Dispersive Wave Focusing on a Shear Current: Part 2—Nonlinear Effects

Document Type

Article

Publication Date

7-1-2024

Abstract

Continuing our recent work [Ellingsen et al., Water Waves (2024)] we investigate the influence of vertically sheared currents on the surface elevation as well as the kinematics of dispersively focusing wave groups up to second order in steepness. The groups are assumed long crested in deep water which may travel at oblique angles with the current, which has a depth-dependent profile in both magnitude and direction. A strong but realistic shear current affects the wave surface elevation only slightly but the wave-induced horizontal velocity beneath the point of focus is very significantly affected, and new phenomena occur at second order. Firstly, a shear current causes wave-induced superharmonic velocity to be nonzero, contributing significantly for moderate wave steepness. At linear order, following (opposing) shear causes horizontal velocities to be amplified (reduced); for crest-focused wave groups, the superharmonic contribution reduces the influence of shear, whereas for trough-focused waves the velocity change from linear and second-order waves add, causing a substantially larger shear-induced effect. Secondly, the sub-harmonic mean flow is not strictly a return flow, but can follow the direction of wave propagation at the depths nearest the surface. Thirdly, unlike the case without shear where the subharmonic mean flow vanishes in the limit of zero bandwidth, it can now tend to a finite value in the narrowband limit. The criterion for this to happen is that the shear current has nonzero curvature.

Identifier

85197367263 (Scopus)

Publication Title

Water Waves

External Full Text Location

https://doi.org/10.1007/s42286-024-00097-z

e-ISSN

25233688

ISSN

2523367X

First Page

413

Last Page

449

Issue

2

Volume

6

Grant

101045299

Fund Ref

European Commission

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