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 seed magnetic field irrespective of plasma being chirally 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 possess 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 .