1. Introduction
Orographic precipitation patterns can be significantly affected by terrain-induced flow patterns due to a variety of mechanisms, such as orographic uplift (Smith and Barstad 2004; Smith 1979, among others), windward flow blockage (e.g., Jiang 2003), and gravity wave modification of flow patterns aloft (Colle 2004, among others). The identification of moist, nearly neutral atmospheric conditions with alpine orographic rainfall events (e.g., Rotunno and Ferretti 2001, 2003) prompted Miglietta and Rotunno (2005) to conduct a series of idealized numerical simulations with constant wind and a two-layer atmospheric stability structure, that is, a saturated, nearly neutral troposphere capped by a stable stratosphere. The atmospheric profile used in their study is shown in Fig. 1.
Atmospheric structure used in the Miglietta and Rotunno (2005) idealized numerical simulations of moist nearly neutral flow over a 700-m-high and 10-km-wide Witch of Agnesi mountain (adapted from their Fig. 3); Nm is the moist static stability.
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
A striking and unexpected feature appearing in several simulations of Miglietta and Rotunno (2005) was a region of desaturated air expanding upwind of the mountain in conjunction with an upstream-propagating wave mode (their Fig. 5, reproduced in Figs. 2a and 2b). The authors hypothesized that buoyancy forces induced by local changes in stability resulting from desaturation of the air as it moved over the mountain was the driving force in the upwind progression of this disturbance, as no such disturbance occurred with a very small amplitude mountain in which desaturation did not occur.
(a) Vertical velocity 
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
Using similar background atmospheric flow conditions to Miglietta and Rotunno, Steiner et al. (2005) conducted a series of both moist and dry numerical simulations using the Weather Research and Forecasting Model (WRF). Although a windward expanding region of desaturated air was cited as a causal factor in the reduced upwind flow deceleration observed in some of the moist simulations, Steiner et al. did not explicitly investigate the upstream modes, and only long-time statistics were presented. Recently we analyzed the temporal evolution of the flow in these same simulations and found upstream-propagating modes in both moist and dry simulations in some, but not all, cases.
An important aspect of the background atmospheric conditions in these simulations is the large stability jump between the troposphere and stratosphere. It is well known that the stability jump at the tropopause can result in significant wave reflections for steady-state mountain waves in dry flow (e.g., Eliassen and Palm 1961; Klemp and Lilly 1975; Durran 1992; Barstad and Schuller 2011). The tropopause acting as a “leaky” lid can be important for lee waves in sheared flow (e.g., Keller 1994) and for propagating gravity waves generated by convection (Lane and Zhang 2011). As the troposphere − stratosphere stability difference becomes greater, less energy leaks through and the tropopause begins to act more like a rigid lid (e.g., Berkshire and Warren 1970). Stable layers functioning as reflective surfaces similar to rigid lids have also been noted in density current simulations (Xue 2002) and for elevated inversions in downslope windstorm simulations (e.g., Smith and Skyllingstad 2011).
In fact, as will be shown in section 2, characteristics of the wave motion in Fig. 2a resemble the transient waves studied extensively for “channel flow”—that is, a constant wind and stability layer flowing over an obstacle with a rigid upper boundary in place of the tropopause. These transient modes are generated as a consequence of the impulsive startup of the flow from rest. Similar modes are generated by a localized heat source representing convection in an atmosphere at rest (e.g., Nicholls et al. 1991; Mapes 1993; Lane and Reeder 2001; Lane and Zhang 2011).
Transient propagating waves for stratified channel flow over topography have been studied extensively using towing tank experiments (e.g., Baines 1977, 1979; Castro and Snyder 1988), analytic theory (Janowitz 1981; McIntyre 1972, among others), and numerical modeling (e.g., Lamb 1994; Hanazaki 1989; Uchida and Ohya 2001). Nonlinear transcritical flow near resonance, when the incoming flow speed is approximately equal to the speed of an upstream-propagating mode, has been investigated numerically (e.g., Rottman et al. 1996; Hanazaki 1992, 1993) and using various forms of the Korteweg–de Vries equation (Grimshaw and Yi 1991, among others). These studies demonstrate the ubiquitous occurrence of transient upstream-propagating modes in stratified channel flow over topography, and confirm that only the waves with an upstream propagation speed greater than the background flow speed actually appear upwind of the obstacle.
Here we show that transient modes similar to those appearing in constant wind and stability flow over terrain with a rigid upper boundary also dominate in both moist and dry simulations of idealized two-layer troposphere–stratosphere flow. For certain parameter ranges the upstream waves propagate slowly enough to induce vertical velocity perturbations upstream of the mountain that may last for many hours.
For moist flow upstream-propagating modes alter the relative humidity of the air impinging on the mountain and, if the amplitude of the transient wave is large enough, these modes may modify both the intensity and location of terrain-induced rainfall rates. And, the vertical velocity perturbations associated with the rapidly moving upstream modes could trigger convection far upwind of the topography, just as gravity wave modes with similar vertical structure generated by convective heating can induce new convection (e.g., Mapes 1993; Lane and Zhang 2011). Thus it is important to understand the mechanisms associated with the generation and maintenance of these modes. Furthermore, it is important to determine if these upstream modes are a common feature and, if so, to what extent they may be contributing to both observed and numerically simulated orographic precipitation.
