Abstract

Interactions between a turbulent boundary layer and nonlinear mountain waves are explored using a large-eddy simulation model. Simulations of a self-induced critical layer, which develop a stagnation layer and a strong leeside surface jet, are considered. Over time, wave breaking in the stagnation region forces strong turbulence that influences the formation and structure of downstream leeside rotors. Shear production is an important source of turbulence in the stagnation zone and along the interface between the stagnation zone and surface jet, as well as along the rotor edges. Buoyancy perturbations act as a source of turbulence in the stagnation zone but are shown to inhibit turbulence generation on the edges of the stagnation zone.

Surface heating is shown to have a strong influence on the strength of downslope winds and the formation of leeside rotors. In cases with no heating, a series of rotor circulations develops, capped by a region of increased winds. Weak heating disrupts this system and limits rotor formation at the base of the downslope jet. Strong heating has a much larger impact through a deepening of the upstream boundary layer and an overall decrease in the downslope winds. Rotors in this case are nonexistent. In contrast to the cases with surface warming, negative surface fluxes generate stronger downslope winds and intensified rotors due to turbulent interactions with an elevated stratified jet capping the rotors. Overall, the results suggest that for nonlinear wave systems over mountains higher than the boundary layer, strong downslope winds and rotors are favored in late afternoon and evening when surface cooling enhances the stability of the low-level air.

1. Introduction

Downslope windstorms and associated rotor events are very important in determining aviation hazards in the lee of major mountain ranges. Turbulence generated by mountain wave breaking and rotor dynamics can create severe conditions for aircraft operations. The onset and decay of such events are frequently tied to synoptic-scale processes in the atmosphere, such as the passage of a front over a mountain range; however, downslope windstorms can also occur on days with little change in synoptic conditions. In the absence of strong synoptic forcing, boundary layer processes may affect the strength and duration of downslope windstorm and rotor events and may explain observations suggesting preferred times for windstorm and rotor formation. The topic of this paper centers on understanding the formation and dynamics of turbulence generated by downslope wind storms and leeside rotors. In particular, we focus on the influence of boundary layer turbulence on mountain wave and rotor formation, and on the generation and interaction of turbulence forced by mountain wave breaking.

Although synoptic-scale features that aid in forecasting strong downslope winds have been studied by a number of researchers (Colle and Mass 1998a,b; Brinkmann 1974; Czyzyk and Runk 2006), there have been few observational studies that have focused on the influence of the upstream boundary layer stability and depth on mountain waves. Recent observations from the Terrain-Induced Rotor Experiment (T-REX) field campaign (Grubisic and Kuettner 2003) suggest a tendency for downslope windstorms to occur preferentially in the late afternoon and early evening (Grubisic and Xiao 2006) in the lee of the Sierra Nevada. Observations from Argentina reported in Seluchi et al. (2003) and Norte (1988) also show an increased frequency of downslope windstorms in the afternoon. These results are suggestive of a dynamical effect of the boundary layer depth and stratification on lee flow regimes; however, studies in other regions show a propensity for downslope windstorms to occur at times other than late afternoon. For example, Brinkmann (1974) found that the most likely time of day for downslope windstorms to occur in Boulder, Colorado was roughly between midnight and 0700 LT in the morning during the winter months.

Because of the difficulty of observing localized and transient mesoscale atmospheric events, there have been few direct observations of mountain wave breaking and associated turbulence. The scarcity of observations has been cited as a major reason why mountain wave breaking events are still so poorly understood (Doyle et al. 2005). One prominent and well-studied case is the 11 January 1972 windstorm along the Front Range of the Rockies (Lilly and Zipser 1972; Lilly 1978). The dynamics of this case have been examined in a number of modeling experiments (Clark and Peltier 1977; Clark and Peltier 1984; Durran 1986). Other important observations of mountain wave breaking events include those over Greenland (Doyle et al. 2005), documentation of clear air turbulence over the Front Range of the Rockies (Clark et al. 2000) and the Alps (Jiang and Doyle 2004), rotor generation over the Falkland Islands (Mobbs et al. 2005), and the importance of upstream inversions and internal gravity wave breaking to the Bora (Glasnovic and Jurcec 1990; Klemp and Durran 1987; Gohm and Mayr 2005; Belusic et al. 2007; Klemp et al. 1997; Gohm et al. 2008).

As with observational studies, very few mountain wave modeling experiments have focused on turbulence and upstream boundary layer stability controls on lee flow regimes. In a modeling study of the diurnal variation of lee wave regimes, Ying and Baopu (1993) point out that classical mountain flow theory does not account for the thermal stratification of the atmospheric boundary layer. They examine the effects of boundary layer stability on flow blocking, mountain wave amplitudes, and lee slope jet speeds. Other researchers have also found that boundary layers tend to reduce the amplitude of mountain waves and in some cases can inhibit internal gravity wave breaking (Olafsson and Bougeault 1997). Richard et al. (1989) note that the inclusion of a boundary layer in their numerical experiments leads to shooting flows that, compared with experiments without a boundary layer, have a spatial extent that is significantly reduced on the lee slope. Peng and Thompson (2003) extend the analysis to the case of narrow mountains where internal gravity wave amplitude may be increased by the presence of a boundary layer. They suggest that the boundary layer height plus the terrain height acts as an effective terrain barrier that determines the lower boundary condition for wave launching.

Vosper (2004) examined the importance of sharp upwind temperature inversions on flow over mountains and the role of surface friction in the formation of rotors. Lee slope flow regimes were found to be sensitive to inversion strengths and heights, which were for the most part greater than the hill height. Although physically relevant for small hills and the Falkland Islands, this situation is generally not representative of realistic conditions near major mountain ranges such as the Front Range in Colorado and the Sierra Nevada in California.

