A framework for ocean surface waves atop a background flow of depth dependence in varying water depth

We present a theoretical and numerical framework for simple, efficient and accurate evaluation of surface wave models allowing presence of a shear current of arbitrary depth dependence, and a slowly varying bathymetry. The framework, which is named Direct Integration Method (DIM), furthermore yields the full-linearized flow field solution throughout the water column. The DIM is of wide applicability in an oceanography setting. We demonstrate this in four different aspects. The DIM is applied to strong and turning shear profiles and slowly varying bathymetry, for which existing approaches either cannot handle or show unsatisfactory performance. Compared to existing approaches in terms of applicability and computational cost, the DIM is, in our opinion, favorable for a wide range of practical applications. The DIM is moreover employed to obtain the full flow field beneath a 2D ring wave upon a near-surface wind-driven exponential shear current, revealing striking qualitative differences compared to no shear, of potential importance to thermal mixing and transport in the upper ocean.