The paper is organized as follows. Fundamental features of the analytic solution for the linear, time-dependent, hydrostatic rigid-lid case are discussed in section 2. Dry numerical simulations of these upstream-propagating modes for both the single-layer and a two-layer troposphere–stratosphere atmospheric profile using a gravity wave model are provided in section 3. The temporal evolution of the upstream-propagating modes in both moist and dry WRF simulations from Steiner et al. (2005) is discussed in section 4. Summary and conclusions are provided in section 5.
2. Linear transient analytic solution
This section discusses fundamental features of the upstream modes in the two-dimensional, time-dependent, hydrostatic, linear analytic solution for constant wind and stability flow over topography in which the tropopause has been replaced by a rigid lid (channel flow). These modes will then be compared with the transient modes observed in the nonhydrostatic numerical simulations of idealized troposphere–stratosphere flow in sections 3 and 4.
The leftward moving modes propagating against the background flow emerge upwind of the mountain only when their propagation speed exceeds the ambient wind speed; that is, c− < 0. Thus for a given K only nk modes actually appear upstream, where nk is the largest integer less than K. All other modes are swept downstream. Note that the transient modes in these hydrostatic solutions translate the initial disturbance away from the mountain with no change in amplitude; however, in the nonhydrostatic solution the wave amplitude falls off as t−½ as t → ∞ [Engevik (1971), see Eq. (4.5)] due to dispersive effects.
There are several important points to note about Eq. (1). First, for a given mode n the amplitude of the upstream, leftward moving wave (second term) is larger than the corresponding downstream, rightward-propagating mode (third term). Second, this linear solution is invalid for K ≡ n, which occurs when the incoming flow speed is equal to the linear long wave phase (and group) velocity of mode n. In this resonant case the amplitude of the upstream-propagating mode becomes infinite [second term in Eq. (1)]. A number of papers have focused on nonlinear effects when K ≈ n, that is, near resonance, using nonlinear numerical modeling (e.g., Rottman et al. 1996) or solving the Korteweg–de Vries equation (Grimshaw and Yi 1991, among others), but these effects are not considered here.
The temporal evolution of the vertical velocity upstream of the mountain [from Eq. (1)] for K = 0.85, 1.27, and 2.55, which have zero, one, and two modes, respectively, penetrating upwind of the topography against the background flow, is shown in Fig. 3. Here both the channel depth and atmospheric stability are fixed, with Zt = 10 km and N = 0.004 s−1, and the background wind speed is varied to change K. The mountain profile is a Witch of Agnesi, 
Temporal evolution of the vertical velocity [Eq. (1)] as a function of K for constant wind and stability flow over topography with a rigid lid at 4, 8, and 12 h. Domain height is 10 km and the Brunt Väisälä frequency 0.004 s−1. Background wind speed 
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
As in Fig. 3, but only the steady-state part of the analytic solution [Eq. (1)].
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
Note that for K = 0.85 all upstream-propagating modes are swept downstream (Figs. 3a–c) and the vertical velocity pattern upwind of the mountain peak differs little from the steady state (Fig. 4a) throughout the entire time period. In contrast, as K increases above one (Figs. 3d–i), the vertical velocity pattern upwind deviates considerably from the steady-state solution (Figs. 4b,c) owing to relatively slow-moving transient modes. In the flow regime where 1 < K < 2 only one transient mode has a leftward propagation speed exceeding the background wind speed and is able to penetrate upwind (K = 1.27; Figs. 3d–f). This mode has a vertical wavelength twice the channel depth, with a cell of upward vertical velocity followed by a cell of downward vertical velocity filling the depth of the troposphere upwind of the topography. Also, in this regime the steady-state hydrostatic gravity wave directly over the mountain has a vertical wavelength between 0.5 and 1.0 times the channel depth. This is similar to the wave pattern seen in Miglietta and Rotunno (2005) (reproduced in Fig. 2a). Note, however, that using their background environmental moist stability to calculate K results in a value of K < 1; that is, all modes should have been swept downstream. This is discussed further in section 4.
For K > 2 the first mode moves rapidly upstream (Fig. 3g) and transient mode 2, with positive vertical velocity at lower levels and negative w aloft at the leading edge, dominates directly upwind of the mountain at 8 and 12 h (Figs. 3h,i). Leftward-moving mode 3, which has a propagation speed less than the background wind speed, is apparent downwind of the mountain in Figs. 3h and 3i.
A plot of the nondimensional wave speeds c−/U0 versus K for leftward-propagating modes traveling faster than the background wind—that is, c− < 0—is shown in Fig. 5. The black lines are comprised of data points representing transient wave speeds calculated from the analytic solution. This clearly shows that for K < 1 all leftward-moving wave modes propagate with speeds less than the background wind and hence are swept downstream. As K increases, the number of modes able to penetrate upwind increases and their nondimensional propagation speeds increase as well. Also plotted are points corresponding to the rigid lid and idealized two-layer troposphere–stratosphere dry numerical simulations discussed in section 3. Note that the mode number and propagation speed of the transients for the numerical simulations correspond closely with the linear analytic rigid-lid solution.
Nondimensional wave speed (c−/U0) of modes traveling upstream faster than the background wind speed as a function of K. Black lines calculated from the linear analytic solution [Eq. (1)]. Symbols indicate wave speeds from the gravity wave numerical model for a linear, single-layer with rigid lid (open diamonds), and a linear (gray filled circles) and nonlinear (asterisk) troposphere–stratosphere atmospheric profile.
