Energetic bounds on gyrokinetic instabilities. Part 4. Bounce-averaged electrons

Upper bounds on the growth of instabilities in gyrokinetic systems have recently been derived by considering the optimal perturbations that maximise the growth of a chosen energy norm. This technique has previously been applied to two-species gyrokinetic systems with fully kinetic ions and electrons. However, in tokamaks and stellarators, the expectation from linear instability analyses is that the most important kinetic-electron contribution to ion-scale modes comes from the trapped electrons, which bounce faster than the timescale upon which instabilities evolve. As a result, a fully-kinetic electron response is not required to describe unstable modes in most cases. Here, we apply the optimal mode analysis to a reduced two-species system that consists of fully gyrokinetic ions and bounce-averaged electrons with the aim of finding a tighter bound on ion-scale instabilities in toroidal geometry. This analysis yields bounds that are greatly reduced in comparison to the earlier two-species result. Moreover, if the energy norm is properly chosen, wave-particle resonance effects can be captured, reproducing the stabilisation of density-gradient-driven instabilities in maximum-$J$ devices. The optimal mode analysis also reveals that the maximum-$J$ property has an additional stabilising effect on ion-temperature-gradient-driven instabilities, even in the absence of an electron-free energy source. This effect is explained in terms of the concept of mode inertia, making it distinct from other mechanisms.