Electromagnetic Landau Resonance: Nonlinear Dynamics and Harmonic Generation

In situ particle observations provide useful information for determining the significance of nonlinearity in wave-particle interactions. Here, we examine an electromagnetic Landau resonance event observed by the Magnetospheric Multiscale mission near the dayside magnetopause along this line of thought. During this event, protons with energy similar to ${\sim} $748 eV and pitch angle similar to 25 degrees ${\sim} 25{}<^>{\circ}$ are observed to resonate with electromagnetic-ion-cyclotron waves. Detailed data analysis reveals the existence of a second harmonic signal in proton modulation. Using an observation-based test particle simulation, this signal is explained as a low-resolution manifestation of phase space rolled-up structures, which themselves are not observable due to limited instrument resolution. The presence of these rolled-up structures suggests that the waves induce significant displacement of the protons in phase space, consequently confirming the nonlinear nature of the observed interaction. Based on the observations, we further discuss the possibility of the nonlinear electromagnetic Landau resonance generating second harmonic waves. The nonlinearity of wave-particle interactions becomes nonnegligible when wave-induced phase space displacement of particles is comparable with that caused by the background fields. Nonlinear interactions differ from their linear counterparts in several aspects, including the area of phase space involved in resonance, particle energization efficiency, and the interactions' asymptotic state. This study investigates the nonlinear nature of an electromagnetic Landau resonance event between electromagnetic-ion-cyclotron waves and protons observed by the Magnetospheric Multiscale mission. Data analysis reveals protons with energy similar to ${\sim} $748 eV and pitch angle similar to 25 degrees ${\sim} 25{}<^>{\circ}$ resonate with the waves. Further investigation shows the existence of a second harmonic signal in proton modulation. Based on a test particle simulation set according to the observations, this second harmonic signal is explained as a low-resolution manifestation of phase space rolled-up structures induced by the waves. The presence of these rolled-up structures, which indicates significant phase space displacement of protons induced by the waves, suggests that the observed interaction is nonlinear. Drawing from the observations, we further discuss the possibility of this nonlinear interaction generating second harmonic waves. Second -harmonic modulation is observed in an electromagnetic Landau resonance event between electromagnetic ion cyclotron waves and protons An observation-based test particle simulation attributes this signal to fine rolled-up structures in phase space The presence of rolled-up structures confirms the nonlinear nature of the observed electromagnetic Landau resonance