  The magnetohydrodynamics (MHD) is modified to incorporate the parity
violation in the Standard Model leading to a new instability of magnetic fields
in the electroweak plasma in the presence of nonzero neutrino asymmetries. The
main ingredient for such a modified MHD is the antisymmetric part of the photon
polarization tensor in plasma, where the parity violating neutrino interaction
with charged leptons is present. We calculate this contribution to the
polarization tensor connected with the Chern-Simons term in effective
Lagrangian of the electromagnetic field. The general expression for such a
contribution which depends on the temperature and the chemical potential of
plasma as well as on the photon's momentum is derived. The instability of a
magnetic field driven by the electron neutrino asymmetry for the $\nu$-burst
during the first second of a supernova explosion can amplify a seed magnetic
field of a protostar, and, perhaps, can explain the generation of strongest
magnetic fields in magnetars. The growth of a cosmological magnetic field
driven by the neutrino asymmetry density $\Delta n_{\nu}=n_{\nu} -
n_{\bar{\nu}}\neq 0$ is provided by a lower bound on
$|\xi_{\nu_e}|=|\mu_{\nu_e}|/T$ which is consistent with the well-known Big
Bang nucleosynthesis (upper) bound on neutrino asymmetries in a hot universe
plasma.
