  We study the generation of magnetic field in the primordial plasma of the
standard model (SM) particles at temperature $T>80$~TeV much higher than the
electroweak scale. It is assumed that there is an excess number of right-handed
electrons over left-handed positrons in the plasma. Using the Berry-curvature
modified kinetic theory to incorporate the effect of the Abelian anomaly, we
show that this chiral-imbalance leads to generation of hyper-magnetic field in
the plasma in both the collision dominated and the collisionless regimes. It is
shown that in the collision dominated regime the chiral-vorticity effect can
generate finite vorticity in the plasma together with the magnetic field.
Typical strength of the generated magnetic field is $10^{27}$~Gauss at $T\sim
80$~TeV with the length scale $10^5/T$ whereas the Hubble length scale is
$10^{13}/T$. Further the instability can also generate the magnetic field of
order $10^{31}$~Gauss at typical length scale $10/T$. But there may not be any
vorticity generation in this regime. We show that the estimated values of the
magnetic field are consistent with the bounds obtained from present
observations.
