On the application of WKB theory for the simulation of the weakly nonlinear dynamics of gravity waves

Many gravity wave parameterizations are based on WKB theory, where the amplitude, wavenumber and frequency of the wave field are represented as slowly varying functions of space and time. However, so-called ray tracing models based on conventional WKB theory in its most general form tend to break down due to the caustics problem -- when wavenumber becomes a multivalued function of position -- even for simple, optimally initialized wave fields. This occurs, for example, in the case of waves reflected by a jet and in many cases of a wave packet propagating through a time-dependent background shear flow. It is partly for this reason that parameterization schemes neglect transience in the background even though it has been shown to have a significant effect of the propagation of the waves and their feedback on the background flow.

In this study the caustics problem is avoided by casting the complete WKB equations in the form of a transport equation for wave action density in position-wavenumber "phase-space" coupled to an equation in physical space for the time-dependent large scale flow. Two numerical implementations of these equations will be presented. The first solves the wave-action density equation using a finite-volume method and the other using an efficient "phase-space ray tracer" which exploits the area-preserving property of the phase-space flow.

Results will be presented from case studies of a packet of small-scale gravity waves propagating upward through a background with vertically varying stratification and through a wind jet. The results from the WKB models are in good agreement with simulations using a weakly nonlinear wave-resolving model as well as with the fully nonlinear large eddy simulation code INCA (http://www.inca-cfd.org).