1. Introduction
Gravity waves, which are frequently generated when air flows over a ridge, are associated with vertical fluxes of horizontal momentum. A decelerative force is exerted on the cross-mountain flow in regions of vertical momentum flux divergence. As an example, mountain waves frequently break down in the lower stratosphere, where the momentum-flux divergence associated with this breaking produces “gravity wave drag.” The important influence of gravity wave drag on the large-scale flow over mountains has long been recognized (Sawyer 1959; Lilly 1972; Smith 1979), and this effect is parameterized in all coarse-resolution weather and climate models [see Kim et al. (2003) for a review].
The accurate parameterization of gravity wave drag is difficult because, among other things, its magnitude can be a sensitive function of nonlinear processes (Durran 1992) and boundary layer structure (Smith 2007). Another potential source of difficulty was suggested by Chen et al. (2005, hereafter CDH05), who noted that the momentum flux can depend on the temporal variations in the cross-mountain flow. They simulated mountain waves above an isolated ridge generated by the passage of a large-scale barotropic jet in which N and U were constant with height at any given x, y, and t. Even in cases where the cross-mountain flow varied on slow multiday time scales, CDH05 found that the mountain-wave momentum fluxes were significantly enhanced during the period of large-scale flow acceleration and diminished during deceleration. Ray tracing and WKB analysis showed that the enhancement of the momentum flux during the accelerating phase was produced by the tendency of wave packets launched when the flow was stronger to have higher vertical group velocities than packets launched when the winds were slower. As a consequence of their higher group velocities, those packets launched later overtook the packets launched earlier, thereby producing an accumulation of wave packets and intense momentum fluxes several kilometers above the surface.
To examine nonsteady mountain waves and momentum fluxes in a more realistic but still idealized environment, Menchaca and Durran (2017, hereafter MD17) conducted simulations of a midlatitude cyclone growing in a baroclinically unstable flow encountering an isolated 3D ridge. The mountain-wave momentum fluxes develop differently in these new simulations: the strongest fluxes occur near the surface after the passage of the strongest large-scale cross-mountain winds. The key factor producing the difference between the momentum fluxes in the current simulation and those in CDH05 is the vertical shear in the large-scale cross-mountain wind. In the following we document this difference and analyze the role of mean shear using ray tracing and WKB theory.
A brief overview of the numerical simulations is presented in section 2. The vertical distribution of the mountain-wave momentum flux triggered by the baroclinically unstable jet is discussed in section 3. A ray-tracing and WKB analysis of the influence of the vertical shear on the accumulation of wave packets above the mountain during periods of mean-flow acceleration is given in section 4. Section 5 contains the conclusions.
2. Simulation details and large-scale flow
The large-scale flow and the initiation of the cyclone are described in MD17, along with the shape of the isolated ridge, whose approximate x and y extents are 80 and 640 km, respectively. Our focus is on the simulation with the lower 500-m-high ridge for which the waves do not break and a quasi-linear analysis is appropriate. This simulation uses an outer grid on which
Figure 1 is a y–z cross section showing isotachs of the westerly wind component and isentropes of potential temperature in the initial unperturbed baroclinically unstable shear flow. The north–south position of the mountain is shown by the white bar above the y axis, which is just south of the core of the jet. As discussed in MD17, a cyclone is triggered by an isolated PV perturbation; its evolution between 2.5 and 7.5 days is illustrated by the surface isobars and θ fields plotted in Fig. 2. Note that the cold front arrives at the ridge at around 4.5 days. The full extent of the nested mesh is shown by the dashed red box in Fig. 2a.

North–south cross section through the background shear flow: θ (color fill at 10-K intervals) and zonal velocity (contoured in black at 2.5 m s−1 intervals). The white bar extending from −800 ≤ y ≤ −160 km shows the north–south extent of the ridge. Data are not plotted in the wave-absorbing layer.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0231.1

Surface isobars (black lines at 8-hPa intervals) and surface θ (color fill at 5-K intervals) for the developing cyclone at (a) 2.5, (b) 3.5, (c) 4.5, (d) 5.5, (e) 6.5, and (f) 7.5 days. The mountain is depicted by the black vertical bar at x = 0 km in all panels. The nested grid is shown by the red dashed lines in (a). Lows and highs are labeled by “L” and “H”, respectively.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0231.1
The mountain waves that develop during and after the interaction of the cold front with the mountain are shown in Fig. 3 by contours of the vertical velocities and isentropes in an x–z vertical cross section through the center of the mountain at 4.5, 5.5, 6.5, and 7.5 days (Fig. 3a–d, respectively). The waves are clearly strongest at 6.5 days, with vertical-velocity extrema of roughly ±1.5 m s−1, but even at this time there is no wave breaking. There are nontrivial north–south variations in the structure of these waves, which are illustrated in Figs. 5 and 6–9a,b of MD17, but the temporal variation in the spatially averaged strength of the wave response is well illustrated in Fig. 3.

