Resistive Magnetohydrodynamics Simulations of the Ideal Tearing Mode

Abstract

We study the linear and nonlinear evolution of the tearing instability on thin current sheets by means of twodimensional numerical simulations, within the framework of compressible, resistive MHD. In particular we analyze the behavior of current sheets whose inverse aspect ratio scales with the Lundquist number S as S 1 3-. This scaling has been recently recognized to yield the threshold separating fast, ideal reconnection, with an evolution and growth that are independent of S provided this is high enough, as it should be natural having the ideal case as a limit for S  ¥. Our simulations confirm that the tearing instability growth rate can be as fast aswhere A t is the ideal Alfvénic time set by the macroscopic scales, for our least diffusive case with S 10 7 = . The expected instability dispersion relation and eigenmodes are also retrieved in the linear regime, for the values of S explored here. Moreover, in the nonlinear stage of the simulations we observe secondary events obeying the same critical scaling with S, here calculated on the local, much smaller lengths, leading to increasingly faster reconnection. These findings strongly support the idea that in a fully dynamic regime, as soon as current sheets develop, thin, and reach this critical threshold in their aspect ratio, the tearing mode is able to trigger plasmoid formation and reconnection on the local (ideal) Alfvénic timescales, as required to explain the explosive flaring activity often observed in solar and astrophysical plasmas.

Fast reconnection in relativistic plasmas: the magnetohydrodynamics tearing instability revisited

“…As the simulation proceeds, we monitor the growth of each Fourier modes as in Landi et al (2015). We find that the fastest growing modes are those with m = 3 and m = 4 (from m = Ly/λmax), in perfect agreement with theory.…”

“…The full dispersion relation γ = γ(ka) has been also computed via accurate single-mode simulations, showing that the most unstable wavenumber behaves as predicted (kmaxa ≃ 1.4 S −1/6 or kmaxL ≃ 1.4 S 1/6 ). Finally, fully nonlinear and multi-mode simulations show the evolution of the ideal tearing instability, with exponential growth of the various modes, the subsequent merging of plasmoids and secondary reconnection events as in Landi et al (2015). In particular, here we find a stage dominated by a single X-point feeding a major plasmoid through fast magnetosonic jets of Petschek-type, shocking directly where the plasmoid has formed.…”

“…A z component of the field, concentrated near the sheet axis, is also needed to ensure a rotation in the y − z plane across the sheet preserving an overall constant magnitude B0. This is the case ζ = 1 of the more general equilibrium used by Landi et al (2015). In the present work we then assume…”

“…In this work we shall present for the first time relativistic MHD simulations of the tearing instability in current sheets with thickness a = S −1/3 L, extending the analysis by Landi et al (2015) to strongly magnetized plasmas with cA ≃ c. A linear analysis of the instability will be first performed, retrieving the appropriate scalings with S and the full dispersion relation, both analytically and numerically. Then, the nonlinear phase of the instability will be followed up to a quasi-steady Petschek-type configuration.…”

“Ideal” tearing and the transition to fast reconnection in the weakly collisional MHD and EMHD regimes

“…Note that the occurrence of a secondary, ideal tearing mode developing as a consequence of a primary tearing in a large aspect ratio current sheet in the resistive RMHD regime was first numerically evidenced by Landi et al [] and further discussed in depth in Tenerani et al []. However, for the discussion here the primary tearing mode has a ∼ L , so the primary reconnection rate cannot be estimated with that of the most unstable mode ; rather, the specific LD or SD regime in which the unstable wave number falls must be taken into account.…”