The development of a high-accuracy, broadband, Green–Naghdi model for steep, deep-water ocean waves

The focus of the research presented here is the development of an efficient analytical model for the time-domain simulation of the evolution of a train of a three-dimensional steep random waves and its associated flow. Of particular interest here is the development of a tool that not only accurately predicts the surface elevation history of this wave train, but also the kinematics within the waves, particularly near the free surface. Fenton (Advances in coastal and ocean engineering, vol 5, World Scientific, Singapore, pp 241–324, 1999 ) reviews the rich literature of various computational models for water waves and is divided between analytical approaches that develop systems of equations to describe the evolution of waves in space and time, and computational approaches such as CFD (which will not be the focus here). What is required to analyze properly the practical ocean engineering problem described in the motivation below is a theory that predicts the three-dimensional evolution of waves that simultaneously has four essential characteristics: deep-water, random broadband seaways, steep waves (close to breaking), and demonstrated accuracy in both wave shape and near-surface wave kinematics. The number and variety of theories that satisfy some but not all of these characteristics is too voluminous to reference here; dynamic theories that exhibit the confluence of all four characteristics are apparently nonexistent. The intent of this paper is to make a step in the direction of filling this void by extending the Green–Naghdi theory of deep-water waves. It is shown that higher level GN models using distributed directors do have a bandwidth that is significantly larger than former GN models and have the same computational effort as using the traditional directors. The bandwidths achieved with the new approach are large enough to be useful in the context of many ocean engineering problems. Applications of these models to random wave situations will be reported in a subsequent article.