## 1. Introduction

Rotors usually form in conjunction with high-amplitude mountain waves (Holmboe and Klieforth 1957). Until recently, these horizontal or quasi-horizontal circulations have been largely neglected as an important component of the mountain-wave system. Rotors are known to be an aviation hazard due to frequent moderate, severe, or sometimes extreme turbulence, as well as the possibility of strong, shifting surface winds (Carney et al. 1997).

When high-amplitude mountain waves are present, widespread turbulence is almost always found in the lower levels to the lee of elevated terrain. The Lower Turbulent Zone LTZ; Lester and Fingerhut 1974) describes the turbulent layer downwind of elevated terrain, above the surface layer, and under an undulating stable layer (e.g., an inversion) comprising the waves themselves. Turbulence within the LTZ has also been referred to as rotor-zone turbulence (WMO 1973). We note that the aviation community often uses the term “rotor” when referring to the LTZ. The spatial scale of rotors within the LTZ range from a few kilometers to a few tens of meters. Intense small-scale rotors within a larger turbulent circulation have been called subrotors (Doyle and Durran 2007).

Based on observations, at least two types of mountain-wave/rotor systems (MW/R) are thought to exist (Kuettner 1959; Lester and Fingerhut 1974). The first type is associated with MW/R comprising trapped waves. The second type is associated with a MW/R resembling a hydraulic jump.

Recently, high-resolution numerical model studies have begun to elucidate rotor dynamics. Doyle and Durran (2002) found that positive horizontal vorticity in the surface layer is lifted as the boundary layer separates due to wave-induced adverse pressure gradients along the lee slope. This provides a horizontal-vorticity source for rotors associated with trapped waves. Jiang et al. (2007) found that simulated boundary layer separation can be induced by trapped waves, internal bores (associated with low-level wave breaking), and undular jumps. In another numerical modeling study, Vosper et al. (2006) found that the boundary layer flow separation depends on the lee-wave-induced pressure field, along with lee-wave wavelength and surface roughness.

Hertenstein and Kuettner (2005, hereafter HK) performed two dimensional, high-resolution simulations using topography with a steep lee slope representing the Sierra Nevada in California. In their study, initial wind and stability profiles were based on data from the 1950s Sierra Wave Project (SWP; Holmboe and Klieforth 1957; Grubiśiĉ and Lewis 2004). Two simulated MW/R types were found to depend on the magnitude of vertical shear within a near-mountaintop, upstream inversion. Weak shear within the inversion leads to MW/R resembling a hydraulic jump, while stronger shear leads to MW/R comprised of trapped waves. Results from a recent field campaign in the Sierra Nevada [Terrain-induced Rotor Experiment (T-REX; Grubiśiĉ and Kuettner 2003; Grubiśiĉ et al. 2008)] should further elucidate the structure of MW/R.

The role of near-mountaintop inversions in mountain waves has been recognized for decades (e.g., Kuettner 1938, 1939; Corby and Wallington 1956). More recently, Vopser (2004) presented results from two-dimensional simulations in which he investigated the effect of inversions on mountain lee waves and rotors. The simulations used a height-independent initial wind profile and a variety of inversion strengths and heights. When an upwind internal Froude number falls below a certain value, resonant waves form with rotors under wave crests. As the Froude number is further reduced, the resonant waves are replaced by a hydraulic jump. Results from numerous simulations were summarized in a regime diagram showing how flows depend on inversion strength and height of the inversion.

Mobbs et al. (2005) used surface and radiosonde observations to study flows in the lee of a small mountain range in the Falkland Islands. They found that a strong upwind inversion was a necessary factor in the formation of rotors and strong surface winds. Subsequently, Sheridan and Vosper (2006) applied the Vosper (2004) regime diagram to realistic topography representing that on the Falkland Islands.

Many parameters influence two-dimensional MW/R formation, for instance, mountain height, lee-slope aspect ratio, wind speed at mountaintop, as well as wind speed, shear, and stability within different layers through the troposphere. The goal of this study is to examine the effect of an upstream inversion on MW/R type. We expand on the work of Vopser (2004) and HK by considering environmental wind profiles with vertical shear as well as a large topographical barrier with a steep lee slope. It was found that the strength and height of the inversion has a profound influence on the type of MW/R that develops.

In the next section, we describe the numerical model and experimental design. Results from simulations are presented in section 3, while formation mechanisms are discussed in section 4. Conclusions appear in section 5.

## 2. Numerical model and experimental design

Two-dimensional simulations were performed using version 4.3.0 of the nonhydrostatic, fully compressible Regional Atmospheric Modeling System (RAMS). A terrain-following vertical coordinate in RAMS has been found effective for modeling atmospheric flows in steep terrain (e.g., Poulos et al. 2000). Complete details regarding model equations, vertical coordinate, grid structure, time and space differencing, as well as available parameterizations (e.g., radiation, turbulence, and microphysics) have been elaborated upon in Pielke et al. (1992), Cotton et al. (2003), and numerous references therein.

The model configuration is similar to HK. A single model grid with 150-m horizontal spacing was employed over 1700 points. Vertical spacing ranged from 20 m at the lowest level stretching to 150 m over 145 points to a maximum height of 19.7 km. Five vertical grid points span the 600-m-deep inversion upstream of the mountain. Horizontal and vertical grid spacing are sufficient to resolve gross features of the MW/R. Each simulation was initialized with a single vertical temperature and wind profile representing the atmospheric state over the entire domain.

