# Spontaneous critical layer formation and robustness beneath rotational waves

## TL;DR

This paper investigates the formation and robustness of critical layers beneath rotational waves with constant vorticity, revealing how Kelvin cat eye structures adapt to bottom topography and spontaneously form under certain conditions.

## Contribution

It introduces a non-stationary framework for analyzing rotational waves, demonstrating spontaneous Kelvin cat eye formation and their robustness over complex bottom topography.

## Key findings

- Kelvin cat eye structures adjust dynamically over topography.
- Spontaneous formation of Kelvin cat eye occurs under pressure forcing.
- Critical layers exhibit a new form with connected stagnation points.

## Abstract

Non-stationary rotational surface waves are considered, where the underlying current has constant vorticity. A study is presented on the robustness of a critical layer in the presence of a bottom topography, as well as on its spontaneous formation for waves generated from rest. The restriction, from previous studies, to a traveling-wave formulation is removed leading to a non-stationary set of equations. In this setting streamlines are not necessarily pathlines. Particle-trajectories are found evolving the respective submarine dynamical system with a cloud of tracers. Pathlines are then visualized and the respective submarine structures identified. Robustness is illustrated through surface waves interacting with topographic undulations. The respective Kelvin cat eye structure dynamically adjusts itself over the bottom topography without loosing its integrity. On the spontaneous formation of a Kelvin cat eye structure, the surface is initially undisturbed and waves are generated from either the current-topography interaction or by a surface pressure distribution suddenly imposed. Under the pressure forcing, an isolated Kelvin cat eye spontaneously forms. The two extreme critical points of the cat eye structure are connected with a stagnation segment, thus exhibiting a new form of a critical layer.