Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids

Rotating fluids support waves. These inertial waves propagate obliquely through the fluid, with an angle that is fixed with respect to the rotation axis. Upon reflection, their wavelength is unchanged only when the wall obeys the local reflectional symmetry, that is, when it is either parallel or perpendicular to the rotation axis. For internal gravity waves in a density-stratified fluid, sloping boundaries thus break the symmetry of ray paths, in a two-dimensional container, predicting their focusing upon attractors: particular paths onto which the wave rays, and hence the energy, converge, and to which the wave energy returns after a small number of refections. Laboratory observations, presented here, show that, despite the intrinsic three-dimensionality of inertial waves, attractors still occur. The intensified wave energy on the attractor encourages centrifugal instabilities, leading to a mean flow. Evidence of this comes from dye spreading, observed to develop most rapidly over the location where the attractor reflects from the sloping wall, being the place where focusing and instabilities occur. This mean flow, resulting from the mixing of angular momentum, accompanying the intensification of the wave field at that location, has geophysical implications, because the ocean, atmosphere and Earth's liquid outer core can be regarded as asymmetrically contained. The relevance of wave focusing in a rotating, spherical shell, the modifications due to the addition of radial stratification, and its implications for observed equatorial current patterns and inertial oscillations are discussed. The well-known universality of oceanic, gravito-inertial wave spectra might reflect complementary, divergent (chaotic) wave-ray behaviour, which occurs in containers obeying the reflectional symmetry, but in which symmetry is broken in the horizontal plane. Periodic orbits still exist, but now repell.