A sponge layer starting at 15.2 km was applied to allay wave reflection at the upper boundary, while a no-slip lower boundary condition was used. At the lateral boundaries, the schemes of Klemp and Lilly (1978) and Durran and Klemp (1982) were applied, allowing energy to pass from the model domain. The scheme of Louis (1979) was employed to represent turbulence in the surface layer, specifically, vertical fluxes of horizontal momentum. Surface roughness over the entire domain was set to 0.1 m. The Deardorff (1980) scheme was used to parameterize turbulence above the surface layer. As reported in HK, results were found to be insensitive to the turbulence parameterization used.

Topography was modeled using the same two-sided Witch of Agnesi profile as in HK: a 40-km half-width on the upstream side and 5-km half-width on the lee side roughly approximate the profile of the Sierra Nevada, especially the steep lee slope. The mountaintop in the model was set to 2500 m, which corresponds to the relief between the Sierra Nevada crest and the leeside terrain (Owens Valley). Simulations did not use a soil model or radiation parameterization (i.e., no heat was transferred from the surface) and were run with no moisture. High Rossby number flows justify neglect of the Coriolis force.

Simulations were designed to explore the effect of inversion strength and height. We define inversion strength (Δ*θ*) as the difference between the isentrope at the inversion top and bottom. In our simulations, the minimum Δ*θ* is 1 K, while the maximum is 20 K. We define inversion height (*z _{i}*) as the height of the lowest initial inversion isentrope above the surface well away from the mountain. Simulations used

*z*= 2700 m (200 m above the mountaintop as in HK), 3225 m, 4050 m, or 5400 m. These

_{i}*z*were chosen to facilitate comparisons with previous studies (e.g., Vosper 2004), though quantitative comparisons were difficult because of differences between studies such as mountain height, lee-slope aspect ratio, and the presence of a sheared environmental flow. The same inversion depth of 600 m is used in all simulations.

_{i}Two wind profiles were used, based on observations taken during the SWP (Fig. 1). Both wind profiles featured constant shear (4.29 × 10^{−3} s^{−1}) above and below the inversion but differ in the magnitude of shear within the inversion, referred to as internal shear. The first wind profile, with no internal shear, produced type 2 MW/R in the simulations of HK, whereas the second profile, with 13.33 × 10^{−3} s^{−1}, internal shear led to type 1 MW/R. The wind speed is constant near the tropopause between 11.2 and 14 km in our simulations, with negative shear (−3.9 × 10^{−3} s^{−1}) above this level.

Choosing a representative set of simulations with sheared flow presents special challenges. Changing one parameter (e.g., *z _{i}*) affects other parameters (e.g., wind speed at mountain height) if the shear is held constant. Based on observations from the SWP, the wind speed at

*z*is 19 m s

_{i}^{−1}for the majority of simulations. Using the same magnitude of vertical shear below the inversion leads to a mountaintop wind speed of 18 m s

^{−1}for

*z*= 2700 m, dropping to 13 m s

_{i}^{−1}for

*z*= 4050 m. An additional series of simulations was run for

_{i}*z*= 3225 m and 4050 m, with the mountaintop wind speed of 18 m s

_{i}^{−1}, which (using the same vertical shear above

*z*) results in a higher wind speed at

_{i}*z*. As expected, there are differences in the wave response, but in all cases tested, the MW/R type did not change. Sensitivity experiments with a 25% increase or reduction of average wind speed below

_{i}*z*are reported upon in section 3c.

_{i}A long-known requirement for mountain-induced gravity wave formation is a minimum wind speed at mountain crest level (e.g., WMO 1960). Simulations with *z _{i}* = 5400 m required increasing winds at all levels by 5 m s

^{−1}to avoid wind speeds at mountaintop that are so low that mountain-induced gravity waves formation would not be expected.

## 3. Results

In this section we present simulations of the two primary MW/R. A diagram summarizing numerous simulations run with a variety of inversion strengths and heights is presented, followed by a discussion of maximum LTZ height and turbulent intensity.

### a. Two distinct MW/R

Figure 2 shows a simulation at 3-h model integration time with *z _{i}* = 2700 m

^{1}, no internal shear, and a relatively strong Δ

*θ*= 11 K, resulting in what we refer to as a type 2 MW/R. Distinguishing features include a spreading inversion (Fig. 2a), as well as high-reaching and widespread turbulence. Horizontal vorticity,

*η*= (∂

*u*/∂

*z*) − (∂

*w*/∂

*x*), is shown in Fig. 2b. A large area of negative

*η*(counterclockwise rotation in this reference frame) is associated with the spreading inversion, while above and downstream of this feature, a rather chaotic pattern with both positive and negative

*η*occurs. Downslope winds (not shown) reach 52 m s

^{−1}under the inversion near

*x*= 12 km, with speeds exceeding 40 m s

^{−1}extending more than 30 km from the mountain crest. Between 2.5- and 3.0-h model integration time, the type 2 MW/R experiences maximum turbulent kinetic energy (TKE) ranging from 10 to almost 50 m

^{2}s

^{−2}. Maximum TKE generally occurs along the upwind side of the LTZ between 4 and 5 km, but at times it is found well within the LTZ, for example, near

*x*= 35 km and

*z*= 3 km. Large shear at the leading edge is likely an effective region for spawning subrotors (Doyle and Durran 2007). Indeed, in our type 2 MW/R, eddies (a few kilometers in diameter) develop along the leading edge and propagate downwind near the LTZ top, as seen by localized areas of maximum positive

*η*in Fig. 2b. The scale of these eddies are on the order of a few kilometers, while subrotors may be as small as a few tens of meters across [see Doyle and Durran (2007, their Fig. 1) or HK (their Fig. 2), taken from a well-known SWP case day].

