We investigate the stability of stratified fluid layers undergoinghomogeneous and periodic tidal deformation. We first introduce a local modelwhich allows to study velocity and buoyancy fluctuations in a Lagrangian domainperiodically stretched and sheared by the tidal base flow. While keeping thekey physical ingredients only, such a model is efficient to simulate planetaryregimes where tidal amplitudes and dissipation are small. With this model, weprove that tidal flows are able to drive parametric subharmonic resonances ofinternal waves, in a way reminiscent of the elliptical instability in rotatingfluids. The growth rates computed via Direct Numerical Simulations (DNS) are invery good agreement with WKB analysis and Floquet theory. We also investigatethe turbulence driven by this instability mechanism. With spatio-temporalanalysis, we show that it is a weak internal wave turbulence occurring at smallFroude and buoyancy Reynolds numbers. When the gap between the excitation andthe Brunt-V\"ais\"al\"a frequencies is increased, the frequency spectrum ofthis wave turbulence displays a -2 power law reminiscent of the high-frequencybranch of the Garett and Munk spectrum (Garrett & Munk 1979) which has beenmeasured in the oceans. In addition, we find that the mixing efficiency isaltered compared to what is computed in the context of DNS of stratifiedturbulence excited at small Froude and large buoyancy Reynolds numbers and isconsistent with a superposition of waves.
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