(Color online) (a) Contour plot of showing the two branches of isolated poles for . (b) Magnified view showing the lowest frequency damped modes.
(Color online) Example of double-beam bump-on-tail distribution function (red dashed line) in a Maxwellian background (blue dotted line). Black solid line represents the total ion population.
(Color online) Dependence of the linear threshold on the injection velocity and fast ions temperature. The case of a resonant velocity is considered here. Parameters used in the simulation of section IV to recover the linear threshold are indicated by an asterisk in this figure.
(Color online) Dependence of the linear threshold on critical and birth energies in the case of , and a resonant velocity . Parameters of the simulation are indicated by an asterisk.
(Color online) Threshold of for GAM excitation as a function of the birth energy and at .
(Color online) (a) Derivative of the equilibrium distribution function with respect to the energy as a function of the perpendicular energy. (b) Left-hand side (solid line) and right-hand side (dashed line) of Eq. (25). When the solid line is above the dashed line, excitation of GAMs by fast ions is possible.
(Color online) Safety factor profiles used for the bump-on-tail and the slowing down distribution functions.
Damping of at the GAM frequency without fast particles (left) and FFT of the mode (right). GAM frequency is represented by the vertical dashed lines.
(Color online) Steady-state oscillations of at the EGAM frequency with fast particles. In this example, the parameters of the bump-on-tail distribution function have been set to , and .
(Color online) Steady-state oscillations of at the EGAM frequency with fast particles. In this example, the parameters of the slowing-down distribution function have been set to , , , , .
(Color online) Radial structure of the mode for a bump-on-tail (left) and a slowing-down (right) distribution functions.
(Color online) Dependence of the real frequency (left) and the linear growth rate (right) on the fast ion concentration for both classes of distributions considered in this paper. We observe that the real frequency (resp. the linear growth rate) decreases (resp. increases) with the concentration.
(Color online) Time evolution of the Fourier components for bump-on-tail (left) and slowing-down (right). Linear excitation occurs up to . After that time, saturation appears for all the modes, but no energy exchange between the modes exists.
(Color online) Level of saturation for the two distribution functions considered in this paper.
(Color online) For a bump-on-tail instability. Left: density of exchanged energy per volume and time unit (solid curve) and its time average (horizontal lines). The imaginary part of the mode is represented here to compare the oscillations frequency. All quantities are normalized. Right: dependence of the derivative of the equilibrium distribution function at the resonance on the adiabatic invariant for various instants of the simulation.
(Color online) For a slowing-down instability. Density of exchanged energy (left) and derivative of the equilibrium distribution function at the resonance for different instants of the simulation (right).
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