Stratified turbulence generated by internal gravity waves

We present numerical simulations of randomly forced internal gravity waves in a uniformly stratified Boussinesq fluid, and compare the resulting vertical wavenumber energy spectra with the saturation spectrum Ez(kz)\,=\,c\,N²kz⁻³ ( N is the Brunt–Väisälä frequency) observed in the atmosphere and ocean. Overall, we have been unsuccessful at reproducing the observed spectrum in our simulations. Our spectra are shallower than kz⁻³ , although they steepen towards it with increasing stratification as long as wave breaking (in the form of static instability) is resolved. The spectral amplitude increases like N1.1 rather than N² . For a single stratification, our spectrum agrees well with the saturation spectrum with c\,=\,0.1 , but only because it is spuriously steepened by insufficient resolution. We show that overturning occurs when the length scale lc\,=\,uᵣₘₛ/N is larger than the dissipation scale, where uᵣₘₛ is the root mean square velocity. This scale must be at least three times larger than the dissipation scale for the energy spectrum to be independent of Reynolds number in our simulations. When this condition is not satisfied, the computed energy spectrum must be interpreted with caution. Finally, we show that for strong stratifications, the presence of vortical energy can have a dramatic effect on the spectrum of wave energy due to the efficiency of interactions between two waves and a vortical mode. Any explanation of the energy spectrum involving resonant interactions must take into account the effect of vortical motion.