As one of the low-frequency, long-wavelength instabilities, rotating spokes are commonly observed in the E×B plasma discharge devices, such as the magnetrons and Hall thrusters. In Hall thrusters， the characteristic of rotating spokes are the bright luminous regions located in the discharge channel and rotating in the azimuthal direction. The space potential will be distorted by the rotating spokes, and the axially electrons transport increase as the possibility of electrons to reach the anode which drift alone the equipotential lines is enhanced. However, the excitation mechanism of the rotating spoke and its influencing factors remain ambiguous. To address this, numerical simulations and linear stability analysis are employed in this study to investigate the effect of the magnetic field gradient on the driving mechanisms and mode characteristics of the rotating spoke instability. In this paper, aparticle-fluid two-dimensional hybrid model in the axial-azimuthal plane is employed to numerical 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 incorporates the effects of plasma density and magnetic field gradient. The output profiles of ions 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 m = 1 rotating spoke slightly increasing as the magnetic field gradient in the discharge channel decreases. However, varying the magnetic field gradient in the discharge channel does not affect the propagation direction and intrinsic characteristics of the rotating spoke. More specifically, when the value of α1 increases from 1.1 to 1.7 (indicating a decrease the magnetic field gradient in the discharge channel), the mode frequency rises from 6.2kHz to 7.5kHz, remaining within the frequency range of the rotating spoke instability. At the same time, the phase velocity also increases form 1013m/s to 1225m/s, which is consistent with the propagation velocity of the rotating spoke instability, and is still propagation along the E×B direction. Dispersion relation analysis indicates that the rotating spoke arises from an azimuthal drift instability which 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 magnetic field alterations 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 altering the magnetic field distribution in the discharge channel does not affect its propagation direction and mode number. The results provide theoretical support for clarifying the excitation mechanism of the rotating spoke and key influencing factors.
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