Chiral Battery, scaling laws and magnetic fields

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.