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
3. Dry numerical simulations
a. Model description
The time-dependent, nonhydrostatic gravity wave model used for the dry simulations discussed in this section is described in detail in Sharman and Wurtele (1983). This model integrates the two-dimensional Boussinesq equations of motion, and the topography is introduced by prescribing the vertical velocity at the lower boundary—that is, w = U0dη/dx at z = 0—rather than using terrain-following coordinates. Thus the lower boundary condition is strictly linear, making it consistent with the analytic solution [Eq. (1)], and the mountain can be raised slowly, eliminating effects of a shock start. Sensitivity studies varying the number of time steps over which the mountain was raised were conducted, and it was found that slowly increasing the mountain height had only a slight effect on the amplitude of the transient mode. Thus, the mountain was impulsively raised at t = 0 in the simulations presented here. The linear lower boundary formulation in this model has the added advantage that the channel depth between the lower boundary and the tropopause does not change due to a finite mountain height, making direct comparison with analytic theory more consistent. As in section 2, the mountain profile is a Witch of Agnesi with a half-width of 20 km and a height of 10 m for the linear simulations. A useful feature of this model is the ability to turn off the nonlinear terms in the equations of motion, that is, to use advection terms in linearized form. This makes it an ideal choice for comparing numerical simulations with linear analytic solutions, as well as comparing linear and nonlinear simulations.
b. Background atmospheric profile
The numerical model was used to simulate both a finite-depth tropospheric layer capped by a rigid lid, to compare with the analytic solution, and two-layer troposphere–stratosphere flow. The background atmospheric flow for the troposphere–stratosphere simulations follows that used in Miglietta and Rotunno (2005) and Steiner et al. (2005)—that is, a two-layer stability profile with constant wind U0 (Fig. 1)—except the flow is dry. The background wind speed U0, tropopause depth Zt, and tropospheric stability N1 were varied, with particular emphasis on values yielding a similar dynamical regime to that in Fig. 2a, which has one transient mode penetrating upwind; that is, K = NZt/πU0 primarily between 1 and 2.
c. Transient wave modes
A comparison of transient modes in the linear single-layer hydrostatic analytic solution and the nonhydrostatic numerical simulations for both the troposphere-only and troposphere–stratosphere atmospheric profile at 5, 10, and 15 h is shown in Fig. 6. Here U0 = 10 m s−1, N1 = 0.004 s−1, N2 = 0.0228 s−1, and the tropopause depth/rigid-lid height is 9 km, yielding K = 1.15.
Vertical velocity for K = 1.15 at 5, 10, and 15 h for (a)–(c) the linear, hydrostatic analytic rigid-lid solution [Eq. (1)], (d)–(f) the linear nonhydrostatic numerical model with rigid lid at the tropopause, and (g)–(i) the linear nonhydrostatic numerical model with the two-layer troposphere–stratosphere atmospheric profile. Background wind speed is 
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
In Fig. 6 transient mode 1, with a single cell of upward vertical velocity filling the depth of the troposphere, moves upstream at about the same speed in the analytic solution and in both simulations. This slow-moving mode, with a vertical wavelength twice the tropopause depth and a maximum velocity in the midtroposphere, creates a broad region of uplift upstream of the mountain which lasts the entire 15 h of simulation. The region of maximum uplift of the slowly upstream-propagating vertical velocity cell is located at about x = −40, −65, and −90 km upstream of the mountain peak at 5, 10, and 15 h, respectively. Closer to the mountain a cell of downward velocity can be seen expanding upstream with time. Nonhydrostatic dispersion, which is included in the numerical model but not the hydrostatic analytic solution, results in a slight decrease in amplitude of the transient mode as it propagates upwind in the rigid-lid simulations (Figs. 6d–f). In addition, in the troposphere–stratosphere simulation the amplitude of the upstream-propagating mode visibly decreases further as the wave moves upwind owing to leakage of energy into the stratosphere (Figs. 6g–i). Leftward-propagating transient mode 2, which has a propagation speed less than the incoming wind speed, is clearly evident downstream at x = +90 km at 5 h in the troposphere–stratosphere simulation (Fig. 6g). This mode has vertical wavelength Zt and upward (downward) motion in the lower (upper) troposphere. The other transient modes have much smaller amplitudes and do not contribute significantly to the overall wave pattern in the region shown.
To further explore the behavior of the transient modes for the idealized troposphere–stratosphere atmospheric profile, K was varied by altering both the tropopause depth and the background wind speed. As in the single-layer analytic solution, the speed of the upstream-propagating mode in the troposphere–stratosphere simulations increases as K increases. In Fig. 7 the atmospheric stability is fixed (N1 = 0.004 s−1, N2 = 0.0228 s−1), the tropopause depth Zt = 10 km, and U0 is 10 m s−1 (Figs. 7a–c) and 5 m s−1 (Figs. 7d–f), yielding K = 1.27 and 2.55, respectively. The corresponding single-layer analytic solution for these K values was presented in Figs. 3d–i.
Vertical velocity as a function of K for the linear, nonhydrostatic simulations of troposphere–stratosphere atmospheric profile over topography at 4, 8, and 12 h. Tropopause depth is 10 km and tropospheric Brunt Väisälä frequency 0.004 s−1. Background wind speed 
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
Although the atmospheric wind and stability in Figs. 7a–c are the same as in Figs. 6g–i, the minor increase in tropopause depth (from 9 to 10 km) results in a slight increase in K and a concomitant increase in speed of the transient waves. Now downward vertical velocity dominates the troposphere between −50 and −100 km upwind of the mountain by 12 h (Fig. 7c), whereas in Figs. 6h and 6i upward vertical velocity persists in this region throughout the time period from 10 to 15 h.
