Linear theory of visco-resistive tearing instability

A new linear theory of a tearing instability is shown, where the modified LSC (Loureiro, Schekochihin, and Cowley) theory [T. Shimizu, arXiv:2209.00149 (2022)] is extended to visco-resistive MHD. In contrast to the original LSC theories [Loureiro et al., Phys. Plasmas 14, 100703 (2007)], in the modified LSC theory, the upstream open boundary condition is implemented at a finite point xi(c). At this point, the original LSC theories are solved for xi(c)=+infinity. This paper first studies when the resistivity and viscosity are uniform in space. In addition, some variations in the non-uniformity are studied. It is shown that the non-uniformity can enhance the linear growth rate, and the tearing instability can occur even in an unlimitedly thin current sheet. Unexpectedly, it suggests that the forward cascade process of the plasmoid instability (PI) does not stop, i.e., the finite differential MHD simulations fail. To stop the forward cascade, viscosity is required not only in the inner region of the current sheet but also in the outer region. When the uniform viscosity is assumed, the critical condition is predicted to be 2P(m)/(S xi(c))=0.06, beyond which the tearing instability, i.e., the forward cascade, stops. Here, S is the Lundquist number, and P-m is the magnetic Prandtl number. According to the critical condition, the resistivity and viscosity employed in most high-S MHD simulations of PI are too small to stop the forward cascade. This critical condition may be also applicable for the trigger problem of the current sheet destabilization.