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We study the relativistic hydrodynamics with chiral anomaly and dynamicalelectromagnetic fields, namely Chiral MagnetoHydroDynamics (CMHD). We formulateCMHD as a low-energy effective theory based on a generalized derivativeexpansion. We demonstrate that the modification of ordinaryMagnetoHydroDynamics (MHD) due to chiral anomaly can be obtained from thesecond law of thermodynamics and is tied to chiral magnetic effect. We furtherstudy the real-time properties of chiral fluid by solving linearized CMHDequations. We discover a remarkable "transition" at an intermediate axialchemical potential $\mu_{A}$ between a stable Chiral fluid at low $\mu_{A}$ andan unstable Chiral fluid at large $\mu_{A}$. We summarize this transition in a"phase diagram" in terms of $\mu_{A}$ and the angle of the wavevector relativeto the magnetic field. In the unstable regime, there are four collective modescarrying both magnetic and fluid helicity, in contrary to MHD waves which areunpolarized. The half of the helical modes grow exponentially in time,indicating the instability, while the other half become dissipative.

MagnetoHydrodynamics with chiral anomaly: phases of collective excitations and instabilities