A Framework for Modeling Linear Surface Waves on Shear Currents in Slowly Varying Waters

We present a theoretical and numerical frameworkwhich we dub the direct integration method (DIM)for simple, efficient, and accurate evaluation of surface wave models allowing presence of a current of arbitrary depth dependence and where bathymetry and ambient currents may vary slowly in horizontal directions. On horizontally constant water depth and shear current the DIM numerically evaluates the dispersion relation of linear surface waves to arbitrary accuracy, and we argue that for this purpose it is superior to two existing numerical procedures: the piecewise-linear approximation and a method due to Dong and Kirby (2012, ). The DIM moreover yields the full linearized flow field at little extra cost. We implement the DIM numerically with iterations of standard numerical methods. The wide applicability of the DIM in an oceanographic setting in four aspects is shown. First, we show how the DIM allows practical implementation of the wave action conservation equation recently derived by Quinn et al. (2017, ). Second, we demonstrate how the DIM handles with ease cases where existing methods struggle, that is, velocity profiles U(z) changing direction with vertical coordinate z and strongly sheared profiles. Third, we use the DIM to calculate and analyze the full linear flow field beneath a 2-D ring wave upon a near-surface wind-driven exponential shear current, revealing striking qualitative differences compared to no shear. Finally we demonstrate that the DIM can be a real competitor to analytical dispersion relation approximations such as that of Kirby and Chen (1989, ) even for wave/ocean modeling. Plain Language Summary Ocean surface wave models are important to seek solutions of key questions in a wide range of real-life practices, for example, exchange of energy, mass, and momentum between ocean and atmosphere and the spread of nutrients as well as pollutants. In coastal waters, surface waves propagate in complex environments. For instance, they coexist with depth-dependent currents and varying water depth. When developing those wave models, we need to consider complicated scenarios and meanwhile offer useful and practical implementations of the models. In the paper, we have developed a theoretical and numerical framework for simple, efficient, and accurate evaluation of surface wave models allowing presence of a current of arbitrary depth dependence and a slowly varying water depth. Our method has wide applicability in particular in an oceanographic setting. It is moreover favorable to existing approaches for a wide range of practical applications.