  Spiral density waves dominate several facets of accretion disc dynamics ---
planet-disc interactions and gravitational instability (GI) most prominently.
Though they have been examined thoroughly in two-dimensional simulations, their
vertical structures in the non-linear regime are somewhat unexplored. This
neglect is unwarranted given that any strong vertical motions associated with
these waves could profoundly impact dust dynamics, dust sedimentation, planet
formation, and the emissivity of the disc surface. In this paper we combine
linear calculations and shearing box simulations in order to investigate the
vertical structure of spiral waves for various polytropic stratifications and
wave amplitudes. For sub-adiabatic profiles we find that spiral waves develop a
pair of counter-rotating poloidal rolls. Particularly strong in the nonlinear
regime, these vortical structures issue from the baroclinicity supported by the
background vertical entropy gradient. They are also intimately connected to the
disk's g-modes which appear to interact nonlinearly with the density waves.
Furthermore, we demonstrate that the poloidal rolls are ubiquitous in
gravitoturbulence, emerging in the vicinity of GI spiral wakes, and potentially
transporting grains off the disk midplane. Other than hindering sedimentation
and planet formation, this phenomena may bear on observations of the disk's
scattered infrared luminosity. The vortical features could also impact on the
turbulent dynamo operating in young protoplanetary discs subject to GI, or
possibly even galactic discs.
