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We investigate hydromagnetic turbulence of primordial magnetic fields using magnetohydrodynamics ͑MHD͒ in an expanding universe. We present the basic, covariant MHD equations, find solutions for MHD waves in the early universe, and investigate the equations numerically for random magnetic fields in two spatial dimensions. We find the formation of magnetic structures at larger and larger scales as time goes on. In three dimensions we use a cascade ͑shell͒ model that has been rather successful in the study of certain aspects of hydrodynamic turbulence. Using such a model we find that after ϳ10 9 times the initial time the scale of the magnetic field fluctuation ͑in the comoving frame͒ has increased by 4-5 orders of magnitude as a consequence of an inverse cascade effect ͑i.e., transfer of energy from smaller to larger scales͒. Thus at large scales primordial magnetic fields are considerably stronger than expected from considerations which do not take into account the effects of MHD turbulence. ͓S0556-2821͑96͒02712-9͔ PACS number͑s͒: 95.30.Qd, 04.40.Nr, 98.62.En

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We consider primordial spectra with simple power behaviours and show that in the NavierStokes 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 k α , then in the "inertial" range, for α > −3 there is an inverse cascade, whereas for α < −3 there is a forward cascade.