Inertial Waves Above an Obstacle in an Unbounded, Rapidly Rotating Fluid

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 = ν/2Ω L^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.