The theory treats naturally the transition from a scatter-dominated sheath adjacent to the shock where the wave intensity is enhanced to the nearly scatter-free ion transport in interplanetary space. The plateau is formed by ions that are extracted from the outer edge of the scatter-dominated sheath by magnetic focusing and escape into interplanetary space; it corresponds quantitatively to the "streaming limit" identified and interpreted in gradual events by D. V. Reames and C. K. Ng. The ion energy spectra at the shock have the standard power-law form dependent on shock strength, which is expected for diffusive shock acceleration, with a high-energy cutoff whose form is determined self-consistently by the ion escape rate. The increased shock strength, magnetic field magnitude, and injection energies close to the Sun account for the observed predominance of high-energy ions early in the event. The downstream ion transport is determined under two extreme assumptions: (i) vanishing diffusive transport and (ii) effective diffusive transport leading to small ion spatial gradients. The latter assumption reproduces the invariant spectra, spatial gradients, and exponential temporal decay observed in the late phase of many events. The minor ion distributions exhibit fractionation due to rigidity-dependent transport and acceleration. However, their energy spectra, spatial gradients, and high-energy cutoffs do not reproduce observed forms and lead to excessive fractionation. The origin of these discrepancies is probably the neglect of nonlinear processes. Although not easily incorporated in the theory, these processes could substantially modify the predicted wave intensity. An illustrative calculation assuming an arbitrary power-law form for the wave intensity demonstrates the sensitive dependence of ion fractionation on the power-law index.