  We study the generation and evolution of magnetic field in the presence of
chiral imbalance and gravitational anomaly which gives an additional
contribution to the vortical current. The contribution due to gravitational
anomaly is proportional to $T^2$ which can generate a seed magnetic field
irrespective of plasma being hirally charged or neutral. We estimate the order
of magnitude of the magnetic field to be $10^{30}$~G at $T\sim 10^9$ GeV, with
a typical length scale of the order of $10^{-18}$ cm, which is much smaller
than the Hubble radius at that temperature ($10^{-8}$ cm). Moreover, such a
system possesses scaling symmetry. We show that the $T^2$ term in the vorticity
current along with scaling symmetry leads to more power transfer from lower to
higher length scale as compared to only chiral anomaly without scaling
symmetry.
