[Paper Review] High Lundquist Number Resistive MHD Simulations of Magnetic Reconnection: Searching for Secondary Island Formation

C. S. Ng, Srivathsan Ragunathan|arXiv (Cornell University)|Jun 2, 2011

Ionosphere and magnetosphere dynamics3 citations

TL;DR

This study performs high-resolution resistive MHD simulations of magnetic reconnection at Lundquist numbers up to $2 \times 10^5$ to investigate secondary island (plasmoid) formation. Without external noise, reconnection follows Sweet-Parker scaling ($S^{-1/2}$) and no plasmoids form; however, with added small-scale noise, plasmoids emerge and reconnection becomes independent of $S$, indicating noise-induced turbulent reconnection mechanisms can enable fast reconnection in resistive MHD.

ABSTRACT

Recently, secondary island formation due to the tearing instability of the Sweet-Parker current sheet was identified as a possible mechanism that can lead to fast reconnection (less sensitive dependence on Lundquist number $S$) both in numerical simulations using Particle-in-Cell (PIC) method [Daughton et al. 2009], as well as using resistive magnetohydrodynamics (MHD) [Lapenta 2008; Bhattacharjee et al. 2009]. This instability is thought to appear when $S$ is greater than a certain threshold. These recent results prompt us to perform more resistive MHD simulations of a basic reconnection configuration based on the island coalescence instability, using much higher resolutions and larger $S$. Our simulations are based on a fairly standard pseudo spectral code, which has been tested for accuracy, convergence, and compared well with codes using other methods [Ng et al. 2008]. In our simulations, formation of plasmoids were not found, except when insufficient resolution was used, or when a small amount of noise was added externally. The reconnection rate is found to follow the Sweet-Parker scaling when no noise is added, but increases to a level independent of $S$ with noise, when plasmoids form. Latest results with $S$ up to $2 imes 10^5$ will be presented.

Motivation & Objective

• To investigate whether secondary island (plasmoid) formation occurs naturally in high-Lundquist-number resistive MHD simulations of magnetic reconnection.
• To determine whether the observed fast reconnection rates in prior studies are due to numerical artifacts or physical mechanisms.
• To assess the role of numerical resolution and external noise in triggering plasmoid formation and enhancing reconnection rates.
• To compare reconnection dynamics in noise-free versus noise-injected simulations to isolate the conditions under which fast reconnection emerges.
• To evaluate the validity of Sweet-Parker scaling in high-resolution resistive MHD under controlled conditions.

Proposed method

• Simulations use a pseudo-spectral code solving 2D incompressible resistive MHD equations in normalized form, with $\partial_t \Omega + [\phi, \Omega] = [A, J] + \nu \nabla_\perp^2 \Omega$ and $\partial_t A + [\phi, A] = \eta \nabla_\perp^2 A$.
• The Lundquist number $S$ is controlled via resistivity $\eta$, with $S$ values up to $2 \times 10^5$ achieved using $\eta \sim S^{-1}$.
• Numerical resolution reaches $8192^2$ grid points to ensure convergence and minimize spurious plasmoid formation from insufficient resolution.
• External random noise is added in spectral space with controlled amplitude (energy ratio ~$10^{-6}$) and correlation time (~0.25) to mimic turbulent perturbations.
• Reconnection rate is quantified via $dA_0/dt$, where $A_0$ is the maximum flux function magnitude, and averaged over the reconnection phase.
• Simulations are compared across cases with and without noise at varying $S$ and resolution to isolate the effect of noise on plasmoid formation and reconnection rate.

Experimental results

Research questions

• RQ1 Does the Sweet-Parker current sheet become unstable to tearing modes at high Lundquist numbers ($S \sim 10^5$) in well-resolved resistive MHD simulations?
• RQ2 Can secondary island (plasmoid) formation occur naturally in high-resolution resistive MHD simulations without external perturbations?
• RQ3 What is the role of numerical noise or external perturbations in triggering plasmoid formation and enabling fast reconnection?
• RQ4 How does the reconnection rate scale with $S$ in the absence and presence of added noise?
• RQ5 To what extent do simulation results depend on resolution and the inclusion of small-amplitude noise?

Key findings

• In high-resolution simulations without external noise, reconnection rates follow the Sweet-Parker scaling of $S^{-1/2}$, with no plasmoid formation observed even at $S = 2 \times 10^5$.
• Plasmoid formation and ejection are only observed when small-amplitude external noise is added to the system, indicating a threshold-like trigger for instability.
• With noise added, the reconnection rate becomes approximately independent of $S$, reaching levels significantly higher than the Sweet-Parker prediction at large $S$.
• The addition of noise leads to fluctuating but enhanced reconnection rates, with averaged $dA_0/dt$ remaining constant across $S$ values from $1.25 \times 10^4$ to $2.5 \times 10^4$.
• The results suggest that plasmoid formation in resistive MHD is not an intrinsic feature of the Sweet-Parker current sheet at high $S$, but rather a consequence of external perturbations or numerical noise.
• The study highlights the critical importance of resolution and the presence of small-scale perturbations in determining whether fast reconnection mechanisms emerge in simulations.