Internal gravity waves propagating within homogeneous stratified turbulence are the subject of the present study. A spatiotemporal analysis is carried out on the results of direct numerical simulations including a forcing term, with the aim of showing the energy content of the simulations as a function of frequency, $\omega$, and wave-vector inclination to the horizontal, $\theta$. Clear signatures of the dispersion relation of internal gravity waves, $\omega =\pm Ncos\theta$, where $N$ is the Brunt-Väisälä frequency, are observed in all our simulations, which have low Froude number, ${\text{Fr}}_h\ll 1$, and increasing buoyancy Reynolds number up to ${\text{Re}}_b\approx 10$. Interestingly, we observe the presence of high-frequency waves with $\omega \sim N$ and a corresponding low-frequency vortex mode, both containing a non-negligible amount of energy. These waves are large-scale waves, their energy signature being found at scales larger than the forcing scales. We also observe the growth of energy in the shear modes, constituting a horizontal mean flow, and we show that their continuous growth is due to an upscale energy transfer, from the forcing scales to larger horizontal as well as vertical scales. These shear modes are found to be responsible for Doppler shifting the frequency of the large-scale waves. When considering the wave energy across the simulations at varying ${\text{Re}}_b$, such energy is seen to reduce as ${\text{Re}}_b$ is increased and the flow enters the strongly stratified turbulence regime. The classical wave-vortex decomposition, based on a purely spatial decomposition of instantaneous snapshots of the flow, is analyzed within the current framework and is seen to correspond relatively well to the “true” wave signal identified by the spatiotemporal analysis, at least for the large-scale waves with $\omega \sim N$. Distinct energy peaks in $\theta \text{−}\omega$ space highlight that the waves have preferential directions of propagation, specifically $\theta ={45}^{\circ }$ and $\theta \approx {55}^{\circ }$, similar to observations in studies of wave radiation from localized regions of turbulence. This suggests that the same wave-generation mechanisms may be relevant for homogeneous and inhomogeneous stratified turbulent flows.

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• Hydrodynamic waves
• Stratified geophysical flows
• Turbulence