Trapped lee wave interference over double bell-shaped obstacles in the presence of surface friction is examined. Idealized high-resolution numerical experiments with the nonhydrostatic Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) are performed to examine the influence of a frictional boundary layer and nonlinearity on wave interference and the impact of interference on wave-induced boundary layer separation and the formation of rotors.
The appearance of constructive and destructive interference, controlled by the ratio of the ridge separation distance to the intrinsic horizontal wavelength of lee waves, is found to be predicted well by linear interference theory with orographic adjustment. The friction-induced shortening of intrinsic wavelength displays a strong indirect effect on wave interference. For twin peak orography, the interference-induced variation of wave amplitude is smaller than that predicted by linear theory. The interference is found to affect the formation and strength of rotors most significantly in the lee of the downstream peak; destructive interference suppresses the formation and strength of rotors there, whereas results for constructive interference closely parallel those for a single mountain. Over the valley, under both constructive and destructive interference, rotors are weaker compared to those in the lee of a single ridge while their strength saturates in the finite-amplitude flow regime.
Destructive interference is found to be more susceptible to nonlinear effects, with both the orographic adjustment and surface friction displaying a stronger effect on the flow in this state. “Complete” destructive interference, in which waves almost completely cancel out in the lee of the downstream ridge, develops for certain ridge separation distances but only for a downstream ridge smaller than the upstream one.
Generation of gravity waves by multiple barriers has received limited attention (e.g., Tampieri and Hunt 1985; Kimura and Manins 1988; Grisogono et al. 1993; Mayr and Gohm 2000; Lee et al. 2006) compared with a sizeable body of research on waves generated by a single barrier (Baines 1995). The topic that has received some more attention is interference of lee waves generated by double bell-shaped obstacles (Lee et al. 1987; Vosper 1996; Scorer 1997; Gyüre and Jánosi 2003; Grubišić and Stiperski 2009, hereafter GS09). Following Scorer (1997), the flow field in the lee of a double mountain can be viewed as a linear superposition of two trapped lee-wave trains emanating from each obstacle. In this model, placing a secondary obstacle of equal height at an odd number of half-wavelengths downstream of the first would lead to cancellation of waves downstream; placing it at an even number of half-wavelengths would lead to the wave amplitude doubling. However, numerical studies (Vosper 1996; GS09) and laboratory experiments (Lee et al. 1987; Gyüre and Jánosi 2003) with twin peak orography show that neither doubling nor complete cancellation occurs in the lee of the second peak. Rather, wave amplitude and gravity wave drag vary around some mean state and are determined by the valley width to wavelength ratio. Based on linear numerical modeling results, Vosper (1996) attributes these variations to the constructive and destructive interference of a partially trapped part of the wave spectrum excited by the two mountains. In GS09 trapped lee waves excited by twin peaks are nonlinear. Yet the interference occurs and displays itself in the amplitude and gravity wave drag variations. The excited waves adjust to the spectrum of the underlying topography, giving rise to an integral number of wavelengths over the valley. The amplitude in the lee of the downstream peak is always larger than the one within the valley—in agreement with Lee et al. (1987) but opposite to Gyüre and Jánosi (2003)—yet never double.
The above studies, with the exception of Lee et al. (1987) and Gyüre and Jánosi (2003), neglect surface friction. The presence of a frictional boundary layer significantly influences the trapped lee wave field (e.g., Smith et al. 2002; Jiang et al. 2006; Smith et al. 2006; Jiang et al. 2008) and allows for the development of an additional set of flow realizations. The boundary layer is shown to move the mountain wave pattern upstream, weaken the wave amplitude, and reduce gravity wave drag and momentum flux (Jiang et al. 2008). Because of frictional attenuation the wave amplitude decreases exponentially with downstream distance (Smith et al. 2002; Jiang et al. 2006) while the horizontal wavelength is reduced (Smith et al. 2006). Surface friction also facilitates boundary layer separation and formation of reversed flow (rotors) underneath the lee wave crests. For sufficiently large wave amplitudes boundary layer separation is forced by adverse wave-induced pressure gradients (Doyle and Durran 2002, 2004). Recent studies of rotor flows show that the development of rotors is strongly influenced by the structure and evolution of the overlying mountain waves and the underlying boundary layer (Vosper 2004; Hertenstein and Kuettner 2005; Vosper et al. 2006; Doyle and Durran 2007; Sheridan et al. 2007; Jiang et al. 2007). The study of coupling between the waves, rotors and the boundary layer is the core scientific objective of the Terrain-Induced Rotor Experiment (T-REX) (Grubišić et al. 2008).
The idealized two-dimensional (2D) free-slip simulations of trapped lee waves in an idealized T-REX environment in GS09 show that the development of lee wave interference significantly affects the overall flow field, both within the valley and, in particular, in the lee of the second obstacle. Following on GS09 here we investigate the mountain wave–rotor–boundary layer coupling over double bell-shaped obstacles by examining the interaction between trapped lee wave interference and the frictional boundary layer. We set out to test the applicability of linear interference theory for small- and finite-amplitude flow regimes. We expect the critical mountain height, for which boundary layer separation first occurs, to be affected by interference. For constructive interference one expects rotors to develop for smaller mountain heights, and to be stronger compared to rotors that form in the lee of a single mountain of equal height. The opposite is expected under destructive interference. To test this hypothesis we carried out two sets of idealized 2D numerical simulations with double bell-shaped mountains. The T-REX idealized (TI) set of experiments follows on the simulations in GS09, in which both the upstream atmospheric profiles and orography are based on the T-REX conditions. To better understand the underlying dynamical mechanisms and interactions a set of highly idealized (HI) simulations was designed in which the upstream profiles were further simplified. In the HI simulations the focus is placed on trapped lee waves over double obstacles of varying heights, extending the analysis from the small- to finite-amplitude flow regimes.
