Gyrokinetic equations for the non-linear simulation of toroidal tearing modes

The standard gyrokinetic ordering is given by $\omega/\Omega_i \sim k_\|/k_\perp \sim e\phi/T_e \sim \rho_i/L_n \sim \delta B/B \sim \mathcal{O}(\epsilon)$, $k_\perp \rho_i \sim \mathcal{O}$$(1)$. We derive equations with a modified electromagnetic gyrokinetic ordering appropriate for the description of tearing modes in toroidal geometry. While the radial wave-number of the perturbation remains of the same order as the ion gyroradius, the perpendicular variation \textit{within} the magnetic surface is one order lower in $\epsilon$. An `inner' solution to these equations, in the region of the rational surface, is matched to and `external' MHD solution encapsulated by a quantity $\Delta'$ [Furth \textit{et al} 1963]. These equations will form the basis of numerical simulations of magnetic island evolution. The eventual application of this formulation will be the study of the non-linear interaction of turbulence and evolving island structures.