Stochastic Proton Acceleration by Cascading Alfven Waves in Impulsive Solar Flares

We present a model for the acceleration of protons from thermal to relativistic energies on impulsive flare timescales. Long-wavelength Alfvén waves are assumed to be generated during the initial flare energy release by, for example, large-scale restructuring of the magnetic field. These waves nonlinearly cascade to high wavenumbers and eventually reach the dissipation range, whereupon they are damped by protons in the tail of the thermal distribution through gyroresonant interactions. The protons, in turn, are energized out the tail and, since longer wavelength waves are already present in the wave spectrum, are promptly accelerated to substantially higher energies. We find that for turbulence energy densities much smaller than the ambient magnetic field energy density and near the lower limit of that required by the total observed energy content of the protons, the acceleration of a sufficient number of protons to gamma-ray line-producing energies occurs on observed timescales essentially independent of the scale at which the waves are generated. Simple scaling arguments are presented in support of, and supported by, detailed numerical results.