The paper is organized as follows. The numerical model and experimental setup are presented in section 2, with the diagnostic parameters defined in section 3. Sections 4 and 5 deliver the results of the TI and HI experiments. These results are further discussed in section 6. Section 7 presents the summary and concludes the paper.
2. Numerical model and experimental setup
The numerical simulations were performed using the atmospheric component of the Naval Research Laboratory (NRL) Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS; Hodur 1997). This is a limited area, nonhydrostatic, fully compressible model that includes a suite of physical parameterizations. In 2D, the turbulent kinetic energy (TKE) e = (u′2 + w′2)/2 is a prognostic variable governed by the equation (Mellor and Yamada 1982)
In (1) KM and KH are eddy mixing coefficients for momentum and heat, given by KM,H = SM,Hlm(2e)1/2, where SM,H are functions of the local Richardson number, Γ is a constant, lm is the mixing length formulated according to Mellor and Yamada (1982) and Thompson and Burk (1991), and De represents subgrid-scale TKE mixing and horizontal smoothing. The subgrid-scale mixing of momentum and heat fluxes is parameterized as and . In “free-slip simulations” vertical fluxes of horizontal momentum are set to zero at the lower boundary. In “no-slip simulations” surface friction is parameterized through vertical fluxes of horizontal momentum between the ground and the lowest model level, following Louis (1979) and Louis et al. (1982). If not specified otherwise, in no-slip simulations the surface roughness is set to 0.1 m. In all simulations vertical fluxes of heat and moisture are set to zero at the lower boundary.
All simulations in this study are 2D, irrotational, and dry. As in GS09 the basic terrain profile is defined as a superposition of two single witch of Agnesi obstacles:
where H1 and H2 are heights of the first (upstream) and second (downstream) obstacle, a is the mountain half-width, and V is the ridge separation distance (i.e., the valley width). The range of values for these terrain parameters is V ∈ [30, 85] km, H1, H2 ∈ [0, 3000] m for the TI experiments, and H1, H2 ∈ [10, 1500] m for the HI experiments. Also, a is set to 5 km.
The numerical domain setup is identical to that described in GS09 except that the vertical grid spacing is refined at the lowest model levels, increasing from 30 m at the ground to 100 m at 300 m.
3. Numerical experiments and diagnostic parameters
The upstream atmospheric structure is represented with a simplified version of the remarkably similar atmospheric soundings obtained upstream of the Sierra Nevada at 0000 UTC 26 March 2004 during the Sierra Rotors Project (SRP) intensive observing period (IOP) 8 and 1400 UTC 25 March 2006 during T-REX IOP 6 (Fig. 1). The idealized upstream sounding of the TI experiments is identical to the baseline sounding described in GS09, in which the crest level vertical wind shear and stability profile from the above two atmospheric soundings is retained. For the HI experiments the TI sounding was further simplified. The HI stability profile has two layers: a stably stratified “troposphere” with the buoyancy frequency N = 0.012 s−1, extending up to 11.5 km, topped by a “stratosphere” with N = 0.022 s−1. In the lowest 5 km the corresponding vertical profile of the barrier-normal wind component is characterized by positive vertical wind shear of S = 6 m s−1 km−1, equal to the vertical wind shear in the TI experiments. Further aloft a layer of constant wind speed extends up to the stratosphere where the wind speed decreases with height. The parameters that describe the initial upstream soundings for all numerical experiments in this study are given in Table 1.
The experiments were run until a quasi-steady state (defined below) was achieved. For the HI experiments this took on average 6 h; for the TI experiments the required time was longer. Consequently, the HI simulations were run for 10 h and the TI simulations for 17 h. All diagnostic parameters, defined further in this section, represent time averages over a 2-h-long diagnostic period within the quasi-steady period of each simulation.
The steadiness of the simulations was affected by several factors. The first is the fact that in 2D simulations with surface friction it is impossible to achieve a balanced boundary layer. Consequently, the boundary layer depth continues to increase throughout the simulations. This increase, however, is slowed down in the later stages of the simulations (Fig. 2a). The change in the horizontal wind speed due to the evolution of the boundary layer with time is small compared to the magnitude of the trapped lee wave flow (<2%). Furthermore, the time scale for the evolution of the boundary layer parameters (such as TKE) is an order of magnitude larger than the simulation time: TKEmaxΔt/ΔTKEmax ∼ 50 h, while the advection and wave propagation time scale is T = a/Uo ∼ 0.3 h for Uo = 5 m s−1. The steadiness is additionally affected by the degree of flow nonlinearity. As a measure of unsteadiness we use here the variability of wave amplitude and drag determined during the quasi-steady periods of the simulations. For HI simulations the maximum variability is 10% whereas in the TI runs it reaches 24%. The flow over one mountain (average variability of 2.3%) is in general steadier than the flow over two mountains, in particular the flow downstream of the second mountain (maximum of 6.9%). For twin peaks lower than 1000 m the flow within the valley displays the same degree of variability (<2.9%) as in the lee of a single mountain; for higher twin peaks the variability increases. Simulations with the lower second mountain were found to be particularly unsteady.
In summary, for the same mountain height the free-slip simulations are steadier than the no-slip simulations. Furthermore, the effect of the boundary layer was found to decrease as the nonlinearity of the flow increases (Jiang et al. 2008). The nonlinear transients in the otherwise steady flow, the type discussed in Nance and Durran (1998), do not appear to play a role in our simulations. The upstream columnar modes do not develop for any of the examined mountain heights in the HI simulations (NH/Uc < 1, where Uc is the wind speed at crest level). In the finite-amplitude TI simulations, as in GS09, upstream blocking is present. The columnar modes propagate upstream in form of small-amplitude waves (Baines 1987) that only weakly affect the flow in the lee of the mountains.
