  The chiral vortical effect is a chiral anomaly induced transport phenomenon
characterized by an axial current in a uniformly rotating chiral fluid. It is
well-understood for Weyl fermions in high energy physics, but its realization
in condensed matter band structures, including those of Weyl semimetals, has
been controversial. In this work, we develop the Kubo response theory for
electrons in a general band structure subject to space- and time-dependent
rotation or vorticity relative to the background lattice. We recover the chiral
vortical effect in the static limit; in the transport or uniform limit, we
discover a new effect that we dub the gyrotropic vortical effect. The latter is
governed by Berry curvature of the occupied bands while the former contains an
additional contribution from the magnetic moment of electrons on the Fermi
surface. The two vortical effects can be understood as analogs of the
well-known chiral and gyrotropic magnetic effects in chiral band structures. We
address recent controversies in the field and conclude by describing device
geometries that exploit Ohmic or Seebeck transport to drive the vortical
effects.
