Understanding the growth rate patterns of ion Bernstein instabilities driven by ring-like proton velocity distributions
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Fast magnetosonic waves in Earth's inner magnetosphere, which have as their source ion Bernstein instabilities, are driven by hot proton velocity distributions (f(p)) with partial derivative f(p)(v(perpendicular to))/partial derivative v(perpendicular to) > 0. Two typical types of distributions with such features are ring and shell velocity distributions. Both have been used in studies of ion Bernstein instabilities and fast magnetosonic waves, but the differences between instabilities driven by the two types of distributions have not been thoroughly addressed. The present study uses linear kinetic theory to examine and understand these differences. It is found that the growth rate pattern is primarily determined by the cyclotron resonance condition and the structure of the velocity distribution in gyroaveraged velocity space. For ring-driven Bernstein instabilities, as the parallel wave number (k(parallel to)) increases, the discrete unstable modes approximately follow the corresponding proton cyclotron harmonic frequencies while they become broader in frequency space. At sufficiently large k(parallel to), the neighboring discrete modes merge into a continuum. In contrast, for shell-driven Bernstein instabilities, the curved geometry of the shell velocity distribution in gyroaveraged velocity space results in a complex alternating pattern of growth and damping rates in frequency and wave number space and confines the unstable Bernstein modes to relatively small k(parallel to) In addition, when k(parallel to) increases, the unstable modes are no longer limited to the proton cyclotron harmonic frequencies. The local growth rate peak near an exact harmonic at small k(parallel to) bifurcates into two local peaks on both sides of the harmonic when k(parallel to) becomes large.