Given the above, the 2-h diagnostic period for each simulation was selected as the period within which a quasi-steady trapped lee wave pattern is well established (Fig. 2b), the increase of the boundary layer depth has slowed down, and the variations of amplitude and drag are at a minimum. Several diagnostic parameters, used in the presentation and analysis of results in the following sections, were determined during these diagnostic periods. Apart from the gravity wave drag D and amplitudes downstream of the first A1 and the second mountain A2, used in GS09, here we also diagnose the minimum surface horizontal wind speed Uimin, where i = 1, 2 determined at the lowest model level (15 m) underneath the first lee wave crest in the lee of each obstacle. Minimum surface wind speed of less than zero is taken as indicative of a rotor. Furthermore, wave spectra were determined using fast Fourier transform (FFT) of vertical wind speed. The FFT results were found not to be sensitive to the distance above the mountain and below the inversion from which the vertical wind speed data is obtained. In the following sections, horizontal wavelength λ refers to the primary peak of a given lee wave spectrum; the primary peak of the lee wave spectrum generated by a single mountain λs defines the intrinsic wavelength. The latter is the characteristic of a given upstream atmospheric profile. Additionally, primary wavelengths from the surface perturbation pressure spectra λpsfc downstream of a single mountain were also calculated.
As in GS09 we make use of several nondimensional ratios. Apart from the mountain height ratio Hn = H2/H1, amplitude ratio A2/A1, and nondimensional wavelength V/λ, introduced in GS09, these include the nondimensional amplitude A/As, nondimensional minimum horizontal wind speed Umin/Usmin, and nondimensional drag D/Ds, where the variable with the subscript s denotes the respective value in the lee of a single mountain. All diagnostic parameters are listed in the appendix.
4. TI experiment results
Figure 3 shows the vertical cross sections of potential temperature, horizontal wind speed, and horizontal perturbation pressure gradient from the no-slip double mountain experiments using the T-REX idealized sounding (TI). The height of the first peak is H1 = 3000 m. The height of the downstream peak, placed at V = 30 km, varies between 0 and 3000 m (0 ≤ Hn ≤ 1). A train of large-amplitude trapped lee waves develops in the lee of a single mountain (Hn = 0). The wave amplitude decreases downstream from the mountain due to a combined effect of partial leakage of wave energy through the inversion, evident in the upstream tilt of phase lines over the first mountain, and the boundary layer friction (Smith et al. 2006). Positive vertical wind shear also contributes to the decrease of wave amplitude directly over the mountain (Jiang et al. 2006). The frictionally modified λs equals 17 km. The wave amplitude is sufficiently large to induce boundary layer separation and flow reversals underneath the lee wave crests. The flow within the rotors is neither steady nor organized in a well-defined circulation. Rather, the reverse flow appears in smaller patches of negative horizontal wind speed. At the top of the boundary layer, shear instabilities develop.
As in GS09 the flow is strongly dependent on the presence and height of the second mountain. For Hn = ⅓ the effect is positive (Fig. 3b)—wave amplitudes are larger, the flow is smoother, and the rotor in the lee of the first mountain better organized compared to the flow past a single peak (Fig. 3a). Particularly interesting is the Hn = ⅔ case (Fig. 3c), where one finds a large-amplitude wave over the valley only. Upon encountering the second obstacle the waves almost completely cancel out; farther downstream only short-wavelength small-amplitude oscillations remain. Compared to the other flow realizations, this particular solution is rather unsteady, suggesting that complete cancellation is not a favored state of the system.
As in GS09, increasing the height of the downstream obstacle leads to the increase of wave amplitude in the lee of the second mountain, reflected in the general increase of A2/A1 with Hn (Fig. 4). For the same ridge separation distance (V = 30 km in Fig. 4), it takes a higher downwind mountain in presence of surface friction for the wave amplitude in its lee to exceed the wave amplitude over the valley (A2/A1 > 1). Thus, with friction a stronger wave response is expected over the valley for a larger span of downstream mountain heights.
A sharp minimum in A2/A1 for Hn = ⅔ corresponds to the “complete” destructive interference. This phenomenon is not unique to the no-slip simulations; the same result is obtained in the free-slip runs but at somewhat larger ridge separation distances. The difference in valley width for which the complete destructive interference occurs in no-slip and free-slip runs stems from the shortening of horizontal wavelengths by surface friction. Removing the potential temperature inversion from the upstream temperature profile (cf. Table 1) does not eliminate the wave cancellation for Hn = ⅔ (Fig. 5a). The inversion does seem to have a weak positive impact on the wave cancellation as the lee waves downstream of the second obstacle appear to be better defined in the absence of the inversion. Nevertheless, regardless of the lower boundary condition (free slip or no slip) and the presence or absence of an inversion, the amplitude ratio under complete destructive interference is nearly the same for all TI simulations and equals A2/A1 ≈ 0.2–0.3.
5. HI experiment results
To better understand the interplay between surface friction and lee wave interference, as well as the influence of nonlinearity on this interplay, in HI experiments we have further simplified the upstream atmospheric profiles and lowered the heights of the two mountains (Fig. 1; Table 1). To isolate pure dynamical effects of the frictional boundary layer on wave interference, in addition to the no-slip (nS) simulations, we have carried out two sets of free-slip HI simulations (Table 1). In the first set (fS) the upstream profiles are identical to the undisturbed profiles used to initialize the no-slip simulations (Figs. 1c,d). The upstream profiles used to drive the second set of free-slip (fnS) simulations correspond to profiles from the fully developed no-slip simulations with a layer of neutral stability and strong positive shear near the lower boundary (Figs. 1c,d).
