We study relativistic hydrodynamics with chiral anomaly and dynamical electromagnetic fields, namely chiral magnetohydrodynamics (CMHD). We formulate CMHD as a low-energy effective theory based on a generalized derivative expansion. We demonstrate that the modification of ordinary magnetohydrodynamics (MHD) due to chiral anomaly can be obtained from the second law of thermodynamics and is tied to the chiral magnetic effect. We further study the real-time properties of a chiral fluid by solving linearized CMHD equations. We discover a remarkable “transition” at an intermediate axial chemical potential μ A between a stable chiral fluid at low μ A and an unstable chiral fluid at high μ A . We summarize this transition in a “phase diagram” in terms of μ A and the angle of the wave vector relative to the magnetic field. In the unstable regime, four collective modes carry both magnetic and fluid helicity, in contrary to MHD waves, which are unpolarized. Half of the helical modes grow exponentially in time, indicating the instability, while the other half become dissipative.
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