An example of the more familiar mountain wave and LTZ, which we refer to as a type 1 MW/R, is shown in Fig. 3. The simulation parameters are as before, with the important exception of a weaker inversion (Δ*θ* = 5.5 K). Maximum TKE associated with the type 1 MW/R is weaker (between 9 and 11 m^{2} s^{−2} from 2.5- to 3.0-h model integration time), much less temporally variable, and (in contrast to the type 2 MW/R) is always confined to the leading edge of the LTZ (Fig. 3a). The turbulent LTZ under the wave crest is more organized; in this simulation two regions of positive *η* separate one smaller area of negative *η*. Downslope winds (not shown) are strong, but weaker than the type 2 MW/R, reaching 38 m s^{−1} under the inversion between *x* = 14 and 17 km. Other quantities associated with the two MW/R types, such a horizontal winds and vertical velocity, were shown in HK.

### b. Shedding lee eddies

Very weak Δ*θ*, especially when combined with internal shear and high *z _{i}*, leads to a third structure, which we refer to as shedding lee eddies. Figure 4 shows potential temperature and TKE at three times for a simulation with internal shear, Δ

*θ*= 2.8 K, and

*z*= 2700 m. The eddies form directly along the lee slope in the near-neutral layer directly above the mountain, then separate from the slope and propagate downstream. As can be seen by individually numbered eddies, the downstream propagation speed is between 4 and 14 m s

_{i}^{−1}and is slowest near the lee slope, increasing as the eddies move farther from the slope. With

*z*= 5400 m (twice the mountain height) much weaker and smaller lee eddies form in the deep near-neutral layer below the inversion.

_{i}Eddies range in diameter from 4 to 8 km and are composed mostly of positive *η*. Maximum TKE is at times stronger than the type 1 MW/R, between 6 and 20 m^{2} s^{−2}. Once the shedding lee eddies form, they effectively change the shape of the lee slope, and any steady wave response due to the terrain is lost. Instead, highly transient waves are excited in the overlying stable layer in response to the downstream-propagating eddies.

Shedding lee eddies are not MW/R in the usual sense since a steady-state mountain wave with LTZ is never achieved. Rather, the eddies resemble rotor streaming (Förchtgott 1949; WMO 1960). However, they differ from rotor streaming in two ways. First, as pointed out by Förchtgott (1949), rotor streaming occurs in an environment in which the maximum wind speed occurs in the vicinity of the mountaintop, and then decreases with height. Such a wind profile is sometimes associated with the bora (e.g., Grubiśiĉ 1989; Gohm et al. 2008). Second, rotor streaming comprised counterrotating eddies, whereas the eddies simulated here are comprised of horizontal vorticity of the same sign.

It is likely that shedding lee eddies require a sufficiently high and steep lee slope for the boundary layer separation to occur there. A thorough investigation of the necessary parameters for their formation is left for future work.

### c. Inversion strength and height

In Fig. 5a we summarize 42 simulations run with no internal shear, but with varying inversion strength (Δ*θ* = 1 to 16 K), and height (*z _{i}* = 2700, 3225, 4050, and 5400 m). At

*z*= 2700 and 3225 m, the transition from type 1 (stars) to type 2 (diamonds) MW/R occurs for inversions with Δ

_{i}*θ*> 8 K, while at

*z*= 4050 m, the transition occurs with a slightly weaker Δ

_{i}*θ*= 7 K. As

*z*increases further to 5400 m, lee eddies (triangles) are simulated for all Δ

_{i}*θ*. Thus, one favorable criterion for the formation of type 2 MW/R is a sufficiently strong inversion above and relatively close to the mountaintop. Our results are agreement with the regime diagram of Vosper (2004) who found that as Δ

*θ*increases, systems transition from lee-wave rotor to hydraulic jump.

We emphasize that the average wind speed below each *z _{i}* is not the same. Thus, we cannot say that the decreasing Δ

*θ*with increasing

*z*in Fig. 5 during transition between type 1 and type 2 MW/R is a function of

_{i}*z*and Δ

_{i}*θ*alone. The Δ

*θ*at which transition from type 1 to type 2 MW/R occurs will undoubtedly vary depending on many factors. As shown by HK (see their Fig. 16), for a given internal shear MW/R tend toward type 2 as wind speed at

*z*decreases. In the present study, additional simulations were run at

_{i}*z*= 2700 m to elucidate the effect of the wind profile. Wind speed at

_{i}*z*was first increased to 25.9 m s

_{i}^{−1}(representing a 25% increase in the average wind speed between the surface and

*z*) for several simulations with various Δ

_{i}*θ*. As indicated by the plus symbol in Fig. 5a, with stronger wind speeds, the transition from type 1 to type 2 MW/R occurs at a higher Δ

*θ*= 10 K. In agreement with the results of HK, as well as the regime diagram of Vopser (2004), when the average wind speed is reduced by 25% (12.5 m s

^{−1}wind speed at

*z*) the transition occurs at a lower Δ

_{i}*θ*= 6 K, as indicated by the minus symbol in Fig. 5a. Other factors (e.g., the mountain height and lee-slope aspect ratio) will also likely affect the Δ

*θ*at which transition from type 1 to type 2 MW/R occurs.