In the following we examine first the effects of surface friction and nonlinearity on lee wave interference by focusing on the flow over twin peak orography. This is followed by an investigation of the impact a lower downstream mountain has on the flow characteristics.
a. Lee wave interference
1) Effects of surface friction
In this section we limit our attention to the flow over twin peak orography with mountain heights of 300 m. While not entirely negligible, finite-amplitude effects are relatively small, allowing us to isolate the key effects of friction on lee wave interference. Vertical cross sections of horizontal wind speed, potential temperature, and horizontal perturbation pressure gradient from the nS simulations show trapped lee waves emanating from each peak and being attenuated as they propagate downstream (Fig. 6). The degree of attenuation varies with the ridge separation distance, suggesting possible lee wave interference. In addition to the downstream amplitude A2, the variation is evident also in the total drag D; both of these vary around their respective values for a single mountain (As and Ds; Fig. 7). For all three sets of HI simulations, nS, fS, and fnS, the curves that describe variations of the normalized downstream amplitude A2/As and D/Ds with the ridge separation distance collapse to a single curve for the ridge separation distance normalized by the respective intrinsic horizontal wavelength of lee waves (cf. Table 2). The pattern that emerges is in perfect agreement with the prediction of linear interference theory, with constructive interference occurring for V/λs = n, n ≥ 1 and destructive interference for V/λs = (2n − 1)/2, n ≥ 2. This illustrates two important points: (i) the lee wave interference occurs regardless of surface friction, and (ii) the reduction of lee wave wavelength due to friction has strong implications for lee wave interference. The second point arises from the fact that interference is governed by the phase with which the wave generated by the upstream peak encounters the downstream one (i.e., by the relation of V to λs) (Scorer 1997).
The intrinsic wavelengths obtained in the nS, fnS, and fS simulations are listed in Table 2. The difference between these wavelengths shows that for double obstacles it is not possible to establish a direct connection between the free-slip wave amplitudes and no-slip rotor strength as was done in Doyle and Durran (2002).
The values of U2min from the three sets of simulations, when normalized in the analogous manner as A2 and D, do not collapse to a single curve (Fig. 7b). The interference pattern is clearly evident in the no-slip results only; in the two frictionless cases there is little evidence for interference in the near-surface velocity. Thus, when present, friction represents the key coupling mechanism between the overlaying trapped waves and the boundary layer, which therefore also reflects interference. This conclusion is further reinforced by a strong anticorrelation (r = −0.95) between A2 and U2min, revealing a direct relationship between the amplitude of the overlying trapped waves and the deceleration of surface flow (even though rotors do not form for mountains as low as these ones and Umin > 0).
Furthermore, for the fnS and nS runs the variation of drag appears to be governed by an alternative, longer wavelength (λpsfc; Table 2). This is in contrast to the fS results as well as those in GS09, where the downstream amplitude and drag are strongly correlated and the interference is governed by a single intrinsic horizontal wavelength.
To explain the nature of a separate wavelength for λpsfc that controls the drag variation, in Fig. 8 we display the power spectra of vertical velocity and perturbation pressure for lee waves in the lee of a single mountain for the nS, fS, and fnS runs. The spectra were determined from the entire lee wave field (Fig. 8, left) and from a downstream portion of the domain, starting at the distance of three mountain half-widths from the obstacle (Fig. 8, right). The results clearly show the reduction of intrinsic horizontal wavelength under the influence of surface friction (cf. Table 2). For the fS results the spectra of perturbation pressure and vertical velocity display the maxima at the same wavelength regardless of what portion of the domain one examines. This is not the case for the nS (and fnS) results, where vertical velocity shows a maximum at a shorter wavelength (21.8 km) than perturbation pressure (23.4 km) for the portion of the wave solution in the immediate vicinity of the mountain. Farther downstream, no difference in intrinsic wavelength of the spectral maxima is obtained, even in the no-slip runs. Sensitivity tests with surface roughness reduced by an order of magnitude, to 0.01 m, do not show any difference in wavelength between surface pressure and vertical velocity. In both of these cases the intrinsic wavelength is equal to that from the fnS run (23.4 km). Thus, a different horizontal wavelength for surface pressure is clearly a frictional effect.
For mountains as low as these (H = 300 m), the flow within the valley is almost unaffected by the presence of the secondary obstacle and matches that in the lee of a single mountain, in agreement with the findings of Lee et al. (1987) for gentle hills. Still, there appears to be a weak increase in amplitude of the first wave over the valley (A1) at V = nλ, suggesting that the flow within the valley will be slightly stronger under constructive interference (not shown).
Figure 9 shows the potential temperature, surface wind speed, and perturbation pressure distribution around the upstream and downstream obstacles for free-slip and no-slip results. Here we limit our discussion to the H = 300 m case only; results for H = 1500 m will be discussed in the next section. Over the upstream mountain one finds a friction-induced shift of the surface wind speed maximum, from the immediate lee, where it lies in the free-slip runs, to over the mountain crest. The pressure perturbation minimum remains in the lee, and the potential temperature profile retains its asymmetric shape—similar to what one finds for a single mountain (Smith et al. 2006; Vosper et al. 2006). Over the downstream peak the flow structure and phase relations depend on interference (two sets of gray lines in Fig. 9), with destructive interference showing more sensitivity to surface friction. Under constructive interference (fS: V = 56 km and nS: V = 44 km), the wind speed, pressure perturbation, and potential temperature perturbation show asymmetry with respect to the mountain peak, in both the free-slip and no-slip runs. Consequently, the respective profiles in Fig. 9 resemble to a large degree those over the upstream mountain. For destructive interference (fS: V = 42 km and nS: V = 33 km) the potential temperature distribution is, to a large degree, symmetric both in the free and no-slip runs (observed also in GS09). This is not the case for the perturbation pressure and surface wind speed profiles that display symmetry in the nS runs and asymmetry under the fS conditions. The symmetric shape of the pressure perturbation, present also in the fnS runs, appears to result from the alteration of the upstream profile due to the boundary layer effects (not shown). The symmetry of the surface wind speed, on the other hand, results from the direct influence of surface friction on the low-level flow.
The difference in shapes of the above profiles under constructive and destructive interference is evident also in the flow solutions. Under destructive interference the wave crests sit atop the downstream peak, forming a vertical phase line with the mountain crest. Furthermore, the amplitude of that short “wave” decays in the vertical, reminiscent of evanescent waves (Fig. 6a). In contrast, under constructive interference, the same phase line has an upstream tilt, and the flow resembles that over the upstream mountain, associated with propagation of wave energy into the vertical. The observed variation of drag as well as minimum horizontal wind speed under different interference conditions (cf. Fig. 7c) reflect to a large degree these differences in flow structure over the downstream peak.
