Bound on the slope of steady water waves with favorable vorticity

TL;DR

This paper establishes an upper bound of 45 degrees on the inclination angle of steady water waves with favorable vorticity, extending understanding of wave overturning limits beyond irrotational cases.

Contribution

It proves a new upper bound on wave inclination angles for a broad class of waves with favorable vorticity, including constant vorticity, and derives pressure inequalities related to wave overturning.

Findings

Maximum inclination angle is 45 degrees for waves with favorable vorticity.

Overturning waves must have a pressure sink within the fluid.

The result applies to waves with constant vorticity of favorable sign.

Abstract

We consider the angle $\theta$ of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. Although $\theta$ may surpass 30$^\circ$ for some irrotational waves close to the extreme wave, Amick [Ami87] proved that for any irrotational wave the angle must be less than 31.15$^\circ$. Is the situation similar for periodic or solitary waves that are not irrotational? The extreme Gerstner wave has infinite depth, adverse vorticity and vertical cusps ($\theta = 90^\circ$). Moreover, numerical calculations show that even waves of finite depth can overturn if the vorticity is adverse. In this paper, on the other hand, we prove an upper bound of 45$^\circ$ on $\theta$ for a large class of waves with favorable vorticity and finite depth. In particular, the vorticity can be any constant with the favorable sign. We also prove a series of general inequalities on the pressure within the fluid, including the fact that any overturning wave must have a pressure sink.