Rotational flow underlying coupled surface and internal waves. I: Eulerian perspective

TL;DR

This paper investigates the complex flow patterns caused by coupled surface and internal water waves in a two-fluid system, highlighting the effects of vorticity and nonlinear dynamics on streamline behavior.

Contribution

It introduces a phase-plane analysis approach to understand the qualitative behavior of flow in a two-layer fluid with rotational and irrotational regions, revealing phenomena like critical layers and stagnation points.

Findings

Identification of critical layers and stagnation points in the flow.

Analysis of streamline behavior across different wave and vorticity regimes.

Demonstration of nonlinear dynamical systems governing linear wave motion.

Abstract

In this paper we examine the flow generated by coupled surface and internal small-amplitude water waves in a two-fluid layer model, where we take the upper layer to be rotational (constant vorticity) and the lower layer to be irrotational. The presence of vorticity greatly complicates the underlying analysis, yet it generates a rich array of otherwise unobservable phenomena such as the presence of critical layers, and stagnation points, in the fluid interior. We employ a phase-plane analysis to elucidate the qualitative behaviour of streamlines for a range of different coupled-wave, and vorticity, regimes. Although the water waves considered are linear in the fluid dynamics sense, the dynamical systems which govern their motion are nonlinear.