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Description
Instabilities driven by energetic particles are central to the physics of a burning plasma. The majority of kinetic simulations and reduced models assume that the unstable distribution is already fully established when energetic-particle-driven modes grow unstable. In realistic scenarios, however, energetic particles may accumulate in the resonance on an effective timescale comparable to the growth rate of the instability, meaning that the formation of the resonant distribution and the growth of the unstable mode must be treated concurrently. We study the behavior of these instabilities in the presence of such a dynamically forming distribution, evaluating two distinct metrics which measure how close a mode is to its linear stability threshold and how close a mode remains to its nonlinear stability threshold. It is found that saturation at large $\omega_b/\nu_\text{eff}$ (where $\omega_b$ is the bounce frequency of deeply trapped particles and $\nu_\text{eff}$ is the effective scattering rate at a resonance), normally associated with strongly driven excitation, can be achieved even if dynamically the mode remains at all times near its nonlinear stability threshold. We extend existing analytic models for near-marginal and far from marginal modes allowing for a time-dependent linear growth rate, deriving explicit expressions for the mode amplitude evolution. These formulas are shown to agree with nonlinear kinetic simulations. The discrepancies between the case of a dynamically forming distribution and the case of a fully formed distribution are shown to be particularly pronounced for energetic particle distributions which relax diffusively.