The boundary layer theory for non-hydrostatic mountain waves presented in Part II is extended to include upward propagating gravity waves and trapped lee waves. To do so, the background wind with constant shear used in Part II is smoothly curved and becomes constant above a “boundary-layer” height d which is much larger than the inner layer scale δ. As in Part II, the pressure drag stays well predicted by a gravity wave drag when the surface Richardson number J < 1 and by a form drag due to non separated sheltering when J < 1. As in Part II also, the sign of the Reynolds stress is predominantly positive in the near neutral case (J < 1) and negative in the stable case (J > 1) but situations characterized by positive and negative Reynolds stress now combine when J ∼ 1. In the latter case, and even when dissipation produces positive stress in the lower part of the inner layer, a property we associated with non separated sheltering in Part II, negative stresses are quite systematically found aloft. These negative stresses are due to upward propagating waves and trapped lee waves, the first being associated with negative vertical flux of pseudo-momentum aloft the inner layer, the second to negative horizontal flux of pseudo-momentum downstream the obstacle. These results suggest that the significance of mountain waves for the large-scale flow is more substantial than expected and when compared to the form drag due to non separated sheltering.

The non-hydrostatic version of the mountain flow theory presented in Part I is detailed. In the near neutral case, the surface pressure decreases when the flow crosses the mountain to balance an increase in surface friction along the ground. This produces a form drag which can be predicted qualitatively. When stratification increases, internal waves start to control the dynamics and the drag is due to upward propagating mountain waves as in part I. The reflected waves nevertheless add complexity to the transition. First, when stability increases, upward propagating waves and reflected waves interact destructively and low drag states occur. When stability increases further, the interaction becomes constructive and high drag state are reached. In very stable cases the reflected waves do not affect the drag much. Although the drag gives a reasonable estimate of the Reynolds stress, its sign and vertical profile are profoundly affected by stability. In the near neutral case the Reynolds stress in the flow is positive, with maximum around the top of the inner layer, decelerating the large-scale flow in the inner layer and accelerating it above. In the more stable cases, on the contrary, the large-scale flow above the inner layer is decelerated as expected for dissipated mountain waves. The structure of the flow around the mountain is also strongly affected by stability: it is characterized by non separated sheltering in the near neutral cases, by upstream blocking in the very stable case, and at intermediate stability by the presence of a strong but isolated wave crest immediately downstream of the ridge.

A hydrostatic theory for mountain waves with a boundary layer of constant eddy viscosity is presented. It predicts that dissipation impacts the dynamics over an inner layer whose depth is controlled by the inner-layer scale δ of viscous critical-level theory. The theory applies when the mountain height is smaller or near δ and is validated with a fully nonlinear model. In this case the pressure drag and the wave Reynolds stress can be predicted by inviscid theory, if one takes for the incident wind its value around the inner-layer scale. In contrast with the inviscid theory and for small mountains the wave drag is compensated by an acceleration of the flow in the inner layer rather than of the solid earth. Still for small mountains and when stability increases, the emitted waves have smaller vertical scale and are more dissipated when traveling through the inner layer: a fraction of the wave drag is deposited around the top of the inner layer before reaching the outer regions. When the mountain height becomes comparable to the inner-layer scale, nonseparated upstream blocking and downslope winds develop. Theory and the model show that (i) the downslope winds penetrate well into the inner layer and (ii) upstream blocking and downslope winds are favored when the static stability is strong and (iii) are not associated with upper-level wave breaking.

The trapped mountain waves produced when the incident wind near the surface is small compared with its value aloft are analyzed with a theory adapted from Long (1953) and compared with fully nonlinear simulations performed with the Weather Research and Forecasting model (WRF). Although small near‐surface incident winds occur naturally in fronts via a combination of the thermal wind balance and the boundary layer, they pose at least two problems in mountain meteorology: zero surface incident winds produce no wave in the fully linear case; they also correspond to places where mountain waves have a critical level. Despite these problems, theory and WRF show that, for small mountains, trapped lee waves (a) can occur and (b) are favored when the surface Richardson number J = N 2/(∂u/∂z)2 is small. This last result is related to the theoretical fact that the surface absorption of stationary gravity waves increases when J increases. The relation with flow stability is corroborated further by the fact that the trapped lee waves resemble the Kelvin–Helmholtz (KH) modes of instability that exist when J < 0.25. For medium mountains, some aspects of the theory still hold but need to be adapted, the more intense winds and foehn that occur along the lee side of the mountain having a tendency to increase the surface flow stability. For “initially” small J, this can limit the onset of trapped lee waves, again consistent with the fact that mountain‐wave surface absorption increases with surface flow stability. For large J, the dynamics produces wave breaking on the lee side, destabilizing the flow in the wake of the mountain. In the region where the Richardson number is small, trapped waves develop despite the fact that the surface Richardson number can be quite large, suggesting that the trapped lee waves now result from an absolute instability of the wake.