Evolution of weak, homogeneous turbulence subject to rotation and stratification: the weakly dispersive case

This article follows on from Scott & Cambon ( J. Fluid Mech. , vol. 979, 2024, A17) and Scott ( Phys. Rev. E , vol. 111, 2025, 035101). Like those articles, it concerns weak, decaying homogeneous turbulence in a rotating, stably stratified fluid with constant Brunt–Väisälä frequency, N . The difference is that here we consider the case in which β =2 /N is close to $1$ , where is the rotation rate. Because this renders inertial-gravity waves only weakly dispersive, wave-turbulence theory, which played a prominent role in the earlier studies, no longer applies. Indeed, wave-turbulence analysis does not appear here. Nonetheless, much of the analytical framework, based on modal decomposition, carries over, as do many of the conclusions. The flow is expressed as a sum of wave and non-propagating (NP) modes and their weak-turbulence mode-amplitude evolution equations are derived for small β -1 . The NP component is found to evolve independently of the wave one, following an amplitude equation which is precisely that of the previous studies in the limit β → 1 . The NP component induces coupling between wave modes and, without it, the wave component has purely linear decay. The mode-amplitude equations are integrated numerically using a scheme similar to that of classical direct numerical simulation and results given. We find an inverse energy cascade of the NP component, whereas the presence of that component induces a forward cascade, hence significant dissipation, of the wave component. Detailed results are given for the energy, energy spectra and energy fluxes of the two components.