Doyle and Durran (2002) examine the effect of lee slope heating on the structure and development of rotors associated with resonant mountain waves using numerical simulations. They demonstrate that the inclusion of surface friction is a requirement of modeling realistic rotors and that increasing the surface heat flux downstream from the mountain crest increases the vertical extent of the rotor circulation and the strength of the turbulence but decreases the magnitude of the reversed rotor flow. Studies such as those by Jiang et al. (2006) and Smith et al. (2006) examine the absorption of trapped lee waves by the atmospheric boundary layer downstream from a ridge. They show that the decay of lee waves is sensitive to surface roughness and heat flux and that the stable nocturnal boundary layer is more efficient in absorbing trapped waves than a turbulent convective boundary layer.

Other contributions have concentrated on the dynamics and energetics of mountain wave breaking. Afanasyev and Peltier (1998), using a direct numerical simulation (DNS), emphasized the inherent three-dimensional nature of internal gravity wave breaking and the role of Kelvin–Helmhotz (K–H) billows above the downslope jet along the lee side of a simulated obstacle. Gheusi et al. (2000) performed a similar modeling study representing a laboratory tank experiment, examining the formation of three-dimensional vortical disturbances generated in the wake breaking region. They emphasized the production of turbulent kinetic energy (TKE) by shear at the bottom of the mixed layer. A series of papers (Laprise and Peltier 1989a,b,c) concerning the energetics and stability of breaking mountain waves developed a conceptual framework for the transition of a mountain wave lee flow regime to that of a self-induced critical layer and the dominance of a resonant mode producing shooting flow along the lee slope. As noted by the authors, these experiments were performed without the benefit of a turbulence closure scheme. Addressing this issue, Epifanio and Qian (2008) used a large-eddy simulation (LES) to analyze the momentum and turbulence budgets for a breaking mountain wave and found that the mean wave energy dissipation was mostly due to turbulent momentum fluxes, which act counter to the mean flow disturbance wind.

In the majority of mountain wave modeling studies, upstream conditions are prescribed and typically do not consider how the stability and turbulence in the upstream boundary layer affect mountain wave systems. The focus of this paper is on the role of the upstream boundary layer and surface heating in controlling the strength of mountain-induced internal waves, low-level internal gravity wave breaking, and downslope windstorms and rotors. We define rotors as a low-level circulation in the lee of a ridge, about an axis parallel to the ridge (Glickmann 2000). In this sense, the existence of negative streamwise surface velocities in the lee of the ridge is not a requirement of the existence of a rotor. We concentrate on the effect of turbulence on lee flow response by applying a large-eddy simulation model designed to directly simulate turbulent eddies that control the boundary layer dynamics as well as eddies forced by mountain wave breaking. Application of LES also allows us to simulate the turbulent boundary layer with arguably more fidelity than models with parameterized turbulence closure (e.g., Mellor and Yamada–type closures) because a portion of the energy-carrying turbulent eddies are resolved in the model, whereas most other simulations completely parameterize turbulent fluxes.

One other prominent modeling study (Epifanio and Qian 2008) that used the LES approach for lee flow regimes focused on the dynamics of internal gravity wave breaking generated by a self-induced critical layer and on orographic generation of potential vorticity. We consider a scenario very similar to their simulations, except that we use a different surface boundary condition (limited slip in ours versus free slip in theirs), simulate a well-developed upstream boundary layer, apply a higher model resolution (10 m in our simulation versus 180 m in theirs), and focus more on downstream rotors.

The paper is structured as follows: A description of the LES model is presented in section 2 along with an outline of the experiments performed in our study. Results are presented next in section 3 for a basic experiment that includes no surface heating. Next we compare the basic no heating simulation to simulations that include weak surface heating, strong surface heating, and surface cooling. Then, we further explore the effects of both upstream and downstream heating in the weak heating case. A summary and conclusions are given in section 4.

2. Model introduction and setup

Experiments were performed using a modified version of the LES model described in Skyllingstad (2003) and used in Smith and Skyllingstad (2005). This model is based on the Deardorff (1980) equation set, with the subgrid-scale model described by Ducros et al. (1996). Pressure in the model is calculated using the anelastic approximation with a conjugate residual iterative pressure solver (Smolarkiewicz and Margolin 1994). Terrain in the LES model is prescribed using a shaved cell approach described in Adcroft et al. (1997) and Steppeler et al. (2002). This approach was selected over the more commonly used terrain-following coordinate methods to avoid problems with the LES filtering assumptions. Comparisons between mountain waves simulated using terrain-following coordinates and the shaved cell approach show only minor differences (Skyllingstad and Wijesekera 2004). Further tests (not shown) confirm that simulations using this model compare quite well with previously reported results (Steppeler et al. 2002, their Figs. 2 and 3).

Simulations were conducted using a narrow channel domain with periodic boundaries in the cross-slope direction, an Orlanski (1976)-type boundary condition at the outflow boundary, and a recirculating upstream boundary (Fig. 1). For the upwind boundary, the recirculating scheme was developed following Mayor et al. (2002) and provides a method for generating a fully developed boundary layer upstream from the mountain. The scheme was implemented by treating a portion of the domain as a periodic subdomain and then using the velocity and scalar fields as boundary values on the upstream edge of the open boundary channel. The length of the periodic subdomain (from y = 0 km to y ≈ 1 km, where y is in the streamwise direction) was chosen to be large enough so that there were no signatures of the subdomain length in the eddy structures in the boundary layer.

