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 1, to a first approximation, the transport equations for thermal energy and momentum decouple. Thus, a large amplitude temperature wave then has little effect on the fluid velocity. Under such conditions in the atmosphere, a small amplitude "turbulent burst" is observed, transporting momentum rapidly and seemingly randomly. A regular perturbation scheme from a base state of a passing temperature wave and no velocity disturbance is developed here. Small thermal energy convection-momentum transport coupling is taken into account. The elements of forcing, wave dispersion, (turbulent) dissipation under strong shearing, and weak nonlinearity lead to this dynamical equation for the amplitude A of the turbulent burst in velocity: Aξ = 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.
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