Extremely large waves atop a depth-dependent background flow

A Nonlinear Schrödinger (NLS) equation-based theoretical model is derived in Li & Chabchoub (2024) for deepwater waves in the presence of a background flow. The flow propagates in the horizontal plane with its profile magnitude and direction being depth-dependent (see, e.g., Li & Ellingsen 2019). A new interpretation of the roles of Stokes drift, Eulerian return flow, and background vertically sheared current in the modulational instability (MI) of Stokes waves has been provided. The results in particular show that a current opposing a long-crested wave group can enhance the oblique modulational instability while suppressing completely the modulational instability which arises from sideband waves in the directions parallel to the wave group.  This provides clear physical insights into the roles of a background flow on rogue waves, owing to that the MI has been well recognized as a plausible cause to their generation. Moreover, the relevance of the background flow suppression or enhancement of the modulation instability process to the Craik-Leibobvich type 2 instability in the presence of Langmuir circulation is discussed and quantified. This relevance suggests a plausible physical mechanism for the energy transfers between waves and currents in the open ocean.   References: Li, Y., Ellingsen, S. ̊A.: A framework for modelling linear surface waves on shear currents in slowly varying waters. Journal of Geophysical Research:Oceans 124, 2527–2545 (2019). Li, Y., Chabchoub, A.: How currents trigger extreme sea waves. The roles of Stokes drift, Eulerian return flow, and a background flow in the open ocean (2024). (submitted to Geophys. Res. Lett.).