Shock-like focusing of inertial waves - the localized generation of turbulence

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This project aims at developing a theory on the evolution of an initially axisymmetric rotating flow containing conical inertial waves that emerge from a vibrating torus and meet in a focal point. The understanding of this archetypal flow raises several questions pertaining to its topology and dynamics. We will address these questions by combining symmetry group theory and numerical simulations, progressing from the simpler linear model to the more complex turbulent one. First, the symmetry properties of the flow will be exhaustively understood for low forcing amplitudes for which linear wave propagation occurs. Then, this will permit to tackle the weakly nonlinear regime at increasing wave amplitude, where a local shock-like phenomenon occurs at the focal point of the inertial waves. This triggers complex energy transfers between waves but also a yet to be explained transfer to large-scale motion. The analytic basis will be an equation derived from the Euler equation using singular asymptotics in the limit of high rotation rates and valid for large wave amplitudes. The structural symmetry breaking will be explained by a combination of stability theory and group theory, which we plan to relate to dynamical arguments drawn from a statistical analysis of triadic interactions of inertial waves. In addition to this system approach, we will study local phenomena, such as the mechanism limiting the core of non-linearity to a localized region in the flow. After the comprehensive study of the wave-turbulence regime, we will consider the fully turbulent regime in which nonlinearities are strong so that transfers in the flow are mediated by both inertial waves exchanges and by classical turbulent ones, thus producing more complex couplings. Our original approach will be to perform the symmetry analysis of two-point statistical equations, and to relate this to the isotropy-breaking in Direct Numerical Simulations with and without helicity. In the helical case, additional invariants have to be considered. The role of the specific geometry will also be evaluated by a parametric investigation of the cone-shaped inertial waves with and without confinement, also by Direct Numerical Simulations. The most original aspect of our project is thus to integrate in a single study a new theory based on the symmetries of rotating turbulent flows, and a dynamical point of view for anisotropic transfers between scales.