East–west vertical cross sections of w (color fill at 20 cm s−1 intervals) and θ (black lines at 5-K intervals) across the centerline of the mountain at (a) 4.5, (b) 5.5, (c) 6.5, and (d) 7.5 days.
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0231.1
3. Vertical momentum flux distribution









The horizontally integrated momentum flux M′ triggered by the interaction of the baroclinically unstable jet with the 500-m-high ridge is plotted as a function of time and height in Fig. 4. Prior to day 5.5, which is approximately the time of maximum low-level cross-mountain flow, and after day 6.8, M′ is almost constant with height. Between 5.5 and 6.8 days the momentum flux is strongest near the surface and decays with height up to roughly z = 5 km. This low-level vertical momentum flux divergence is not due to wave breaking, which does not occur over the comparatively low 500-m-high mountain. Vertical momentum flux gradients were also evident in the idealized simulations of time-dependent flow over an isolated ridge in CDH05, but the pattern in that case (their Fig. 4a) was very different, with maximum momentum fluxes aloft before the time of maximum wind.

The momentum flux M′ (contour intervals of 0.5 × 1010 N) generated by the baroclinically unstable jet passing over a 500-m-high mountain as a function of z and t. At heights zl below the top of the mountain, the pressure drag due to the portion of the ridge extending above zl is added to the fluxes computed at the same level in the free air (see text).
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0231.1
Since there is no wave breaking, the wave packet accumulation mechanism responsible for the enhancement aloft during the period of flow acceleration in CDH05 might be expected to produce a similar response in the current simulation. Why is the momentum flux distribution in Fig. 4 different? The key difference is the vertical shear in the large-scale cross-mountain flow, which was zero in CDH05, but, at 6.5 days in the present simulation, is roughly 20 m s−1 between the surface and 5 km.
4. The influence of vertical shear

























The mountain profile is taken parallel to the x axis along the center line of the 3D mountain [i.e., is given by Eq. (1) in MD17 evaluated at
The maximum acceleration of the large-scale flow, as well as the greatest potential for wave packet accumulation, occurs at 12.5 h




Momentum fluxes
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0231.1
In summary, as the vertical shear is increased in the 2D simulations shown in Fig. 5, the time-height distribution of








5. Conclusions
We numerically simulated the mountain waves generated as an idealized cyclone growing in a baroclinically unstable flow passes over an isolated 3D 500-m-high ridge. This experimental design avoids artificial start-up transients in the mountain-wave field, thereby facilitating the analysis of the temporal evolution of the momentum fluxes associated with these waves. The mountain height is low enough that there is no wave breaking during the simulation. The momentum fluxes remain relatively constant with height until about the time when the large-scale cross-mountain winds reach their maximum speed. The strongest vertical momentum fluxes occur after the time of maximum winds, roughly between 5.5 and 7 days, when the momentum fluxes decrease with height between the surface and 5 km.
These results differ from the behavior documented in CDH05 for a barotropic jet crossing a similar isolated ridge in which wave packets launched later during the accelerating phase of the large-scale flow accumulate aloft, thereby maximizing the momentum flux during the period of flow acceleration and creating a layer at low levels throughout which the momentum fluxes increase with height. Ray tracing and WKB analysis suggest that the vertical momentum flux profile evolves differently in the current simulation because the vertical shear in the baroclinic jet prevents the accumulation of wave packets aloft. This analysis is supported by additional 2D simulations of waves in shear flows that vary periodically with time. When the vertical shear in the environmental wind is zero, the momentum flux evolution in the 2D simulation was similar to that in CDH05. On the other hand, in the case with the strongest shear, the momentum flux evolution was similar to that generated by the isolated ridge in the baroclinically unstable shear flow.
The weaker sensitivity of the momentum flux to the past history of the flow in large-scale environments where the cross-mountain winds increase with height is good news for those attempting to improve gravity wave-drag parameterizations over mountains exposed to the midlatitude westerlies. Nevertheless, flow transience can still exert an important influence on the momentum flux in such flows by setting the magnitude of the flux. The importance of transience in regulating the momentum flux (and the surface pressure drag) was demonstrated in Menchaca and Durran (2018) by comparing the simulation with the evolving baroclinic jet with other 3D simulations in which the mountain waves were forced by the same isolated 500-m-high ridge in steady large-scale flows representative of the instantaneous near-mountain environment at four successive times in the evolving flow. At 6.5 days, when M′ reaches its extremum, and at 7.5 days, the evolving baroclinic jet generated momentum fluxes that were roughly 50% larger than those for simulations with the corresponding steady large-scale flows. On the other hand, there is some evidence that a precise representation of the mountain-wave dynamics can be avoided in gravity wave drag parameterizations on sufficiently large scales. In particular Smith and Kruse (2018) have had success estimating surface pressure drags over the entire south Island of New Zealand using a very simple linear mountain-wave model and a more sophisticated representation of the effective smoothness of the terrain under different wind speeds.
This research is funded by NSF Grant AGS-1545927. Author Menchaca was also supported by a National Science Foundation Graduate Research Fellowship. Three anonymous reviewers provided very helpful comments. We gratefully acknowledge high-performance computing support from Yellowstone (ark:/85065/d7wd3xhc) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation.
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Because of the WRF Model’s vertical coordinate, the height of the model top varies slightly in space and time.
The sum of these two quantities is the total vertical momentum flux through the bottom of a volume bounded by z = zl and the penetrating mountain top.
Consistent with the linear analysis, here we take z = 0 as the elevation of the surface.
The vertical wavenumber is positive because m = N/U(0, tl) when the packet is launched.