Trichel pulses are a self-pulsing oscillatory phenomenon in negative corona discharges, characterized by periodic current pulses and intrinsic nonlinear dynamics. In this paper, a two-dimensional fluid model is developed to investigate the formation of Trichel pulses in an atmospheric-pressure negative needle-to-plane corona discharge. The simulations reveal that the rapid ionization near the needle tip generates positive space charge, which shields the local electric field and quenches the discharge, while subsequent space-charge transport and decay enable field recovery and initiate the next pulse. From a nonlinear dynamics perspective, this oscillatory behavior arises from the competition between growth and suppression. To further explore the nonlinear dynamics, a reduced-order model is developed based on the dominant physical processes. The model demonstrates that self-sustained oscillations can emerge from a low-dimensional nonlinear dynamical system.