Fig. 1.

Schematic showing the channel flow configuration used in the simulations.

Fig. 1.

Schematic showing the channel flow configuration used in the simulations.

A limited-slip lower boundary for some of the simulations was set by assuming a neutral log scaling and setting the subgrid momentum flux to

 
formula
 
formula

with a surface roughness z0 = 0.01 m, where 〈υw″〉 is the average subgrid-scale momentum flux, CD is the drag coefficient, υ is the velocity in the streamwise direction, w is the velocity in the vertical direction, κ = 0.4 is the von Kármán constant, and δz = Δz/2 is half of the vertical grid spacing. The turbulent viscosity Km is specified in the lowest grid cell only following similarity theory for neutrally stable boundary layers:

 
formula

where u* is the friction velocity defined in the usual manner. The top boundary condition included a sponge layer following Durran and Klemp (1983) with a depth of 2500 m, which was greater than the wavelength of the dominant waves generated by the mountain in our model. We chose the sponge layer depth to eliminate significant energy reflection off of the upper boundary and sponge layer. Mass conservation in the model was ensured by calculating the difference between the mean outflow momentum flux on the downstream boundary and the prescribed inflow momentum flux, and then setting the model-top vertical velocity to maintain constant mass.

A two-dimensional ridge obstructing the flow was based on the Witch of Agnesi profile (4),

 
formula

for a mountain of height h = 400 m and width a = 4000 m. The nondimensional mountain height, Nh/υ = 1.2, and nondimensional mountain width, Na/υ = 12, were calculated using the prescribed inflow conditions in the free atmosphere above the surface boundary layer where N is the Brunt–Väisälä frequency; that is,

 
formula

where g is the acceleration due to gravity and θ is the potential temperature.

An idealized initial state is prescribed with υ set to 5 m s−1 and N = 0.015 s−1. Both are constant with height, in contrast to previous modeling studies, which have generally used a trapping mechanism of decreasing N with height or increasing U with height (Doyle and Durran 2002; Hertenstein and Kuettner 2005) to generate rotors. In this study we instead use a wave breaking mechanism in which a self-induced critical layer acts as a reflector of wave energy in order to generate rotors. Although computational issues restrict us to modeling flow over a ridge that is small in comparison to the Sierra Nevada or the Rocky Mountains, our ridge height is comparable to the rotor-inducing terrain on the Falkland Islands (Mobbs et al. 2005). Moreover, dynamical similarity, based on nondimensional mountain height and nondimensional width, can be used for comparisons. The domain size was set to 100 × 4560 × 300 grid points in the along-slope, cross-slope, and vertical directions respectively, with grid resolution of 10 m in all directions below 1.5 km. Above 1.5 km, the vertical grid spacing stretches from 10 to 80 m. The total domain size was 1.0 km × 45.6 km × 9.8 km. The mountain was centered at y = 27 km, slightly past the center point in the streamwise direction.

Surface heating was applied throughout the entire domain unless otherwise indicated. The surface heating characteristics and names of the simulations are summarized in Table 1. In all simulations, surface heating of 25 W m−2 was applied for the first 20 min to initiate boundary layer turbulence. Heat fluxes for the weak and strong heating runs were set to 25 and 200 W m−2, respectively, and held constant after the initial 20 min. In the surface cooling run, a surface heat flux of −50 W m−2 was applied after a 1-h spinup period. All of the cases began with a neutrally stratified boundary layer height zi of 200 m.

Table 1.

List of all simulations and their characteristics.

List of all simulations and their characteristics.
List of all simulations and their characteristics.

3. Results

a. No heating run

We first present results from the no heating case, which provides a baseline for assessing the effects of surface fluxes. Cross-sectional plots of streamwise velocity, vertical velocity, and average resolved eddy TKE overlaid with contours of potential temperature are presented in Fig. 2 for three different times. In this and subsequent plots, we have chosen to present results at dynamically important times in the flow evolution. After 3 h of integration, in all cases the flow has reached a quasi-steady state. All variables presented in the plots are instantaneous at a single cross section in the model domain unless otherwise specified as being averaged. For the purposes of this study, we define the average of a variable ϕ as follows:

 
formula

where the overbar denotes the average. In this equation imax is the number of points in the spanwise direction and nmax is the number of time steps over the 5-min averaging time. The spanwise mean,

 
formula

is used to calculate perturbations about the spanwise mean

 
formula

In Fig. 2c we present average resolved eddy TKE, where resolved eddy TKE is defined as

 
formula

the perturbation velocities are calculated as in (7) and (8), and the averaging is done in the spanwise direction as well as temporally as in (6). Computing averages in this way removes the large-scale internal waves generated by the mountains; however, smaller-scale waves are still treated as “turbulence.” Nevertheless, this method yields turbulence fields that are consistent with buoyant and shear production of turbulence as shown below.

Fig. 2.

Close-up view of (a) streamwise velocity (shading in m s−1), (b) vertical velocity (shading in m s−1), and (c) average TKE (shading in m2 s−2) and potential temperature (lines of constant °C) for the no heating case at t = (left) 180, (middle) 205, and (right) 235 min.

Fig. 2.

Close-up view of (a) streamwise velocity (shading in m s−1), (b) vertical velocity (shading in m s−1), and (c) average TKE (shading in m2 s−2) and potential temperature (lines of constant °C) for the no heating case at t = (left) 180, (middle) 205, and (right) 235 min.

