The transition to turbulence within internal tide boundary layers in the abyssal ocean.

The presence of a sloping boundary with no-slip, impermiable, and adiabatic boundary conditions in an otherwise quiescent, stably stratified, Boussinesq flow generates baroclinic vorticity within a diffusive boundary layer. Such are the conditions that may describe the frictional boundary layers that are generated as low wavenumber internal tides heave isopycnals up and down adiabatic abyssal slopes. We investigate the dynamics of the transition to turbulence of this flow within non-dimensional parameter space typical of the M$_2$ tide and smooth mid-latitude abyssal slopes by using a combination of analytical solutions, Floquet stability theory, and direct numerical simulations. The flow dynamics depend on four non-dimensional variables: the Reynolds number for Stokes' second problem, the Prandtl number, a frequency ratio that accounts for the resonance conditions (criticality) of the buoyant restoring force and the tidal forcing, and the Rossby number. We find that for unity Prandtl number, Reynolds numbers less than 1680 (approximately corresponding to tide amplitudes of less than 2 cm/s), the boundary layers are stabilized by the stratification during the downslope oscillation phase, both shear and gravitational instabilities can initiate turbulent bursts, that the presence of the Coriolis force is destabilizing for low slope Burger number flows on subcritical slopes, and that turbulent bursts occur even on far-from-critical slopes. The mixing efficiency of the intermittently turbulent boundary layers is small relative to that typical of internal wave breaking processes and the time-mean boundary layer stratification is increased. The results suggest that weakly stratified boundary layers observed on abyssal slopes arise from other sources of turbulent buoyancy flux divergence than boundary layer gravitational instabilities. However, even in the absence of small scale roughness and high wavenumber internal wave breaking, buoyancy flux divergence occurs within turbulent Stokes-Ekman layers on subcritical slopes that form in relatively low Reynolds number simulations, corresponding to tide amplitudes of approximately 1 cm/s.