Mathematical analysis of inertial waves in rectangular basins with one sloping boundary

TL;DR

This paper mathematically analyzes inertial waves in rectangular basins with sloping boundaries, discovering new solutions that expand understanding of wave types and energy transfer in geophysical fluid dynamics.

Contribution

It provides explicit solutions to the Poincaré-Sobolev equation in trapezoidal domains, revealing a new type of inertial wave associated with the continuous spectrum.

Findings

Existence of new inertial wave solutions with unique properties

These waves are progressive and differ from traditional inertial waves

Results help explain past geophysical experimental data

Abstract

Here we consider the problem of small oscillations of a rotating inviscid incompressible fluid. From a mathematical point of view, new exact solutions to the two-dimensional Poincar\'e-Sobolev equation in a class of domains including trapezoid are found in an explicit form and their main properties are described. These solutions correspond to the absolutely continuous spectrum of a linear operator that is associated with this system of equations. For specialists in Astrophysics and Geophysics the existence of these solutions signifies the existence of some previously unknown type of inertial waves corresponding to the continuous spectrum of inertial oscillations. A fundamental distinction between monochromatic inertial waves and waves of the new type is shown: usual characteristics (frequency, amplitude, wave vector, dispersion relation, direction of energy propagation, and so on) are not applicable to the last. Main properties of these waves are described. In particular it is proved that they are progressive. Main features of their energy transfer are described. The existence of such inertial waves enables us to explain in a new way a lot of experimental data that were obtained in Geophysics in the past two decades and to predict the occurrence of such oscillations in natural waters.