Two‐Layer Solutions to Long's Equation For Vertically Propagating Mountain Waves: How Good Is Linear Theory?

Finite‐amplitude solutions for infinitely deep stratified flow over orography are obtained for upstream conditions with uniform wind and a two‐layer static‐stability structure. the flow within each layer is governed by Long's equation. the governing equations are solved using a semi‐analytic model with an open (radiation) boundary condition in the upper layer. A simplified, computationally efficient form of the finite‐amplitude radiation upper‐boundary condition is presented and utilized in the two‐layer calculations. When the higher stability is in the lower layer, finite‐amplitude effects may produce large‐amplitude waves in circumstances where linear theory predicts a weak response. the relationship between the continuously stratified system and shallow‐water hydraulic theory is explored, and an expression is derived for the effective Froude number in unbounded continuously stratified flow. In those cases producing the strongest finite‐amplitude response, the flow undergoes a transition from subcritical to supercritical just to the lee of the mountain crest. When the higher stability is in the upper layer, the finite‐amplitude response is also sensitive to the location of the stability interface. the dependence of the solution on the height of the interface is, once again, different from that predicted by linear theory. the interface height that produces the strongest response increases as the mountain height increases. These results support the idea, originally suggested through the analysis of linear multi‐layer models, that the vertical phase shift between the ground and the tropopause has an important influence on the strength of downslope windstorms. the optimal phase shift for finite‐amplitude waves cannot, however, be reliably determined from linear theory.