The interference pattern affects also the flow steadiness. Constructive interference leads to steadier flows; within the averaging periods the variation of D and A2 remain within 2% for constructive interference, compared with 10% under destructive interference. More pronounced unsteadiness of destructive interference is likely the reflection of orographic adjustment (GS09), in which the trapped lee wave solution, in its approach to a steady state, has to adjust to the terrain shape by selecting one of the harmonics from the terrain spectrum that is closest to the intrinsic wavelength of lee waves for a given atmospheric profile.
2) Finite-amplitude effects
To investigate the influence of nonlinearity on interference the no-slip simulations were conducted for an additional range of H ∈ [10, 1500] m. The results are presented in Fig. 10 showing the dependence of amplitudes and minimum horizontal wind speed on mountain height. For a single mountain three critical mountain heights (Hs, Hc, and Hh) can be inferred from the dependence of Umin on H. The first critical mountain height Hs (∼370 m) separates the linear from nonlinear regimes and represents the point at which boundary layer separation in the lee of a single mountain first occurs. In agreement with Vosper et al. (2006), the minimum horizontal velocity Umin underneath the first wave crest decreases almost linearly with increasing mountain height up to Hs. For H > Hs the rate of decrease of Umin is reduced, especially after the second critical mountain height Hc (∼500 m) is exceeded. The amplitude growth remains nearly linear up to the third critical mountain height Hh (∼1000 m). For H > Hh the increase of amplitude and rotor strength, as measured by Umin, are substantially reduced.
The significance of these critical mountain heights becomes more evident for the flow over a double mountain (Fig. 10). Up to Hc both A1 and reversed flow strength U1min over the valley have the same values as for a single mountain, as suggested by Lee et al. (1987). For H > Hc, the two sets of values start to diverge; the wave amplitudes and rotor strength within the valley are weaker than those in the lee of a single mountain. This is in agreement with the free-slip simulations of GS09 and the findings of Lee et al. (1987). For H > Hh the rotor strength over the valley (U1min) asymptotically approaches a constant value despite the continued and almost linear increase of the wave amplitude. This suggests that in the finite-amplitude regime there is a limit to the rotor strength within the valley.
In the lee of the second peak we see a clear distinction between constructive and destructive interference (Fig. 10d). The strength of the reversed flow in the lee of the second peak (U2min < 0) and the critical mountain height required for its occurrence strongly depend on the nature of interference. Under constructive interference rotors form (U2min < 0) for the same mountain heights as for a single mountain (Hs), whereas under destructive interference it takes a substantially higher mountain (H > Hc) for this to occur. Thus, contrary to our starting hypothesis, constructive interference does not promote formation of rotors over twin peak orography. Rotors do not form for lower mountains nor is their strength enhanced compared to the single mountain case; up to Hh the strength of the reversed flow (U2min) under constructive interference is the same as for the single mountain. It is only for H > Hh that rotors are enhanced beyond Usmin (by 22% at H = 1000 m). Under destructive interference, on the other hand, rotors are significantly weaker than those in the lee of a single mountain, particularly for H < Hh. In contrast to the rotors that form over the valley, there appears to be no limit to the rotor strength in the lee of the second peak as U2min continues to decrease as H is increased.
Based on the flow characteristics and relations between A and Umin at different critical mountain heights, we identify four regimes: (i) H < Hs, (ii) Hs < H < Hc, (iii) Hc < H < Hh, and (iv) H > Hh (Fig. 10). Regime 1 (H < Hs) is a small-amplitude wave regime with no boundary layer separation occurring anywhere. Regime 2 (Hs < H < Hc) can be viewed as a transitional regime. The boundary layer separation occurs downstream of twin peaks under constructive interference; under destructive interference the flow still displays linear characteristics. In this regime separation occurs also downstream of a single mountain. In regime 3 (Hc < H < Hh) and regime 4 (H > Hh) the boundary layer separation occurs under all scenarios. In addition, the rate of increase of rotor strength no longer matches a nearly linear increase in amplitudes with mountain height. The main difference between the latter two regimes is in the degree to which frictional impact is felt by the flow, with the impact being weaker the stronger the finite-amplitude effects are; that is, there is a weaker frictional impact in regime 4 compared to regime 3.
The interference characteristics for different mountain heights within the above regimes are illustrated in Fig. 11. In regimes 2 and 3 (for H < Hh) the interference pattern follows the linear prediction. The amplitude downstream of the second peak displays a symmetric variation around As for constructive and destructive interference for all H < Hh. The variation in U2min for positive and negative interference decreases as the mountain height is increased. Also, this variation is strongly asymmetric: the surface flow deceleration downstream of twin peaks is more strongly suppressed under destructive interference (to the point that no separation occurs in regime 2; i.e., U2min > 0) than it is enhanced under constructive interference.
In regime 4 (for H > Hh) the finite-amplitude effects lead to the decrease in variability in both the amplitude and rotor strength downstream of twin peaks. In this regime, the interference pattern is still evident in the amplitude variation; however, A2 > As for all ridge separation distances. Also, the amplitudes over the valley are smaller than the amplitudes in the lee of the second peak so that A2/A1 > 1 for all valley widths (not shown), in agreement with the free-slip simulations of GS09 and findings of Lee et al. (1987). The flow field over twin peaks in regime 4 is shown in Fig. 12. Weak upstream blocking starts to develop windward of the first mountain for 1500-m-high mountains. The nonlinear wave interactions that cause the amplitude reduction over the valley (Fig. 10) could be viewed as the “upstream” influence of the second obstacle. Yet the wave amplitudes over the valley are still small enough to prevent blocking there. The diminished influence of the frictional boundary layer with increasing nonlinearity of the flow (Jiang et al. 2008) is clearly evident in our results. Consequently, the interference pattern is no longer controlled by the no-slip intrinsic horizontal wavelength (Fig. 11; Table 2); instead, it goes back to being governed by the free-slip intrinsic horizontal wavelength (27.7 km) with strong positive correlation between D and U2min evident in Fig. 11 (r = 0.9).
