Focusing of inertial waves by a vertically oscillating annular forcing*

We investigate the focusing phenomenon of inertial waves generated by a vertically oscillating slender torus in a uniformly rotating fluid. Based on the linearized Navier-Stokes equations, we obtain the analytical solution of the velocity field. It is shown that, under the axisymmetric annular forcing, the wave rays form a double cone symmetric about the plane on which the torus is located. At the vertex of the cone, the waves are focused in a shock-like manner, causing localized energy surges. After focusing, the waves continue their propagation and form a new inverted cone with the same cone angle. These purely theoretical results are in good agreement with the experimental and numerical study by M. Duran-Matute et al. (Phys. Rev. E87, 041001(R) 2013). Furthermore, when friction effects occur, the wave away from the focal point is significantly attenuated so that the symmetry about the focal point is broken. Under the same viscous condition, the amplitude at the focal point reaches its maximum at a forcing frequency close to the angular velocity of the rotating flow. Additionally, a preliminary analysis of the nonlinear problem shows that inertial waves with higher frequency appear and affect the energy transfer at the focal point.