  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, we may trace the intermediate evolution history of the magnetic field and
clarify whether each process governing its evolution is frozen or not, which is
essential to follow the evolution of relatively weak magnetic fields.