Additional distinction between regime 4 and the other two finite-amplitude regimes is the development of flow symmetry over both the upstream and downstream peaks under constructive interference. This results in the surface wind speed maximum and the perturbation pressure minimum being collocated right at the mountain peak [cf. section 5a(1) and Fig. 9]. For both of these fields essentially the same profiles are obtained for a single mountain and double mountain under constructive and destructive interference; the extrema, however, are achieved for destructive interference (Fig. 9).
b. Lower downstream mountain
The TI simulations from section 4, as well as the free-slip simulations of GS09, suggest that there is a certain mountain height ratio for which the flow field in the lee of the second peak is most sensitive to interference, so that amplitude variations attain maximum values (Fig. 16 in GS09). To investigate the impact of downstream obstacle height on this sensitivity under no-slip lower boundary condition, a subrange of HI simulations was conducted with Hn < 1 for different heights of the upstream mountain H1 ∈ [400, 1500] m. The flow was found to be most sensitive to Hn for the valley width that corresponds to destructive interference (not shown); therefore V = 35 km was selected for these experiments.
Figure 13 shows that the dependence of A2/A1 on Hn in the HI experiments is similar to that in the TI experiments discussed earlier (A2/A1 ≈ 0.2 for Hn = ⅔; cf. Fig. 4). The amplitude ratio decreases with Hn until reaching a minimum for a critical mountain height ratio Hnc for which the flow field in the lee of the second obstacle almost completely cancels out—that is, Hn for which complete destructive interference occurs (Fig. 5b). Further increase in Hn leads to increase in the amplitude ratio. A similar pattern is obtained for the surface wind ratio U2min/U1min (Fig. 13b) and for a range of different heights of the upstream obstacle H1. The critical mountain height ratio for which the minima occur shows a weak variation with the height of the first mountain, except for the highest mountains simulated in these experiments (Fig. 13) for which Hnc decreases as H1 increases. This implies that for higher upstream mountains the second obstacle must be substantially lower than the first to cause complete cancellation.
The minimum amplitudes A2 at Hnc, although significantly reduced, are not equal to zero; rather, they are larger the higher the mountains are. The amplitude ratio A2/A1, however, is virtually the same for all mountain heights and remains close to 0.2, as in the TI experiments (Fig. 14). This signifies that the wave amplitudes are always reduced to the same degree (∼80%) irrespective of the upstream profile or the degree of nonlinearity. The same is not true for rotor strength: U2min/U1min at Hnc increases monotonically with H1. That means that rotors are more strongly attenuated for lower mountains (Fig. 13). The fact that Hnc is a function of H1 and does not attain the value of ⅔ as in the TI experiments suggests that Hnc is also dependent on the characteristics of the upstream profile.
The potential temperature distribution around the downstream peak at Hnc for destructive interference is shown in Fig. 15. It is clear that a symmetric potential temperature distribution is not conducive to complete cancellation. Rather, the wave field is cancelled out when the potential temperature perturbation due to the first obstacle is nearly 3π/4 out of phase with that due to the second obstacle. Still, the exact phase depends on the mountain height and is larger for lower mountains.
6. Linear superposition
The question arises as to why no complete wave cancellation develops downwind of the second obstacle in case of destructive interference over twin peaks. To answer this question we evoke a linear superposition principle, which states that trapped lee waves generated by twin peaks θ1+2 can be seen as a linear superposition of wave trains forming in the lee of two single mountains, placed a distance V apart. If the linear superposition applies, as suggested by Scorer (1997), the amplitudes should conform to
Before proceeding any further we can qualitatively see why the waves cannot completely cancel out in the lee of twin peaks in our simulations. For no-slip lower boundary conditions, surface friction leads to an exponential decay of wave amplitude downstream of each obstacle (Jiang et al. 2006; Smith et al. 2006) so that the amplitude and the attendant pressure perturbation of waves generated by the first mountain, even if exactly in phase with the waves generated by the second obstacle, are not large enough to counterbalance pressure perturbations associated with the second mountain. The additional source of wave attenuation is imperfect wave trapping and energy leakage in the vertical. Based on that, one should not expect complete destructive interference for trapped waves at Hn = 1 for upstream profiles used in this study, regardless of whether frictional boundary layer is present or not.
To test the above qualitative arguments we examined two sets of constructive and destructive interference cases within the linear regime, for 50- and 300-m-high twin peaks, under fS and nS lower boundary condition. Difference plots between the potential temperature simulated by the model and that obtained by linear superposition over twin peaks are presented in Fig. 16. The results show that the linear superposition principle applies well to constructive interference under free-slip conditions; for both mountain heights there is an excellent agreement between the linear superposition and the simulation results, in terms of wavelength, amplitude, and the phase of the disturbance (Figs. 16a,c, left). The differences in amplitude between linear superposition and simulations are only 1% for 50-m-high mountains and up to 5% for 300-m-high ones. For destructive interference under free-slip conditions, the difference in amplitude between the linear superposition and the simulation results is greater than for constructive interference even for very low mountains; however, it does not exceed 5% for H = 50 m (Fig. 16, right; even H < 10 m, not shown here). For 300-m-high mountains, the difference in amplitude is more significant (17%). Thus, it appears that destructive interference is more susceptible than constructive interference to finite-amplitude effects, which display themselves for very low mountain heights in case of the former. The reason for this could lie in the orographic adjustment, which has a stronger impact on the solutions under conditions leading to destructive interference.
