Two-dimensional effects on electrostatic instabilities in Hall thrusters. II. Comparison of particle-in-cell simulation results with linear theory dispersion relations

In Paper I, we successfully used an external circuit to significantly damp the Breathing Mode (BM) oscillations in 2D particle-in-cell selfconsistent simulations of the axial-azimuthal plane of a Hall thruster. We also introduced the two-point power spectral density reconstruction method (PSD2P) used to analyze electrostatic instabilities and generate dispersion diagrams in azimuthal and axial directions, at various times during the BM period. Here, a 3D Dispersion Relation (DR) for electrostatic modes is calculated by linearizing the continuity/momentum fluid equations for electrons and ions. We show that by taking the appropriate limits, this relation can be simplified to derive the DRs of some well-known E x B instabilities, such as the electron cyclotron drift instability and its evolution to the Ion Acoustic Wave (IAW), and the Ion Transit-Time Instability (ITTI). The PSD2P diagrams demonstrate the importance of considering the 2D nature of the IAW and ITTI, which have been previously considered to be mono-dimensional (azimuthal and axial, respectively). In particular, we show that the IAW grows near the maximum of the magnetic field and due to its axial components propagates toward both the anode and the cathode (in addition to the well-known azimuthal propagation). The resulting wavefront is, therefore, bent. By analogy to the propagation of acoustic waves in gases, it is proposed that the cause of the IAW wavefront bending is the strong electron temperature gradients in the axial direction. We also show that the ITTI has a strong positive growth rate when a small azimuthal component is present. Finally, we observe that the ITTI significantly affects the discharge current.