  Motivated by observations of turbulence in the strongly stratified ocean
thermocline, we use direct numerical simulations to investigate the interaction
of a sinusoidal shear flow and a large-amplitude internal gravity wave. Despite
strong nonlinearities in the flow and a lack of scale separation, we find that
linear ray tracing theory is qualitatively useful in describing the early
development of the flow as the wave is refracted by the shear. Consistent with
the linear theory, the energy of the wave accumulates in regions of negative
mean shear where we observe evidence of convective and shear instabilities.
Streamwise-aligned convective rolls emerge the fastest, but their contribution
to irreversible mixing is dwarfed by shear-driven billow structures that
develop later. Although the wave strongly distorts the buoyancy field on which
these billows develop, the mixing efficiency of the subsequent turbulence is
similar to that arising from Kelvin-Helmholtz instability in a stratified shear
layer. We run simulations at Reynolds numbers of 5000 and 8000, and vary the
initial amplitude of the internal gravity wave. For high values of initial wave
amplitude, the results are qualitatively independent of $Re$. Smaller initial
wave amplitudes delay the onset of the instabilities, and allow for significant
laminar diffusion of the internal wave, leading to reduced turbulent activity.
We discuss the complex interaction between the mean flow, internal gravity wave
and turbulence, and its implications for internal wave-driven mixing in the
ocean.
