Internal wave attractors and spectra of zeroth-order pseudo-differential operators

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. Microlocal analysts have recently related this behaviour to the spectral properties of an underlying nonlocal zeroth-order pseudo-differential operator that characterizes the dynamics of this problem. In this work, we analyze this phenomenon 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 non-square-integrable. Then, we use similar tools to discretize the corresponding eigenvalue problem. Since the eigenvalues are embedded in a continuous spectrum, their computation is based on viscous approximations. Finally, we explore the effect that the embedded eigenmodes have in the long-term evolution of the system.