Microscopic scales of linear tearing modes: a tutorial on boundary layer theory for magnetic reconnection

Abstract

We revise in detail and in a pedagogical way the analysis of the boundary layer theory of warm tearing modes in slab, reduced magnetohydrodynamics (MHD), when magnetic reconnection is driven by electron inertia and/or resistivity, and ion-sound Larmor radius effects are included. By comparison with the numerical solution of the corresponding eigenvalue problem, we interpret these results by means of a heuristic approach, which in the warm-electron regime, we show to be in general not feasible without knowledge of the scaling of the gradient of the magnetic flux function, differently from what happens in the cold-electron regimes. We put in evidence for a non-trivial relation between the first derivative of the magnetic flux function and of the velocity parallel to the neutral line, evaluated in its proximity, by thus providing insight to the multiple boundary layer analysis that Pegoraro & Schep (Plasma Phys. Control. Fusion, vol. 28, 1986, p. 647) first showed to be required in warm-tearing regimes. In this way, we also suggest and justify a general operational definition of the reconnecting layer width and we discuss the linear appearance of microscopic scales related to the gradients of the eigenfunctions of the tearing modes.