Hydrodynamics of topological Dirac semi-metals with chiral and ${\mathbb{Z}}_2$ anomalies

We consider the hydrodynamical model of topological Dirac semi-metal possessing two Dirac nodes separated in momentum space along a rotation axis. It has been argued that the system in question, except the chiral anomaly, is endowed with the other one ${\mathbb{Z}}_2$. In order to model such a system we introduce two $U(1)$-gauge fields. The presence of the additional ${\mathbb{Z}}_2$ anomaly leads to the non-trivial modifications of hydrodynamical equations and to the appearance of new kinetic coefficients bounded with the vorticity and the magnetic parts of Maxwell and auxiliary $U(1)$-gauge fields.