On the vorticity threshold for steady water waves

Abstract

This paper presents a comprehensive analysis of two-dimensional water waves characterized by a significant adverse constant vorticity over flows without stagnation points. Surprisingly, we discover qualitative distinctions between this scenario and those featuring favorable vorticity. Our findings reveal that in the presence of large adverse vorticity, neither Stokes waves nor solitary waves can approach extreme waves exhibiting surface stagnation points. Conversely, we demonstrate that global bifurcation curves always lead to unidirectional Stokes and solitary waves with precisely one stagnation point located at the bottom right beneath the crest. In contrast, it is known that any favorable or even small adverse vorticity eliminates the possibility of bottom stagnation, leading to extreme waves exhibiting surface singularities. Additionally, we establish several novel facts, including new bounds for both amplitude and the Froude number.