Modulation of Wind Ripples by Long Surface Waves via the Air Flow: a Feedback Mechanism

The evolution of a short-wave (SW) spectrum along a long wave (LW) is studied. The evolution of the SW spectrum variation is treated in the relaxation time approximation. The variation of the SW spectrum is caused by the LW orbital velocities and by the variation of the wind stress along the surface of a LW. The latter is due to the distortion of the flow by a LW, and to the variation of the roughness induced by the modulated short waves. This introduces a feedback mechanism: more SWs give rise to a larger roughness, which by increasing the local stress stimulates the growth of more SWs. It is shown that this aerodynamic feedback effect dominates the modulation of the SW spectrum for moderate and strong winds. The feedback mechanism is most effective for SWs in the gravity-capillary range, increasing its dominance with increasing windspeed and decreasing frequency of a LW. The maximum of the SW amplitude modulation is situated at the crest of a LW. The results are in agreement with laboratory and field measurements of the short-wave modulation.