An asymptotic theory for the generation of nonlinear surface gravity waves by turbulent air flow

Abstract

Based on a previous linear theory (van Duin & Janssen 1992), turbulent air flow over a surface gravity wave of finite amplitude is studied analytically by the methods of matched asymptotic expansions and multiple-scale analysis. In particular, an initial-value problem for weakly nonlinear waves is solved, where the initial conditions are prescribed by a single Stokes wave, displacing the water surface. The water is inviscid and incompressible, and there is no mean shear current. Wave-wave interactions are not taken into account. The validity of the theory is restricted to slow waves and small drag coefficient. We investigate in detail the change of the mean air flow with the evolution of the wave, with a prescribed order of magnitude of the initial wave slope. The rate of change of this flow is fully determined by an evolution equation for the wave slope, which is obtained from the continuity condition for the normal stress at the air-water interface. This equation also determines the amplitude-dependent rate of growth or damping of the wave, for which a closed-form expression is derived. It turns out that nonlinear effects reduce the rate of energy transfer from the mean air flow to the growing wave, which implies that nonlinearity has a stabilizing effect. For sufficienty large time scales, the slope of the growing wave becomes so large that the original evolution equation, which is approximately a Landau equation, ceases to be valid. For such relatively large wave slope, an alternative evolution equation is derived, which presumably describes the further evolution of the wave until the occurrence of wave breaking. The relative effects of nonlinearity, which can be characterized by a single parameter, increase with increasing wave slope and decreasing wave frequency.