Near-inertial waves in the ocean: beyond the ‘traditional approximation’

Abstract

The dynamics of linear internal waves in the ocean is analysed without adopting the ‘traditional approximation’, i.e. the horizontal component of the Earth's rotation is taken into account. It is shown that non-traditional effects profoundly change the dynamics of near-inertial waves in a vertically confined ocean. The partial differential equation describing linear internal-wave propagation can no longer be solved by separation of spatial variables; it was however pointed out earlier in the literature that a reduction to a Sturm–Liouville problem is still possible, a line that is pursued here. In its formal structure the Sturm–Liouville problem is the same as under the traditional approximation, but its eigenfunctions are no longer normal vertical modes of the full problem. The question is addressed of whether the solution found through this reduction is the general one: a set of eigenfunctions to the full problem is constructed, which depend in a non-separable way on the two spatial variables; these functions are orthogonal and form, under mild assumptions, a complete basis.In the near-inertial range, non-traditional effects act as a singular perturbation; this is seen from the sub-inertial short-wave limit, which is present whenever the ‘non-traditional’ terms are there, but disappears under the traditional approximation. In the dispersion relation the sub-inertial modes represent a smooth continuation of the super-inertial ones. The combined effect of the horizontal component of rotation and a vertical inhomogeneity in the stratification is found to play a crucial role in the dynamics of sub-inertial waves. They are trapped in waveguides localized around minima of the buoyancy frequency. The presence of horizontal inhomogeneities in the effective Coriolis parameter (such as shear currents or beta effect) are shown to enable a transition from super-inertial to sub-inertial waves (and thus effectively an irreversible transformation of large-scale into small-scale motions). It is suggested that this transformation provides a mechanism for mixing in the deep ocean.The notion of critical reflection of internal waves at a sloping bottom is also modified by non-traditional effects, and they strongly increase the probability of critical reflection in the near-inertial to tidal range.