On the possibility of wave-induced chaos in a sheared, stably stratified fluid layer

Abstract

Shear flow in a stable stratification provides a waveguide for internal gravity waves. In the inviscid approximation, internal gravity waves are known to be unstable below a threshold in Richardson number. However, in a viscous fluid, at low enough Reynolds number, this threshold recedes to Ri = 0. Nevertheless, even the slightest viscosity strongly damps internal gravity waves when the Richardson number is small (shear forces dominate buoyant forces). In this paper we address the dynamics that approximately govern wave propagation when the Richardson number is small and the fluid is viscous. When Ri ξ = λ1A + λ2Aξξ + λ3Aξξξ + λ4AAξ + b(ξ) where ξ is the coordinate of the rest frame of the passing temperature wave whose horizontal profile is b(ξ). The parameters λi are constants that depend on the Reynolds number. The above dynamical system is know to have limit cycle and chaotic attrators when forcing is sinusoidal and wave attenuation negligible.