Effects of Line-tying on Resistive Tearing Instability in Slab Geometry

The effects of line-tying on magnetohydrodynamic instabilities are an important issue for astrophysical plasmas, such as the solar corona or astrophysical jets. Recently, several laboratory experiments aimed at studying line-tying effects have been initiated. This work studies the effect of line-tying on the resistive tearing instability in the slab geometry. A strong guide field perpendicular to the conducting boundary is assumed, therefore the system is described by the well-known reduced magnetohydrodynamic (RMHD) equations. The linearized eigenvalue problem is solved numerically. It is found that line-tying has a stabilizing effect. The tearing mode is stabilized when the system length $L$ is shorter than a critical length $L_{c}$, which is independent of the resistivity $\eta$. When $L$ is not too much longer than $L_{c}$, the growthrate $\gamma$ is proportional to $\eta$ . When $L$ is sufficiently long, the tearing mode scaling $\gamma\sim\eta^{3/5}$ is recovered. The transition from $\gamma\sim\eta$ to $\gamma\sim\eta^{3/5}$ occurs at a transition length $L_{t}\sim\eta^{-2/5}$.