Inertial waves above an obstacle in an unbounded, rapidly rotating fluid

Abstract

The slow transverse motion of an obstacle of horizontal dimension L , in a rapidly rotating flow (such that the Rossby number, Ro , is small), is shown to cause an inertial-wave disturbance above and behind the obstacle. This disturbance spreads over an ever-increasing downstream region with increasing vertical distance above the obstacle, but is constrained laterally to lie within a wedge-shaped region. The discussion of Cheng (1977) is shown to give the diffraction pattern due to a point source, and his analysis is modified to allow for arbitrary obstacle shapes. Such a modification shows that the amplitude of the disturbance decays downstream at a rate deter­mined by the Fourier transform of the obstacle shape as noted by Johnson (1982) for ridge-like topography. The inclusion of viscosity is shown to damp out the disturbance on a vertical scale of L/Ek (where Ek = v /2 ΩL 2 is an Ekman number for the flow). At moderate Reynolds numbers visco­sity reduces the amplitude of the disturbance in the neighbourhood of the caustic at the wedge boundary, the attenuation increasing with distance downstream. Internal dissipation removes the short-wavelength com­ponents of the disturbance and thus removes the strong dependence of the wave pattern on the precise shape of the obstacle.