We explore the strong stratification regime of stably stratified turbulence and the intermediate regime towards the viscosity-affected stratified state. Three-dimensional velocity-density fields from direct numerical simulations are decomposed into internal gravity waves (IGWs) and eddy motion based on Riley's decomposition (Riley et al., AIP Conf. Proc., vol. 76, issue 1, 1981, pp. 79–112) extended to account for the space–time properties of waves, their modification by vertically sheared horizontal flow and the vertical mixing by eddies (Lam et al., Atmosphere, vol. 11, issue 4, 2020, p. 420). We establish the evolution equations for the IGW and eddy parts separately. Up to buoyancy Reynolds number considered, the mixing due to eddies is four times that due to waves.

We consider the separation of motion related to internal gravity waves and eddy dynamics in stably stratified flows obtained by direct numerical simulations. The waves’ dispersion relation links their angle of propagation to the vertical $\theta$ , to their frequency $\omega$ , so that two methods are used for characterizing wave-related motion: (a) the concentration of kinetic energy density in the $(\theta ,\omega )$ map along the dispersion relation curve; and (b) a direct computation of two-point two-time velocity correlations via a four-dimensional Fourier transform, permitting to extract wave-related space-time coherence. The second method is more computationally demanding than the first. In canonical flows with linear kinematics produced by space-localized harmonic forcing, we observe the pattern of the waves in physical space and the corresponding concentration curve of energy in the $(\theta ,\omega )$ plane. We show from a simple laminar flow that the curve characterizing the presence of waves is distorted differently in the presence of a background convective mean velocity, either uniform or varying in space, and also when the forcing source is moving. By generalizing the observation from laminar flow to turbulent flow, this permits categorizing the energy concentration pattern of the waves in complex flows, thus enabling the identification of wave-related motion in a general turbulent flow with stable stratification. The advanced method (b) is finally used to compute the wave-eddy partition in the velocity–buoyancy fields of direct numerical simulations of stably stratified turbulence. In particular, we use this splitting in statistics as varied as horizontal and vertical kinetic energy, as well as two-point velocity and buoyancy spectra.

Direct numerical simulations are performed in rotating turbulence for different regimes at various Rossby and inertial Reynolds numbers ( $\textit {Re}_I$ ). A new algorithm, adapted from stratified turbulence (Lam et al., J. Fluid Mech., vol. 923, 2021, A31) to rotating turbulence, permits to separate the three-dimensional velocity field into three parts: inertial waves (IWs), eddies and a geostrophic mode (GM). It uses the space–time properties of waves and their advection by the GM to filter the IWs from the rest of the motion. We obtain balance equations for the separate energies of waves, eddies and the GM. Their mutual interactions are evaluated and analysed via Sankey diagrams that provide a global picture of energy exchanges. When the flow is forced at large scale, it mainly feeds the wave part and the multiple interactions lead to energy dissipation in eddy and GM motion. We also show that, in addition to the wave/wave interaction that feeds the GM, corresponding to different mechanisms described in the literature, other non-documented interactions feed it, as the eddy/wave interaction or the eddy/eddy interaction at moderate $\textit {Re}_I$ . We propose a scale-by-scale analysis of the transfer to the GM: we show that transfers from wave or eddy occur at large scale, that they either inject or remove energy, and that this occurs with or without direct cascade depending on the kind of interaction, wave/wave, eddy/wave or eddy/eddy. The self-interaction of the GM is an inverse cascade for its horizontal component, shaping it into a very large-scale flow.

Séparation des ondes et des tourbillons dans des écoulements stratifiés ou en rotation: We propose a new separation technique of internal gravity waves (resp. inertial waves) and eddies in stratified (resp. rotating) turbulence. The separation is based on the dispersion relation of waves modified by the advection of a vertically sheared horizontal flow (resp. the geostrophic mode) called sweeping effect. Different regimes are studied with a low Froude number Fr<<1 (resp. Rossby number Ro<<1) but with a varying buoyancy Reynolds number Reb (resp. inertial Reynolds number ReI). We observe that the distribution of energy between waves and eddies follow the Fr (resp. Ro) number. We establish the evolution equations for the waves and eddies parts separately. Generally, we observe a large transfer of energy from waves to eddies. We also observe in the rotating case that it is mostly waves that transfer energy to the geostrophic mode. A few inverse cascades are observed for particular types of transfers. The dissipation and mixing due to waves and eddies against the different parameters are calculated. Finally, 2D velocity fields are decomposed into their wave and eddy parts.