Fig. 6

Nonlinear wave growth theory. a Linear growth rates (dash–dot lines) and nonlinear growth rates at the threshold amplitudes (dashed lines) and the nonlinear growth rates at the optimum amplitudes (solid lines). A subtracted Maxwellian for the energetic electrons is assumed with \(\beta = 0.3\), \(\rho = 1.0\), \(U_{t \parallel } = 0.25 c\), and \(U_{t \perp } = 3.0 c\). b Optimum and threshold amplitudes in solid and dashed lines for different values of the plasma frequency. \(\omega _{\text{pe}} =\)2, 4, 8 \(\Omega _{\text{e}}\) in blue, green, and red, respectively. The plasma frequency of hot electrons is kept constant as \(\omega _{\text{ph}} = 0.1789\) corresponding to \(n_\text{h}/n_\text{c} = 2\times 10^{-3}\) for \(\omega _{\text{pe}} = 4 \Omega _{\text{e}}\)