  In certain circumstances, chiral (parity-violating) medium can be described
hydrodynamically as a chiral fluid with microscopic quantum anomalies. Possible
examples of such systems include strongly coupled quark-gluon plasma, liquid
helium 3He-A, neutron stars and the Early Universe. We study first-order
hydrodynamics of a chiral fluid on a vortex background and in an external
magnetic field. We show that there are two previously undiscovered modes
describing heat waves propagating along the vortex and magnetic field. We call
them the Thermal Chiral Vortical Wave and Thermal Chiral Magnetic Wave. We also
identify known gapless excitations of density (chiral vortical and chiral
magnetic waves) and transverse velocity (chiral Alfven wave). We demonstrate
that the velocity of the chiral vortical wave is zero, when the full
hydrodynamic framework is applied, and hence the wave is absent and the
excitation reduces to the charge diffusion mode. We also comment on the
frame-dependent contributions to the obtained propagation velocities.