### d. Internal shear

The effect of internal shear on MW/R formation was discussed in detail by HK. Their simulations used a single Δ*θ* = 11 K, relatively strong internal shear lead to type 1 MW/R, while weak internal shear lead to type 2. To further understand the role of internal shear, an additional 41 simulations were run with winds increasing 8 m s^{−1} through the 600-m-deep inversion (i.e., internal shear = 1.33 × 10^{−3} s^{−1} as in HK) and a range of inversion strengths and heights. Figure 5b shows that simulations with internal shear can produce type 2 MW/R, but they require substantially stronger Δ*θ* than simulations with no internal shear, especially at *z _{i}* = 2700 and 3225 m. We note that Δ

*θ*> 15 K is not commonly found in the atmosphere—other than perhaps a near-surface, nocturnal inversion—but these strong inversions are included here to more completely explore the parameter space. Thus, in environments with internal shear, type 2 MW/R are unlikely. Simulations with internal shear also lead to shedding lee eddies when inversions are weak (Δ

*θ*< 3 to 5 K) or high (

*z*= 5400 m). The modifying role of internal shear will be further explored in section 4.

_{i}### e. Wave amplitude and turbulence within the LTZ

We now investigate the wave amplitude and turbulence associated with the two simulated MW/R. Wave amplitude is approximated by measuring the vertical displacement of the isentrope at the inversion top between its lowest point in the lee of the mountain and its highest point in the first wave crest. Wave amplitude can be used as a surrogate for the LTZ height: an increase in one implies an increase in the other. Graphical output of model data at 3-h model integration time was examined to determine wave amplitude. As an example, in Fig. 2a, type 2 wave amplitude is 4100 m, while in Fig. 3a, type 1 wave amplitude is 2700 m. Turbulence is determined from maximum TKE coincident with the wave amplitude.

Figures 6a and 6b show wave amplitude for *z _{i}* = 2700 and 4050 m for simulations with (dashed lines) and without (solid lines) internal shear. For type 1 MW/R (stars), wave amplitude increases as Δ

*θ*increases for both

*z*, regardless of the presence or absence internal shear. Our results are consistent with those of Corby and Wallington (1956), who showed that resonant-wave amplitude increased as lower-layer stability (in a two- or three-layer model) tended toward a sharp inversion. By contrast, for type 2 MW/R (diamonds), wave amplitude

_{i}*decreases*as Δ

*θ*increases with or without internal shear (note that only one simulation with internal shear produced a type 2 MW/R for

*z*= 2700 m).

_{i}Turbulence within the LTZ (maximum TKE, Figs. 6c,d) generally increases with increasing Δ*θ*. With two exceptions, simulations with no internal shear result in greater TKE than those with internal shear for the same Δ*θ*. In addition, most, but not all, type 2 MW/R are associated with greater turbulence. However, type 1 MW/R with internal shear and strong inversions produce the strongest TKE.

At *z _{i}* = 3225 m, trends of wave amplitude and maximum TKE as a function of Δ

*θ*are qualitatively similar to

*z*= 2700 m. Overall, results indicate that for type 1 MW/R, stronger inversions lead to a higher-amplitude wave response (and higher-reaching LTZs). By contrast, for type 2 MW/R, stronger inversions lead to a lower-amplitude wave response. Regardless of MW/R type, or the presence of internal shear, stronger inversions are associated with stronger turbulence.

_{i}## 4. Discussion

We now further explore the formation of the type 2 MW/R. Our results indicate that, regardless of internal shear, processes within the inversion itself play an important role. For each *z _{i}*, depending on Δ

*θ*either a type 2 or type 1 MW/R develops early in the simulation.

*η*within the inversion. The relationship between baroclinic generation and

*η*is clearly seen by considering the two-dimensional, Boussinesq system, with Coriolis and compressibility neglected (Doyle and Durran 2002; HK). The total tendency of

*η*is given by

*B*

_{x}= −

*g*∂(

*θ*′/

*θ*

*x*,

*θ*=

*θ*

*z*) +

*θ*′, and

*D*represents subgrid-scale turbulent stresses. The first term on the right-hand side represents baroclinic generation through horizontal buoyancy gradients. As the evolution of the mountain wave continues, the inversion becomes further distorted; negative

*η*continues to be baroclinically generated and is advected down the slope.

The magnitude of *η* within the inversion is determined almost entirely by strong negative vertical shear (∂*u*/∂*z*), with the horizontal wind decreasing by almost 20 m s^{−1} through the inversion depth. The strongest downslope flow is largely confined below the lowest isentrope of the inversion, which, due to surface friction, produces a thin sheet of positive *η* associated with strong positive vertical shear in the surface layer.