Figure 2 spans the time period when a self-induced critical level at ∼1 km becomes buoyantly unstable and generates a breaking wave with considerable turbulence. The first time is at t = 180 min, which is just before the breaking event begins. Streamwise velocity at this time shows a typical nonlinear amplified gravity wave response with a large zone of stagnant air with near-zero streamwise velocity over the lee of the mountain at 1–1.5-km height. Beneath this stagnant layer is a jet of increased winds along the surface of the lee of the mountain. The surface jet extends down the slope until boundary layer separation occurs at y = ∼34 km and the jet is lofted into the air by the first of a series of rotors, which are in the initial stage of formation. Turbulence at t = 180 min is generally limited to the boundary layer beneath the evolving rotor circulation.

At 205 min the isotherms in the stagnation zone (near y = 30 km) begin to overturn, indicating the onset of internal gravity wave breaking. As the isotherms overturn, wave breaking generates strong vertical motion and increased turbulence aloft. Over time, turbulence generated aloft is advected throughout the stagnation zone and over the rotor system. Vertical velocities associated with the rotors nearest the mountain are intensified during the breaking process and the rotor heights increase. The rotors also propagate upstream with the first rotor moving from y = 34 km at 180 min to y = ∼32 km at t = 235 min. Negative streamwise velocities associated with the rotor nearest the mountain are quickly destroyed by transport and mixing in the breaking stagnation zone. Rotors further downstream, which have not yet interacted with the turbulence advected downstream from the breaking wave region, are shorter and less turbulent and still contain negative streamwise velocities at the surface.

Factors controlling TKE in the simulations can be diagnosed using the TKE budget equation,

 
formula

where Uj represents the spanwise average velocity components, ɛ is the dissipation of turbulence, ρ is the density, p′ is the perturbation pressure, and overbars represent a spanwise average. Terms in (10) are defined as TKE storage (I), horizontal advection (II), buoyant production/destruction (III), shear production (IV), turbulent transport (V), pressure transport (VI), and dissipation (VII).

Formation of TKE in the mountain wave system is dominated by the combined action of the buoyancy and shear production terms as presented in Fig. 3 along with contours of potential temperature. At the onset of wave breaking aloft (at t = 215 min), buoyant perturbations associated with the overturning of stable isotherms and shear production act to generate turbulence in the stagnation zone. Buoyant suppression on the edges of the stagnation zone is offset by shear production of TKE between the surface and 0.5 km. Shear production in the lower layer is associated with vertical gradients of streamwise velocity at the top of the surface jet, which can clearly be seen in the velocity field (Fig. 2). Concentrated TKE in the first rotor at 215 min is associated with both buoyant production and shear production of TKE in the rotor updraft region. Eventually, TKE from this updraft source region is advected throughout the rotor system. These results are in general agreement with Epifanio and Qian (2008) and Gheusi et al. (2000), who show that shear production from the top of the surface jet is the dominant source of TKE and that buoyant production is much less important than shear production of TKE.

Fig. 3.

Close-up view of average TKE budget terms of (a) buoyant production of TKE and (b) shear production of TKE (shading in m2 s−3) and potential temperature (lines of constant °C), for the no heating case at t = (left) 215 and (right) 235 min.

Fig. 3.

Close-up view of average TKE budget terms of (a) buoyant production of TKE and (b) shear production of TKE (shading in m2 s−3) and potential temperature (lines of constant °C), for the no heating case at t = (left) 215 and (right) 235 min.

Vertical gradients of streamwise velocity above the stagnation layer decrease much faster than they do below the stagnation layer. Consequently, by 235 min the shear production at the top of the stagnation zone is significantly less than it is at the interface of the stagnation zone and the surface jet. Some of the shear production at the top of the stagnation zone might be due to the elevated jet near 2 km (Fig. 2). The existence of this elevated jet is thought to be at least partially due to the leakage of some portion of wave energy through the stagnation zone. Although the secondary breaking zone might be strictly a numerical artifact of an upstream velocity profile that does not change with height, observational evidence provides little insight into the existence of secondary wave breaking regions.

Shear and buoyant production at 235 min continue to play dominant roles throughout the slope flow and rotor system, and turbulence generated in the stagnation zone has largely advected downstream above the trapped lee wave system. Animations of this simulation reveal that most of the TKE within the rotors is generated locally, especially along the rotors updrafts, and not advected into the rotors from the stagnation zone.

b. Influence of surface heat fluxes

In this section we compare results from cases where surface heat flux has been applied to the models. These simulations were designed to examine the effects of weak surface heating (25 W m−2), strong surface heating (200 W m−2), and surface cooling (50 W m−2) on downslope winds and rotor formation. A list of all experiments and their abbreviated names can be found in Table 1. For the weak and strong heating cases, fluxes are applied for the entire period of model simulation. In the surface cooling case, a positive surface flux of 25 W m−2 is applied for the first hour of simulation time to generate a well-mixed upstream boundary layer. This is followed by surface cooling of 50 W m−2, simulating the transition from afternoon to evening in the upstream boundary layer.

Upstream profiles of velocity, potential temperature, and TKE are presented in Fig. 4 for each case after 200 min, demonstrating the effects of the heat flux on the boundary layer structure. We note that the cooling case generates a weakly stratified, shallow boundary layer that is not well resolved by the model. Our intent in simulating this case was to examine how reduced surface turbulence and increased low-level stratification affect the mountain wave response.

Fig. 4.

(a) Inflow near-surface velocity, (b) potential temperature, and (c) TKE profiles at y = 4 km for the experiments at t = 200 min.

Fig. 4.

(a) Inflow near-surface velocity, (b) potential temperature, and (c) TKE profiles at y = 4 km for the experiments at t = 200 min.

