Kinetic Theory of Oblique Collisionless Tearing Instabilities

Recent 3D PIC and linear Vlasov-Maxwell simulations [1] have shown that conventional kinetic tearing mode theory [2] vastly overestimates the growth rate of oblique modes, i.e., those with nonzero $\phi = \textrm{arctan}(k_z/k_y)$ where $k_z$ is the wavenumber in the guide field direction and $k_y$ is the wavenumber perpendicular to this in the plane of the current sheet. Thus, doubts have been cast on the validity of asymptotic boundary-layer analysis in collisionless kinetic theory. We show that this disagreement is a consequence of a strong guide field assumption made in conventional theories. We relax this assumption and obtain a dispersion relation that accounts for oblique angles. Focusing on a Harris equilibrium, Ref. 1 discusses modifications in the ideal MHD region outside the $\mathbf{k} \cdot \mathbf{B}_o = 0$ surface [1]. Here we complete the necessary modifications by correcting the inner layer solution. These show that obliquity has a stabilizing effect in all but the strongest guide field cases. Our theoretical predictions are compared to the previous simulation results. [1] W. Daughton \textit{et al}., Nature Phys. {\bf 7}, 539 (2011). [2] J. Drake and Y. C. Lee, Phys. Fluids {\bf 20}, 1341 (1977).