Development of Tearing Instability in a Current Sheet Forming by Sheared Incompressible Flow
Sweet-Parker current sheets in high Lundquist number plasmas are unstable to tearing, suggesting they will not form in physical systems. Understanding magnetic reconnection thus requires study of the stability of a current sheet as it forms. Formation can occur due to sheared, sub-Alfvenic incompressible flows which narrow the sheet. Standard tearing theory (Furth et al. 1963; Coppi et al. 1976; Rutherford 1973) is not immediately applicable to such forming sheets for two reasons: first, because the flow introduces terms not present in the standard calculation; second, because the changing equilibrium introduces time dependence to terms which are constant in the standard calculation, complicating the formulation of an eigenvalue problem. This paper adapts standard tearing mode analysis to confront these challenges. In an initial phase when any perturbations are primarily governed by ideal MHD, a coordinate transformation reveals that the flow compresses and stretches perturbations. A multiple scale formulation describes how linear tearing mode theory (Furth et al. 1963; Coppi et al. 1976) can be applied to an equilibrium changing under flow, showing that tearing eigenfunctions grow slowly due to the flow and undergo exponential growth at a rate given by standard scalings with time dependence added. In the nonlinear Rutherford stage, the coordinate transformation shows that standard theory can be adapted by adding to the stationary rates time dependence and an additional term due to the strengthening equilibrium magnetic field. Overall, this understanding supports the use of flow-free scalings with slight modifications to study tearing in a forming sheet.