In Fig. 5 we present streamwise velocities, vertical velocities, and averaged TKE and contours of potential temperature for the no heating, weak heating, strong heating, and cooling cases, respectively, at t = 220 min. Compared with the no heating case, weak heating produces small differences in the downstream lee wave structure with a reduction in lee wave strength beyond the first rotor and the absence of negative streamwise velocities under all lee waves. The absence of negative streamwise velocities beneath the lee waves means that rotors are not present at all in the weak heating case. However, the strength of the downslope jet is roughly the same in these two cases. In contrast, both the lee wave structure and downslope jet in the strong heating case are nonexistent relative to the weak heating and no heating cases. Surface flow reversal in this case is much less prevalent than it is in the no heating case. In addition to differences in the lee wave structure, the leeside surface jet is significantly shorter and higher in the strong heating case. Stratification in the lowest 0.5 km above the ground is essentially unstable, which leads to a much more diffuse surface jet. The stagnation zone is also smaller in the strong heating case, with most of the boundary layer turbulence resulting from convective forcing rather than turbulence from mountain wave breaking.

Fig. 5.

Close-up view of (a) streamwise velocity (shading in m s−1), (b) vertical velocity (shading in m s−1), and (c) average TKE (shading in m2 s−2) and potential temperature (lines of constant °C) at t = 220 min for the (left) no heating case, (second column) weak heating case, (third column) strong heating case, and (right) cooling case.

Fig. 5.

Close-up view of (a) streamwise velocity (shading in m s−1), (b) vertical velocity (shading in m s−1), and (c) average TKE (shading in m2 s−2) and potential temperature (lines of constant °C) at t = 220 min for the (left) no heating case, (second column) weak heating case, (third column) strong heating case, and (right) cooling case.

In contrast to the heating cases, the surface cooling case shows a series of well-developed and distinct lee wave rotors. The first rotor begins slightly upslope from the location of the first rotor in the no heating case (y = 32 km versus y = 33 km), a result consistent with the numerical experiments of Poulos et al. (2000). Unlike the no heating case, each rotor is capped by a stably stratified, undulating jet with relatively high velocities. Negative streamwise velocities under each rotor crest are more pronounced in the cooling case than in the other simulations. In comparison with the no heating case, rotor wavelengths in the cooling case are shorter (∼1.9 km versus ∼2.5 km). This result is consistent with Jiang et al. (2006), who find that surface cooling results in shorter wavelengths for trapped lee waves. An additional simulation (not shown here), run with surface cooling but no mean streamwise flow velocity, develops a katabatic flow of small magnitude (roughly 2 m s−1) over the slope, less than ∼50 m in depth, which is significantly less than the depth of the shooting flow on the lee slope. Because of the small height and width of our mountain, we found katabatic flow velocities on the lee slope to be significantly smaller than did previous researchers who studied the interaction of katabatic flow and mountain waves (Poulos et al. 2000). The increased near-surface stratification, however, as shown below, does play a dynamically significant part in the structure of the lee rotors.

Plots of TKE for the no surface heating, weak surface heating, strong surface heating, and surface cooling cases are also presented in Fig. 5. Both the weak and no heating cases show very little difference in the stagnant flow region above the leeside jet. Surface heating is not strong enough in the weak heating case to affect the stratification of the stagnation zone, which begins over half a kilometer above ground level on the lee slope. However, turbulence in the boundary layer is significantly different between the two cases. In the no heating case, turbulence in the rotors and boundary layer is confined to the lee of the mountain, whereas turbulence is located upstream of the mountain and throughout the entire flow downstream of the mountain in the weak heating case. Indeed, some of the upstream boundary layer turbulence is advected over the mountain and into the rotors themselves (although as shown below, turbulence transport for the weak heating case is not the dominant factor inhibiting rotors). Increased boundary layer turbulence is one reason why the rotor circulations downstream of the mountain are more diffuse and weaker in the weak heating case.

Boundary layer turbulence plays a very prominent role in the dynamics of the strong surface heating case. Here, convection located both upstream and downstream of the mountain generates strong turbulence that is transported over the mountain. The well-mixed upstream turbulent boundary layer is approximately twice the height of the mountain in this case (see Fig. 4). Increased turbulence in the boundary layer weakens the surface jet, decreasing the amount of shear-produced turbulence at the bottom of the stagnation zone and top of the surface jet. Both the stagnation zone and dispersive waves play a significantly smaller role in the strong heating case.

For the surface cooling case, the dynamics of the stagnation zone and associated internal gravity wave breaking are similar to the no heating case. However, surface cooling enhances the strength of near-surface stratification upstream from the mountain. Increased surface stratification leads to a strengthening of the potential temperature gradient at the top of the rotor zone and an extension of the leeside jet over the rotors farther downstream in comparison with the no heating case. Increased streamwise velocities associated with the leeside jet and flow over the rotors fuels increased TKE through shear production. In the no heating case, the largest concentrations of TKE in the boundary layer decrease rapidly as a function of downstream distance.

Contour plots of TKE budgets for the no heating, strong heating, and cooling cases at t = 220 (Fig. 6) show significantly less shear production of TKE aloft in the strong heating case. As noted before, TKE from the surface acts to diffuse and weaken the surface jet, which is the primary source of shear-produced TKE in the lower stagnation zone. Consequently, shear production in the strong heating case is spread over a larger area as compared to the no heating and cooling cases. Convection in the strong heating case generates a deep mixed layer upstream from the mountain, which effectively prevents a strong mountain internal wave response.

