A Kinematic Conservation Law in Free Surface Flow

TL;DR

This paper reveals a new kinematic conservation law in the Green-Naghdi model for shallow water waves, linking surface tangent velocity evolution to a conserved quantity, enhancing understanding of nonlinear wave dynamics.

Contribution

It introduces a novel conservation law in the Green-Naghdi system, derived from the tangent velocity evolution, providing new insights into wave surface dynamics.

Findings

Identifies a fourth conservation law in the Green-Naghdi system.

Connects the law to the evolution of tangent velocity at the free surface.

Shows the law's derivation from the full Euler equations.

Abstract

The Green-Naghdi system is used to model highly nonlinear weakly dispersive waves propagating at the surface of a shallow layer of a perfect fluid. The system has three associated conservation laws which describe the conservation of mass, momentum, and energy due to the surface wave motion. In addition, the system features a fourth conservation law which is the main focus of this note. It is shown how this fourth conservation law can be interpreted in terms of a concrete kinematic quantity connected to the evolution of the tangent velocity at the free surface. The equation for the tangent velocity is first derived for the full Euler equations in both two and three dimensional flows, and in both cases, it gives rise to an approximate balance law in the Green-Naghdi theory which turns out to be identical to the fourth conservation law for this system.