Weakly nonlinear analysis of wind-driven gravity waves

Abstract

We study the weakly nonlinear development of shear-driven gravity waves induced by the physical mechanism first proposed by Miles, and furthermore investigate the mixing properties of the finite-amplitude solutions. Linear theory predicts that gravity waves are amplified by an influx of energy through the critical layer, where the velocity of the wind equals the wave phase velocity. As the wave becomes of finite amplitude nonlinearities have to be taken into account. In this paper we derive asymptotic solutions of finite-amplitude waves for weak wind and strong gravitation $U^2 \ll gl$, applicable to many astrophysical scenarios. Because of the presence of a critical layer, ordinary weakly nonlinear methods fail; in this paper, we use rescaling at the critical layer and matched asymptotics to derive the amplitude equations for the most unstable wave, under the assumption that the physical domain is periodic. These amplitude equations are compared with the equations derived by Reutov for the small-density-ratio case (applicable to oceanography); and after numerically integrating these equations, we also analytically derive their quasi-steady limit. As in other analyses of critical layers in inviscid parallel flow, we find that the initial exponential growth of the amplitude $A$ transitions to an algebraic growth proportional to the viscosity, $A \sim \nu t^{2/3}$. We also find that the weakly nonlinear flow allows superdiffusive particle transport within the critical layer, with an exponent $\sim 3/2$, consistent with Venkataramani's results.