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

Inverse cascading for initial MHD turbulence spectra between Saffman and Batchelor