New linear stability parameter to describe low-$\beta$ electromagnetic microinstabilities driven by passing electrons in axisymmetric toroidal geometry

In magnetic confinement fusion devices, the ratio of the plasma pressure to the magnetic field energy, $\beta$, can become sufficiently large that electromagnetic microinstabilities become unstable, driving turbulence that distorts or reconnects the equilibrium magnetic field. In this paper, a theory is proposed for electromagnetic, electron-driven linear instabilities that have current layers localised to mode-rational surfaces and binormal wavelengths comparable to the ion gyroradius. The model retains axisymmetric toroidal geometry with arbitrary shaping, and consists of orbit-averaged equations for the mode-rational surface layer, with a ballooning space kinetic matching condition for passing electrons. The matching condition connects the current layer to the large scale electromagnetic fluctuations, and is derived in the limit that $\beta$ is comparable to the square root of the electron-to-ion-mass ratio. Electromagnetic fluctuations only enter through the matching condition, allowing for the identification of an effective $\beta$ that includes the effects of equilibrium flux surface shaping. The scaling predictions made by the asymptotic theory are tested with comparisons to results from linear simulations of micro-tearing and electrostatic microinstabilities in MAST discharge #6252, showing excellent agreement. In particular, it is demonstrated that the effective $\beta$ can explain the dependence of the local micro-tearing mode (MTM) growth rate on the ballooning parameter $\theta_0$ -- possibly providing a route to optimise local flux surfaces for reduced MTM-driven transport.