Under no-slip conditions the agreement between the simulation results and the linear superposition is fairly good for constructive interference. This is despite the phase being slightly underestimated and amplitude slightly overestimated (Fig. 16b, left), likely due to weak nonlinearities excited by the frictional boundary layer (Jiang et al. 2006). For mountain heights in excess of 100 m the wave in the immediate lee of the second obstacle experiences a significant upstream phase shift and the reduction in wavelength compared to the linear superposition result. As the finite-amplitude effects increase, the linear superposition principle becomes less valid under no-slip lower boundary condition, regardless of the interference pattern (Fig. 17). The amplitude is overestimated by linear superposition even for constructive interference. The phase lag also increases, particularly for destructive interference, for which the simulation results display a significant upstream phase shift compared to the linear superposition (Figs. 16c,d, right). As expected, the influence of finite-amplitude effects is more pronounced in the no-slip simulations because of the presence of the frictional boundary layer.
In both the free-slip and no-slip simulations we find that the orographic adjustment of GS09 leads to a change of wave phase under destructive interference, which adjusts the waves to the orography and disrupts a possible cancellation of the wave field. A limited set of simulations was also performed for trapped waves generated in the atmosphere with a two-layer stability and a constant vertical wind profile under free-slip lower boundary conditions [profiles such as used in, e.g., Doyle and Durran (2002, 2007) and Jiang et al. (2007)] for which the wave amplitude does not decrease with downstream distance. Orographic adjustment prevents wave cancellation in those cases also, irrespective of linear superposition theory predictions. Furthermore, A2/A1 is always greater than unity (not shown).
For the same reasons that linear superposition is unable to predict the occurrence of destructive interference for twin peaks, it is also unable to predict complete cancellation for Hnc for both the free- and no-slip simulation results (Fig. 18). First, complete cancellation does not develop when both waves are completely out of phase. Rather, this occurs, as was previously shown, when the phase difference is close to 3π/4. Second, because of these phase differences, the amplitude of the disturbance is also greatly overestimated by the linear superposition.
7. Summary and conclusions
In this study we have investigated the influence of a frictional boundary layer on trapped lee wave interference over double obstacles, and the impact of interference on the formation of rotors. Two sets of idealized 2D numerical simulations were performed: the TI simulations in an idealized T-REX environment and the HI simulations within a highly idealized framework.
The results demonstrate that lee wave interference over a double mountain in presence of surface friction occurs through the mechanism analogous to that identified in the free-slip simulations in GS09. That is, linear interference theory is found to predict well the occurrence of wave interference over a double obstacle regardless of surface friction. For two mountains a distance V apart, it is the ratio of the ridge separation distance to the intrinsic horizontal lee wave wavelength λs that determines the interference pattern: constructive interference occurs for V/λs = n, n ≥ 1 and destructive interference for V/λs = (2n − 1)/2, n ≥ 2. Surface friction is found to have a strong modifying effect on interference that enters through λs. Characteristic of the given upstream atmospheric structure, the intrinsic horizontal wavelength is the wavelength of lee waves generated by the individual mountains. Under the effect of surface friction the intrinsic wavelength is reduced (Smith 2007). Given that the interference outcome depends on the phase with which the wave generated by the upstream peak encounters the downstream one (Scorer 1997), this has strong implications for wave interference. Despite its success in predicting the interference pattern, the linear interference theory fails at predicting the wave amplitudes, which downstream of a double obstacle show neither doubling nor complete cancellation under, respectively, constructive and destructive interference. The reason for this is wave attenuation with downstream distance due to imperfect trapping, as well as orographic adjustment, identified in GS09, through which the lee waves generated by twin peaks adjust to the terrain by selecting for the horizontal wavelength λ one of the peaks of the terrain spectrum that is closest to the intrinsic wavelength, insuring V/λ is an integer. The orographic adjustment has an overall stronger impact on destructive interference than constructive one, leading to less steady solutions in the former case.
Surface friction represents a strong coupling mechanism between the wave field aloft and the near-surface flow. As shown by Doyle and Durran (2002, 2004), for sufficiently large lee wave amplitude wave-induced adverse pressure gradient will be sufficiently strong to induce boundary layer separation in the lee of a mountain. This leads to the formation of rotors characterized by the appearance of negative (mean flow opposing) velocities within the boundary layer, which we take in this study to define the rotor strength. Surface friction also introduces an additional intrinsic wavelength for surface pressure perturbation. While detectable only in the immediate vicinity of the obstacles, this wavelength governs the variation of the total drag due to interference, in the manner analogous to which the intrinsic horizontal wavelength of lee waves controls the interference pattern evident in the wave amplitude.
Three critical mountain heights (Hs, Hc, Hh) emerge from our study, separating different flow regimes for lee waves over twin peaks in presence of surface friction. In the small-amplitude regime (H < Hs) there is no boundary layer separation anywhere. The flow over the valley is nearly insensitive to the presence of the second mountain, in agreement with the findings of Lee et al. (1987). Downstream of the twin peaks the interference develops. The deceleration of the boundary layer flow there mirrors the characteristic interference variation of the wave amplitudes and the total drag of the wave field aloft. Under constructive interference, the wave amplitude and total drag are higher and the boundary layer flow more strongly decelerated than in the lee of a single mountain; the opposite is true for destructive interference. Under constructive interference, the flow over the downstream peak resembles that over a single mountain, including an upstream tilt of the phase line over the peak and the attendant asymmetric flow structure around it. For destructive interference, the phase line over the downstream peak is vertical, the attendant flow structure symmetric, and the perturbation evanescent with height.
Assuming this linear behavior extends well into the finite-amplitude regimes, the following would imply (i) lower critical mountain heights required to achieve separation by lee waves downwind of a double mountain under constructive interference, and (ii) stronger rotors under constructive interference, compared to those past an isolated peak for the same mountain heights. Our results show that, for the most part, this does not hold true for large-amplitude lee waves over double mountains.
