The role of Lagrangian drift in the generation of surface waves by wind

In this work, an asymptotic analysis is performed in the Lagrangian frame to gain a new perspective on the well-studied problem of the generation of surface waves by wind. A prevailing theory for the generation of waves by wind holds that waves develop due to a resonant interaction at a critical layer within a background shear flow (Miles 1957). While this framework has seen numerous refinements, these theories largely overlook any influence of the wave's own induced current on further growth. This leads to a fundamental question: does the Lagrangian drift—the velocity a fluid parcel actually experiences—play a role in the resonance mechanism underlying wave growth?

To answer this question, we conduct a nonlinear stability analysis entirely in the Lagrangian frame. We first recover the classic Miles growth rate before extending the analysis to third order in the wave slope to derive a modified growth rate, which is altered by the leading-order wave-induced mean flow. We find that growth is suppressed with increasing wave steepness for realistic wind profiles, qualitatively consistent with observations. Notably, new airborne current sensors observe the Lagrangian drift, providing a direct observational pathway to account for this feedback in growth estimates. More broadly, this approach provides a new methodology for analyzing shear instabilities and a direct path toward refining wind-stress parameterizations.