As K increases above 2 (Figs. 7d–f) transient mode 1, with vertical wavelength 2 Zt, moves quickly upstream and the second leftward-moving mode, with vertical wavelength Zt, dominates just upwind of the mountain, as in the analytic solution for channel flow presented in section 2 (Figs. 3g–i). This creates a persistent pattern of upward velocity in the lower troposphere and downward velocity in the upper troposphere upstream of the topography for many hours. Leftward-propagating mode 3, with vertical wavelength (2/3)Zt, is swept downstream and is clearly visible downwind at 8 and 12 h (Figs. 7e,f).
Thus, the linear numerical simulations show that for this idealized two-layer troposphere–stratosphere atmospheric profile the stability jump at the tropopause, acting like a leaky lid, selects the same transient modes in the troposphere as the rigid lid does in channel flow. And, as with channel flow, minor changes in atmospheric parameters that result in slight changes in K values can markedly alter the horizontal extent and persistence of the vertical velocity pattern upstream, which in turn could have substantial effects on the upstream relative humidity profile when moist air is impinging on the mountain.
The speed of the transient modes moving upwind in the numerical simulations agrees well with linear analytic theory. As discussed in section 2, Fig. 5 shows a plot of nondimensional wave speed c−/U0 versus K. Transient speeds calculated from the analytic solution comprise the black lines, and wave speeds measured in the linear numerical simulations for a single layer with a rigid lid and the troposphere–stratosphere atmospheric profile are indicted by the open diamonds and gray filled circles, respectively. To determine how nonlinearities affect the transient wave speeds, further simulations were run with similar atmospheric conditions but increasing the mountain height and switching on the nonlinear advection terms in the model. The same modes dominate upstream, and propagate at speeds close to the speed predicted by linear theory, as indicated by the asterisk symbols in Fig. 5.
Vertical velocity patterns from simulations using the nonlinear version of the gravity wave model and increasing the mountain height to 1 km for K = 2.55, which has two transients penetrating upwind, are shown in Fig. 8. In these simulations tropospheric stability is 0.008 s−1, U0 = 10 m s−1, and the tropopause depth is 10 km. Also shown in Fig. 8 is the effect of varying the stratospheric stability; the stratospheric stability is increased from 0.0114 s−1 in Figs. 8a and 8b, to 0.0228 s−1 in Figs. 8c and 8d, to 0.0456 s−1 in Figs. 8e and 8f. Note that the amplitude of the transient modes increases as the stability jump at the tropopause increases, as would be expected.
Vertical velocity at 1.5 and 5 h for troposphere–stratosphere simulations using the nonlinear gravity wave model and varying stratospheric stability. Mountain height and half-width are 1 km and 20 km, respectively. Tropospheric K = 2.55 (N1 = 0.008 s−1, U0 = 10 m s−1, and tropopause depth is 10 km). Stratospheric stability is (a),(b) 0.0114, (c),(d) 0.0228, and (e),(f) 0.0456 s−1. Contour interval is 0.05 m s−1.
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
d. Nonhydrostatic effects
In contrast to the hydrostatic linear analytic solution presented in section 2, the numerical simulations, even with this gently sloped and broad 20-km half-width mountain, include nonhydrostatic effects that impact both the steady-state and transient wave solutions.
Nonhydrostatic effects on the steady-state wave solution produce lee waves downstream of the topography in the numerical simulations (contrast Figs. 6b,c with Figs. 6e,f and Figs. 6h,i for 0 < x < +100 km). Linear stationary lee waves have a horizontal wavelength determined by the vertical variations in wind and stability (Wurtele et al. 1987, 1996; Keller 1994, among others) and amplitude dependent on the mountain forcing. In our case the lee waves are trapped by the rigid lid in the single-layer simulations and partially trapped by the large stability jump at the tropopause in the two-layer simulations. The amplitude of the downstream lee wave is reduced due to leakage into the stratosphere in the troposphere–stratosphere simulation. Note that transient modes swept downstream can constructively or destructively interfere with the stationary lee wave pattern, resulting in temporal variability in the standing wave pattern.
Nonhydrostatic effects on the transient modes create a train of dispersive traveling waves. Since the magnitude of the group velocity is smaller than the phase velocity in the nonhydrostatic case, the wave pattern in the nonhydrostatic transient solution has an additional oscillatory behavior that moves more slowly upstream of the mountain, trailing behind the hydrostatic upwind-propagating disturbance (e.g., Hanazaki 1989). Pandya et al. (1993) have nicely illustrated these nonhydrostatic effects for thermally generated gravity waves in an atmosphere at rest.
Note that the hydrostatic approximation is valid for a disturbance when the horizontal scale is much larger than the vertical scale (e.g., Lin 2007, p. 19). For the atmospheric structure investigated here the tropopause depth, which plays a key role in the wave development, is the appropriate vertical length scale. Using the mountain half-width for the horizontal scale, the flow response is approximately hydrostatic when the parameter a/Zt is large, as noted in previous channel flow studies (e.g., Rottman et al. 1996). This is in contrast to an infinitely deep, constant wind and stability atmosphere where the vertical scale is U0/N, related to the vertical wavelength, and the parameter Na/U0, a measure of the advective time scale compared to the period of a buoyancy oscillation (e.g., Smith 1979), is used to determine hydrostaticity. Various authors have used different cutoff values of Na/U0 for the validity of the hydrostatic assumption; for example, in Lin and Wang (1996) Na/U0 > 7.5 and in Laprise and Peltier (1989) Na/U0 > 10. In all simulations discussed in this section Na/U0 is greater than or equal to 8.
