| Summary: | We present a new description of cosmological evolution of the primordial magnetic field under the condition that it is non-helical and its energy density is larger than the kinetic energy density. We argue that the evolution can be described by four different regimes, according to whether the decay dynamics is linear or not, and whether the dominant dissipation term is the shear viscosity or the drag force. Using this classification and conservation of the Hosking integral, we present analytic models to adequately interpret the results of various numerical simulations of field evolution with variety of initial conditions. It is found that, contrary to the conventional wisdom, the decay of the field is generally slow, exhibiting the inverse transfer, because of the conservation of the Hosking integral. Using the description proposed here, one can trace the intermediate evolution history of the magnetic field and clarify whether each process governing its evolution is frozen or not. Its applicability to the early cosmology is important, since primordial magnetic fields are sometimes constrained to be quite weak, and multiple regimes including the frozen regime matters for such weak fields.
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