Nonlinear evolution of perturbatively driven kinetic instabilities far from marginal stability

Kinetic instabilities develop when a system's distribution function deviates from thermal equilibrium in such a way that allows free energy from that distribution to drive resonant modes.

These instabilities occur in many systems, such as fusion and astrophysical plasmas, neutral fluids, and self-gravitating systems.

Motivated by the case of Alfvénic instabilities driven by minority populations of energetic particles in tokamak plasmas, we consider a kinetic instability in the presence of sources and sinks and with perturbative drive far from its instability threshold, where a theoretical description of the time evolution has not yet been established.

In cases with steady-state saturation levels, we find that the mode first evolves linearly, driven by the positive distribution gradient, then undergoes a fast, strongly nonlinear transition where the distribution function slope is completely flattened around the resonance.

The system then evolves in a weakly nonlinear regime driven by the balance of the wave drive and dissipation until reaching saturation.

The strongly nonlinear transition is sufficiently fast that it can be treated as occurring instantaneously, and by considering a time-local approximation for the distribution after the flattening has occurred, we find a closed-form analytical solution for the mode amplitude in the weakly nonlinear phase.

A compact piecewise-continuous solution for the entire time evolution of the mode amplitude is therefore constructed.

This result is shown to agree closely with nonlinear kinetic simulations and is derived within a framework common to other physical systems such as galactic discs and viscous fluids.