Initial perturbed fast particle distributions, , for values of ranging from 0.05 to 0.25.
Initial perturbed fast particle distribution, , for , plotted as a function of the phase space variables . The separatrix is highlighted by the red curve.
Plot of the initial mode amplitude as a function of . For large , the curve tends asymptotically to 0.
Frequency evolution of hole with a linear equilibrium distribution profile, , and . The solid line is the simulation result and the dashed line, included here for comparison, is the square root scaling (23) .
Amplitude evolution corresponding to the spectrogram in Figure 4 .
Snapshots of for the hole evolution in Figures 4 and 5 . The curves correspond to, from top to bottom, , 0.05, 0.1, 0.2, 0.5, 1.
Phase space plot of for the hole at in Figure 6 .
Initial amplitude evolution for holes with varying , simulated with a constant slope. The dashed line has , and the solid lines have, from top to bottom, , 0.2, 0.292, and 0.4.
Frequency evolution of holes and clumps in the absence of drag collisions.
Amplitude evolution of holes in the absence of drag collisions. These simulations are performed with the critical slope, .
Amplitude evolution of clumps in the absence of drag collisions. These simulations are run with a constant slope, .
Lifetimes of holes and clumps as functions of , in log-log scale. The solid lines are holes (black line) and clumps (gray line), and the dashed line is the previous theoretical estimate .
Unperturbed fast particle distribution function, , corresponding to for and for tends asymptotically to for large .
Frequency evolution of holes at the threshold between monotonic and hooked non-monotonic sweeping in a constant equilibrium slope. The collision frequency for drag is set to and equals, from the top to bottom curve, 2.2, 2.3, 2.4, 2.5. The transition occurs at .
Amplitude evolution corresponding to the curves in Figure 14 .
Parameter domains in -space for the three classes of solutions: Monotonically increasing frequency and amplitude (gray area), hooked non-monotonic regime (blank area) and asymptotic steady states (dark area). The dashed, straight line is an approximate analytical boundary of the steady state domain, given by Eq. (44) .
Frequency evolution of holes when and varies across the steady state domain in Figure 16 . The monotonically increasing spectral line has , the steady states have, from top to bottom, , 6.5, 6.6, 6.8, and the hook has .
Amplitude evolution corresponding to the spectral lines in Figure 17 .
Eigenvalues of the nonlinear harmonic oscillator (33) .
The approximate power balance residual (42) for and . The curves start at and asymptote to , with a maximum at . As increases the curves are stretched upwards, with solutions to emerging at .
Steady state frequency as function of when . The end values in Figure 17 are shown as squares for comparison.
Steady state amplitude as function of when . The end values in Figure 18 are shown as squares for comparison.
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