Fig. 6.

Close-up view of average TKE budget terms of (a) buoyant production of TKE and (b) shear production of TKE (shading in m2 s−3) and potential temperature (lines of constant °C) at t = 220 min for the (left) no heating, (middle) strong heating, and (right) cooling cases.

Fig. 6.

Close-up view of average TKE budget terms of (a) buoyant production of TKE and (b) shear production of TKE (shading in m2 s−3) and potential temperature (lines of constant °C) at t = 220 min for the (left) no heating, (middle) strong heating, and (right) cooling cases.

In the surface cooling case, shear production of TKE is significant along the rotor interfaces, especially farther downslope from the ridge, because of the undulating jet discussed above. Buoyant production of TKE in the rotor updrafts is also a source of TKE, but not as dominant as shear production. The maintenance of the strongly stratified undulating jet above the rotors is largely due to buoyant suppression of TKE along the crest of the rotors and in the rotor downdrafts. This is also important in maintaining the rotor circulation by preventing turbulence advected from the stagnation zone from destroying the stratified layer that defines the top of the rotors. The increased amount of shear production along the bottom of the especially strong and distinct stably stratified undulating jet also generates more turbulence within the rotors themselves. In contrast, convective mixing in the strong heating case prevents the formation of the stratified layer and accompanying rotors.

c. Upstream versus downstream weak surface heating

Overall, our results suggest that boundary layer turbulence can have a dramatic impact on the strength of leeside winds and trapped waves. However, it is not clear if turbulence generated locally by convection is more or less important than turbulence generated upstream from mountains and transported downstream by the mean flow in cases with weak heating. We next explore the effects of upstream versus downstream heating by conducting experiments where weak surface heating (25 W m−2) is applied either upstream of the ridge crest only or downstream of the ridge crest only. A separate set of simulations (not shown here), run with no mean streamwise flow velocity and surface heating on half of the domain only, show a surface horizontal flow speed of less that 1 m s−1. Thus, the mean flow in the simulations presented dominates the horizontal flow that results from the differential heating of the surface.

In Fig. 7 we present streamwise velocities, vertical velocities, and averaged TKE and contours of potential temperature for the upstream heating and downstream heating cases at t = 230 min. Although there is a significant amount of TKE transported over the ridge in the upstream heating case, the downstream heating case has more turbulence in the boundary layer in the lee of the ridge. The rotors in the upstream heating case are fairly well defined, whereas the rotors in the downstream heating case are not as distinct. The downstream heating case shows a lee wave train in which there are virtually no negative streamwise velocities. In contrast, the upstream heating case bears a strong resemblance to the no heating case (Fig. 5), except that the rotors in the upstream heating case are slightly taller (as in Doyle and Durran 2002) and the streamwise wavelength is smaller.

Fig. 7.

Close-up view of (a) streamwise velocity (shading in m s−1), (b) vertical velocity (shading in m s−1), and (c) average TKE (shading in m2 s−2) and potential temperature (lines of constant °C) at t = 230 min for the (left) upstream and (right) downstream heating cases.

Fig. 7.

Close-up view of (a) streamwise velocity (shading in m s−1), (b) vertical velocity (shading in m s−1), and (c) average TKE (shading in m2 s−2) and potential temperature (lines of constant °C) at t = 230 min for the (left) upstream and (right) downstream heating cases.

Turbulence aloft and at the surface jet–stagnation zone interface is stronger in the downstream heating case in comparison with the upstream heating case. This is largely because of an increase in the shear production of TKE (see Fig. 8, contour plots of the TKE budget terms). Transport of TKE over the mountain in the upstream heating case is not as important as locally generated TKE in the downstream heating case. Most of the TKE within the rotors in the upstream heating case is generated by shear production in the rotors themselves and at the foot of the surface jet where the rotors begin. In the downstream heating case, shear production of TKE is more vigorous at the interface of the surface jet and stagnation zone and continues to a lower height than in the upstream heating case, allowing for more efficient entrainment of TKE from the foot of the jet into the rotors (near y = 33 km). The increased shear production of TKE results in weaker and more diffuse lee wave rotors in the downstream heating case. Hence, surface heating affects the rotor and lee wave structure indirectly through increased shear production of TKE, rather than through direct buoyant production of TKE.

Fig. 8.

Close-up view of average TKE budget terms of (a) buoyant production of TKE and (b) shear production of TKE (shading in m2 s−3) and potential temperature (lines of constant °C) at t = 230 min for the (left) upstream and (right) downstream heating cases.

Fig. 8.

Close-up view of average TKE budget terms of (a) buoyant production of TKE and (b) shear production of TKE (shading in m2 s−3) and potential temperature (lines of constant °C) at t = 230 min for the (left) upstream and (right) downstream heating cases.

Further insight into the role of upstream versus downstream heat fluxes is presented in Fig. 9, which shows the streamwise flux of integrated turbulence as a function of downstream distance at t = 195 min, just after the onset of wave breaking aloft. Integrated turbulence flux is defined as a vertical integral of the product of streamwise velocity with turbulence e,

 
formula

where the integral is taken over the entire domain. We concentrate on locations with y < 30 km and y > 40 km, since these do not include the direct effects of wave breaking aloft. Integrated turbulence flux values in the weak heating case upstream of the mountain are almost exactly the same as those of the upstream heating case, whereas both the no heating case and the downstream heating case show almost no integrated turbulent flux upstream of the mountain. However, in the lee of the mountain and behind the area affected by the stagnation zone breaking (y > 40 km) the integrated turbulence flux in the upstream heating case is very small, whereas the integrated turbulence flux in the downstream heating case is roughly the same as the weak heating case. This points to the role of indigenously generated turbulence in the lee of the mountain as a controlling factor in the formation of rotors. In the case of weak surface heating, where the flux of turbulence from the upstream boundary layer does not completely dominate the dynamics of the lee flow, surface heating downstream of the ridge crest plays a more important role in determining lee flow dynamics than does surface heating upstream of the ridge crest. For larger upstream surface heating (and higher boundary layer heights relative to mountain heights), the effect of upstream surface heating on lee rotors is much more pronounced, as shown by the strong heating case. Nevertheless, even weak heating on the lee side of the mountain is able to increase low-level turbulence and prevent strong rotor formation.

