Modeling "crossing sea state" wave patterns in layered and stratified fluids

Abstract

Free surface waves with two or more spectral peaks propagating at an oblique angle to each other are commonly termed "crossing sea states". Such crossing patterns have been suggested as possible causes for rogue waves and maritime accidents. Modulation instabilities of plane waves using coupled Schrodinger or Zakharov equations have been adopted as theoretical models in the literature. Here, extensions to layered and stratified fluids are conducted. For a two-layer fluid with long-wave-short-wave resonance, crossing patterns with two short waves will enhance instability compared with the single-wave case. Analytical treatment beyond the linear instability regime is elucidated by a cascading mechanism. Growth of the higher-order harmonics eventually leads to finite-amplitude pulsating modes or breathers. Breathers subsequently exhibit a Fermi-Pasta-Ulam-Tsingou type recurrence. The time for the first formation of breathers predicted by the cascading mechanism attains excellent agreements with the full numerical simulations. A similar study is performed for a continuously stratified fluid with constant buoyancy frequency. Triad resonance with two components as a pair of oblique waves also produces enhanced instability and a preferred inclination of maximum growth rate. These crossing patterns will likely play critical roles in many wave-propagation configurations in fluid mechanics.