Kinetic energy properties of irrotational deep-water Stokes waves

TL;DR

This paper analyzes the kinetic energy distribution and properties of irrotational deep-water Stokes waves using complex analysis and conformal mappings, revealing energy convexity, streamline behavior, and particle drift characteristics.

Contribution

It introduces a conformal hodograph transform to study kinetic energy properties and proves new qualitative results about energy convexity and streamline dynamics in deep-water Stokes waves.

Findings

Kinetic energy integral means are convex.

Streamline time-periods are non-increasing and independent of initial data.

Particles exhibit positive drift with no closed paths in deep-water Stokes flow.

Abstract

We investigate kinetic energy properties of an irrotational deep-water Stokes wave. By introducing a conformal hodograph transform, we perform complex analysis in the new variables, and show that the existence of the streamline time-period for deep-water Stokes waves. Relying on the conformal mappings, we prove some qualitative results about the kinetic energy such as the convexity property of integral means of kinetic energy, the streamline time-period being non-increasing and independent of initial data and the monotonicity and logarithmic convexity of the total kinetic energy over a streamline time-period. Moreover, taking advantage of the non-increasing nature of the streamline time-period, we show that the streamline time-period strictly larger than the wave period, which suggests the fact that there are no closed paths for all particles in a deep-water Stokes flow and the drift of any streamline is positive. These results allow us to quantify the observation that the kinetic energy and the streamline time-period are larger near the free surface and decrease with increasing the depth.