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Rotating spokes, as one of the low-frequency, long-wavelength instabilities, are commonly observed in the ${\boldsymbol{E}} \times {\boldsymbol{B}}$ 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 $m = 1$ 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 ${\alpha _1}$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 ${\boldsymbol{E}} \times {\boldsymbol{B}}$ 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.
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Keywords:
- Hall thruster /
- rotating spokes /
- density gradient /
- magnetic gradient /
- dispersion relation
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图 1 磁场(a)和电子温度(b)的轴向分布
Figure 1. Axial distribution of magnetic field (a) and electron temperature (b).
图 2 ${\alpha _1} = 1.1$时离子数密度(a)、电离率(b)、中性原子数密度(c)、电势(d)、轴向电场(e), 角向电场(f)在一个周期内不同时刻的分布
Figure 2. Distribution of ion number density (a), ionization rate (b), neutral particle number density (c), electrical potential (d), axial electric field (e), and azimuthal electric field (f) at different time in one period for ${\alpha _1} = 1.1$.
图 3 归一化的离子数密度在角向上的分布随时间的演化
Figure 3. Time history of azimuthal distribution of normalized ion number density.
图 4 $\theta = {180^ \circ }$时, 离子数密度(上方)和电离率(下方)的轴向分布随时间的演化
Figure 4. Time history of axially distribution for ion number density (upper panel) and ionization rate (lower panel) at $\theta = {180^ \circ }$.
图 5 ${\alpha _1} = 1.1$, 1.3, 1.5, 1.7时的(a)离子密度分布, (b)空间电势分布, (c)电场分布, (d)密度梯度${\kappa _{\text{N}}}$, 以及(e)磁场梯度${\kappa _{\text{B}}}$
Figure 5. The axial distribution of (a) ion density profile, (b) space potential, (c) electric field, and (d) the density gradient ${\kappa _{\text{N}}}$, (e) the magnetic field gradient ${\kappa _{\text{B}}}$, for ${\alpha _1} = 1.1$, 1.3, 1.5, 1.7, respectively.
图 6 由密度梯度和磁场梯度驱动的不稳定性的增长率(a), 频率(b), 以及相速度(c); (b), (c)中的阴影区域表示旋转辐条不稳定性典型的频率范围和相速度范围, 分别为5—25 kHz和1200—2800 m/s
Figure 6. The instability growth rate (a), frequency (b), and phase velocity (c) induced by density and magnetic gradient. The shaded area in (b), (c), are the typical spoke frequency and phase velocity, range from 5–25 kHz and 1200–2800 m/s, respectively.
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