Calculations in the theory of tearing instability
  Recent studies have suggested that the tearing instability may play a
significant role in magnetic turbulence. In this work, we review the theory of
the magnetohydrodynamic tearing instability in the general case of an arbitrary
tearing parameter, which is relevant for applications in turbulence. We discuss
a detailed derivation of the results for the standard Harris profile and
accompany it by the derivation of the results for a lesser known sine-shaped
profile. We devote special attention to the exact solution of the inner
equation, which is the central result in the theory of tearing instability. We
also briefly discuss the influence of shear flows on tearing instability in
magnetic structures. Our presentation is self-contained; we expect it to be
accessible to researchers in plasma turbulence who are not experts in magnetic
reconnection.