Nonhydrostatic dispersion producing a train of transient waves is illustrated in Fig. 9 for rigid-lid numerical simulations. Here K is fixed at 1.5 and U0 = 10 m s−1, so the one transient mode penetrating upwind propagates at the same speed according to hydrostatic theory in all three simulations. The mountain half-width is 20 km, and the layer depth is changed to yield an increasingly more nonhydrostatic solution; that is, a/Zt = 6.7, 3.3, and 1.7 in Figs. 9a–c, respectively. Nonhydrostatic effects are particularly evident in Fig. 9c, where a train of waves can be seen propagating upwind, with alternating up and downdrafts. Note that the position of the maximum amplitude of the first cell advances slightly more slowly as nonhydrostatic effects become more dominant. Although the stability is also changing as Zt varies, in order to hold K fixed, Na/U0 ≥ 8 in Fig. 9.
Nonhydrostatic effects on upstream-propagating transient wave mode 1 in linear numerical simulations of a single layer capped by a rigid lid for K = 1.5 at 11 h; 
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
Interestingly, in the regime K < 1, in which all leftward-moving transient modes have speeds less than the background wind and hence are swept downstream, there are no stationary nonhydrostatic lee waves possible (e.g., Lamb 1994). Now leftward-propagating mode one is moving the most slowly downstream and may dominate downwind of the mountain for hours. When nonhydrostatic effects are significant, the pattern produced by this dispersive packet of waves associated with this transient mode may resemble a lee wave train (Fig. 10). Since the nonhydrostatic waves travel more slowly against the flow than the hydrostatic mode, they are swept downwind more quickly and actually precede the hydrostatic component of the transient wave downstream. Thus, nonsteady lee wave patterns may be generated not only by mechanisms such as nonlinearities or unsteadiness in the background flow (e.g., Nance and Durran 1997, 1998; Ralph et al. 1997), but also by these slowly moving nonhydrostatic transient wave packets.
Vertical velocity contours for a single-layer rigid-lid simulation showing the nonhydrostatic wave packet associated with leftward-moving transient mode 1 being swept downstream at (a) 20 and (b) 35 h. Tropopause depth 
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
In summary, for the two-layer dry simulations the stability jump at the tropopause acts as a leaky lid in the troposphere–stratosphere simulations, selecting the same transient modes in the troposphere as the rigid lid in channel flow. The wave amplitude depends on the stability jump between troposphere and stratosphere, with less energy leakage for larger jumps. The same modes dominate in both linear and nonlinear numerical simulations. When nonhydrostatic effects become important, the transient mode becomes a traveling wave packet and, when these modes are swept downstream, they may resemble an unsteady lee wave.
4. Moist numerical simulations
Steiner et al. (2005) conducted a suite of both moist and dry numerical simulations with background atmospheric flow conditions similar to those of Miglietta and Rotunno (2005) using the Weather Research and Forecasting model (Skamarock et al. 2005), version 1.3. However, in that study the upstream modes were not explicitly investigated and only long-time (30 h) statistics were presented. We analyzed the temporal evolution of the flow in these simulations and found upstream-propagating modes in both moist and dry simulations in some, but not all, cases. Representative simulations of this two-layer flow over a 500-m-high Witch of Agnesi mountain with a half-width of 20 km are discussed here.
Upstream-propagating modes in the dry troposphere–stratosphere WRF simulations have the same characteristics as the modes discussed in section 3, with the number of upwind modes and relative propagation speeds increasing with increasing tropospheric K (Fig. 11). Here tropospheric stability N1 = 0.004 s−1, the tropopause depth is 11 km, and the wind is decreased from 20 m s−1 in the top panel, to 10 m s−1 in the middle panel, and down to 5 m s−1 in the bottom panel, yielding a K of 0.7, 1.4, and 2.8, respectively. Thus there are zero, one, and two transient modes able to propagate upwind of the mountain in Figs. 11a,b, 11c,d, and 11e,f, respectively. Higher transient modes that are swept downstream are evident downwind of the mountain in Figs. 11c,d (mode 2) and Fig. 11f (mode 3). Thus the existence of these transient modes is model independent—that is, despite differences in topography representation and model initialization procedures, traveling wave modes are a persistent feature in dry simulations for the two-layer troposphere–stratosphere atmospheric profile.
Vertical velocity contours from dry troposphere–stratosphere WRF simulations described in Steiner et al. (2005). Mountain is a 500-m Witch of Agnesi with a 20-km half-width. Tropospheric stability 
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
Similar transient modes were found in the moist-saturated troposphere–stratosphere simulations. Figure 12 shows vertical velocity (contour lines) and relative humidity (color) at hours 4 and 8 from three moist flow simulations in which the moist tropospheric stability Nm, calculated following Miglietta and Rotunno (2005), is 0.002 s−1 and the tropopause depth is 11 km. As in the dry simulations above, the background wind speed is decreased from 20 m s−1 in the top panel, to 10 m s−1 in the middle panel, and down to 5 m s−1 in the bottom panel, resulting in an increasing Km (=NmZt/U0) from top to bottom. Thus Figs. 11 and 12 are comparable in the sense that the flow regimes have zero, one, and two modes propagating upwind in panels (a) and (b), (c) and (d), and (e) and (f), respectively.
