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

Waves induce a Lagrangian current at the sea surface, which has the potential to modify air-sea interactions. A prevailing theory for the generation of waves by wind holds that waves develop due to resonance at a critical layer with respect to a background shear flow (Miles 1957), a framework that has seen numerous refinements. Yet, 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. Our analysis first recovers the classic Miles growth rate from linear theory before extending the analysis to third order in the wave slope to derive a modified growth rate. We find that the leading-order wave-induced mean flow alters the instability growth rate. This modification manifests as a suppression of growth, particularly at high wavenumbers, with increasing wave steepness for realistic wind profiles—a result qualitatively consistent with observations. We interpret this suppression physically: the wave-induced current alters the coupling between the phase speed and the mean flow at the critical level, reducing the efficiency of momentum transfer. 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 elucidating the physical mechanisms driving shear instabilities and a direct path toward refining wind-stress parameterizations.