Inverse cascading for initial MHD turbulence spectra between Saffman and Batchelor

In decaying magnetohydrodynamic (MHD) turbulence with a strong magnetic field, the spectral magnetic energy density is known to increase with time at small wavenumbers $k$, provided the spectrum at low $k$ is sufficiently steep. This process is called inverse cascading and occurs for an initial Batchelor spectrum, where the magnetic energy per linear wavenumber interval increases like $k^4$. For an initial Saffman spectrum that is proportional to $k^2$, however, inverse cascading has not been found in the past. We study here the case of an intermediate $k^3$ spectrum, which may be relevant for magnetogenesis in the early Universe during the electroweak epoch. This case is not well understood in view of the standard Taylor expansion of the magnetic energy spectrum for small $k$. Using high resolution MHD simulations, we show that also in this case there is inverse cascading with a strength just as expected from the conservation of the Hosking integral, which governs the decay of an initial Batchelor spectrum. Even for shallower $k^\alpha$ spectra with spectral index $\alpha>3/2$, our simulations suggest a spectral increase at small $k$ with time $t$ proportional to $t^{4\alpha/9-2/3}$. The critical spectral index of $\alpha=3/2$ is related to the slope of the spectral envelope in the Hosking phenomenology. Our simulations with $2048^3$ mesh points now suggest inverse cascading even for an initial Saffman spectrum.