Magnetic Prandtl number dependence of plasmoid-mediated reconnection

The paper investigates the dependence of the plasmoid-mediated magnetic reconnection rate on the magnetic Prandtl number (PrM) using two-dimensional magnetohydrodynamic simulations of two coalescing magnetic islands. The authors use the Pencil Code to simulate compressible isothermal visco-resistive MHD equations, initializing the system with two coalescing islands following the configuration of Huang & Bhattacharjee (2010), with a current sheet width following Sweet-Parker scaling (a = Ly S−1/2). The Lundquist number is defined as S = VA L/η, with characteristic length Ly = 1, Alfvén speed VA = 1, so S = 1/η. The sound speed is set to 10VA, giving plasma beta β = 50.

The reconnection rate is measured through the rate of depletion of magnetic flux, defined as Vrec = -(1/VA Bamp) d/dt [max x Az(x,0,t) - max y Az(0,y,t)], which measures the magnetic flux contained within an island.

For Lundquist numbers below the onset of the plasmoid instability (S < Sc ≈ 105), the authors find that the reconnection rate follows the expected Sweet-Parker scaling Vrec ∝ S−1/2. For the PrM dependence in this regime, they verify the theoretical prediction Vrec ∝ S−1/2 PrM−1/4, first derived by Park et al. (1984). The measured scaling appears slightly shallower, with an exponent between-1/4 and-1/5, which they attribute to limited dynamic range in PrM and possible incomplete asymptotic behavior.

For Lundquist numbers above the plasmoid instability threshold (S ≳ 105), the authors find that the reconnection rate exhibits only weak dependence on PrM across the range 1 ≤ PrM ≤ 50. In the fully plasmoid-mediated regime, they find reconnection rates that remain nearly independent of PrM. The paper states: We find that the reconnection rate exhibits only a weak dependence on PrM across this range, remaining broadly consistent with a PrM-independent scaling.

The authors identify that the largest reconnection rates are associated with strongly non-linear phases involving plasmoid interactions and mergers. They show this through a detailed analysis of the S = 5 × 105, PrM = 40 run, where the reconnection proceeds in a quasi-cyclic manner. During fast-reconnection phases, the current sheet contains multiple plasmoids of appreciable size that interact and undergo mergers. During quiescent phases, the layer is dominated by a single comparatively small plasmoid with much less dynamical activity. They note: This suggests that interactions and coalescence between plasmoids may play an important role in sustaining enhanced reconnection rates.

The authors compare their results with simulations of the boundary-driven Taylor problem, following the setup of Comisso et al. (2015, CGW15). In this setup, the equilibrium magnetic field is By = x with perfectly conducting boundaries, stable to tearing, and reconnection is driven externally through boundary perturbations. They perform two simulations with the same resistivity η = 4 × 10−7 and PrM = 10 and 40. In the Taylor problem, they find that during the Sweet-Parker-like phase, the reconnection rate collapses when rescaled by PrM1/4, consistent with the visco-resistive Sweet-Parker scaling. During the plasmoid-dominated phase, a collapse is obtained when rescaled by PrM1/2, consistent with CGW15 and Loureiro et al. (2013).

The authors identify two key differences between the setups that may explain the differing PrM scalings. First, in the Taylor problem, the equilibrium current sheet is stable to tearing and reconnection is initiated externally through boundary perturbations (forced reconnection), whereas the coalescing-tubes setup undergoes spontaneous reconnection arising self-consistently from non-linear evolution. Second, in CGW15 and their reproductions, the plasmoid-mediated reconnection rate is measured during the early plasmoid phase before substantial mergers occur, whereas in the coalescing-tubes setup, interactions and mergers between multiple plasmoids appear essential for attaining large reconnection rates that become nearly independent of PrM.

The paper also includes an appendix discussing an important subtlety regarding pressure equilibrium of the initial condition. The authors compare their full-equilibrium setup with the pressure-equilibrium-only configuration used by Vicentin et al. (2026, VKDL26), who reported an intermediate Lundquist number regime (104 ≲ S ≲ 105) with S−1/3 scaling. The authors attribute this discrepancy to the residual force imbalance from magnetic tension in the pressure-equilibrium-only setup, which becomes dynamically important at low β. They demonstrate that the S−1/3 scaling appears only in the pressure-equilibrium configuration and in the same Lundquist number range reported by VKDL26.

