Resistive Scaling in the Magnetic Helicity-Driven Inverse Cascade

The inverse cascade in MHD turbulence plays a crucial role in various astrophysical processes such as galaxy cluster formation, solar and stellar dynamo mechanisms, and the evolution of primordial magnetic fields in the early universe. A standard numerical approach involves injecting magnetic helicity at intermediate length scales to generate a secondary, time-dependent spectral peak that gradually propagates toward larger scales. Previous simulations have already suggested a resistive dependence of inverse transfer rates and demonstrated the significant influence of magnetic helicity flux density $ε_\mathrm{H}$ on this process. On dimensional grounds, we have $E_\mathrm{M}(k,t)=C_\mathrm{H} ε_\mathrm{H}^{2/3} k^{-1}$ where $C_\mathrm{H}$ represents a potentially universal dimensionless coefficient analogous to the Kolmogorov constant. We present a summary of the 25 distinct simulations conducted with the \textsc{Pencil Code}, systematically varying the forcing wavenumber $k_\mathrm{f}$, magnetic Prandtl number $Pm$, grid resolution $N^3$, and Lundquist number $Lu$. We obtained $C_\mathrm{H}$ and corresponding error bars by calculating the compensated spectrum and investigated its dependence with $Lu$ and $k_\mathrm{f}$. For the $C_\mathrm{H}$ - $Lu$ relationship, we observe strong correlations with power-law exponents of 1 and 2/3. In contrast, we find no significant correlation between $C_\mathrm{H}$ and $k_\mathrm{f}$.