  If vector type perturbations are present in the primordial plasma before
recombination, the generation of magnetic fields is known to be inevitable
through the Harrison mechanism. In the context of the standard cosmological
perturbation theory, non-linear couplings of first-order scalar perturbations
create second-order vector perturbations, which generate magnetic fields. Here
we reinvestigate the generation of magnetic fields at second-order in
cosmological perturbations on the basis of our previous study, and extend it by
newly taking into account the time evolution of purely second-order vector
perturbations with a newly developed second-order Boltzmann code. We confirm
that the amplitude of magnetic fields from the product-terms of the first-order
scalar modes is consistent with the result in our previous study. However, we
find, both numerically and analytically, that the magnetic fields from the
purely second-order vector perturbations partially cancel out the magnetic
fields from one of the product-terms of the first-order scalar modes, in the
tight coupling regime in the radiation dominated era. Therefore, the amplitude
of the magnetic fields on small scales, $k \gtrsim 10~h{\rm Mpc}^{-1}$, is
smaller than the previous estimates. The amplitude of the generated magnetic
fields at cosmological recombination is about $B_{\rm rec} =5.0\times
10^{-24}~{\rm Gauss}$ on $k = 5.0 \times 10^{-1}~h{\rm Mpc}^{-1}$. Finally, we
discuss the reason of the discrepancies that exist in estimates of the
amplitude of magnetic fields among other authors.
