The wave-mean current problem with a continuous near surface solution.

Commonly applied wave propagation theory employs a velocity potential function obtainedfor averaged mean sea levels (MSL). The basic linear wave-current problem is composed of asuperposition of the velocity potential of finite amplitude waves without current and thevelocity potential of a constant current (U0x). Nevertheless, such a superposition poses aninherent problem as the current’s potential fits the boundary conditions of the MSL but notthe one of the changing surface. The required adaptation to the changing surface is of linearorder in wave slope (ka). In the common approach, even with no ambient current, the dynamic and kinematicboundary conditions are expanded around the MSL to the first order using Taylorapproximation. This process does not provide an accurate behavior in the near surface region(see [1]). For the wave current problem, this limitation poses even a higher discrepancy as thecurrent itself also oscillates in the upper layer in order to account for the changing surfaceelevation. In order to account for this discrepancy and understand this fundamentalwave-current interaction problem, this work presents a continuous solution for the wave andcurrent velocity potential function given for a 2D curvilinear {ξ, ζ} coordinate system thatfollows monochromatic waves in the upper layer, and decays to {x, z} coordinates withdepth. Unlike the traditional approach, the linear problem is solved without a verticalTaylor approximation but rather with a longitudinal one. This suggested solutionsolved analytically and considered to be more realistic in the near surface as itagrees better with the continuous nature of the flow. It provides an alternative to Airysolution without current and is generalized to account for a mean wave-currentproblem. Changes in other basic properties of the flow such as Stokes drift and Lagrangian orbitalvelocities are also discussed. The importance of this fundamental model to interpretations ofwave-current measurements using Acoustic Doppler Current Profilers and Acoustic DopplerVelocimeters will also be discussed. References [1] P. B. Smit, T. T. Janssen, T. H. C. Herbers, P. B. Smit, T. T. Janssen, and T. H. C.Herbers. Nonlinear Wave Kinematics near the Ocean Surface. Journal of PhysicalOceanography, 47(7):1657–1673, 2017.