Drift Resonance Between Particles and Compressional Toroidal ULF Waves in Dipole Magnetic Field

We examine the drift-resonant particle dynamics for toroidal ultralow frequency (ULF) waves in a pure dipole background geomagnetic field. We confirm that the resonant condition originally believed to apply only for poloidal ULF waves, m omega d=omega, also applies for compressional toroidal waves. The predicted phase relationships have been confirmed from Van Allen Probes observations. Their good agreement provides the first observational evidence for the drift resonance condition controlled by the compressional toroidal ULF wave. Moreover, we extend the drift resonance theory into a nonlinear regime. The resulting particle motion can be described by a modified pendulum equation with solutions depending on the wave number m. For high-m toroidal waves, the resonant islands become asymmetric to perturb the particle trajectories within each potential well and consequently increase the trapping widths in both energy and L-shell. We further carry out test-particle simulations to show the evolution of electron distribution functions when they interact with either toroidal or poloidal waves. These findings demonstrate that toroidal ULF waves, like their poloidal counterparts, play an important role in magnetospheric particle dynamics.