Linear growth rate of compressible tearing instability in a force-free current sheet

We investigate the linear growth rate of tearing instability in a resistive force-free current sheet using the linearised fully compressible magnetohydrodynamics (MHD) equations. Our results show that, as the plasma beta (

), which is proportional to the ratio of thermal pressure to magnetic pressure, increases, the dispersion relation approaches that of the incompressible tearing mode. As

beta β $\beta$

decreases, the growth rate increases, but this increase remains finite even in the limit

beta right arrow 0 β → 0 $\beta \rightarrow 0$

. The relative increase in the growth rate from

beta equals plus normal infinity β = + ∞ $\beta = +\infty$

to

beta equals 0 β = 0 $\beta =0$

depends on the Lundquist number, with lower Lundquist numbers showing a more pronounced increase. These findings indicate that, while compressibility has a negligible effect on the growth rate of tearing instability in high-Lundquist-number regimes, its effect becomes significant at low Lundquist numbers. This suggests that compressibility may influence the critical Lundquist number for the transition from laminar to plasmoid-dominated magnetic reconnection. Moreover, in low-

beta β $\beta$

regimes (

beta less than 1 β < 1 $\beta \lt 1$

), the generated density perturbation can be substantial, comparable to or even larger than the magnetic field perturbation, even though the growth rate is only slightly modified by the compressibility.