[Paper Review] Compression of the current sheet and its impact into the reconnection rate

С. И. Ваинштейн, Z. Mikić+1 more|ArXiv.org|Nov 11, 2007

Solar and Space Plasma Dynamics1 references3 citations

TL;DR

This paper demonstrates that strong compressibility in force-free magnetic configurations, such as those in stellar atmospheres, drives rapid collapse of the current sheet, leading to dramatic increases in the magnetic reconnection rate. Through numerical MHD simulations, the authors show that compression reduces the current sheet width below Sweet-Parker predictions, enabling reconnection rates that exceed standard models by orders of magnitude, particularly in highly compressible plasmas with low $ B_z/B_ot $ ratios.

ABSTRACT

Numerical simulations of strongly compressible MHD corresponding to a stellar atmosphere with substantial gravity and near force-free magnetic fields show that the current sheet collapses (its width decreasing substantially). As a result, the reconnection rate increases dramatically.

Motivation & Objective

• To investigate how compressibility in force-free magnetic configurations affects the magnetic reconnection rate in astrophysical plasmas.
• To resolve the discrepancy between slow Ohmic reconnection timescales and the rapid timescales observed in solar flares and stellar activity.
• To determine whether current sheet compression due to gravity and magnetic pressure imbalance can significantly enhance reconnection beyond the Sweet-Parker model.
• To quantify the scaling of reconnection rate and compression with Lundquist number and compressibility parameters in strongly compressible MHD simulations.

Proposed method

• Numerical simulations of strongly compressible magnetohydrodynamics (MHD) in a 1D geometry with gravity and force-free magnetic fields.
• Use of a one-dimensional model where magnetic field components $ B_y(x,t) $ and $ B_z(x,t) $ depend only on $ x $, with $ B_y $ odd and $ B_z $ even, satisfying $ B_y^2 + B_z^2 = \text{const} $.
• Analysis of the continuity equation $ \partial_t B_z + \nabla \cdot (\mathbf{v} B_z) = \eta \nabla^2 B_z $ at $ x=0 $, showing that finite resistivity leads to $ \nabla \cdot \mathbf{v} < 0 $, indicating compression.
• Definition of current sheet width $ \delta $ using the Sweet-Parker scaling $ \delta_{SP} = L / S^{1/2} $, and comparison with simulated $ \delta $ to quantify compression.
• Introduction of dimensionless reconnection rate $ \Omega_R = t_A / t_R = (L / \delta) / S $, and compression factor $ \rho / \rho_0 = \delta_{SP} / \delta = \Omega_R S^{1/2} $.
• Use of integral measure $ M \approx B^2 L^2 / (t_R \eta) \sim \Omega_R S $ to assess Ohmic dissipation and validate scaling laws.

Experimental results

Research questions

• RQ1 How does compressibility in force-free magnetic configurations influence the width and evolution of the current sheet?
• RQ2 To what extent does current sheet compression enhance the magnetic reconnection rate beyond the Sweet-Parker model?
• RQ3 What is the scaling of the reconnection rate with Lundquist number $ S $ in highly compressible plasmas?
• RQ4 How does the ratio $ B_z / B_\perp $ control the compressibility and reconnection efficiency in MHD simulations?
• RQ5 Can the observed reconnection rates be explained by a compression-driven mechanism rather than anomalous resistivity or Hall effects?

Key findings

• Strong compressibility leads to significant collapse of the current sheet width, reducing it below the Sweet-Parker scale $ \delta_{SP} $, with compression factor $ \delta_{SP}/\delta \sim S^{0.23 \pm 0.05} $, consistent with simulation data.
• The reconnection rate $ \omega_R = 1/t_R $ increases dramatically under compression, scaling as $ \Omega_R \sim S^{0.23} $, which is significantly faster than the Sweet-Parker scaling $ \Omega_{SP} \sim S^{-1/2} $.
• In the limit of low $ B_z/B_\perp $ (high compressibility), the reconnection rate becomes nearly flat with respect to $ S $, indicating a saturation of enhancement due to extreme compression.
• For $ B_z/B_\perp = 0.01 $, the reconnection rate scaling is nearly flat (power-law exponent ~0), while for $ B_z/B_\perp = 0.4 $, the system behaves incompressibly and reverts to Sweet-Parker scaling.
• Ohmic dissipation $ M \sim \Omega_R S $ scales approximately as predicted, with values consistent with simulations, validating the theoretical framework.
• The simulations confirm that even in the absence of anomalous resistivity or Hall effects, strong compression alone can drive fast reconnection, resolving the long-standing timescale problem in solar and stellar flares.