  We investigate the stability of stratified fluid layers undergoing
homogeneous and periodic tidal deformation. We first introduce a local model
which allows to study velocity and buoyancy fluctuations in a Lagrangian domain
periodically stretched and sheared by the tidal base flow. While keeping the
key physical ingredients only, such a model is efficient to simulate planetary
regimes where tidal amplitudes and dissipation are small. With this model, we
prove that tidal flows are able to drive parametric subharmonic resonances of
internal waves, in a way reminiscent of the elliptical instability in rotating
fluids. The growth rates computed via Direct Numerical Simulations (DNS) are in
very good agreement with WKB analysis and Floquet theory. We also investigate
the turbulence driven by this instability mechanism. With spatio-temporal
analysis, we show that it is a weak internal wave turbulence occurring at small
Froude and buoyancy Reynolds numbers. When the gap between the excitation and
the Brunt-V\"ais\"al\"a frequencies is increased, the frequency spectrum of
this wave turbulence displays a -2 power law reminiscent of the high-frequency
branch of the Garett and Munk spectrum (Garrett & Munk 1979) which has been
measured in the oceans. In addition, we find that the mixing efficiency is
altered compared to what is computed in the context of DNS of stratified
turbulence excited at small Froude and large buoyancy Reynolds numbers and is
consistent with a superposition of waves.
