1. Introduction
Banner clouds are clouds in the lee of steep mountains or sharp ridges on otherwise cloud-free days (Glickman 2000). They appear like a banner attached to the leeside face of the mountain flapping in the wind, hence their name (Schween et al. 2007; Wirth et al. 2012). Their formation requires a region of strong uplift on the leeward side of the mountain; given suitable moisture conditions, this may then lead to banner cloud formation (Voigt and Wirth 2013). Banner clouds are noteworthy, because flow past orography usually implies uplift and, hence, cloud formation on the windward rather than the leeward side of the orography. This suggests that they occur under special conditions only, which would explain why they are a fairly rare phenomenon. Prestel and Wirth (2016) have argued that favorable conditions for banner cloud formation include a rather steep mountain and weak stratification of the ambient atmosphere.
During the past decades aspects of banner cloud formation were investigated with the help of numerical simulations (Voigt and Wirth 2013; Schappert and Wirth 2015; Prestel and Wirth 2016; Wirth et al. 2020). In all these studies, the orography was represented by a square pyramid with a vertical aspect ratio of around 1 protruding from a plain. In addition, in most of these studies the ambient atmosphere was chosen to be neutrally stratified up to the summit of the orography. On the one hand this allowed the authors to clarify a number of fundamental questions regarding key mechanisms and preferred flow conditions of banner cloud formation; on the other hand, it remained unclear to what extent the results depended on the idealized nature of the model configuration.
This state of affairs motivates the present study, in which we consider flow past realistic rather than idealized orography. In addition, we also account for a shallow boundary layer with its top lying below the summit of the mountain, since this is a situation that may occur in nature. The geographic region that we consider is the Mount Matterhorn in the Swiss Alps including its near environment. The Matterhorn is one of the most suitable mountains in the Alps to study the phenomenon, as it features a relatively frequent occurrence of banner clouds with sometimes near-iconic appearance (e.g., Fig. 1 in Wirth et al. 2020). We will carry through large-eddy simulations similar to previous studies, except that the use of realistic orography requires a considerably larger model domain, which makes the simulations computationally more expensive. In addition, we consider two configurations with more idealized orography: the topmost part of the Matterhorn orography protruding from a flat surface, and a quadratic pyramid protruding from a flat surface. The former represents an intermediate stage, allowing us to study the transition from a highly idealized pyramid to the realistic Matterhorn orography. We will investigate the differences of the flow associated with these three orographies in order to identify features that are generic to any steep mountain and distinguish them from features that are special due to particular properties of the orography. We pay particular attention to the existence and properties of leeward vortices, to the windward–leeward asymmetry in uplift, and to the flow path of streamlines reaching the banner cloud. Another focus will be on the question to what extent the stratification of the ambient atmosphere influences banner cloud formation, and whether the mountain, in turn, has an impact on the stratification. We intend to learn to what extent the results from previous studies with idealized model configurations carry over to more realistic scenarios.
The paper is organized as follows. First, in section 2 we present the setup of the numerical model. Subsequently, our diagnostic tools will be introduced in section 3, and the results of our analysis are presented in section 4. Finally, section 5 provides a summary and our conclusions.
2. Numerical model and model configuration
We simulate three-dimensional turbulent flow of dry air in a nonrotating atmosphere. In our earlier work, we used for this purpose the nonhydrostatic anelastic version of the EULAG model (Prusa et al. 2008). Unfortunately, this model version turned out to be unsuitable in connection with the very steep and complex orography of the Matterhorn region, presumably due to the existence of complex orography at the boundaries of the domain. We, therefore, switched to the compressible version of EULAG (Smolarkiewicz et al. 2014; Kurowski et al. 2014, 2015; Smolarkiewicz et al. 2016). This model version incorporates a semi-implicit integrator, which allows for a variable integration step size constrained by the CFL stability criterion independent of the propagation of sound waves.
