Linear boundary layer analysis of the near-critical reflection of internal gravity waves with different sizes of viscosity and diffusivity

TL;DR

This paper investigates the near-critical reflection of internal gravity waves from a slope, analyzing how different scales of viscosity and diffusivity influence boundary layer behavior and providing stable approximate solutions under specific scaling laws.

Contribution

It extends boundary layer analysis to cases with differing viscosity and diffusivity sizes, offering a systematic characterization and stable solutions for the linear reflection problem.

Findings

Boundary layer decay and size depend on viscosity and diffusivity scales.

Constructed an $L^2$ stable approximate solution under specific scaling laws.

Analyzed the impact of different viscosity and diffusivity magnitudes on wave reflection.

Abstract

The aim of this work is to make a further step towards the understanding of the near-critical reflection of internal gravity waves from a slope in the more general and realistic context where the size of viscosity $\nu$ and the size of diffusivity $\kappa$ are different. In particular, we provide a systematic characterization of boundary layers (boundary layer wave packets) decays and sizes depending on the order of magnitude of viscosity and diffusivity. We can construct an $L^2$ stable approximate solution to the linear near-critical reflection problem under the scaling assumption of Dauxois \& Young JFM 1999, where either viscosity of diffusivity satisfies a precise scaling law in terms of the criticality parameter.