Invariant and approximately invariant solutions of non-linear internal gravity waves forming a column of stratified fluid affected by the Earth's rotation

Abstract

The exact solutions of three-dimensional equations of motion for internal gravity waves in cylindrical coordinates in unbounded media are found by means of approximate transformation groups of equations with a small parameter. Introduction of the small parameter has been motivated by justifying the analogy of the Kelvin hypothesis on vanishing the component of the velocity ur normal to wall for the rectilinear motion. In the present case of the cylindrical domain, ur is non-zero and achieves its maximum in the interior, which also agrees with analytical predictions in [6]. However, as linear analysis shows, ur can be considered to be small in the limiting case when the aspect ratio σ=H/r0 is small, in which H and r0 are the basin's depth and radius respectively. As a particular applications to the ocean and atmospheric modeling, in terms of linear modeling, the time series of the energy density were visualized as spinning patterns that appear to be rotating in an anticlockwise sense when looking from above the North Pole. Such spinning patterns were compared with the flow around a low-pressure area that is usually being linked with a modeling of hurricanes. In terms of zeroth-order approximate transformations, the invariant solutions were visualized as funnels having something in common with the geometric structure of oceanic whirlpools.