Inverse cascades and primordial magnetic fields

We consider primordial spectra with simple power behaviours and show that in the Navier-Stokes and magnetohydrodynamics equations without forcing, there exists systems in three dimensions with a subsequent inverse cascade, transferring energy from small to large spatial scales. This can have consequences in astrophysics for the evolution of density fluctuations, for primordial magnetic fields, and for the effect of diffusion. In general, if the initial spectrum is kα, then in the “inertial” range, for α > −3 there is an inverse cascade, whereas for α < −3 there is a forward cascade.