Our general model setup closely follows the approach used in previous work (Schappert and Wirth 2015; Wirth et al. 2020). For the representation of subgrid-scale turbulence we apply a TKE prognostic equation closure based on Schumann’s parameterization (Schumann 1991). Orography is represented through an immersed boundary approach (Mittal and Iaccarino 2005; Smolarkiewicz et al. 2007), which is effectively like a no-slip condition (Goldstein et al. 1993; Smolarkiewicz et al. 2007). In the flat portion of the lower boundary, which only exists for the idealized mountain configurations, the surface stress is parameterized as
We use three distinct configurations of the model domain and its orography, ranging from highly idealized to fully realistic (Fig. 1). The configuration with realistic orography will be referred to as “Reference Matterhorn.” The domain was extracted from a Cartesian topographic dataset of the region around the Matterhorn at a native resolution of 25 m (Swisstopo 2017). The orography within the model domain ranges from the Zermatt valley at zmin = 1615.9 m to the Matterhorn top with zmax = 4478 m. At the other extreme we consider a pyramid protruding from a flat elevated plain (called “Pyramid”) like in several previous publications. As our intermediate configuration, we use the realistic orography from the Matterhorn summit protruding from a flat plain, and this configuration will be referred to as “Isolated Matterhorn.” The Pyramid and Isolated Matterhorn have the same mountain height of H = 1478 m and range from zmin = 3000 m to zmax = 4478 m. The geometry of the Pyramid was chosen such that its cross-sectional profile along the center of the mountain (y = 0) has a slope of α = 48.7°, which roughly corresponds to the average slope of the Mount Matterhorn along this axis. For the Pyramid and Isolated Matterhorn, the model domain extends 11.6 km in the streamwise direction (coordinate x), 9.2 km in the spanwise direction (coordinate y), and 4.5 km in the vertical direction (coordinate z). Sensitivity studies regarding larger domain sizes show negligible effects on the results. The Reference Matterhorn configuration covers a slightly larger domain with extensions of 12.8 km × 9.6 km × 7 km in the x, y, and z directions, respectively. This domain size represents a reasonable compromise between domain size and impact of the surrounding orography. To be sure, there is no “correct” domain size for the realistic orography due to the fact that the complex orography continues in each direction for many dozens of kilometers. However, it turned out to be important to ensure that the high ridges located west of Mount Matterhorn are included within the domain. We slightly smoothed the orography at the northern boundary to avoid numerical instabilities. For all three configurations the grid spacing is equidistant with δx = δy = δz = 25 m. We are aware that large-eddy simulations of the atmospheric boundary layer depend on resolution. However, we assume that for our purpose the grid spacing is small enough because we only study neutral boundary layers without surface heat fluxes and the flow is highly resolved.
Model orography for (a) the “Pyramid” configuration, (b) the “Isolated Matterhorn” configuration, and (c) the “Reference Matterhorn” configuration.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
The inflow boundary is located at xin = −4550 m for the Pyramid and the Isolated Matterhorn case, and at xin = −6325 m for the Reference Matterhorn case. At the inflow boundary we prescribe profiles of wind u = (u, υ, w) and potential temperature θ = T(p0/p)κ, where p denotes pressure, p0 = 1000 hPa is a constant reference pressure, κ = R/cp, R is the gas constant for dry air, and cp is the specific heat at constant pressure. The inflow wind profiles have a component uin(z) in the x direction only. They are organized into two groups as in Wirth et al. (2020) (see our Fig. 2a): 1) profiles with a constant wind of strength U = 5, 10, and 20 m s−1; and 2) profiles with linear shear, defined through vanishing wind at lowest point of the inflow boundary and increasing to a magnitude of U = 5, 10, or 20 m s−1 at the summit of the mountain (z = H); the shear extends another 500 m above the summit and transits into a constant wind at higher altitudes.
