On gravito-inertial surface waves

TL;DR

This paper provides a geometric framework for understanding gravito-inertial surface waves in rotating, stratified fluids within bounded domains, highlighting boundary concentration and attractor phenomena.

Contribution

It introduces a novel geometric description of gravito-inertial surface waves, including boundary behavior and the role of domain shape, using a reduced pseudo-differential boundary equation.

Findings

Wave energy concentrates on the boundary for large covectors.

Surface wave attractors can form in generic domains.

In ellipsoids, waves reduce to spherical harmonics and are square-integrable.

Abstract

In geophysical environments, wave motions that are shaped by the action of gravity and global rotation bear the name of gravito-inertial waves. We present a geometrical description of gravito-inertial surface waves, which are low-frequency waves existing in the presence of a solid boundary. We consider an idealized fluid model for an incompressible fluid enclosed in a smooth compact three-dimensional domain, subject to a constant rotation vector. The fluid is also stratified in density under a constant Brunt-V\"ais\"al\"a frequency. The spectral problem is formulated in terms of the pressure, which satisfies a Poincar\'e equation within the domain, and a Kelvin equation on the boundary. The Poincar\'e equation is elliptic when the wave frequency is small enough, such that we can use the Dirichlet-to-Neumann operator to reduce the Kelvin equation to a pseudo-differential equation on the boundary. We find that the wave energy is concentrated on the boundary for large covectors, and can exhibit surface wave attractors for generic domains. In an ellipsoid, we show that these waves are square-integrable and reduce to spherical harmonics on the boundary.