In regime 2 (Hs < H < Hc) the surface flow is particularly sensitive to interference. Under constructive interference, boundary layer separation occurs and rotors form both within the valley and in the lee of the downstream peak; their strength over the valley does not exceed those in the lee of a single mountain and only marginally so those in the lee of the downstream peak. Under destructive interference, rotors do not form in the lee of the downstream peak and the flow there remains linear. In regimes 3 (Hc < H < Hh) and 4 (H > Hh), rotors form both over the valley and in the lee of the downstream peak, irrespective of the interference pattern. In regime 3 the rotors over the valley are weaker compared to those in the lee of a single mountain; also, their strength does not vary significantly with interference. Downstream of the second peak the rotor strength depends on interference; rotors are stronger under constructive interference but again their strength does not exceed that in the lee of a single mountain. In regime 4, the rotor strength over the valley saturates, no longer following the continued linear increase of wave amplitude with mountain height, as in Doyle and Durran (2002) and Vosper et al. (2006). This implies the existence of an upper limit to the rotor strength over the valley. In the lee of the second mountain the rotor strength keeps increasing with mountain height (and amplitude); also, it is in this regime that the rotor strength under constructive interference starts to systematically exceed that in the lee of a single obstacle.
In the presence of surface friction, the velocity minimum in the lee of a double obstacle U2min for the two interference patterns is not symmetric with respect to the reference value for a single mountain; in general, there is a shift toward less deceleration, leading to rotors that are as strong as or weaker than those in the lee of a single mountain. The symmetry gets restored as the flow nonlinearity increases and the influence of the frictional boundary layer diminishes (Jiang et al. 2008). The reason for the absence of stronger rotors under constructive interference in regimes 2 and 3 in our experiments appears to be leakage of wave energy into the vertical, evident through stronger upstream phase line tilts and stronger downstream wave attenuation compared to the flow over a single mountain. As the degree of energy leakage into the vertical depends on the atmospheric structure, this result might not generalize to all trapped lee waves. The strong decrease in rotor strength for destructive interference is due to the evanescent nature of wave solutions over the downstream mountain, independent of the atmospheric structure.
Our results for the linear wave superposition show that destructive interference is more susceptible to finite-amplitude effects than is constructive interference. For the latter, the difference between the full wave solution and the linear superposition continues to be small for mountain heights beyond the small-amplitude regime, particularly in the absence of surface friction. For destructive interference, on the other hand, the full wave solution departs from the linear superposition already for very small mountain heights, both in terms of phase, because of both the orographic adjustment and surface friction, and in terms of amplitude, because of nonlinear wave interactions.
While complete wave cancellation under destructive interference does not occur downwind of twin peaks, almost complete cancellation occurs if the second obstacle is lower than the first. The critical mountain height ratio Hnc for which this “complete” destructive interference develops, is a function of flow nonlinearity (as measured by the upstream mountain height H1) and the upstream atmospheric profile. While complete destructive interference develops regardless of the presence or absence of frictional boundary layer, surface friction was found to move Hnc to higher values. In all complete destructive interference cases, the wave amplitude was reduced by a factor of approximately 0.7–0.8 regardless of the details of the experiments. To produce complete cancellation, the displacement field around the downstream peak needs to posses a degree of asymmetry. The wave impinging on the second peak should be approximately 3π/4 out of phase with the crest of the second peak.
In closing, our results from the finite-amplitude experiments show that for the terrain and upstream wind and potential temperature profiles that are characteristic of the T-REX environment stronger waves and rotors form over the valley under constructive interference, compared to those that would form in the lee of a single ridge. This is also the regime in which virtually complete destructive interference can occur. The evidence of complete destructive interference, in which trapped waves encountering a lower downstream obstacle cancel out in its lee, can be found in aircraft measurements from the T-REX campaign. During T-REX IOP 6 (25 March 2006) a significant reduction in the wave amplitude is seen downstream of the Inyo Mountains (Fig. 19). The height ratio of the Sierra Nevada and the Inyo mountain ranges is Hn = ⅔, in agreement with the results shown in section 4. A more systematic exploration of wave cancellation in the T-REX aircraft measurements will be presented in future studies.
We wish to thank two anonymous reviewers for their constructive and insightful comments and suggestions, which helped improve the manuscript. This research was motivated by observations collected in the Sierra Rotors Project and the Terrain-Induced Rotor Experiment (T-REX), for which the primary sponsor is the U.S. National Science Foundation (NSF). The first author acknowledges support of the Croatian Ministry of Science through Grant 004-1193086-3036 to the Croatian Meteorological and Hydrological Service. In the initial stages of this work, the second author was supported by the NSF Grant ATM-0524891 to the Desert Research Institute.
List of Abbreviations
A1 Amplitude in the lee of the upstream mountain
A2 Amplitude in the lee of the downstream mountain
As Amplitude in the lee of a single mountain
D Total gravity wave drag
Ds Total gravity wave drag over a single mountain
fS Free-slip HI simulations
fnS Free-slip HI simulations with no-slip upstream profile
H Mountain height
H1 Height of the upstream mountain
H2 Height of the downstream mountain
Hc Second critical mountain height = 500 m
Hh Third critical mountain height = 1000 m
Hn Mountain height ratio
Hnc Critical mountain height ratio at which “complete” destructive interference occurs
Hs First critical mountain height = 370 m
HI Highly idealized set of experiments
nS No-slip HI simulations
TI T-REX idealized set of experiments
T-REX Terrain-Induced Rotor Experiment
U1min Minimum horizontal wind speed in the lee of the first obstacle
U2min Minimum horizontal wind speed in the lee of the second obstacle
Usmin Minimum horizontal wind speed in the lee of a single obstacle
λ Horizontal trapped lee wave wavelength calculated from vertical velocity
λs Intrinsic trapped lee wave wavelength calculated in the lee of a single mountain from vertical velocity
λpsfc Intrinsic trapped lee wave wavelength calculated in the lee of a single mountain from perturbation pressure
V Valley width
V/λ Nondimensional mountain height