Inflow profiles for the numerical experiments: (a) streamwise wind component uin(z) and (b) potential temperature θin(z). The profile in (a) distinguishes two groups of profiles: one group with a constant wind (black) and another group with vertical shear (blue). In (b), the solid black line depicts the “standard” profile, while the dashed orange lines depict two additional profiles with shallower boundary layers. In both panels, the horizontal dashed line marks the top of the mountain.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
We define our standard potential temperature profile at the inflow boundary to correspond to neutral stratification (∂θ/∂z = 0 K km−1) up to the mountain summit (z = H), and stable stratification with (∂θ/∂z = 4 K km−1) above (see Fig. 2b). For the Reference Matterhorn configuration we consider, in addition, profiles in which the top of the neutrally stratified part is lowered by 500 and 800 m below the summit. In the following, we refer to the level at which the stratification changes from neutral to stable as the top of the boundary layer and denote it by zABL. The additional profiles were introduced because we suspect that in reality very high mountains often reach into the stably stratified free atmosphere. Unfortunately, we are not aware of any systematic observations, so in this study we consider the variation of the boundary layer height as a sensitivity experiment.
In all simulations, we apply a rigid-lid condition at the upper boundary. For the Pyramid and Isolated Matterhorn, we use open conditions at the streamwise boundaries and periodic conditions at the spanwise boundaries. For the Reference Matterhorn configuration, all boundaries are open. In all three mountain configurations a sponge layer was added at the model top in order to minimize wave reflections. For the Reference Matterhorn case, we used additional sponge layers at the streamwise and spanwise boundaries. We tested our main results for robustness and found that they are qualitatively independent of the exact sponge design.
Each model run is split into two parts, a spinup period of duration tsu and a period of analysis of duration tan. The duration of the spinup period and the total simulation time tsim = tsu + tan are given in Table 1 for the different configurations. The spinup period was made just long enough such as to reach a statistically stationary state. The initial conditions for the spinup period are identical to the inflow conditions, with a zero wind extension below the altitude of the lowest point of the inflow boundary. For our analysis, we consider only the time-averaged fields of the period of analysis tan, which increases with decreasing wind speed; using considerably shorter averaging intervals for low wind speeds would result in differences in the exact pattern of the flow geometry.
EULAG model details for the three different mountain configurations.
In the Pyramid setup, we eliminated small spanwise asymmetries in the y direction by computing the arithmetic mean between field values north and south of y = 0 (for details see Schappert and Wirth 2015).
3. Diagnostic tools
a. Cloud occurrence
b. Vortex geometry
Previous studies have established that the strong leeward uplift in case of banner cloud occurrence is associated with specific vortical flow structures (e.g., Fig. 4 in Voigt and Wirth 2013). In addition, in the case of idealized orography the exact shape of these lee vortices turned out to depend sensitively on the wind profile specified at the inflow boundary (Wirth et al. 2020). This result motivates us to diagnose the three-dimensional vortex structures in the immediate vicinity of our three mountain configurations, allowing us to determine whether the occurrence of lee vortices is pyramid specific or a general feature for any orography.
c. Parcel pathway
As detailed above, the occurrence of a banner cloud is estimated through a threshold on Δz. Depending on the flow regime, one may obtain a coherent cloud volume in the lee of the mountain (Prestel and Wirth 2016). However, a more detailed investigation by Schappert and Wirth (2015) indicated that even in that case different parcels contained in the diagnosed cloud volume may have traveled along very different paths and, hence, have very different origins. This motivates us to compute backward trajectories starting in the “cloud volume” as diagnosed from Δz. These backward trajectories correspond to streamlines, because we limit ourselves to the time-averaged flow in this study. The streamlines are integrated backward in time using the classical Runge–Kutta fourth-order scheme with a time step of Δt = 0.25 s for the three mountain configurations. Eventually, in section 4d we identify different classes of streamlines that distinguish various pathways of parcels that end up in the banner cloud.
d. Turbulent kinetic energy budget
4. Results
a. Vortex structure and flow geometry
First, we investigate the changes in the flow past the mountain as we move from the idealized Pyramid toward the fully realistic Matterhorn orography. We pay particular attention to vortical structures in the immediate vicinity of the mountain and their relation to regions of strong upwelling.