The authors discuss implications for reconnection-mediated decay in magnetically dominated turbulence, noting that recent studies (Brandenburg et al. 2024) reported decay timescales largely independent of PrM at high Lundquist numbers, which is consistent with their findings. They also note potential implications for primordial magnetic field evolution, as Hosking & Schekochihin (2023) used the PrM−1/2 scaling in estimating turbulent decay rates for assessing survivability of primordial magnetic fields in cosmic voids.

Caveats acknowledged include: all simulations are two-dimensional, and the explored parameter range remains limited compared to many astrophysical systems. The authors suggest that determining whether the weak PrM dependence survives in fully non-linear 3D plasmoid-mediated reconnection remains an important direction for future work.

Based on this paper, I can improve AI systems in the following specific ways:

• Extract and organize key numerical parameters (Lundquist number S, magnetic Prandtl number PrM, resolution, grid sizes) from tables into structured data for reproducibility

• Identify scaling laws and regimes (Sweet-Parker vs. plasmoid-mediated) from figures and text, enabling automatic classification of reconnection regimes

• Cross-reference citations (e.g., CGW15, HB10, VKDL26) to track how different research groups' findings agree or conflict

• Recommend optimal simulation parameters (resolution, PrM range, S values) based on the paper's Table 1 and convergence tests

• Detect numerical artifacts by flagging runs marked with asterisks (convergence tests) and comparing with lower-resolution counterparts

• Suggest diagnostic methods (flux depletion rate vs. alternative measures in Appendix B) appropriate for different reconnection regimes

• Distinguish between forced (Taylor problem) and spontaneous (coalescing islands) reconnection based on the paper's key finding that PrM scaling differs between them

• Predict expected PrM dependence (−1/4 in Sweet-Parker, weak in plasmoid-mediated, −1/2 in Taylor problem) based on setup characteristics

• Identify when linear vs. nonlinear plasmoid phases dominate using the paper's time-series analysis (Figure 4-5)

• Flag contradictions between this paper's findings (PrM-independent plasmoid reconnection) and CGW15's predictions (PrM−1/2 scaling)

• Reconcile differences by identifying key physical distinctions (spontaneous vs. forced, plasmoid mergers vs. isolated plasmoids)

• Track evolving consensus on reconnection scalings across multiple papers

• Generate structured abstracts following the paper's format (motivation → methods → key findings → implications)

• Suggest appropriate caveats (2D vs. 3D, limited parameter range, numerical termination issues)

• Produce data availability statements with proper DOI and repository links

• Estimate decay timescales for primordial magnetic fields using the paper's weak PrM dependence, potentially revising Hosking & Schekochihin (2023) estimates

• Model reconnection in galaxy clusters using the paper's plasmoid-mediated rates rather than Sweet-Parker predictions

• Incorporate 2D vs. 3D caveats when extrapolating to turbulent astrophysical systems

• Optimize grid resolution using the paper's convergence criteria (e.g., 61442 vs. 40962 for S=105)

• Implement adaptive time-stepping based on the observed quasi-cyclic reconnection behavior (Figure 5)

• Detect plasmoid instability onset automatically from flux evolution curves (Figure 4)

• Create interactive visualizations of the two-island coalescence setup (Figure 1) with adjustable S and PrM

• Generate step-by-step derivations of the reconnection rate diagnostic (Appendix B)

• Produce comparison tables of different reconnection regimes and their scaling laws

These improvements would enable AI systems to better design, analyze, and interpret magnetohydrodynamic reconnection simulations, while also providing more accurate predictions for astrophysical applications and reconciling conflicting results in the literature.

We investigate the dependence of the plasmoid-mediated magnetic reconnection rate on the magnetic Prandtl number using two-dimensional magnetohydrodynamic simulations of two coalescing magnetic islands. For Lundquist numbers below the onset of the plasmoid instability, the reconnection rate follows the expected Sweet-Parker scaling and decreases with increasing magnetic Prandtl number. However, once the current sheet becomes plasmoid unstable, the dependence on the magnetic Prandtl number weakens considerably. In the fully plasmoid-mediated regime, we find reconnection rates that remain nearly independent of the magnetic Prandtl number over the explored parameter range. We show that the largest reconnection rates are associated with strongly non-linear phases involving plasmoid interactions and mergers. We further compare our results with simulations of the boundary-driven Taylor problem, where previous studies reported a stronger magnetic Prandtl number dependence, and provide a possible explanation for the differing scalings obtained in the two setups. These results may have implications for reconnection-mediated decay in magnetically dominated turbulence and related astrophysical systems.