Linear wave equations in which the coefficients vary slowly with space and/or time arise in various branches of physics. Approximate solutions may be obtained using WKB theory, whereby the constant-coefficient solution is locally valid, but the wavelength, frequency and amplitude of the solution vary slowly in space and time.

Internal gravity waves, associated with stably stratified fluids, are of importance in atmospheric dynamics, especially because of their effect on the circulation in the middle atmosphere. Weather and climate models can at best resolve only the longest-wavelength part of the gravity wave spectrum, and the rest must be parameterized. Since the density and wind fields through which the waves propagate vary with space and time, many currently used parameterizations are based on WKB theory, albeit with additional, sometimes strong, simplifying assumptions and many tunable parameters.

We have developed a weakly nonlinear model for describing internal gravity waves based on WKB theory in a position-wavenumber phase-space that solves the unsimplified WKB equations (in one spatial dimension) and couples them to an equation for the evolution of the mean flow through which the waves propagate. In this talk, I will explain the basic ideas of WKB theory as it applies to gravity waves and present results from some deceptively simple experiments using our new "phase-space WKB" model.