Figure 3 visualizes the vertical wind field w and the vortical structures through our curvature measure 〈K〉 for either a constant or a sheared wind profile with an amplitude of 20 m s−1. First, we consider the Pyramid configuration (left column in Fig. 3). In agreement with Wirth et al. (2020), a sheared inflow profile (Fig. 3a) results in two forwardly tilted vortices. A footprint of these two counterrotating vortices also appears in the patterns of the streamlines in Fig. 4a. The same panel also depicts the vertical wind, indicating a volume of strong upwelling on the axis of symmetry in the lee of the mountain. That same volume can be seen in Fig. 3d. It transpires that air parcels being caught in the leeward recirculation region experience a direct vertical transport into the banner cloud (white volume in Fig. 3a). By contrast, for the constant inflow profile the leeside flow pattern and vortex geometry have a considerably different character. Combining Figs. 3g and 4d we can detect an arc-shaped vortex with a strong recirculation between its two counterrotating vortex feet. The vortex is orientated along the leeward face of the mountain which means that it is slightly tilted upstream with altitude. In addition, the region of upwelling is very close to the mountain, which is in strong contrast to the previous behavior for the sheared inflow profile. Taken together, this results in a spiraling upward transport of air parcels into the banner cloud.
Three-dimensional visualizations of vortical structures (yellow) and the vertical wind field (red) in relation to the banner cloud (white). The vortical structures are displayed via the Gaussian-smoothed (σ = 0.6 grid distance) isosurface of 〈K〉 in the lee of the mountain. The banner cloud volume is defined through a threshold on Δz. The respective thresholds for isosurfaces of w, K, and Δz are given in the upper-right corner of the panels. The three columns represent the three different orography configurations: (left) Pyramid, (center) Isolated Matterhorn, and (right) Reference Matterhorn. (a)–(f) Shear inflow and (g)–(l) constant inflow wind setup.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
Vertical wind w (colors; m s−1) and streamlines (blue lines) in a horizontal cross section for (left) the Pyramid, (center) the Isolated Matterhorn, and (right) the Reference Matterhorn configuration. The section is located at z = H − 1000 m for all three mountain configurations. Results are shown for the (top) sheared and (bottom) constant wind profile, both with U = 20 m s−1.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
Next, we consider the Isolated Matterhorn configuration (center columns in Figs. 3 and 4). Overall, our simulations indicate rather substantial differences in the vortex geometry and upwelling structure between the Pyramid and the Isolated Matterhorn configuration. We start our discussion with the sheared inflow profile. There are no coherent, distinct vortex structures in the lee of the mountain (Fig. 3b). Instead, several individual vortex tubes exist whose axes are mostly aligned in streamwise direction. Therefore, in contrast to the pyramid, no vortex feet can be seen in the patterns of streamlines in Fig. 4b. Compared to the Pyramid configuration (Fig. 4a), the leeward recirculation is much closer to the mountain slope. At the same time, there are areas of strong upwelling on the southeast and northeast ridges of the mountain (Figs. 3e and 4b). In these regions, air parcels are lifted and transported to the leeward side before they are engulfed into the banner cloud. The simulation for the constant inflow profile reveals a more distinct geometry of vortex structures and upwelling regions (Figs. 3h,k). A lobe of strong upwelling extends upward from the southeast mountain ridge to the leeward side (Fig. 3k). This dominant upwind region is also visible in Fig. 4e. The streamline patterns shows a considerable spanwise asymmetry, with the leeward recirculation shifted to the south. Consequently, air parcels are mainly transported upward into the banner cloud from the southeastern mountain ridge.
