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

Abstract

The dynamics of internal gravity waves is modelled using WKB theory in position-wavenumber phase space. A transport equation for the phase-space wave-action density is derived for describing one-dimensional wave fields in a background with height-dependent stratification and height- and timedependent horizontal-mean horizontal wind, where the mean wind is coupled to the waves through the divergence of the mean vertical flux of horizontal momentum associated with the waves. The phase-space approach bypasses the caustics problemthat occurs inWKB ray-tracingmodels when the wavenumber becomes a multivalued function of position, such as in the case of a wave packet encountering a reflecting jet or in the presence of a time-dependent background flow. Two numerical models were developed to solve the coupled equations for the wave-action density and horizontal mean wind: an Eulerian model using a finite-volumemethod, and a Lagrangian “phase-space ray tracer” that transports wave-action density along phase-space paths determined by the classical WKB ray equations for position and wavenumber. The models are used to simulate the upward propagation of a Gaussian wave packet through a variable stratification, a wind jet, and the mean flow induced by the waves. Results from the WKB models are in good agreement with simulations using a weakly nonlinear wave-resolving model as well as with a fully nonlinear large-eddy-simulation model. The work is a step toward more realistic parameterizations of atmospheric gravity waves in weather and climate models.