Maximization of inertial waves focusing in linear and nonlinear regimes

We study the propagation of inertial waves (IWs) generated by an axisymmetric torus oscillating at frequency ${\omega }_f$ in a rotating fluid. Inertial waves are emitted from the torus and propagate at an angle ${\theta }_f$ that depends on the ratio of the rotation frequency of the fluid to the forcing frequency of the torus. The waves focus in a neighborhood of the apex of the propagation cone. Using direct numerical simulations, we characterize the flow in this region, within a linear approximation or in the regime where nonlinear interactions between waves produce a turbulent patch. Forcing by the torus is modeled in two ways. The first model represents the effect of the oscillating torus as a local volume force in the form of a Dirac delta function, called the Dirac ring. The second approach aims at a more realistic three-dimensional model of a torus represented by a volume penalization technique. We observe the appearance of a mean flow composed of a central vortex produced by the nonlinear interaction of the IWs. We show that this phenomenon is in agreement with the theory of Davidson et al. [J. Fluid Mech. 557, 135 (2006)] for a rotating fluid. Using Dirac ring forcing in the linear regime, we obtain the dependence on the propagation angle of the vertical kinetic energy at the focal point, which reaches a maximum for ${\theta }_f={35}^{\circ }$, in agreement with the linear theory developed by Liu et al. [Phys. Fluids 34, 086601 (2022)]. A similar angle is observed in the 3D torus forcing case for both linear and nonlinear simulations: the angle ${\theta }_f={30}^{\circ }$ maximizes the vertical velocity and dissipation, attesting an optimal energy transfer from the oscillating source to the focal region. In the nonlinear regime, we obtain the detailed spectral distribution of the kinetic energy in the focal zone, and we develop a spatiotemporal analysis of the velocity field that shows a wide presence of IWs in the flow. Moreover, we identify triadic resonances of IWs that lead to the production of the turbulent patch and of a large-scale mode similar to the geostrophic mean flow.