The energy flux spectrum of internal waves generated by turbulent convection

We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ( $Ra\in [4\times 10^{7},10^{9}]$ ) and large stratification (Brunt–Väisälä frequencies $f_{N}\gg f_{c}$ , where $f_{c}$ is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like $k_{\bot }^{4}\,f^{-13/2}$ for weakly damped waves (with $k_{\bot }$ and $f$ the waves’ horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with $z$ , the distance from the convective region, like $z^{-13/8}$ .