On the calculation of the water particle kinematics arising in a directionally spread wavefield

This paper concerns the calculation of the water particle kinematics generated by the propagation of surface gravity waves. The motivation for this work lies in recent advances in the description of the water surface elevation associated with extreme waves that are highly nonlinear and involve a spread of wave energy in both frequency and direction. To provide these exact numerical descriptions the nonlinear free-surface boundary conditions are time-marched, with the most efficient solutions simply based upon the water surface elevation, η, and the velocity potential, φ, on that surface. In many broad-banded problems, computational efficiency is not merely desirable but absolutely essential to resolve the complex interactions between wave components with widely varying length-scales and different directions of propagation. Although such models have recently been developed, the calculation of the underlying water particle kinematics, based on the surface properties alone, remains a significant obstacle to their practical application. The present paper tackles this problem, outlining a new method based upon an adaptation of an existing approximation to the Dirichlet-Neumann operator. This solution, which is presently formulated for flow over a flat bed, is appropriate to the description of any kinematic quantity and has the over-riding advantage that it is both stable and computationally efficient. Indeed, its only limitation arises from the assumed Fourier series representations. As a result, both η and φ must be single-valued functions and are not therefore appropriate to the description of overturning waves. The proposed method is compared favourably to both existing analytical wave models and laboratory data providing a description of the kinematics beneath extreme, near-breaking, waves with varying directional spread. The paper concludes by investigating two important characteristics of the flow field beneath large waves arising in realistic ocean environments.