Dispersive wave focusing on a shear current. II: Nonlinear effects (Q6593240)

This study continues the authors et al.'s previous work [Water Waves 6, No. 2, 367--411 (2024; Zbl 07901735)] and addresses the complex interaction of dispersive wave focusing with vertically sheared currents, extending the analysis to account for nonlinear effects up to the second order in wave steepness. The problem revolves around understanding how nonlinear waves, when subject to vertically sheared currents, modify surface elevation and kinematics. This research is motivated by the broader phenomenon of rogue waves, which are large and destructive waves, and the recognition that most studies on these waves have typically assumed quiescent water or simple uniform currents. The authors argue that real-world ocean currents exhibit depth-dependent variations, particularly shear, which may significantly impact the behaviour of wave groups, especially in cases of dispersive wave focusing. The key challenge is to quantify how these sheared currents influence the propagation, focusing, and nonlinear evolution of wave groups, especially when the currents vary in both magnitude and direction with depth.\N\NTo address this problem, the authors employ a weakly nonlinear theoretical framework based on second-order perturbation theory. The methodology incorporates numerical simulations of wave groups propagating in the presence of shear currents, accounting for both long-crested waves and arbitrary shear profiles. The wave surface elevation and kinematics are computed using a system of equations derived from Euler's momentum equations for incompressible flows, subject to boundary conditions at the free surface and at great depth. The study adopts a non-dimensionalised formalism, making use of characteristic scales for time, length, and velocity, and solves the boundary value problem through a combination of analytical approximations and numerical methods. The primary method involves prescribing the linear surface elevation as a boundary condition at the focus point and computing second-order corrections.\N\NThe main findings of the paper highlight several nonlinear effects introduced by the presence of vertically sheared currents. Firstly, while the effect of shear on the surface elevation is minor at linear order, the wave-induced horizontal velocity beneath the surface is significantly affected. At second order, the influence of the shear becomes more pronounced. Specifically, the shear current induces a non-zero superharmonic velocity component, which contrasts with the case without shear. The superharmonic velocity reduces the influence of shear for crest-focused waves but enhances it for trough-focused waves. The subharmonic mean flow also exhibits non-trivial behaviour, with the surprising result that, unlike in quiescent water, it may tend to a finite value in the narrowband limit when shear curvature is non-zero. Additionally, shear currents may cause the subharmonic flow to align with the direction of wave propagation near the surface, contradicting the expectation of a simple return flow.\N\NThe significance of this research lies in its detailed exploration of nonlinear wave-current interactions, a topic that has been underexplored in previous studies. The results have important implications for understanding rogue wave formation and for the design of offshore structures, which must withstand the complex hydrodynamic forces introduced by sheared currents. The findings demonstrate the necessity of accounting for both shear and curvature effects when modelling wave behaviour in realistic oceanic conditions. This work advances the field by providing new insights into how nonlinear wave dynamics are influenced by sheared environments, thereby contributing to a more comprehensive understanding of wave-current interactions.