As in HK, we choose an early model integration time to illustrate the salient processes, since by this time the incipient MW/R has already responded to the evolving horizontal vorticity. The magnitude of negative *η* (and negative ∂*u*/∂*z*) within the inversion increases following the flow down the lee slope until overturning (or breaking) of isentropes in a counterclockwise sense begins (Fig. 7a).^{2} The overturning is responsible for the spreading inversion. Onset of overturning occurs in about the same time as boundary layer flow separation, as near-surface positive *η* is lifted due to an adverse pressure gradient along the lee slope (Doyle and Durran 2002).

As overturning begins within the inversion of the type 2 MW/R, the wave amplitude has reached less than half its quasi-stationary value (Fig. 6) and isentropes in the wave flow above the inversion are not steep. Thus, overturning of the isentropes *within* the inversion is not wave breaking in the usual sense of a mountain lee-wave amplitude that increases and becomes steeper to the point of overturning. The evolution of our type 2 MW/R is thus in contrast to the hydraulic jump evolution described by Vosper (2004).

The early evolution of the type 1 MW/R differs in that the weaker initial near-mountaintop inversion results in less baroclinic generation, and hence the magnitude of negative *η* along the lee slope is reduced (Fig. 7b). As near-surface flow separation begins, negative *η* is not strong enough to cause upstream overturning or spreading of the inversion.

From the foregoing analysis, the formation of the MW/R can be qualitatively understood by whether positive or negative *η* dominates as surface-layer separation begins. In the type 2 MW/R, negative *η* overpowers lifted surface-layer positive *η,* leading to overturning in the upstream direction. By contrast, in the type 1 MW/R, lifted surface-layer positive *η* dominates, leading to LTZ with prevailing clockwise rotor circulations. Earlier work by HK also includes a discussion of the early evolution of the two systems. However, they did not fully appreciate the vital role of inversion strength. In contrast to our findings, Jiang et al. (2007) found that type 2 MW/R are not necessarily associated with near-mountaintop inversions. However, direct comparison between the studies is difficult, given differences in initial conditions (e.g., inversion strength and depth, mountain height and profile, along with a height-independent wind profile).

*B*, η, horizontal wind (

_{x}*u*), and the Richardson number,

*z*= 2700 m. For both the type 2 (Δ

_{i}*θ*= 11 K, solid line) and type 1 (Δ

*θ*= 5.5 K, dot-dashed line) MW/R,

*B*reaches a minimum within the inversion (Fig. 8a). However, stronger

_{x}*B*within the type 2 MW/R leads to stronger negative

_{x}*η*(Fig. 8b). The downslope flow below the inversion is stronger for the type 2 MW/R (Fig. 8c) along with stronger negative vertical shear within the inversion. Strong positive shear and

*η*are found in the surface layer. Figure 8d shows

*R*approximately 0.25 near the base of the inversion for the type 2 MW/R, implying dynamical instability (e.g., Stull 1988) while values remain greater than the threshold for turbulence in the type 1 MW/R. In the type 2 MW/R,

_{i}*R*∼ 0.25 occurs mostly in the lower part of the inversion, where the isentropes first become steeper and break (cf. Fig. 7a). Differences in vertical profiles between simulations taken further along the lee slope at 3-h model integration time are qualitatively similar to those shown in Fig. 8. The underlying dynamics determining MW/R type are captured at the early model integration time shown.

_{i}The important role of baroclinic generation continues for the duration of the simulation. Figure 9a shows *η* averaged over the depth of the inversion, while Fig. 9b shows the corresponding *B _{x}* for type 2 (solid) and type 1 (dot-dashed) MW/R. For either MW/R,

*η*becomes negative along the lee slope in response to the increasingly negative

*B*. Both

_{x}*η*and

*B*are of greater magnitude for the type 2 than type 1 MW/R as shown in Fig. 8. Baroclinic generation accounts for most of the −

_{x}*η*. For instance, using an average speed of 30 m s

^{−1}within the inversion along the lee slope, a parcel would require about 170 s to transition from

*x*= 0 to

*x*= 5 km. An average

*B*of −0.00017 s

_{x}^{−2}(from Fig. 9b) over this time yields

*η*= −0.029 s

^{−1}. This is close to

*η*= −0.027 s

^{−1}at

*x*= 5 km from Fig. 9a. Subgrid turbulence (not shown) is much less than

*B*with one exception. Beyond

_{x}*x*= 19 km, downwind of the spreading inversion (see Fig. 2a), subgrid-scale turbulence dominates the flow, fluctuating rapidly in sign downstream and an order of magnitude greater than

*B*. The dominance of subgrid-scale turbulence over

_{x}*B*the LTZ agrees with the findings of Doyle and Durran (2007).

_{x}withinThe effect of internal shear is also clear from the vertical profiles in Fig. 8. The dashed line is taken from a simulation of a type 1 MW/R *with* internal shear and Δ*θ* = 11 K (see Fig. 5). As expected, given the same Δ*θ*, *B _{x}* is similar in magnitude to the type 2 MW/R. However, the presence of positive internal shear in the initial conditions, which must first be eliminated by

*B*, leads to lower magnitude negative

_{x}*η*(Fig. 8b). Other combinations of wind and stability profiles likely modify the leeside inversion distortion. For instance, HK showed that for the same internal shear, a somewhat

*lower*near-mountaintop wind speed favors to the type 2 MW/R (see their Fig. 16). Of course, very weak near-mountaintop winds (less than 5 to 8 m s

^{−1}) cause little distortion of the inversion; hence there will be little or no mountain-wave response. Thus, any MW/R forecast tool based on

*z*and Δ

_{i}*θ*alone would not be adequate.

