Trapped-electron modification of kinetic ballooning instabilities in general geometry

Within the gyrokinetic formalism, we present and analytically study the equations for an explicit treatment of the trapped-electron-modified kinetic ballooning mode (KBM) and the electromagnetic version of the trapped-electron mode, in general geometry. The gradient of the plasma $\beta =8\pi p /B^2,$ the ratio of kinetic to magnetic pressure, is taken to be small enough to avoid including perturbations of the magnetic field strength. Trapped-electron-modified KBMs are first described close to ideal magnetohydrodynamic marginality, retaining the explicit resonant contribution of both ions and trapped electrons, and then in a strongly driven fluid limit. We show that maximum- $\mathcal J$ devices (where $\mathcal J$ is the second adiabatic invariant) enjoy relatively good stability properties at finite $\beta ,$ but the coupling of trapped electron and KBMs might induce modes rotating in the ion direction, thus eluding good maximum- $\mathcal J$ properties. An eigenvalue equation for the finite $\hbox{-}\beta$ trapped-electron-mode is derived and studied. We highlight the possibility of having an electron-temperature-gradient driven electromagnetic instability in regions of bad magnetic curvature. A mechanism for the destabilisation of the trapped-particle-enabled collisionless microtearing mode also is proposed. Our results are general and provide new theoretical ground for the characterisation of several magnetic confinement concepts, such as tokamaks, quasisymmetric and quasi-isodynamic stellarators.