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On the interaction of deep water waves and exponential shear currents. (English) Zbl 1173.76007
Summary: A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method. The magnitude of vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil-Jacotin transformation is used to transfer the original exponentially nonlinear boundary value problem in an unknown domain into an algebraically nonlinear boundary value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current, but also for large amplitude waves on a strong current. We study in detail the nonlinear wave-current interaction. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criterion of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytical method is rather general in principle, and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.

MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76M25 Other numerical methods (fluid mechanics) (MSC2010)
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