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 is an element of [4 x 10(7); 10(9)]) and large stratification (Brunt-Vaisala frequencies f(N) >> 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(perpendicular to)(4) f(-13/2) for weakly damped waves (with k(perpendicular to) 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).