STOCHASTIC GYRORESONANT ELECTRON ACCELERATION IN A LOW-BETA PLASMA .1. INTERACTION WITH PARALLEL TRANSVERSE COLD-PLASMA WAVES

We consider the gyroresonance of electrons with parallel transverse cold plasma waves and derive the Fokker-Planck equation describing the evolution of the electron distribution function in the presence of a spectrum of turbulence. We take into account all such modes, which include nondispersive Alfven as well as dispersive whistler and cyclotron waves, and also employ the full gyroresonance condition. We have identified a new resonance which produces a divergence in the Fokker-Planck coefficients; it results when the electron is in gyroresonance with a wave that has a group velocity equal to the velocity of the electron along the magnetic field. Assuming a power-law spectral density, we numerically calculate the Fokker-Planck coefficients and discuss their complicated momentum and pitch-angle dependence, as well as the influence of various approximations to the dispersion relation, gyroresonance condition, and spectral density. We find that there is no resonance gap at any pitch angle as long as the full gyroresonance condition is used and waves propagating in both directions are present. In solar flares, waves near the electron cyclotron frequency, whistlers, and Alfven waves are the most important resonant waves for the energization of nonrelativistic, relativistic, and ultrarelativistic electrons, respectively. The time scale for acceleration to highly relativistic energies is approximately 1 s for reasonable flare conditions. The precise threshold energy, however, depends upon thermal modifications to the cold plasma dispersion relation around the electron cyclotron frequency, but should lie in the tail of the Maxwellian. In this nonrelativistic region, the pitch-angle scattering and momentum diffusion time scales are comparable, producing a breakdown in the commonly used momentum diffusion equation.