Fig. 9.

Integrated turbulent flux for no heating, weak heating, upstream heating, and downstream heating cases at t = 195 min.

Fig. 9.

Integrated turbulent flux for no heating, weak heating, upstream heating, and downstream heating cases at t = 195 min.

d. Rotor structure

Throughout our analysis we emphasize the significant role of turbulence in controlling the number and strength of trapped waves and rotors in the simulations. For the large-scale structure of rotors, our results are in many ways similar to past studies showing a range of rotor behavior dependent on boundary layer heating and surface roughness. However, our results do not show the organized structures noted in the previous high-resolution rotor study of Doyle and Durran (2007). In their study, small-scale circulations or subrotors are produced along the upstream edge of the rotor and propagate through the rotor circulation. They attribute these subrotors to K–H waves that are triggered by the flow separation at the leading edge of the rotor and amplify at the expense of shear. For comparison, we present plots of spanwise vorticity similar to Doyle and Durran (2007) for the leading rotor in the no heating case (Fig. 10). Spanwise vorticity is defined as

 
formula
Fig. 10.

Close-up view of spanwise vorticity (shading in s−1) and vectors of flow velocity at t = 205, 210, and 215 min for the no heating case.

Fig. 10.

Close-up view of spanwise vorticity (shading in s−1) and vectors of flow velocity at t = 205, 210, and 215 min for the no heating case.

In contrast to their results, our simulations do not show strongly organized subrotors along the edge of the main circulation; rather, they display a more chaotic vorticity field indicative of fully turbulent flow. The flow in the simulations of Doyle and Durran (2007), in which turbulent motions of all length scales are explicitly accounted for by the TKE parameterization, does not contain vorticity perturbations such as are apparent in Fig. 10 along the slope of the ridge. We speculate that grid resolution or subgrid-scale TKE parameterization issues may play a role in this. Small-scale turbulence in our simulations is the likely reason that K–H billows are not generated in the shear layer above the rotor. Resolved turbulence prevents K–H billow formation by disrupting the smooth shear flow upstream from the rotor, preventing the more gradual growth of well-organized instability waves noted in Doyle and Durran (2007).

4. Conclusions

The focus of this study was twofold: to examine low-level internal gravity wave breaking and its interaction with lee wave rotors using an eddy-resolving model and to examine the interaction of an upstream boundary layer modified by surface heat fluxes on downslope winds and rotors.

Our experiments were based on a simple scenario with constant velocity and stratification, a mountain height selected to generate low-level internal wave breaking, and an upstream boundary layer one half the height of the mountain. Simulations with various heat fluxes were conducted and analyzed to understand how rotors interact with internal wave breaking and turbulence. In the case with no surface heating, a typical nonlinear wave response was produced, with a leeside surface jet and a large zone of stagnant air above the lee side of the mountain that eventually resulted in overturning of isotherms and generation of turbulence. Turbulence was also generated by shear production along the edges of the stagnation zone, especially at the interface with the surface jet on the lee of the slope. Over time, a series of trapped waves or rotors formed in the lee of the ridge, with the first rotor representing the separation of the leeside jet from the mountain slope. Downstream rotors maintain a distinct identity as they slowly propagate back toward the ridge. Once sufficiently close to the ridge, the lead rotor decayed because of interaction with turbulence advected from the wave breaking region and turbulence generated indigenously in the rotor zone by gradients in vertical and horizontal velocity.

Increasing surface flux was found to alter the strength and number of rotors. With weak heating of 25 W m−2, there were no longer any negative streamwise velocities under the lee waves, which were strongly modified by turbulence generated in the stagnant wave breaking region and through convective forcing. However, leeside downslope winds in this case were not strongly affected by turbulence. Strong surface heating of 200 W m−2 prevented the formation of rotors and produced a much weaker downslope wind event. In this case, the boundary layer depth increased rapidly to a depth roughly double the mountain height, thereby reducing the stratification of flow forced over the mountain. Simulations with heating confined to either the upstream or downstream side of the mountain [similar to the Doyle and Durran (2002) study] revealed that locally generated turbulence on the downstream side of the mountain was much more important in controlling the rotor behavior than turbulence advected over the mountain from the upstream boundary layer.

In contrast with the surface heating cases, surface cooling of 50 W m−2 forced an enhanced rotor circulation, leading to a train of leeside waves capped by a jet of increased streamwise winds. Increased stratification from surface cooling was found to be important in the formation of a stably stratified undulating jet, which capped the rotors. Shear between this jet and the rotors generated increased turbulence in the rotors themselves, while buoyant destruction of TKE in the rotor downdrafts acted to maintain the rotor circulations for a longer distance downstream from the ridge.

