Characterization of singular flows of zeroth-order pseudo-differential operators via elliptic eigenfunctions: a numerical study

TL;DR

This paper investigates the formation of wave attractors in stratified fluids through numerical analysis of zeroth-order pseudo-differential operators, revealing how embedded eigenmodes influence long-term fluid behavior.

Contribution

It introduces a high-order pseudo-spectral numerical method to analyze eigenvalues within continuous spectra of nonlocal operators related to wave attractors.

Findings

Eigenvalues are computed using viscous approximations.

Embedded eigenmodes significantly affect the system's long-term evolution.

Numerical methods effectively analyze singular flow phenomena.

Abstract

The propagation of internal gravity waves in stratified media, such as those found in ocean basins and lakes, leads to the development of geometrical patterns called "attractors". These structures accumulate much of the wave energy and make the fluid flow highly singular. In more analytical terms, the cause of this phenomenon has been attributed to the presence of a continuous spectrum in some nonlocal zeroth-order pseudo-differential operators. In this work, we analyze the generation of these attractors from a numerical analysis perspective. First, we propose a high-order pseudo-spectral method to solve the evolution problem (whose long-term behaviour is known to be not square-integrable). Then, we use similar tools to discretize the corresponding eigenvalue problem. Since the eigenvalues are embedded in a continuous spectrum, we compute them using viscous approximations. Finally, we explore the effect that the embedded eigenmodes have on the long-term evolution of the system.