  The presence of a no-slip, impermeable, adiabatic, sloped boundary in an
otherwise quiescent, stably stratified, Boussinesq flow generates baroclinic
vorticity within a diffusive boundary layer. Such conditions are typical of the
oscillating boundary layers on adiabatic abyssal slopes, sloped lake
bathymetry, and sloped coastal bathymetry in the absence of high-wavenumber
internal waves, mean flows, far-field turbulence on larger scales, and resonant
tidal-bathymetric interaction. We investigate the linear stability of the
oscillating flow within non-dimensional parameter space typical of the $M_2$
tide and hydraulically smooth, mid-latitude abyssal slopes through Floquet
linear stability analysis. The flow dynamics depend on three non-dimensional
variables: the Reynolds number for Stokes' second problem (Re), the Prandtl
number, and a frequency ratio that accounts for the resonance conditions (C,
criticality) of the buoyant restoring force and the tidal forcing. The Floquet
analysis results suggest that oscillating laminar boundary layers on adiabatic
abyssal slopes are increasingly unstable as Reynolds number, criticality
parameter, and/or spanwise disturbance wavenumber are increased. We also show
that the two-dimensional Floquet linear instability necessarily generates
three-dimensional baroclinic vorticity, which suggests that the evolution of
the gravitational instabilities may be nonlinear as $t\rightarrow\infty$.
