  We present two new results on relativistic hydrodynamics with anomalies and
external electromagnetic fields, "Chiral MagnetoHydroDynamics" (CMHD). First,
we study CMHD in four dimensions at second order in the derivative expansion
assuming the conformal/Weyl invariance. We classify all possible independent
conformal second order viscous corrections to the energy-momentum tensor and to
the U(1) current in the presence of external electric and/or magnetic fields,
and identify eighteen terms that originate from the triangle anomaly. We then
propose and motivate the following guiding principle to constrain the CMHD: the
anomaly--induced terms that are even under the time reversal invariance should
not contribute to the local entropy production rate. This allows us to fix
thirteen out of the eighteen transport coefficients that enter the second order
formulation of CMHD. We also relate one of our second order transport
coefficients to the chiral shear waves. Our second subject is hydrodynamics
with (N+1)-gon anomaly in an arbitrary 2N dimensions. The effects from the
(N+1)-gon anomaly appear in hydrodynamics at (N-1)'th order in the derivative
expansion, and we identify precisely N such corrections to the U(1) current.
The time reversal constraint is powerful enough to allow us to find the
analytic expressions for all transport coefficients. We confirm the validity of
our results (and of the proposed guiding principle) by an explicit
fluid/gravity computation within the AdS/CFT correspondence.
