Gravity Water Waves over Constant Vorticity Flows: From Laminar Flows to Touching Waves

Area of Science:

• Fluid dynamics
• Nonlinear wave phenomena

Background:

• Previous work established periodic steady water waves in infinitely deep, constant vorticity flows.
• These solutions included overhanging profiles and surfaces touching at a point, enclosing air bubbles.

Purpose of the Study:

• Extend existing results to include both infinitely and finitely deep water flows.
• Prove the existence of a continuous curve of water waves connecting laminar flow to touching waves.
• Investigate the behavior of critical layers near the surface.

Main Methods:

• Formulation of a mathematical problem encompassing both finite and infinite depths.
• Proof of existence for a continuous curve of water waves.
• Analysis of critical layers where horizontal velocity vanishes.

Main Results:

• Existence of a continuous curve of water waves connecting laminar flow to touching waves for fixed, nonzero gravity.
• Demonstration of a wave profile with a vertical tangent (breaking wave), which is not overhanging.
• The results hold for arbitrary vorticity and include analysis of critical layers.

Conclusions:

• The study successfully extends the understanding of water wave existence to finite depths and introduces the concept of breaking waves.
• The findings are applicable to a wide range of fluid flows, including those with arbitrary vorticity.
• The research provides new insights into the complex behavior of water waves and their properties.