  Due to the spin-orbit coupling, Dirac fermions, submerged in a thermal bath
with finite macroscopic vorticity, exhibit a spin polarisation along the
direction parallel to the vorticity vector $\boldsymbol{\Omega}$. Due to the
symmetries of the Lagrangian for free massless Dirac particles, there are three
independent and classically conserved currents corresponding to the vector,
axial, and helical charges. The constitutive relations for the charge currents
and the stress-energy tensor at thermal equilibrium, derived in the framework
of quantum field theory at finite temperature, reveal vorticity-induced
contributions that deviate from the perfect fluid form. In this paper, we
consider the mode structure of the corresponding hydrodynamical theory and
derive collective excitations associated with coherent fluctuations of all
three charges. We show that the chirally imbalanced rotating fluid should
possess non-reciprocal gapless waves that propagate with different velocities
along and opposite to the vorticity vector. We also uncover a strictly
unidirectional mode, which we call the Axial Vortical Wave, propagating in the
background of the axial charge density. We point out an unexpected instability
in the limit of degenerate matter and discuss possible solutions when helicity
and axial charge non-conservation is taken into account.
