Inertio-gravity waves under the non-traditional -plane approximation: singularity in the large-scale limit

Abstract

The low horizontal wavenumber limit of waves on a plane under a constant rotation rate (the so-called $f$-plane) is degenerate: all wave frequencies asymptotically approach the inertial frequency. This degeneracy has no serious consequence when the rotation axis is perpendicular to the plane (traditional $f$-plane approximation). However, when the rotation axis is tilted from the vertical direction (non-traditional $f$-plane approximation), we encounter a type of ‘singularity’ in the sense that each term of the Taylor expansion of the wave frequency in the horizontal Coriolis parameter diverges in the limit of low horizontal wavenumber. Such a drastic change of the solution behaviour by adding the horizontal Coriolis parameter in the low horizontal wavenumber limit is rather counter-intuitive, because the conventional scale analysis suggests that the horizontal Coriolis effect is negligible in this limit. However, the degeneracy of the system makes this effect critical with a need for considering higher-order terms. Two possible asymptotic limits are proposed for resolving this degeneracy. One of them, as it turns out, amounts to a representation of the wave frequency by a Taylor expansion in the horizontal wavenumber.