Dispersive wave focusing on a shear current. I: Linear approximations (Q6593239)

The manuscript addresses the complex interaction of dispersive water waves with depth-varying shear currents, providing a detailed analysis grounded in linear approximations. The scientific problem tackled by the paper revolves around the phenomenon of rogue waves, which are unexpected and abnormally large waves posing significant threats to maritime operations. These rogue waves arise from various physical mechanisms, one of which is dispersive focusing -- a process where wave energy concentrates in a small region due to the differing velocities of waves of different wavelengths. The paper's primary focus is to analyse how depth-dependent shear currents affect the formation and dynamics of these dispersively focused waves in deep water, with a specific emphasis on linear wave approximations. The study aims to understand how shear currents, which vary with depth, influence both the envelope and kinematics of wave groups, particularly in the context of extreme wave events such as rogue waves.\N\NTo approach this problem, the authors develop several analytical solutions using linear approximations for various shear current profiles, including linearly and exponentially varying currents. They employ long-crested wave models propagating at arbitrary angles to the current, assuming low wave steepness, which is a reasonable approximation when studying rogue waves. The analysis also includes currents of arbitrary depth profile, approximated using the weak shear framework introduced by \textit{R. H. Stewart} and \textit{J. W. Joy} [Deep Sea Res. Oceanograph. Abstr. 21, No. 12, 1039--1049 (1974; \url{doi:10.1016/0011-7471(74)90066-7})]. The methods focus on deriving approximate closed-form solutions for wave dynamics in the presence of shear currents, offering both general results and specific case studies such as the Columbia River estuary.\N\NThe main findings of the paper indicate that while the presence of shear currents has only a modest effect on the overall envelope of the wave group, it significantly alters the wave kinematics, especially the horizontal orbital velocities near the surface. Depending on whether the shear is following or opposing the wave propagation direction, these velocities can either be amplified or suppressed. For example, in the Columbia River case study, the horizontal velocities are found to increase by a factor of 1.4 when waves propagate against a following shear and decrease by a factor of 0.7 in opposing shear conditions. This has practical implications for maritime safety, as the forces exerted by rogue waves on vessels and structures are strongly influenced by these changes in velocity profiles.\N\NThe significance of this research lies in its contribution to understanding the intricate dynamics of rogue wave formation in the presence of shear currents, a phenomenon that has been less explored in comparison to other mechanisms like modulational instability or constructive interference. The analytical approximations presented in this study provide valuable tools for predicting wave behaviour in real-world situations, where shear currents are common, such as in estuaries and coastal regions. This work enhances the ability to assess risks associated with rogue waves and contributes to the design of safer maritime structures and operational strategies. Overall, the paper makes a significant contribution to the field of wave-current interactions, particularly in advancing the theoretical understanding of wave focusing on sheared currents.\N\NFor Part II, see [the first author et al., Water Waves 6, No. 2, 413--449 (2024; Zbl 07901736)].