Finally, we turn to the orography of the fully realistic Reference Matterhorn configuration (right columns in Figs. 3 and 4). Overall, the leeward flow geometry is again considerably different from the previous two configurations and illustrates the important role of the ambient orography. For both the sheared and the constant wind simulation, there is a region of strong upwelling on the northern side of the Matterhorn orography (Figs. 3f,l). However, air parcels lifted in this region are not carried into the banner cloud (not shown). Instead, the volume of upwelling relevant for banner cloud formation extends from the southern flank into the immediate lee of the Matterhorn. The broad independence of the location of the upwelling region to inflow wind shear is in distinct contrast to the two more idealized orographies discussed before. This feature probably results from the unique and very specific orography in the realistic mountain configuration that serve to channel the flow. For example, the ridges to the southwest and southeast of the Matterhorn extend much farther in the Reference Matterhorn configuration compared to the Isolated Matterhorn configuration (Fig. 1c versus Fig. 1b). The flow pattern in the direct vicinity of the Matterhorn is quite similar for both inflow wind profiles (Figs. 4c,f). The only difference is that the leeward recirculation flow extends much farther downwind for the sheared inflow profile. Consequently, air parcels from the south flow into the banner cloud either via the large upwelling region or they enter the recirculation flow on the leeward side, where they are transported upward into the banner cloud. Overall, it is difficult to identify a clear vortex structure in the lee of the mountain for the Reference Matterhorn configuration. To be sure, our measure for curvature indicates a leeward vortical volume for the sheared wind profile in Fig. 3c, but this is not associated with substantial upwelling and does, therefore, most likely not contribute to the formation of the banner cloud.
b. Leeside plume of vertical displacement
Next, we consider the dependence of the vertical uplift on the upstream wind profile. Figure 5 shows cross sections of Δz through the center of the mountain for the three orography configurations. For all simulations a plume of large Δz is visible on the leeward side. The maximum value of Δz is always larger for the constant inflow profile (bottom row) in comparison with the sheared inflow profile (top row). However, the sheared inflow profile promotes a visually more pronounced windward–leeward asymmetry compared to the constant inflow profile. For the Pyramid and the Isolated Matterhorn configurations, the potential P [see Eq. (3)] for constant wind speeds of 5 and 10 m s−1 is much smaller than P for 20 m s−1 (Fig. 6); it even becomes negative at U = 10 m s−1. The reason for this is strong uplift close to the windward face of the mountain which leads to a shallow layer of large Δz (Figs. 5d,e). This results in a very small windward–leeward asymmetry of vertical displacement in terms of our metric P. At the same time, there is a well-defined plume of large Δz on the leeward side in Figs. 5d and 5e, that does have an appearance which is similar to a banner cloud. We conclude that our simple diagnostic P is not perfect for the idealized orographies. The strongest windward–leeward asymmetry is found in the Reference Matterhorn configuration (right column). Correspondingly, the potential P exhibits the largest values for the Reference Matterhorn configuration for each of the six colored tiles (Fig. 6c) compared to the other two configurations (Figs. 6a,b). We conclude that 1) the realistic Matterhorn orography is particularly conducive to banner cloud formation, that 2) the actual ambient orography is important for this property, and 3) that in this case the banner cloud formation potential is broadly independent of wind shear.
Vertical displacement Δz (color; m) in a vertical section through the center of the mountain (dark gray) for (left) the Pyramid, (center) the Isolated Matterhorn, and (right) the Reference Matterhorn configuration. Results are shown for the (top) sheared and (bottom) constant wind profile, both with U = 20 m s−1. The green line depicts the zero contour. The purple lines delineate areas on the windward and leeward side of the mountain, which are subsequently used to compute the diagnostic P (see text for details).
