Is Nonlinearity Important in Non-Breaking Mountain Waves?

Linear theory has long been utilized to study mountain waves and has been successful in describing much of their behavior. Nevertheless, mountain waves are a finite-amplitude phenomenon, and nonlinear effects can be significant. In the simplest theoretical context, that of two-dimensional steady-state constant Brunt-Väisälä frequency (N) and horizontal wind speed (U) flow, nonlinear effects are relatively minor until wave breaking occurs. However, in more complex profiles, for example those that emulate the presence of the tropopause, more dramatic nonlinear effects occur, even in the absence of wave breaking. To better elucidate these effects, we constructed a linearized version of the originally fully nonlinear time-dependent University of Washington meso12 model. The solutions from this linearized model can then be compared to the those from the original nonlinear model to determine the effects of nonlinearity in flow with realistic upstream profiles of N and U.

We investigated three classes of background atmospheric profiles: (1) uniform U and a two-layer profile with constant N in each layer, where N in the upper layer is twice that of the lower layer; (2) a profile with a similar structure of N but with linear shear from 10 m s^-1 at the surface to 30 m s^-1 at the tropopause, relaxing back to 20 m s^-1 in the stratosphere; and (3) a case with the same wind profile and stratospheric stability as in (2) but with a thin low-level inversion above an almost neutral stability layer in the lower troposphere. For each class, the 2D Boussinesq linear and nonlinear meso12 models were run to approximately steady-state for various combinations of nondimensional mountain heights and tropopause heights. The two-layer no-shear case is consistent with previous studies and features a shift of the peak surface pressure drag to higher tropopause heights as the mountain height is increased. The shear cases also exhibit this general trend, but there is significant amplification of the surface pressure drag with increasing mountain height independent of the trend as well. In the shear case without a low-level inversion, there is approximately a five-fold amplification over the corresponding linear solution in the most extreme case without wave breaking. For the shear case with an inversion, the most extreme case without breaking exhibits a twenty-fold amplification over its linear counterpart.

These results demonstrate that nonlinear effects can play an important role in modulating lee-wave amplitude and gravity-wave drag even when there is no trace of wave breaking.