Focusing of weakly nonlinear inertial waves by a vertically oscillating annular forcing

Abstract

We investigate the focusing phenomenon of inertial waves generated by a vertically oscillating slender torus in a uniformly rotating fluid. Building on the previous research on linear aspects of inertial wave focusing by Liu et al. [“Focusing of inertial waves by a vertically annular forcing,” Phys. Fluids 34, 086601 (2022)], we analyze the focusing phenomenon in a fluid with a small Rossby number using the perturbation method. Our analytical solution to the weakly nonlinear problem reveals that, in addition to the primary inertial waves predicted by the linear theory, secondary inertial waves (SIW) are excited due to wave interaction if the forcing frequency is lower than the fluid rotation rate. Particularly, in our case, the SIWs induced directly by the oscillating torus form a relatively weak secondary focal area on the rotation axis below the primary focal point. The rays of the SIWs form a new double cone with a different angle than the primary cone. In addition, the SIWs are also excited at the primary focal point with a much stronger effect than that generated by the oscillating torus in an inviscid limit, but significantly suppressed even in low-viscosity fluids. A fully nonlinear analysis, especially in the primary focal zone, is necessary for further study.