In general, there are three relevant physical cases, in terms of the ratio of upstream boundary height to ridge crest height, presented in our experiments. When upstream boundary layer height is much less than ridge crest height, upstream surface heat fluxes are largely irrelevant for the downstream flow because upstream turbulence is not advected over the ridge. When upstream boundary layer height is much greater than ridge crest height, upstream turbulence is advected over the ridge crest and can dominate lee flow behavior. Finally, for the case in which upstream boundary layer height is comparable to ridge crest height, both upstream turbulence and surface heat fluxes can affect the lee flow behavior, depending on the strength of surface heat fluxes. For these cases, our simulations suggest that well-developed lee wave trains will be favored at night and in the early morning and will generally become indistinct or washed out on days with strong surface heating.

Our simulations, for the most part, represent conditions that have not been examined in past modeling studies of lee wave systems and rotors. For example, Doyle and Durran (2007) examined rotors and small-scale subrotor circulations for environmental conditions without the low-level internal gravity wave breaking and turbulence generated by a self-induced critical level. They used a sounding that included a low-level inversion and velocity that increased with height, which prevented the low-level internal gravity wave breaking that is so dominant in our simulations. In their simulations, rotors were found to produce well-defined Kelvin–Helmholtz billows. Rotors in our study did not produce the same structures; instead, they were characterized by chaotic eddies more indicative of fully turbulent flow. Differences between our simulations and Doyle and Durran (2007) might be explained by different upstream soundings, the presence or absence of internal gravity wave breaking, differences in grid resolution in our simulations (by roughly a factor of 3), the presence of upstream turbulent eddies, and differing subgrid parameterization of turbulence.

Epifanio and Qian (2008) also used an eddy-resolving model to examine how turbulence affects internal wave breaking for a self-induced critical level, but without considering upstream boundary layer effects or rotor formation. In general, our analysis of turbulence formation in the stagnant region compares favorably with their results. However, our simulations suggest a more prominent role for shear production of TKE along the top of the stagnation zone early on in the breaking process.

Simulations focusing on the effects of surface fluxes and boundary layers on internal waves have largely applied mesoscale models that are not turbulence resolving. For example, Jiang et al. (2006) examined the role of the boundary layer, surface heat fluxes, and drag on trapped waves generated by a two-layer atmospheric structure. Their scenario was quite different from conditions we examined in that trapping was produced by a reduction in stratification with height rather than the nonlinear effects of a self-induced critical level. Consequently, it is not too surprising that their results differ from ours. In particular, they found that the boundary layer acts as a sponge for trapped waves by partially absorbing down going wave energy. For stagnant, stable surface conditions forced by surface cooling, wave absorption was highly efficient because of the formation of a critical level when the flow decreases to zero. We find a directly opposite result: surface cooling in our simulations enhances leeside waves by decoupling the waves from turbulence generated at the surface, whereas surface heating decreases lee wave intensity by disrupting the lee wave system through increased turbulent mixing.

Differences between our study and that of Jiang et al. (2006) are mostly likely tied to the very different lee wave characteristics. In our experiments, the leeside atmospheric structure is similar to an interfacial wave scenario with a two-layer structure divided by a thin, strongly stratified layer. Waves are trapped along the interfacial layer and propagate horizontally along the interface with little vertical propagation. In contrast, waves in the Jiang et al. (2006) case have a vertical structure that spans a stratified fluid depth many times the boundary layer depth and mountain height. Vertical energy propagation in these waves is much larger relative to our cases, allowing for greater loss of wave energy in the boundary layer. Ultimately, waves in our simulations are more strongly affected by turbulent processes whereas waves in Jiang et al. (2006) are governed more by internal wave dynamics.

In a more recent mesoscale modeling study, Jiang and Doyle (2008) examined the effects of surface heating using idealized conditions with a simple ridge, where the internal wave response was strongly controlled by surface fluxes. With no fluxes, their simulations produced a very weak internal wave response. They found that cooling forced a leeside jet similar to a mountain wave response, in partial agreement with our cooling experiment. The effect of heating on the downstream side of the ridge was also examined, similar to our experiment contrasting upstream and downstream heating. However, in their experimental setup, heating on the downstream side of the ridge increased the cross-mountain pressure gradient, resulting in a stronger leeside jet rather than a more turbulent downstream boundary layer. Overall, their experiments suggest that for large-scale mountains (hundreds of kilometers) the effects of heating on buoyancy and cross-mountain pressure gradient are more important than the effects of turbulent mixing. In our case, both the height and width of the mountain are much smaller and so, not surprisingly, the turbulent boundary layer has a larger impact on the flow dynamics.

Many unanswered questions remain regarding the relative importance of environmental factors such the role of upper-level stratification, elevated inversions, and ridge-top level shear in the formation of downslope windstorms and rotors. Although results presented here address some of the outstanding issues regarding the importance of surface heat fluxes in nonlinear trapped lee wave regimes, a significant portion of the physically feasible parameter space (based on upstream soundings taken during T-REX and elsewhere) remains unexplored. Moreover, it is not clear that synoptically based upstream soundings (i.e., once every 12 h) can help to resolve these issues; hence, we intend to continue to explore the relative importance of these factors in the onset and decay of lee flow regimes, especially those with hydraulic jump conditions and rotors, which are considered extremely hazardous by the aviation community.

Acknowledgments

We are pleased to acknowledge the supercomputer time provided by the National Center by Atmospheric Research, which is funded by the National Science Foundation. This research was funded by the National Science Foundation under Grant ATM-0527790. We also wish to thank the anonymous reviewers for their help in strengthening the manuscript.

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Footnotes

Corresponding author address: Craig Smith, COAS, 104 COAS Admin Bldg., Oregon State University, Corvallis, OR 97331. Email: csmith@coas.oregonstate.edu