The mechanism above also describes the MW/R evolution in simulations at *z _{i}* = 3225 and 4050 m. As

*z*increases to 5400 m, an MW/R no longer forms since the near neutral stratification below the inversion is not conducive to mountain-induced gravity waves. Qualitatively, high inversions no longer “feel” the effects of flow over the mountain.

_{i}Other sensitivity experiments were performed to further elucidate the role of the inversion. In one set of simulations, the depth of the inversion was increased from 600 to 1200 m and then 1800 m. In all cases tested, *B _{x}* and hence negative

*η*are reduced to the point that type 2 MW/R were not simulated. In another set of simulations, the inversion was eliminated altogether, and the simulations were run with constant stability and vertical shear over the entire troposphere. Once again, no type 2 MW/R was simulated, even with a very stable, isothermal troposphere. (A deep, very stable layer is “stiff” and resists deformation along the lee slope. A strong but relatively thin inversion bounded above and below by less stable layers is more easily deformed).

The spreading inversion in our simulations shares similarities with recently reported flows downwind of the Austrian Alps. For instance, Armi and Mayr (2007) present a shallow gap flow prototype (their Fig. 19a) in which a stable layer in an environment of negative shear is seen to be spreading. However, as the authors note, the isentropes in their overlying stable layer never overturn as in our type 2 MW/R. Their deep-foehn case also bears some resemblance to our type 2 MW/R, especially stagnant flow downwind of (and descending flow below) a spreading inversion. Important differences include a major downwind barrier (the Nordkette) in their deep-foehn case, a gap in the primary range, and other complex, three-dimensional topography. Nonetheless, dynamic processes similar to our type 2 MW/R are possibly at work in their Alpine cases.

Despite the qualitative resemblance, referring to type 2 MW/R as “hydraulic-jump rotors” (i.e., Hertenstein and Kuettner 2006) may be premature. The dynamics responsible for our type 2 MW/R are more favorably compared with the results of Rotunno and Smolarkiewicz (1995). They show that baroclinically generated horizontal vorticity along a thin boundary between fluids of different density leads to overturning of the interfacial surface.

## 5. Conclusions and final comments

The presence or absence of a near-mountaintop inversion is vital in determining the type of mountain-wave/rotor (MW/R) system that develops. Strong inversions favor type 2 MW/R, moderate inversions favor type 1 MW/R, and very weak or high inversions lead to shedding lee eddies. When no inversion is present, type 2 MW/R do not form, even with a deep stable air flowing down the lee slope.

Type 2 MW/R form due to baroclinically generated horizontal vorticity within the inversion. The vertical shear of the horizontal wind (∂*u*/∂*z*) is the main component of the horizontal vorticity. When the horizontal vorticity (and thus vertical shear) strengthens sufficiently, dynamic instability and overturning (breaking) of internal isentropes occur, leading to a spreading inversion and an LTZ with chaotic turbulence and counterrotating eddies. Thus consideration of the dynamics *within* the near-mountaintop inversion is important to understanding the type 2 MW/R.

When positive vertical shear within the initial upstream inversion is present, baroclinic generation must first reduce ambient positive horizontal vorticity. Unless the inversion (and hence baroclinic generation) is very strong, insufficient negative horizontal vorticity develops for type 2 MW/R formation.

The author knows of no direct measurements of the internal structure of a type 2 MW/R, in particular the spreading inversion, along with chaotic circulations due to the presence of both positive and negative horizontal vorticity within the LTZ. Anecdotal evidence from glider pilots (including the author) and from field campaigns as far back as the 1950s SWP, suggests their existence. Direct measurements of some characteristics—for instance, the spreading of the inversion along the lee slope—are difficult to obtain. The internal structure of LTZs along the Colorado Front Range is being explored by the author and collaborators using an instrumented sailplane (Hertenstein and Martin 2008). Thus far, only type 1 MW/R, with quasi-stationary turbulent, but well defined, up and down branches, have been encountered on research flights with the instrumented sailplane.

Type 2 MW/R are likely infrequent or perhaps even rare. First, although near-mountaintop stable layers are common, strong inversions, such as those leading to our simulated type 2 MW/R, are not. A casual examination of standard, twice-daily soundings near the Colorado Front Range indicates that inversions with Δ*θ* greater than approximately 8 to 10 K occur only a few times during the fall through spring season. Second, the experience of glider pilots indicates that most LTZs have discernible quasi-organized up and down branches characteristic of the type 1 MW/R. In addition, as noted also by Sheridan and Vosper (2006), predicting type 2 MW/R using output from routine numerical weather prediction forecasts will not be easy since the vertical grid spacing necessary to properly resolve strong, relatively thin inversions is still largely lacking.