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
Banner cloud formation potential P (color; m) for (a) the Pyramid, (b) Isolated Matterhorn, and (c) Reference Matterhorn configuration with the standard potential temperature profile. The colored tiles represent the value of P for six different combinations of wind shear and wind strength; the wind strength U is given along the vertical axis (in m s−1). The diagnostic P is calculated via Eq. (3), where the windward and leeward integrals are computed in the respective regions delineated by the purple lines in Fig. 5.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
c. Sensitivity to the top of the boundary layer
So far, we have only examined simulations with neutral stratification up to the mountain top. This was motivated by the results of Prestel and Wirth (2016), which suggested that banner cloud formation with idealized orography is generally associated with weak stratification. However, this may exclude important realistic situations, in which the summit of the mountain extends above the boundary layer top and protrudes into the stably stratified free atmosphere. Therefore, we extended the set of potential temperature inflow profiles and lowered the boundary layer top to an altitude below the mountain summit as shown in Fig. 2b. This allows us to investigate to what extent the ambient stratification has an impact on banner cloud occurrence at Mount Matterhorn.
First, we compare the potential P [see Eq. (3)] for banner cloud occurrence for the different inflow profiles in the top row of Fig. 7. While for the standard potential temperature profile (Fig. 7a) almost all considered combinations of wind shear and wind strength are associated with large positive values of P, this is not the case any longer for the shallower boundary layers in Figs. 7b and 7c; rather, we obtain considerably smaller and even negative values of P in case of a constant wind profile with wind speeds lower than U = 20 m s−1.
(top) Banner cloud formation potential P (color; m) for the Reference Matterhorn configuration with (a) zABL = H, (b) zABL = H − 500 m, and (c) zABL = H − 800 m. The numbers in the colored tiles represent the value of P for six different combinations of wind shear and wind strength; the wind strength U is given along the vertical axis (in m s−1). The diagnostic P is calculated via Eq. (3), where the windward and leeward integrals are computed in the respective regions delineated by the purple lines in Fig. 5. (bottom) The vertical displacement Δz (color; m) in a vertical section through the center of the mountain (dark gray) for the (d) sheared and (e) constant inflow profile with U = 5 m s−1 of the configuration with zABL = H − 500 m.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
This result prompts a more detailed discussion about our diagnostic P for banner clouds occurrence. For this purpose, we consider Figs. 7d and 7e, which show cross sections of the vertical displacement Δz for the configuration with U = 5 m s−1 and a boundary layer top located 500 m below the mountain summit, both for the sheared (Fig. 7d) and the constant wind profile (Fig. 7e). For both configurations, the value of P is about equal with a fairly low value (Fig. 7b). By definition, this corresponds to a weak potential for banner cloud formation for both simulations. However, it is immediately noticeable that for the constant wind profile (Fig. 7e), the key mechanism for banner cloud formation, namely, positive vertical displacement on the leeward side, is completely absent. This is in sharp contrast to the results for a sheared wind profile shown in Fig. 7d. Here, we see high positive values of Δz just below the summit with a weak windward–leeward asymmetry favoring banner cloud occurrence. The comparison shows that despite the positive values of P for a constant wind profile, no cloud formation can be expected. Similarly, our metric P provides a misleading prediction for the configuration with a boundary layer top 800 m below the summit for the constant wind profile with U = 5 m s−1 (Fig. 7c), because no positive vertical displacement is found at the summit (not shown). In both cases the interpretation of our metric P is problematic, and this is indicated by hatching the colored tiles in Fig. 7.
For all other simulations, the results indicate that there is a considerable dependence of P on wind speed in situations in which the boundary layer top of the ambient atmosphere is located below the mountain summit. The question is, Why?
We hypothesize that the underlying reason for this sensitivity to wind speed lies in the fact that the complex orography of the Matterhorn and its immediate environment has a nonnegligible impact on the stratification profile, and that the strength of the impact depends on the wind speed of the incoming flow. To test this hypothesis, we show vertical profiles of potential temperature slightly upstream and downstream of the Matterhorn summit in Fig. 8. First, we can see that in the case of the standard potential temperature profile (darkest color hue in each panel of Fig. 8), the boundary layer extends to higher altitudes on the leeward side than on the windward side independent of the inflow wind profile.
