  In the pre-recombination era, cosmological density fluctuations can naturally
generate magnetic fields through Thomson scatterings. In previous studies, only
the magnetic field generation from the initially-adiabatic fluctuations has
been considered. Here we investigate the generation of cosmological magnetic
fields sourced by the primordial non-adiabatic fluctuations based on the
cosmological perturbation theory, using the tight-coupling approximations
between photon and baryon fluids. It is found that the magnetic fields from the
non-adiabatic fluctuations can arise at the first-order expansion of the tight
coupling approximation. This result is in contrast to the case of adiabatic
initial fluctuations, where the magnetic fields can be generated only at the
second-order. In a general case where the primordial density perturbations
contain small non-adiabatic fluctuations on the top of the dominant adiabatic
ones, we show that the leading source of magnetic fields is given by the
second-order coupling of the adiabatic and non-adiabatic fluctuations. We
calculate the power spectrum of the generated magnetic fields when the
non-adiabatic fluctuations have a blue power spectrum, which has been suggested
by recent cosmological observations.