Finally, two-dimensional simulations cannot explain complexities encountered in real-world three-dimensional topography and detailed wind and stability profiles. Recent observational and modeling studies have documented variability of the flow downwind of mountain ranges due to gaps and other topographical features (such as an individual peak) along mountain ranges, for example, Gohm and Mayr (2004), Sheridan and Vosper (2006), Doyle and Durran (2007), and Grubiśiĉ et al. (2008). Horizontal stretching and tilting are not captured in two-dimensional simulations, but were found to be important for the intensification of subrotors in three-dimensional simulations (Doyle and Durran 2007). It is possible that in some cases, three-dimensional topography alters the flow along the lee slope, leading to complex MW/R responses along the lee of a mountain range.

## Acknowledgments

This research was supported by the National Science Foundation under Grants ATM-0233165 and ATM-0541729. The manuscript benefited from conversations with Drs. Joachim Kuettner, James Doyle, and Richard Rotunno.

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Simulation of a type 2 MW/R for *z _{i}* = 2700 m, Δ

*θ*= 11 K, and no internal shear at 3-h model integration time. (a) Potential temperature (contour interval 1 K) with TKE shaded (blue = 10 m

^{2}s

^{−2}, green = 20 m

^{2}s

^{−2}). (b) Horizontal vorticity

*η*(contour interval 0.015 s

^{−1}, negative dashed). The top and bottom isentrope of the inversion are shown as thicker solid lines.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Simulation of a type 2 MW/R for *z _{i}* = 2700 m, Δ

*θ*= 11 K, and no internal shear at 3-h model integration time. (a) Potential temperature (contour interval 1 K) with TKE shaded (blue = 10 m

^{2}s

^{−2}, green = 20 m

^{2}s

^{−2}). (b) Horizontal vorticity

*η*(contour interval 0.015 s

^{−1}, negative dashed). The top and bottom isentrope of the inversion are shown as thicker solid lines.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Simulation of a type 2 MW/R for *z _{i}* = 2700 m, Δ

*θ*= 11 K, and no internal shear at 3-h model integration time. (a) Potential temperature (contour interval 1 K) with TKE shaded (blue = 10 m

^{2}s

^{−2}, green = 20 m

^{2}s

^{−2}). (b) Horizontal vorticity

*η*(contour interval 0.015 s

^{−1}, negative dashed). The top and bottom isentrope of the inversion are shown as thicker solid lines.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

As in Fig. 2, but for the simulation of a type 1 MW/R for *z _{i}* = 2700 m, Δ

*θ*= 5.5 K, and no internal shear.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

As in Fig. 2, but for the simulation of a type 1 MW/R for *z _{i}* = 2700 m, Δ

*θ*= 5.5 K, and no internal shear.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

As in Fig. 2, but for the simulation of a type 1 MW/R for *z _{i}* = 2700 m, Δ

*θ*= 5.5 K, and no internal shear.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Simulation of lee eddy shedding, for *z _{i}* = 2700 m, Δ

*θ*= 2.8 K, and no internal shear. Contours and shading as in Fig. 2a. Numbers show individual eddies at (a) 2.50-, (b) 2.75-, and (c) 3.00-h model integration time.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Simulation of lee eddy shedding, for *z _{i}* = 2700 m, Δ

*θ*= 2.8 K, and no internal shear. Contours and shading as in Fig. 2a. Numbers show individual eddies at (a) 2.50-, (b) 2.75-, and (c) 3.00-h model integration time.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Simulation of lee eddy shedding, for *z _{i}* = 2700 m, Δ

*θ*= 2.8 K, and no internal shear. Contours and shading as in Fig. 2a. Numbers show individual eddies at (a) 2.50-, (b) 2.75-, and (c) 3.00-h model integration time.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Inversion height (*z _{i}*) vs inversion strength (Δ

*θ*) for simulations (a) with no internal shear and (b) with internal shear. Symbols: stars indicate type 1 MW/R, diamonds show type 2 MW/R, and shedding lee eddies are represented by triangles. The three larger symbols along

*z*= 2700 m in both (a) and (b) are those cases discussed in section 4; plus and minus symbols indicate sensitivity experiments discussed in section 3c.

_{i}Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Inversion height (*z _{i}*) vs inversion strength (Δ

*θ*) for simulations (a) with no internal shear and (b) with internal shear. Symbols: stars indicate type 1 MW/R, diamonds show type 2 MW/R, and shedding lee eddies are represented by triangles. The three larger symbols along

*z*= 2700 m in both (a) and (b) are those cases discussed in section 4; plus and minus symbols indicate sensitivity experiments discussed in section 3c.

_{i}Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Inversion height (*z _{i}*) vs inversion strength (Δ

*θ*) for simulations (a) with no internal shear and (b) with internal shear. Symbols: stars indicate type 1 MW/R, diamonds show type 2 MW/R, and shedding lee eddies are represented by triangles. The three larger symbols along

*z*= 2700 m in both (a) and (b) are those cases discussed in section 4; plus and minus symbols indicate sensitivity experiments discussed in section 3c.

_{i}Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Wave amplitude vs inversion strength (a) at *z _{i}* = 4050 m and (b) at

*z*= 2700 m. Maximum TKE vs inversion strength (c) at

_{i}*z*= 4050 m and (d) at

_{i}*z*= 2700 m. Solid lines are simulations with no internal shear; dashed lines are simulations with internal shear. Symbols are as in Fig. 5.

