Unveiling a New $β$-Scaling of the Tearing Instability in Weakly Collisional Plasmas

TL;DR

The paper addresses how tearing instability behaves in weakly collisional, gyrotropic plasmas, challenging the classic $\beta$-independent picture from resistive MHD. It develops a nonideal gyrotropic MHD model with pressure anisotropy, derives a boundary-layer theory, and validates it with a numerical eigenproblem. The key finding is a $\beta$-dependent stabilization: the maximum growth rate scales as $\sigma_{max} \tau_a \propto \beta^{-1/4}$ in the high-$\beta$ limit, and $k_{max}a \propto \beta^{-3/8}$, with $\tilde{\beta} = \tfrac{1}{2}(\gamma_{\parallel}+\gamma_{\perp}-2)\beta_0$. These results imply slower, larger-scale reconnection in high-$\beta$ plasmas, affecting astrophysical contexts from the solar wind to the intracluster medium. The work highlights the necessity of incorporating pressure anisotropy dynamics into reconnection models.

Abstract

We investigate the linear tearing instability in weakly collisional plasmas using a non-ideal gyrotropic-MHD framework, uncovering a previously unknown scaling relation for the instability growth rate in high-$β$ environments. Even starting from an isotropic equilibrium, our analysis reveals a $β$-dependence, with the maximum growth rate scaling as $σ_\mathrm{max} τ_a \propto β^{-1/4}$, challenging the long-held assumption of $β$-independence inherent in classical MHD formulations. This novel scaling emerges due to self-consistent fluctuations in pressure anisotropy, dynamically induced by perturbations in velocity and magnetic fields. Increasing plasma-$β$ always suppresses the instability, whereas a background pressure anisotropy can either enhance or further suppress it, depending on its sign: for $p_{\parallel,0} < p_{\perp,0}$ the instability is strengthened, while for $p_{\parallel,0} > p_{\perp,0}$ it is weakened. Importantly, this effect is not limited to low-collisionality plasmas at high $β$; it can also manifest in more collisional environments once the strict assumption of pressure isotropy is relaxed. This finding has profound implications for various astrophysical contexts characterized by high $β$ and varying degrees of collisionality, including the solar corona and heliospheric current sheets, planetary magnetospheres, as probed by space missions, and the intracluster medium, where magnetic reconnection critically impacts magnetic field evolution and cosmic ray transport. Our results therefore question the universality of the widely-accepted plasmoid-mediated fast reconnection paradigm and underscore the necessity of incorporating pressure anisotropy effects into reconnection models for accurate representation of astrophysical plasmas.

Figures (3)

• Figure 1: Maximum growth rate (left) and the corresponding maximum wavenumber (right) of the tearing mode instability as functions of the equilibrium plasma-$\beta$ parameter for $\Delta \beta=0$ for different Lundquist numbers $S = 10^4$, $10^5$, and $10^6$, with $Pr_{\rm m}=1$. For the plasma-$\beta$ dependence, the analysis considers both the MHD framework (dashed) and the nonideal gyrotropic MHD model (solid).
• Figure 2: Comparison of maximum growth rates and wavenumbers for different polytropic closures in the nonideal gyrotropic MHD model. Left panel: the normalized maximum growth rate ($\sigma_\mathrm{max}\tau_A$) as a function of plasma-$\beta$. Right panel: the wavenumber corresponding to the maximum growth rate ($k_\mathrm{max}a$) as a function of plasma-$\beta$. The results are shown for a Lundquist number $S=10^4$ and a magnetic Prandtl number $Pr_{\rm m}=1$. The lines represent three distinct polytropic closures: double-adiabatic ($\gamma_\parallel=3, \gamma_\perp=2$; blue), double-polytropic ($\gamma_\parallel=1/2, \gamma_\perp=2$; orange), and double-isothermal ($\gamma_\parallel=1, \gamma_\perp=1$; green). The figure demonstrates that the growth rate's dependence on $\beta$ emerges in nonisothermal models, with a notable suppression in high-$\beta$ regimes, whereas the isothermal model remains independent of $\beta$, consistent with classical MHD theory. The black line indicates the reference slope $\sim\beta^{-1/4}$ for the growth rate scaling at high $\beta$.
• Figure 3: Maximum growth rate (left) and the corresponding maximum wavenumber (right) of the tearing mode instability as functions of the equilibrium pressure anisotropy parameter $\Delta\beta_0$, for $\beta = 1$ and $10$ (upper and lower rows, respectively) for different Lundquist numbers $S = 10^4$, $10^5$, and $10^6$, with $Pr_{\rm m}=1$. The points at $\Delta \beta_0 = 0$ represent $\sigma_\mathrm{max} \tau_a$ and $k_\mathrm{max} a$ obtained from the MHD approximation.