Weakly nonlinear surface gravity waves in a depth-dependent background flow

An open ocean often has a wind driven shear-current near the surface that is able to significantly change the properties of surface waves. This work aims to investigate the effects of a vertically sheared background flow on weakly nonlinear waves with both a statistical study for irregular random waves and a deterministic study for a wave group.

We first extended the theory by Dalzell (1999) to allow for the effects of a horizontal background flow with arbitrary depth dependence. The extended theory is valid up to second order in wave steepness and is applicable for directional-spread waves of a broad bandwidth. The Direct Integration Method (Li & Ellingsen 2019) is used for the linear dispersion relation.

Using the theory, we examine the effects of an opposing and assisting shear, respectively, on the nonlinear properties of a short wave group on deep-water through comparisons to cases without a shear flow. A shear flow leads to wave crests (troughs) being either steepened or flattened, depending mainly on the direction of a shear relative to the propagation direction of the group and the strength of the depth-integrated velocity of a shear relative to the group velocity. We, furthermore, investigated skewness and kurtosis of a time record of the wave elevation for irregular waves in a background sheared flow, compared to a linear Gaussian random sea for surface waves only. We obtained the probability density function and exceedance probability for wave crests. Relevance for rogue wave formation is discussed.

Key words: waves/free-surface flow, ocean surface waves, wave-current interaction

 

References

Dalzell, J. F. "A note on finite depth second-order wave–wave interactions." Applied Ocean Research 21, no. 3 (1999): 105-111.

Li, Y., and Ellingsen, S. Å. "A framework for modeling linear surface waves on shear currents in slowly varying waters." Journal of Geophysical Research: Oceans 124, no. 4 (2019): 2527-2545.