Dynamics of rotating stably stratified flows

In the absence of preexisting organised eddies and without geometric constraints, a turbulent motion in a vertically stably stratified and rotating fluid is analysed as a combination of steady and oscillating normal modes. These modes correspond to Quasi-Geostrophic (QG) and Ageostrophic (AG) motions, the latter reflecting inertia-gravity dispersive waves. Linear and nonlinear dynamics axe then discussed; the amplitudes of the eigenmodes, which are kept constant in the inviscid linear, or RDT, limit, are treated as 'slowly' evolving variables, whose long-time evolution is governed by nonlinear triadic interactions. Statistical multimodal and anisotropic models of EDQNM-type ([10], [17]) are revisited, in close connection with recent Eulerian Wave-Turbulence theories for weak interaction. Emphasis is placed on the case of pure rotation, in which the inertial wave modes form a complete basis for the velocity field (no QG contribution, actually pure wave-turbulence at small Rossby number). Pure inviscid linear dynamics, even if irrelevant for predicting the rise of columnar structures, yield interesting new results for diffusivity of rapidly rotating turbulence, if two-time velocity correlations are calculated and time-integrated [8], according to [27]. Partial two-dimensionalisation, however, is only triggered by nonlinear interactions, with a dominant role of triadic resonances. In the general case with stable stratification, the role of the QG mode appears to be pivotal; it is responsible for the dominantly horizontal diffusivity, that can be analysed in the linear limit, and it drives the energy and anisotropy transfers mediated by nonlinear interactions, at least if its energy is initially significant at large scale. At dominant stratification, the nonlinear tendency to create the layering, or pancake structuring, is explained by a wave-released spectral transfer, which tends to concentrate the kinetic energy towards vertical wave-vectors (this limit correspond to the Vertically Sheared Horizontal Flow, very different from the two-dimensional, or baxotropic, limit, which is concerned in pure rotation). Comparing different studies, including recent DNS [46] and stability analyses, a new insight is given to nonlinear dynamics, including both angular energy drain and inverse cascades in Fourier space, with typical eigenmodes called into play.