_{i}Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Wave amplitude vs inversion strength (a) at *z _{i}* = 4050 m and (b) at

*z*= 2700 m. Maximum TKE vs inversion strength (c) at

_{i}*z*= 4050 m and (d) at

_{i}*z*= 2700 m. Solid lines are simulations with no internal shear; dashed lines are simulations with internal shear. Symbols are as in Fig. 5.

_{i}Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Wave amplitude vs inversion strength (a) at *z _{i}* = 4050 m and (b) at

*z*= 2700 m. Maximum TKE vs inversion strength (c) at

_{i}*z*= 4050 m and (d) at

_{i}*z*= 2700 m. Solid lines are simulations with no internal shear; dashed lines are simulations with internal shear. Symbols are as in Fig. 5.

_{i}Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Horizontal vorticity *η* (contour interval 0.015 s^{−1}, negative dashed) and isentropes above the inversion (contour interval 1 K) at 15-min model integration time, *z _{i}* = 2700, and no internal shear for (a) type 2 MW/R, Δ

*θ*= 11 K, and (b) type 1 MW/R, Δ

*θ*= 5.5 K.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Horizontal vorticity *η* (contour interval 0.015 s^{−1}, negative dashed) and isentropes above the inversion (contour interval 1 K) at 15-min model integration time, *z _{i}* = 2700, and no internal shear for (a) type 2 MW/R, Δ

*θ*= 11 K, and (b) type 1 MW/R, Δ

*θ*= 5.5 K.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Horizontal vorticity *η* (contour interval 0.015 s^{−1}, negative dashed) and isentropes above the inversion (contour interval 1 K) at 15-min model integration time, *z _{i}* = 2700, and no internal shear for (a) type 2 MW/R, Δ

*θ*= 11 K, and (b) type 1 MW/R, Δ

*θ*= 5.5 K.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Vertical profiles at *x* = 6.6 km and 15-min model integration time for three simulations using *z _{i}* = 2700 of: (top) (left)

*B*(10

_{x}^{−4}s

^{−2}) and (right)

*η*(s

^{−1}); (bottom) (left) horizontal wind speed (m s

^{−1}) and (right)

*R*. Solid line is type 2 with Δ

_{i}*θ*= 11 K and no internal shear, dot-dashed is type 1 with Δ

*θ*= 5.5 K no internal shear, and the dashed line is type 1 with Δ

*θ*= 11 K and with internal shear. The top and bottom of the inversion in each case are marked.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Vertical profiles at *x* = 6.6 km and 15-min model integration time for three simulations using *z _{i}* = 2700 of: (top) (left)

*B*(10

_{x}^{−4}s

^{−2}) and (right)

*η*(s

^{−1}); (bottom) (left) horizontal wind speed (m s

^{−1}) and (right)

*R*. Solid line is type 2 with Δ

_{i}*θ*= 11 K and no internal shear, dot-dashed is type 1 with Δ

*θ*= 5.5 K no internal shear, and the dashed line is type 1 with Δ

*θ*= 11 K and with internal shear. The top and bottom of the inversion in each case are marked.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

Vertical profiles at *x* = 6.6 km and 15-min model integration time for three simulations using *z _{i}* = 2700 of: (top) (left)

*B*(10

_{x}^{−4}s

^{−2}) and (right)

*η*(s

^{−1}); (bottom) (left) horizontal wind speed (m s

^{−1}) and (right)

*R*. Solid line is type 2 with Δ

_{i}*θ*= 11 K and no internal shear, dot-dashed is type 1 with Δ

*θ*= 5.5 K no internal shear, and the dashed line is type 1 with Δ

*θ*= 11 K and with internal shear. The top and bottom of the inversion in each case are marked.

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

(a) Average (within the inversion) horizontal vorticity (s^{−1}) at 3-h model integration time for type 2 (solid) and type 1 (dot-dashed) MW/R shown in Figs. 2 and 3, respectively. (b) As in (a), but for baroclinic generation (s^{−2}). In (a) and (b), the vertical line indicates the location of the mountain crest. (c) Topography profile (m).

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

(a) Average (within the inversion) horizontal vorticity (s^{−1}) at 3-h model integration time for type 2 (solid) and type 1 (dot-dashed) MW/R shown in Figs. 2 and 3, respectively. (b) As in (a), but for baroclinic generation (s^{−2}). In (a) and (b), the vertical line indicates the location of the mountain crest. (c) Topography profile (m).

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

(a) Average (within the inversion) horizontal vorticity (s^{−1}) at 3-h model integration time for type 2 (solid) and type 1 (dot-dashed) MW/R shown in Figs. 2 and 3, respectively. (b) As in (a), but for baroclinic generation (s^{−2}). In (a) and (b), the vertical line indicates the location of the mountain crest. (c) Topography profile (m).

Citation: Monthly Weather Review 137, 1; 10.1175/2008MWR2482.1

^{1}

Note that the inversion rises during the model integration, due partly to mixing and partly to updrafts along the upwind side of the relatively high barrier.

^{2}

As pointed out by HK, the results do not depend on the use of an impulsive start versus a gradual increase in the winds. The model integration time shown illustrates the evolution leading to the final states shown in Figs. 2 and 3. Although the exact timing of upstream overturning varies between simulations, the type of MW/R is determined early during the model integration.