Depth-integrated wave-current models. Part 1. Two-dimensional formulation and applications

Depth-integrated mathematical models for simulating waves and currents from deep to shallow water are presented. These models are derived from Euler's equations in the-coordinate system, mapping the total water depth in Cartesian coordinates onto a specified range of-coordinates. The horizontal velocity is approximated as a truncated infinite series of products of prescribed shape functions of and unknown functions of horizontal coordinates and time. Adopting the method of weighted residuals, the new models are obtained by minimizing the residuals of the horizontal momentum equations with either the Galerkin method or the subdomain method. These models' linear and nonlinear water wave properties are investigated. The new models are implemented numerically. A hierarchy of numerical models with different degree of polynomial approximation is developed and checked against several benchmarked experiments and a new set of experiments of self-focusing wave groups. For both the Galerkin and subdomain models, excellent agreements are observed for both the free surface elevations and the velocity profiles. The new models are superior to the existing Boussinesq-Type models for their applicability to a wide range of physical scenarios, including the interactions between a wave package of multiple frequency components and a linearly sheared current. The new Galerkin models have similar characteristics and accuracy as the Green-Naghdi models, but the new models are more efficient computationally. Finally, for the same degree of polynomial approximation the subdomain models perform better than the Galerkin models and require less computational time.