Profiles of potential temperature (top) on the windward side (x = −1500 m) and (bottom) on the leeward side (x = +1500 m) averaged along y = ±250 m for the Reference Matterhorn configuration. In each panel the three color hues represent the three different boundary layer tops as shown in Fig. 2b. The results for the constant and sheared inflow profiles are drawn as solid and dashed, respectively. The horizontal dashed line marks the mountain top. The different columns represent different wind speeds: (left) U = 5 m s−1, (center) U = 10 m s−1, and (right) U = 20 m s−1.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
By contrast, for the shallower boundary layer inflow profiles, the difference between windward and leeward side shows a more complex behavior. In the particular case of U = 20 m s−1 with wind shear (right column of Fig. 8), the boundary layer in the lee is much deeper than on the windward side and extends to just above the mountain summit. In this case our hypothesis verifies in the sense that the orography has substantially modified the stratification while the air has passed the mountain. Looking at the vertical profile of the resolved TKE and the corresponding shear production/loss in Figs. 9c and 9f, we can interpret the modified stratification as the result of shear production of TKE near the Matterhorn summit, which leads to mixing at the top of the boundary layer and consequently to erosion of the lowest part of the free atmosphere; the net effect is an upward shift of the boundary layer top all the way to the summit level.
Profiles of the (top) resolved TKE and (bottom) TKE shear production/loss on the leeward side (x = +1500 m) and averaged along y = ±250 m for the Reference Matterhorn configuration. The three color hues represent the three different inflow boundary layer tops. The results of the constant and sheared inflow profiles are shown as solid and dashed curves, respectively. The horizontal dashed line marks the mountain top. The different columns represent different wind speeds: (left) U = 5 m s−1, (center) U = 10 m s−1, and (right) U = 20 m s−1.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
The question is whether this effective erosion depends on the strength of the turbulence and, hence, on the wind speed of the ambient flow. To answer this question, we turn to the left column in Fig. 8 corresponding to U = 5 m s−1. In this simulation there is much less of a clear upward shift of the boundary layer top as a result of the flow past the mountain. Rather, we see a slightly lower boundary layer top, but at the same time the boundary layer itself turns more neutral. Figures 9a and 9d show that the weaker wind is associated with considerably less turbulence and, hence, less erosion of the free atmosphere. This would explain the rather small or even negative values of P in Fig. 7 for U = 5 m s−1, given that banner cloud formation prefers weak stratification all the way up to the summit level (Prestel and Wirth 2016).
It is also interesting to consider U = 10 m s−1, which is intermediate between the previous two cases. In both Figs. 7b and 7c, there is a strong contrast between the constant and the sheared wind profiles, with much lower or even negative values for the constant wind profile and high positive values for the sheared wind profile. It follows that in this case the wind shear plays an important role for banner cloud occurrence. This is consistent with the profiles of potential temperature in Fig. 8 (middle column) and strong shear production of resolved TKE in Fig. 9e. They show a considerable upward shift of the boundary layer top (Figs. 8b,c) with strong turbulence production at the summit level (Fig. 9e) for the sheared inflow wind profiles only. For the configuration with the boundary layer top located 500 m below the mountain summit and the constant wind profile of U = 10 m s−1, the boundary layer top remains well below the summit on the leeward side (Fig. 8e). In this case the boundary layer top does not experience a vertical shift because (weak) shear production of TKE is confined to the lower part of the boundary layer (Fig. 9e). Accordingly, the relatively low positive potential of the banner cloud occurrence (Fig. 7b) corresponds to a banner cloud that is located very close to the mountain summit and cannot extend far downstream because the boundary layer is too shallow.
There is another way to test our hypothesis regarding the importance of vertical mixing for banner cloud formation in the case of shallow boundary layer height. Whenever there is strong mixing below the mountain summit, the leeward wind profile should be approximately constant with altitude; as a result, one would expect rather strong wind shear close to the summit level, because the wind profile has to continuously join the ambient wind above the mountain. We, therefore, consider profiles of the streamwise wind component in the lee of the mountain (Fig. 10). The figure suggests that we can distinguish two classes of profiles, one with strong shear and one with near-constant wind close to the summit level. As it turns out, those profiles that have strong shear at summit level correspond to a large positive value of P. As argued above, this is consistent with our hypothesis.
