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) 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. 1 Two-dimensional incompressible homogeneous flow This is our basic starting point. We first develop the mathematical equations for this kind of flow and then we consider waves and mean flows in it. 1.1 Mathematical equations We work in a flat, two-dimensional domain with Cartesian coordinates x = (x, y) and velocity field u = (u, v). In the y-direction the domain is bounded at y = 0 and y = D by solid impermeable walls such that v = 0 there. In the x-direction there are periodic boundary conditions such that u(x+ L, y, t) = u(x, y, t). In an atmospheric context we can think of x as the “zonal” (i.e. east–west) coordinate and of y as the “meridional” (i.e. south–north) coordinate. The period length L is then the Earth’s circumference. The flow is incompressible, which means that the velocity field is area-preserving and hence has zero divergence: ∇ · u = 0 ⇔ ux + vy = 0. (1.1)