Effect of magnetic field gradient on motion of ions resonating with ion cyclotron waves

The gradient of the geomagnetic field may appreciably affect the energy that can be exchanged in resonant interactions between charged particles and circularly polarized waves. The equations of motion of a charged particle resonating with a circularly polarized wave in the presence of a field gradient are derived, and some numerical examples of the particle trajectories are given. It is shown that a charged particle moving upward along a magnetic field line may be accelerated to a very high energy while it is resonating with a wave propagating downward along the field line. Conversely, a charged particle moving downward may lose appreciable energy while it is resonating with a wave propagating upward. It is suggested that the energetic O+ ions observed in the magnetosphere during a magnetic storm were principally accelerated by resonating with ion cyclotron turbulence generated by an unstable distribution of ring current protons. It is also suggested that resonant interactions occurring off the equator between ion cyclotron waves and the ring current protons may account for some of the unexplained features in the energy distribution of the protons.