J. Muraschko, M.D. Fruman, U. Achatz, S. Hickel, Y. Toledo (2015)
Quarterly Journal of the Royal Meteorological Society 141: 676-697. doi: 10.1002/qj.2381

The dynamics of internal gravity waves is modelled using Wentzel–Kramer–Brillouin (WKB) theory in position–wave number 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 time-dependent 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 problem that occurs in WKB ray-tracing models when the wave number 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-volume method 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 wave number. 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 parametrizations of atmospheric gravity waves in weather and climate models.


Examples of the caustics problem in physical space: (a) rays (dark grey lines) associated with waves encountering a reflecting jet (light grey line); (b) waves propagating through a slowly varying alternating wind; (c) modulationally unstable waves propagating through their own induced mean flow; and (d) waves encountering a critical level due to a positive jet. Markers are placed along rays at intervals of (a, d) 100 min, (b) 500 min and (c) 33 min.


Horizontally averaged wave energy density of a very long wave packet reflected by a wind jet from (a) the wave-resolving linear model and (b) the WKB models (in linear mode). (c) Mean energy in the layer between 150 and 168 km as a function of time. Once the downward (outgoing) waves balance the upward (incoming) waves, the wave energy reaches a plateau at the value corresponding to a case of a steady wave train reflected by a jet. The energy in the layer predicted by the analytic solution to the steady-state WKB problem is indicated by the dashed horizontal line.


Normalized wave-induced mean flow from the modulational instability experiment simulated with the weakly nonlinear wave-resolving model and both WKB models: (a) stable case (m0 = −1.4k, a0 = 0.12), (b) metastable case (m0 = −0.7k, a0 = 0.21) and (c) unstable case (m0 = −0.4k, a0 = 0.42). Plots are in the reference frame moving with the initial group speed at the centre of the wave packet.