Vertical velocity (lines) and relative humidity (color) from moist WRF simulations at 4 and 8 h. Mountain is a 500-m Witch of Agnesi with a 20-km half-width. Moist stability 
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
In Figs. 12a and 12b all transient modes are swept downstream. There is no region of reduced relative humidity expanding upstream and the upward vertical velocity cell along the upwind slope of the mountain reaches steady state very quickly. The rapidly moving transient wave swept downstream is associated with local changes in humidity and moves out of the domain by 8 h.
When the background wind speed is decreased to 10 m s−1 leftward-propagating mode one is traveling faster than the background flow and is associated with a broad region of reduced relative humidity extending upwind (Figs. 12c,d). By 8 h this region has expanded upwind as far as −200 km (Fig. 12d). Vertical velocity perturbations associated with leftward-propagating mode 2, which is swept downstream, are also associated with changes in relative humidity in the upper troposphere as it moves downwind from the mountain. This flow regime is analogous to dry flow with 1 < K < 2 (e.g., Figs. 6g–i, 7a–c, 9, and 11c,d), with only transient mode 1 penetrating upwind, mode two clearly visible downstream, and a steady-state hydrostatic wave directly above the mountain with a vertical wavelength between 0.5 and 1 times the tropopause depth. This is similar to the wave pattern seen in Miglietta and Rotunno (2005). As the wind is decreased further to 5 m s−1, transient mode 1 moves more quickly upstream of the mountain, and has traveled out of the domain by 8 h (Fig. 12f). At this time mode 2 is visible upwind at approximately −120 km.
Note that simply substituting the moist static stability directly into the nondimensional number K to calculate a corresponding Km, as well as the propagation speed of the upstream modes, does not result in the same cutoff values as in the dry case. In fact, from analyzing a number of moist simulations it appears that the effective stability determining the speed of the modes is actually a value somewhere between the moist and dry stability. This is likely due to the mechanism discussed in Miglietta and Rotunno (2005), in which perturbations induced by the air flowing over the mountain result in regions of subsaturated air that have much larger static stability locally. This increased stability plays a role in conditioning the background flow, allowing the flow to transition between a regime in which there are no modes with a leftward propagation speed exceeding the background flow to a regime conducive to upwind penetration of transient mode one. This effect explains why the wave pattern for the small 50-m-high mountain presented in Miglietta and Rotunno (2005) (their Fig. 4) differs from the wave pattern for the 700-m-high mountain (reproduced in our Fig. 2a).
In the moist-saturated simulations shown in Fig. 12 vertical motions associated with the transient modes induce expanding regions of desaturated air. For the same background flow conditions as in Figs. 12a,b and 12c,d, if the mountain height is increased to 1.5 km, the more vigorous vertical velocities associated with upstream-propagating modes induces areas of precipitation that also move upwind (Fig. 13). In Fig. 13a the background flow is U0 = 20 m s−1 and precipitation remains concentrated on the mountain slope. In contrast, when the background wind is decreased to 10 m s−1, transient mode 1 moves upwind and creates precipitation moving along with the wave (Fig. 13b). Here lines represent model precipitation accumulated over the last 30 min, plotted for hours 2–6, with the accumulation peak shifting leftward with each hour. Furthermore, preliminary investigation of simulations in which the background flow is initially subsaturated—for example, when the relative humidity of the incoming flow is 70%–90% (not shown)—vertical velocity perturbations associated with these modes increases the relative humidity upwind of the mountain, potentially triggering precipitation.
Spatial distribution of 30-min accumulated precipitation as a function of time for WRF simulations with (a) no upstream-propagating mode (U0 = 20 m s−1) and (b) one upstream-propagating mode (U0 = 10 m s−1). Background atmospheric conditions as in Figs. 12a,b and 12c,d. Mountain is centered at x = 0, with a height of 1.5 km and half-width of 20 km. Lines correspond to accumulated precipitation over the past 30 min for hours 2–6.
Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-12-06.1
Thus, the upstream modes in the saturated case are similar to the upstream modes in the dry case in that 1) both are transient perturbations induced by the impulsive start; 2) the number of leftward-moving modes appearing upwind depends on the background stability, wind, and tropopause depth; and 3) either trapping by a rigid lid or partial trapping by the large stability jump at the tropopause is essential. However, moisture alters the propagation of the transient modes in fundamental ways so that the number and propagation speeds of the upwind penetrating transient waves cannot be calculated by simply replacing the dry stability by the moist stability in K. In addition, for the gravest mode propagating into its saturated environment, downward vertical velocity leads the disturbance. These important differences are currently under investigation and will be elucidated in an upcoming paper by the second author and a colleague.
5. Summary and conclusions
In the idealized two-dimensional simulations of saturated, nearly neutral flow over topography, Miglietta and Rotunno (2005) found a wave mode and region of desaturated air expanding upwind of the mountain with time (reproduced in our Fig. 2). The generation and propagation mechanism of this upstream-moving mode was attributed to localized buoyancy variations due to mountain-induced flow patterns creating localized desaturated regions and altered stability.
Here we have shown that this wave mode is one of the transient modes excited by the initialization of the troposphere–stratosphere flow over topography, which induces transient upstream and downstream-propagating wave modes in both moist and dry two-dimensional flow. Modes traveling with speeds faster than the background wind appear upstream of the topography and may at times move slowly enough that their effects persist upwind for hours. These transient modes are similar to the modes observed in dry channel flow—that is, a single constant wind and stability layer in which the tropopause is replace by a rigid lid (Baines 1977; Hanazaki 1989, among others). In moist-saturated air, these traveling waves induce a concomitant upwind-progressing desaturated zone. Also, waves with sufficiently large vertical velocities induce a precipitation pattern that moves upstream with the mode, even if the incoming flow is subsaturated (not shown).
