# Proton velocity ring-driven instabilities in the inner magnetosphere: Linear theory and particle-in-cell simulations

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Linear dispersion theory and electromagnetic particle-in-cell (PIC) simulations are used to investigate linear growth and nonlinear saturation of the proton velocity ring-driven instabilities, namely, ion Bernstein instability and Alfven-cyclotron instability, which lead to fast magnetosonic waves and electromagnetic ion cyclotron waves in the inner magnetosphere, respectively. The proton velocity distribution is assumed to consist of 10% of a ring distribution and 90% of a low-temperature Maxwellian background. Here two cases with ring speeds v(r)/v(A)=1 and 2 (v(A) is the Alfven speed) are examined in detail. For the two cases, linear theory predicts that the maximum growth rate (m) of the Bernstein instability is 0.16(p) and 0.19(p), respectively, and (m) of the Alfven-cyclotron instability is 0.045(p) and 0.15(p), respectively, where (p) is the proton cyclotron frequency. Two-dimensional PIC simulations are carried out for the two cases to examine the instability development and the corresponding evolution of the particle distributions. Initially, Bernstein waves develop and saturate with strong electrostatic fluctuations. Subsequently, electromagnetic Alfven-cyclotron waves grow and saturate. Despite their smaller growth rate, the saturation levels of the Alfven-cyclotron waves for both cases are larger than those of the Bernstein waves. Resonant interactions with the Bernstein waves lead to scattering of ring protons predominantly along the perpendicular velocity component (toward both decreasing and, at a lesser extent, increasing speeds) without substantial change of either the parallel temperature or the temperature anisotropy. Consequently, the Alfven-cyclotron instability can still grow. Furthermore, the free energy resulting from the pitch angle scattering by the Alfven-cyclotron waves is larger than the free energy resulting from the perpendicular energy scattering, thereby leading to the larger saturation level of the Alfven-cyclotron waves.