Ultimate large-$Rm$ regime of the solar dynamo

TL;DR

This study probes how large-scale magnetic fields emerge in rotating turbulent systems at large magnetic Reynolds numbers $Rm$ by using maximally-simplified 3D Cartesian MHD with hemispheric helicity reversal. It identifies three dynamical regimes in the magnetic-helicity budgets—resistive $R$, intermediate $I$, and ultimate $U$—and develops a diffusive helicity-flux model that predicts saturation levels and the thresholds $Rm_{R-I}$ and $Rm_{I-U}$, with $Rm_{I-U}=(6/\xi)(k_{\mathrm{eff}}/\overline{k})^2$. The results indicate that the ultimate regime is reachable only in simplified DNS setups and that current global solar-dynamo simulations likely reside in the intermediate regime due to strong scale separation, requiring $Rm$ well above $5\times10^3$ to approach the ultimate state. The work underscores the limitations of brute-force global modeling for turbulent dynamos and offers diagnostic tools and modeling avenues—such as adjusting forcing scales, developing closures, or leveraging data-driven transports—to bridge toward solar-like cycles. Overall, it clarifies the parameter space structure of large-scale dynamos and motivates targeted strategies to access the ultimate high-$Rm$ regime within feasible computational resources.

Abstract

For more than fourty years, the quest to understand how large-scale magnetic fields emerge from turbulent flows in rotating astrophysical systems, such as the Sun, has been a major thread of computational astrophysics research. Using a parameter scan and phenomenological analysis of maximally-simplified three-dimensional cartesian magnetohydrodynamic simulations of large-scale nonlinear helical turbulent dynamos, I present results in this Letter that strongly point to an asymptotic ultimate regime of the large-scale solar dynamo, at large magnetic Reynolds numbers $Rm$, involving helicity fluxes between hemispheres. I obtained corresponding numerical solutions at both $Pm>1$ and $Pm<1$, and show that they can currently only be achieved in clean, simplified numerical setups. The analysis further strongly suggests that all global simulations to date lie in a non-asymptotic turbulent MHD regimes highly sensitive to changes in kinetic and magnetic Reynolds numbers. Ideas are presented to attempt to reach this ultimate regime in such "realistic" global spherical models at a reasonable numerical cost. Overall, the results clarify the current state, and some hard limitations of the brute-force numerical modelling approach applied to this, and other similar astrophysical turbulence problems.

Figures (6)

• Figure 1: Butterfly diagrams $\overline{B}_{x}(z,t)$ of large-scale non-linear helical dynamo modes as a function of $Re$ and $Rm$. Run S01 ($Pm=0.5$) is in the rightmost plot, run T06 ($Pm=4$) of R21 in the topmost plot. The parameter range spanned by the simulations is shown in the bottom right.
• Figure 2: Time- and $(x,y)$-averaged kinetic and current helicity $z$-profiles in run S01 ($Rm\simeq 1400$, $Re\simeq 2900$).
• Figure 3: Evolutions of kinetic and magnetic energy den-sities in run S01 ($Rm\simeq 1450$, $Re\simeq 2900$). Inset: corresponding energy spectra.
• Figure 4: Helicity budgets for the T and S runs as a function of $Rm$, with corresponding regime tags and qualitative separations between regimes (vertical dashed lines). The lines and full circles correspond to the T runs ($Re\simeq 694$ based on T06) in R21, the empty squares show the same quantities for the new $Re\simeq 2900$, $Pm=0.5$ run S01.
• Figure 5: Turbulent fluxes of large and small-scale magnetic helicity in the T06 run at $Rm=2800$, and their respective diffusive fits. The best fit for the turbulent flux of large-scale helicity gives $\kappa_t=0.6 \,u_\mathrm{rms}/(3\,k_f)$, and that for the flux or small-scale helicity $\kappa_t=0.55 \,u_\mathrm{rms}/(3\,k_f)$.