  Macroscopic evolution of relativistic charged matter with chirality imbalance
is described by the chiral magnetohydrodynamics (chiral MHD). One such
astrophysical system is high-density lepton matter in core-collapse supernovae
where the chirality imbalance of leptons is generated by the parity-violating
weak processes. After developing the chiral MHD equations for this system, we
perform numerical simulations for the real-time evolutions of magnetic and flow
fields, and study the properties of the chiral MHD turbulence. In particular,
we observe the inverse cascade of the magnetic energy and the fluid kinetic
energy. Our results suggest that the chiral effects that have been neglected so
far can reverse the turbulent cascade direction from direct to inverse cascade,
which would impact the magnetohydrodynamics evolution in the supernova core
toward explosion.
