Wave-mean interaction theory

This is an informal account of the fluid-dynamical theory describing nonlinear interactions between small-amplitude waves and mean flows. This kind of theory receives little attention in mainstream fluid dynamics, but it has been developed greatly in atmosphere and ocean fluid dynamics. This is because of the pressing need in numerical atmosphere-ocean models to approximate the effects of unresolved small-scale waves acting on the resolved large-scale flow, which can have very important dynamical implications. Several atmosphere ocean example are discussed in these notes (in particular, see §5), but generic wave-mean interaction theory should be useful in other areas of fluid dynamics as well. We will look at a number of examples relating to the basic problem of classical wave-mean interaction theory: finding the nonlinear O(a ²) mean-flow response to O(a) waves with small amplitude a ≪ 1 in simple geometry. Small wave amplitude a ≪ 1 means that the use of linear theory for O(a) waves propagating on an O(1) background flow is allowed. Simple geometry means that the flow is periodic in one spatial coordinate and that the O(1) background flow does not depend on this coordinate. This allows the use of averaging over the periodic coordinate, which greatly simplifies the problem.