In the moist simulations of Miglietta and Rotunno (2005) the tropospheric depth, moist stability, and wind velocity, which determine the propagation speed of the modes, were such that that no upwind propagation was possible for a very small mountain; however, some desaturation of the air encountering the 700-m-high mountain increased the effective stability to a value between moist and dry stability so that the flow became ideally suited for the generation of a slow-moving mode 1 upwind of the mountain. In addition, the large stability jump at the tropopause reduced the leakage of wave energy into the stratosphere. Although the focus herein has been on a deep, low-stability troposphere, topped by a stable stratosphere, the regime associated with the waves explored here would also include a lower-stability boundary layer capped by a stable layer aloft.
Transient modes, which are a fundamental feature of the time evolution of flow over topography, are part of the solution in any numerical simulation in which the flow is impulsively started from rest at time zero. Often transient features propagate out of the domain during the spinup time of the numerical simulation and do not significantly affect the phenomena being simulated. However, when atmospheric conditions are such that one or more modes propagate slowly upstream of the mountain, their vertical velocity cells may persist upwind for hours, significantly altering the moisture content of the airstream impacting the mountain, as in Miglietta and Rotunno (2005). For moist but unsaturated flow, upstream-propagating waves can increase the relative humidity upwind of the mountain. Thus, just as the propagating wave modes generated by convection with similar vertical structure can trigger new convection (e.g., Mapes 1993; Lane and Zhang 2011), these modes can induce precipitation upwind of topography under certain conditions. This type of feedback loop between the terrain-induced flow dynamics and thermodynamics was diagramed in Steiner and Rotunno (2004).
Whether these modes play a significant role in nature needs to be determined. It is possible that upstream-propagating transient gravity waves could be excited whenever there is a rapid change in the background flow, such as the passage of a front, as this would be similar to an impulsive start. Also, for saturated flow, transients can be generated when orographic flow perturbations result in moisture changes that alter the effective stratification, resulting in a transition to a dynamical regime conducive to transient gravity waves penetrating upwind of the mountain.
The findings from this study suggest that caution must be taken when simulating moist flow over topography, as transient wave modes can alter rain patterns and relative humidity upwind of the mountain. To what extent these transient modes contribute to rainfall rates observed upwind of topography both in nature and in numerical simulations needs to be investigated further.
Acknowledgments
The initial research for this study was supported by the National Science Foundation (NSF) under Grants ATM-9906012 and ATM-0223798 to Princeton University. We thank Dr. Todd Lane for helpful comments on an earlier version of this paper. We also thank the anonymous reviewers whose comments on an earlier version of this paper helped improve the presentation.
REFERENCES
Baines, P. G., 1977: Upstream influence and Long’s model in stratified flows. J. Fluid Mech., 82, 147–159.
Baines, P. G., 1979: Observations of stratified flow over two-dimensional obstacles. Tellus, 31, 351–371.
Baines, P. G., and F. Guest, 1988: The nature of upstream blocking in uniformly stratified flow over long obstacles. J. Fluid Mech., 188, 23–45.
Barstad, I., and F. Schuller, 2011: An extension of Smith’s linear theory of orographic precipitation: Introduction of vertical layers. J. Atmos. Sci., 68, 2695–2709.
Berkshire, F. H., and F. W. G. Warren, 1970: Some aspects of linear lee wave theory for the stratosphere. Quart. J. Roy. Meteor. Soc., 96, 50–66.
Castro, I. P., and W. H. Snyder, 1988: Upstream motions in stratified flow. J. Fluid Mech., 187, 487–506.
Colle, B. A., 2004: Sensitivity of orographic precipitation to changing ambient conditions and terrain geometries: An idealized modeling perspective. J. Atmos. Sci., 61, 588–606.
Durran, D. R., 1992: Two-layer solutions to Long’s equation for vertically propagating mountain waves: How good is linear theory? Quart. J. Roy. Meteor. Soc., 118, 415–433.
Eliassen, A., and E. Palm, 1961: On the transfer of energy in stationary mountain waves. Geofys. Publ., 22 (3), 1–23.
Engevik, L., 1971: On the flow of a stratified fluid over a barrier. J. Eng. Math., 5, 81–88.
Grimshaw, R., and Z. Yi, 1991: Resonant generation of finite-amplitude waves by the flow of a uniformly stratified fluid over topography. J. Fluid Mech., 229, 603–628.
Hanazaki, H., 1989: Upstream advancing columnar disturbances in two-dimensional stratified flow of finite depth. Phys. Fluids A, 1, 1976–1987.
Hanazaki, H., 1992: A numerical study of nonlinear waves in a transcritical flow of stratified fluid past an obstacle. Phys. Fluids A, 4, 2230–2243.
Hanazaki, H., 1993: On the nonlinear internal waves excited in the flow of a linearly stratified Boussinesq fluid. Phys. Fluids A, 5, 1201–1205.
Janowitz, G. S., 1981: Stratified flow over a bounded obstacle in a channel of finite depth. J. Fluid Mech., 110, 161–170.
Jiang, Q., 2003: Moist dynamics and orographic precipitation. Tellus, 55A, 301–316.
