Focusing of inertial waves by a vertically annular forcing

Abstract

A theory of inertial wave focusing generated by a vertically oscillating slender torus immersed in a uniformly rotating fluid is presented. The analytical solution of the velocity field shows that, under an axisymmetric annular forcing in inviscid rotating fluid, the wave rays form a double cone symmetric about the plane on which the torus is located. At the two vertices of the double cone, the waves are in a shock-like manner focus causing localized surges of energy. After focusing, the waves continue their propagation and form a new inverted cone with the same cone angle, such that both cones are symmetric about the focal point. These results are in good agreement with the experimental and numerical study by Duran-Matute et al. [Phys. Rev. E 87, 041001(R) (2013)]. When friction effects occur, the wave pattern changes substantially and the wave away from the focal point is significantly attenuated so that the symmetry about the focal point is broken. As a consequence, the wave beam widens and the focusing effect becomes weaker with increasing Ekman number ( Ek), which indicates the ratio of the viscous force to the Coriolis force. Furthermore, for the same [Formula: see text], the focusing effect tends to disappear when the forcing frequency is close to zero or twice the angular velocity of rotating flow. For forcing frequency close to the angular velocity of a rotating fluid, the amplitude of the vertical velocity at the focal point reaches its maximum, which corresponds to a wave propagation angle of 60 degrees.