  We demonstrate via direct numerical simulations that a periodic, oscillating
mean flow spontaneously develops from turbulently generated internal waves. We
consider a minimal physical model where the fluid self-organizes in a
convective layer adjacent to a stably stratified one. Internal waves are
excited by turbulent convective motions, then nonlinearly interact to produce a
mean flow reversing on timescales much longer than the waves' period. Our
results demonstrate for the first time that the three-scale dynamics due to
convection, waves, and mean flow is generic and hence can occur in many
astrophysical and geophysical fluids. We discuss efforts to reproduce the mean
flow in reduced models, where the turbulence is bypassed. We demonstrate that
wave intermittency, resulting from the chaotic nature of convection, plays a
key role in the mean-flow dynamics, which thus cannot be captured using only
second-order statistics of the turbulent motions.