Keller, T. L., 1994: Implications of the hydrostatic assumption on atmospheric gravity waves. J. Atmos. Sci., 51, 1915–1929.
Klemp, J. B., and D. K. Lilly, 1975: The dynamics of wave-induced downslope winds. J. Atmos. Sci., 32, 320–339.
Lamb, K. G., 1994: Numerical simulations of stratified inviscid flow over a smooth obstacle. J. Fluid Mech., 260, 1–22.
Lane, T. P., and M. J. Reeder, 2001: Convectively generated gravity waves and their effect on the cloud environment. J. Atmos. Sci., 58, 2427–2440.
Lane, T. P., and F. Zhang, 2011: Coupling between gravity waves and tropical convection at mesoscales. J. Atmos. Sci., 68, 2582–2598.
Laprise, R., and W. R. Peltier, 1989: On the structural characteristics of steady finite-amplitude mountain waves over bell-shaped topography. J. Atmos. Sci., 46, 586–595.
Lin, Y. L., 2007: Mesoscale Dynamics. Cambridge University Press, 630 pp.
Lin, Y. L., and T. A. Wang, 1996: Flow regimes and transient dynamics of two-dimensional stratified flow over an isolated mountain ridge. J. Atmos. Sci., 53, 139–158.
Mapes, B. E., 1993: Gregarious tropical convection. J. Atmos. Sci., 50, 2026–2037.
McIntyre, M. E., 1972: On Long’s hypothesis of no upstream influence in uniformly stratified or rotating fluid. J. Fluid Mech., 52, 209–243.
Miglietta, M. M., and R. Rotunno, 2005: Simulations of moist nearly neutral flow over a ridge. J. Atmos. Sci., 62, 1410–1427.
Nance, L. B., and D. R. Durran, 1997: A modeling study of nonstationary trapped mountain lee waves. Part I: Mean-flow variability. J. Atmos. Sci., 54, 2275–2291.
Nance, L. B., and D. R. Durran, 1998: A modeling study of nonstationary trapped mountain lee waves. Part II: Nonlinearity. J. Atmos. Sci., 55, 1429–1445.
Nicholls, M. E., R. A. Pielke, and W. R. Cotton, 1991: Thermally forced gravity waves in an atmosphere at rest. J. Atmos. Sci., 48, 1869–1884.
Pandya, R. E., D. R. Durran, and C. S. Bretherton, 1993: Comments on “Thermally forced gravity waves in an atmosphere at rest.” J. Atmos. Sci., 50, 4097–4101.
Ralph, F. M., P. J. Neiman, T. L. Keller, D. Levinson, and L. Fedor, 1997: Observations, simulations, and analysis of nonstationary trapped lee waves. J. Atmos. Sci., 54, 1308–1333.
Rottman, J. W., D. Broutman, and R. Grimshaw, 1996: Numerical simulations of uniformly stratified fluid flow over topography. J. Fluid Mech., 306, 1–30.
Rotunno, R., and R. Ferretti, 2001: Mechanisms of intense alpine rainfall. J. Atmos. Sci., 58, 1732–1749.
Rotunno, R., and R. Ferretti, 2003: Orographic effects on rainfall in MAP cases IOP2b and IOP8. Quart. J. Roy. Meteor. Soc., 129, 373–390.
Sharman, R. D., and M. G. Wurtele, 1983: Ship waves and lee waves. J. Atmos. Sci., 40, 396–427.
Skamarock, W. C., J. B. Klemp, J. Dudhia, D. O. Gill, D. M. Barker, W. Wang, and J. G. Powers, 2005: A description of the Advanced Research WRF version 2. NCAR/TN-468+STR, 88 pp. [Available online at http://www.mmm.ucar.edu/wrf/users/docs/arw_v2.pdf.]
Smith, C. M., and E. D. Skyllingstad, 2011: Effects of inversion height and surface heat flux on downslope windstorms. Mon. Wea. Rev., 139, 3750–3764.
Smith, R. B., 1979: The influence of mountains on the atmosphere. Advances in Geophysics, Vol. 21, Academic Press, 87–230.
Smith, R. B., and I. Barstad, 2004: A linear theory of orographic precipitation. J. Atmos. Sci., 61, 1377–1391.
Steiner, M., and R. Rotunno, 2004: Airflow within complex terrain. Bull. Amer. Meteor. Soc., 85, 1059.
Steiner, M., R. Rotunno, and W. C. Skamarock, 2005: Examining the moisture effects on idealized flow past 2D hills. Preprints, 11th Conf. on Mesoscale Processes, Albuquerque, NM, Amer. Meteor. Soc., 5M.7. [Available online at https://ams.confex.com/ams/32Rad11Meso/techprogram/paper_95767.htm.]
Uchida, T., and Y. Ohya, 2001: Numerical study of stably stratified flows over a two-dimensional hill in a channel of finite depth. Fluid Dyn. Res., 29, 227–250.
Wurtele, M. G., R. D. Sharman, and T. L. Keller, 1987: Analysis and simulations of a troposphere–stratosphere gravity wave model, Part I. J. Atmos. Sci., 44, 3269–3281.
Wurtele, M. G., R. D. Sharman, and A. Datta, 1996: Atmospheric lee waves. Annu. Rev. Fluid Mech., 28, 429–476.
Xue, M., 2002: Density currents in shear flows: Effects of rigid lid and cold-pool internal circulation, and application to squall line dynamics. Quart. J. Roy. Meteor. Soc., 128, 47–73.
