The Steepening of Hydrostatic Mountain Waves

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Abstract
The nonlinear effects in the flow of a Boussinesq stratified fluid over two-dimensional sinusoidal topography are examined in this paper using second-order perturbation theory. The waves generated by finite-amplitude topography are found to steepen in the forward sense in a way which is periodic with height. The nonlinear lower boundary condition has an important influence on the steepening as well as on wave breakdown and the production of severe downslope winds. These nonlinear effects are shown to be important for moderate and large mountain even if the surface slopes are small. Using the results of linear theory, other causes of steepening such as mountain asymmetry, vertical variation of wind, stability, density and mountain isolation can be estimated. The nonlinear effects associated with partial reflection from the tropopause are found to be very important in certain situations. The theoretical ideas of wave steepening are in qualitative agreement with previous case studies of flow over large mountains and with seasonally averaged measurements of stratosphere turbulence. The theory suggests that for large mountains light wind conditions (as in the summer in mid-latitudes) will lead to wave breaking just above the mountain, while the stronger winds (e.g., winter-time) the mountain waves will propagate vertically through the troposphere—finally breaking down in the lower stratosphere near 17 km.

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