G. G. Plunk

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.

Optimised stellarators and other magnetic-confinement devices having the property that the average magnetic curvature is favourable for all particle orbits are called maximum-$J$ devices, and have recently been shown to be immune to trapped-particle instabilities driven by the density gradient. Gyrokinetic simulations reveal, however, that another instability can arise, which is also associated with particle trapping but causes less transport than typical trapped-electron modes. The nature of this instability is clarified here. It is shown to be similar to the "ubiquitous mode" in tokamaks, and is driven by ion free energy but requires trapped electrons to exist.