Analytical Model of a Toroidal Mode Field Line Resonance and Its Drift‐Resonant Interaction With Energetic Electrons

Abstract Ultra‐low‐frequency (ULF) waves play a critical role in magnetospheric dynamics, yet their transient growth and resonant electron interactions remain poorly understood. We develop a first‐principle model of toroidal mode field line resonances that captures wave growth, saturation, and phase mixing. The model reproduces magnetospheric multiscale (MMS) observations of a microinjection event (4 August 2016), explaining wavefield beat patterns as phase mixing signatures. Guiding‐center test‐particle simulations reveal two distinct drift‐resonance types: Type A (energy‐dispersive) and Type B (gradient‐driven, non‐dispersive) resonance islands. These structures trap energetic electrons, producing repetitive flux enhancements matching MMS energy‐time spectrograms. Energy‐dependent phase shifts align with the 90° lag predicted by drift resonance theory. We demonstrate that local wave‐particle interactions can generate microinjections without remote substorm injections, potentially resolving a long‐standing ambiguity in magnetospheric physics. By bridging magnetohydrodynamic theory and spacecraft observations, we provide a framework for diagnosing ULF wave‐electron coupling with direct implications for radiation belt modeling and space weather forecasting.