Influence of magnetic field gradient on rotating spoke instability

Rotating spokes, as one of the low-frequency, long-wavelength instabilities, are commonly observed in the <inline-formula><tex-math id="M4">\begin{document}${\boldsymbol{E}} \times {\boldsymbol{B}}$\end{document}</tex-math></inline-formula> plasma discharge devices, such as the magnetrons and Hall thrusters. In Hall thrusters, the rotating spokes, which are located in the discharge channel and rotate in the azimuthal direction, feature the bright luminous regions. The space potential will be distorted by the instability of rotating spokes, thereby increasing the possibility for electrons to reach the anode and enhancing their drift along the equipotential lines. However, the excitation mechanism of the rotating spoke and its influencing factors remain ambiguous. In order to address this problem, we conduct numerical simulations and linear stability analysis to investigate the effects of the magnetic field gradient on the driving mechanism and mode characteristics of the rotating spoke instability. In this work, a particle-fluid two-dimensional hybrid model in the axial-azimuthal plane is employed to numerically study the effect of axial magnetic field gradient in the discharge channel on the rotating spoke. The numerical simulation results are analyzed using a dispersion relation derived from fluid theory, which combines the effects of plasma density and the magnetic field gradient. The output profiles of ion density, potential, and electric field from the numerical simulation serve as input parameters for the dispersion relation used in the linear stability analysis. The simulation results show that the frequency and propagation velocity of the <inline-formula><tex-math id="M5">\begin{document}$m = 1$\end{document}</tex-math></inline-formula> rotating spoke slightly increase as the magnetic field gradient in the discharge channel decreases. However, changing the magnetic field gradient in the discharge channel does not affect the propagation direction nor intrinsic characteristics of the rotating spoke. More specifically, when the value of <inline-formula><tex-math id="M6">\begin{document}${\alpha _1}$\end{document}</tex-math></inline-formula>increases from 1.1 to 1.7, which means a decrease of the magnetic field gradient in the discharge channel, the mode frequency rises from 6.2 kHz to 7.5 kHz, remaining within the frequency range of the rotating spoke instability. At the same time, the phase velocity also increases form 1013 m/s to 1225 m/s, which is consistent with the propagation velocity of the rotating spoke instability, and the rotating spoke instability still propagates along the <inline-formula><tex-math id="M7">\begin{document}${\boldsymbol{E}} \times {\boldsymbol{B}}$\end{document}</tex-math></inline-formula> direction. Dispersion relation analysis indicates that the rotating spoke arises from an azimuthal drift instability which is located near downstream region of the thruster exit, and it is excited by the plasma density and magnetic field gradient effects. The axial position of the azimuthal drift instability, responsible for the rotating spoke formation, is slightly modulated by density profile variations caused by the change of magnetic field in the discharge channel. However, it remains near the downstream region of the thruster exit. The results indicate that the rotating spoke does not originate from ionization instabilities, and changing the magnetic field distribution in the discharge channel does not affect its propagation direction nor mode number. The research results provide theoretical support for explaining the excitation mechanism and key influencing factors of rotating spoke.