The Amplitude of Lee Waves Forming on the Boundary Layer Inversion

Lee waves are a special type of gravity wave response in stratified flows over mountains, characterized by vertical orientation of phase lines and horizontal propagation of wave energy. The typically observed horizontal wavelengths of atmospheric lee waves are 3-15 km. Interfacial waves are the most frequently observed type of atmospheric lee waves. Similar to waves on a free water surface, interfacial lee waves form on a sharp discontinuity of the vertical density profile, such as the boundary-layer capping inversion. Their wavelength can be described accurately using linear interfacial gravity-wave theory. This theoretical framework, however, fails to properly predict the lee-wave amplitude. Since it is well known that large-amplitude lee waves may lead to low-level turbulence, which poses a potential hazard for aviation, this property of interfacial lee waves deserves further attention.

In this study, we develop a simple analytical model for the amplitude of lee waves at the boundary-layer inversion. Using linear internal gravity-wave theory, we derive a relationship between the wave amplitude and the horizontal wave energy flux. Numerical simulations suggest that this relationship is accurate even for large-amplitude lee waves. Given that measurements of wave energy flux are not commonly available, this quantity needs to be parameterized. In this study, we do this by employing hydraulic theory. Assuming mass and momentum conservation, it can be shown that the energy flux in the perturbed flow in the lee of a mountain is larger than that farther downstream in the unperturbed flow. As a consequence, hydraulic theory predicts energy accumulation at a location where the flow readjusts back to its undisturbed state. This local energy accumulation can be compensated either by dissipation, as in hydraulic jumps, or by downwind transport of wave energy by a train of lee waves. Assuming that the excess energy is entirely radiated in the lee wave train, the lee wave amplitude can be estimated without measuring the wave energy flux explicitly.

The verification of the amplitude model with idealized numerical simulations gives excellent results. In addition, the model is applied to analyze water tank measurements of density-stratified two-layer flow over two-dimensional topography, taken during the HYDRALAB experiments. Reasonable agreement between the measured wave amplitudes and those predicted by our model is found.