图  1  质子伯恩斯坦波不稳定性线性增长率$ \gamma $随波数$ k $和传播角$ \psi $的变化. (a)接近$ 3{{\varOmega }}_{\text{cp}} $的第三谐波模式, (b)接近$ 3.5{{\varOmega }}_{\text{cp}} $时的半倍频波模

Figure  1.  Variation of the proton Bernstein instability linear growth rate ($ \gamma $) with wave number ($ k $) and wave normal angle ($ \psi $). Left panel (a) Third-harmonic mode near $ 3{{\varOmega }}_{\text{cp}} $, (b) Half-harmonic wave mode near $ 3.5{{\varOmega }}_{\text{cp}} $

图  2  传播角为88.6°时动理学色散关系求解器计算得到的质子伯恩斯坦波模的色散关系与磁流体力学中快磁声波对应的冷等离子体色散关系曲线对比

Figure  2.  Comparison between the proton Bernstein wave dispersion relation obtained from the kinetic dispersion relation solver and the cold plasma dispersion relation of fast magnetosonic waves in magnetohydrodynamics at the wave normal angle of 88.6°

图  3  不同背景氦离子占比条件下质子伯恩斯坦波不稳定性各波模的最大增长率所在传播角的变化

Figure  3.  Wave normal angles of the most unstable proton Bernstein waves at various harmonics under different background helium ion concentrations

图  4  不同背景氦离子占比条件下固定传播角为89°时, 动理学色散关系求解器计算出的质子伯恩斯坦波模的色散关系与磁流体力学中对应于快磁声波的冷等离子体色散关系曲线对比(左列)以及各不稳定波模增长率γ随频率ω的变化情况(右列)

Figure  4.  Comparison between the proton Bernstein wave dispersion relation obtained from the kinetic dispersion relation solver and the cold plasma dispersion relation of fast magnetosonic waves in magnetohydrodynamics (left column) and the wave growth rate γ versus frequency ω (right column) under different background helium ion concentrations and at a fixed wave normal angle of 89°

图  5  不同背景氧离子占比条件下固定传播角为89°时, 动理学色散关系求解器给出的质子伯恩斯坦波模的色散关系与磁流体力学中对应于快磁声波的冷等离子体色散关系曲线对比(左列)以及各不稳定波模增长率γ随频率ω的变化情况(右列)

Figure  5.  Comparison between the proton Bernstein wave dispersion relation obtained from the kinetic dispersion relation solver and the cold plasma dispersion relation of fast magnetosonic waves in magnetohydrodynamics (left column) and the wave growth rate γ versus frequency ω (right column) under different background oxygen ion concentrations and at a fixed wave normal angle of 89°

图  6  不同背景氦离子与氧离子占比条件下固定传播角为89°时, 对应于快磁声波的冷等离子体色散关系曲线与第一类贝塞尔函数$ J_{n}^{2}(\dfrac{{k}_{\bot }{v}_{\text{s}}}{{{\varOmega }}_{\text{cp}}}) $的第一峰值位置(圆圈)对比

Figure  6.  Comparison between the cold plasma dispersion relations of fast magnetosonic waves and the locations of the first peaks of the squared Bessel function of the first kind, $ J_{n}^{2}(\dfrac{{k}_{\bot }{v}_{s}}{{{\varOmega }}_{\text{cp}}}) $ (marked by circles), under different background helium and oxygen ion concentrations and at a fixed wave normal angle of 89°