Streamwise wind profiles on the leeward side of the mountain at x = 1500 m averaged along y = ±250 m for the Reference Matterhorn configuration. The three color hues represent the three different inflow boundary layer tops. The results of the constant and sheared inflow profiles are shown as solid and dashed curves, respectively. The horizontal dashed line marks the mountain top. The different columns represent different wind speeds: (a) U = 5 m s−1, (b) U = 10 m s−1, and (c) U = 20 m s−1.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
d. Pathways into the banner cloud
In section 4a we found that the step from the highly idealized Pyramid to the Isolated Matterhorn configuration was associated with considerable differences in the flow geometry in the direct vicinity of the mountain. Furthermore, the fully realistic Reference Matterhorn configuration featured a unique region of upwelling on the southern face of the Matterhorn, which extends to the leeward side. One can assume that these differences have a pronounced impact on the Lagrangian behavior of air parcels flowing into the banner cloud. This subsection is meant to shed some light into the issue.
Pathways of air parcels leading into the banner cloud were previously investigated by Schappert and Wirth (2015) using an idealized pyramid. They identified two distinct classes. The first class was associated with parcels flowing directly past the mountain summit and into the banner cloud; the second class was associated with parcels that take a much longer path around the mountain, recirculate toward the leeward face, and finally get lifted into the cloud. We perform a similar analysis using our three configurations for the model orography with the standard potential temperature inflow profile.
The banner cloud is diagnosed using a subjectively chosen, case-specific threshold of Δz, namely, 900 m (370 m) for the Pyramid and the Isolated Matterhorn, and 450 m (360 m) for the Reference Matterhorn configuration for a constant (sheared) inflow profile. These threshold values, which are independent of wind speed, were motivated by the desire to obtain a situation where the cloud is broadly confined the leeward side with only very little cloudiness on the windward side—since this corresponds best with the definition of a banner cloud (Schween et al. 2007). To distinguish the different pathways, we computed backward trajectories as described in section 3c; thereafter, we classified them according to the orientation of the parcels’ path relative to the mountain summit during the uplift phase on their way into the cloud. The backward integration was limited to 200 min, and we eliminated those trajectories which terminated within the orography or were caught in regions of near-zero wind speed. The fraction of “survived” trajectories for each model configuration is given in Table 2. On average, 83.9% of the trajectories for the Pyramid, 77.3% for the Isolated Matterhorn, and 81% for the Reference Matterhorn configuration were retained by our algorithm.
Overview of the number of backward trajectories used in the analysis for various model configurations with the standard potential temperature inflow profile. No backward trajectories were calculated for the Matterhorn reference orography with a constant inflow wind profile of U = 5 m s−1, because in this case the maximum vertical displacement Δz was well below the selected threshold of 450 m.
An example for the Reference Matterhorn configuration is shown in Fig. 11. Apparently, there are different classes of pathways referred to as “Windward,” “Lee,” and “South.” A corresponding pathway “North” does not exist in this case.
Pathways of parcels on their way into the banner cloud for the Reference Matterhorn orography with a sheared inflow wind profile with U = 20 m s−1. The color distinguishes different classes of pathways: “Windward” (blue), “Lee” (green), and “South” (yellow). For reasons of clarity, the figure shows only a limited selection of trajectories for each pathway.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
We now present a systematic comparison of the relative importance of the pathways between the different mountain configurations in Fig. 12, where “relative importance” is measured by the fraction of backward trajectories that belong to a certain class. First, we have a look at the idealized Pyramid configuration (left column). In accordance with Schappert and Wirth (2015) only the classes “Windward” and “Lee” make substantial contributions, while the classes “South” and “North” are much less important by comparison. For a constant inflow wind profile (Fig. 12a), the “Windward” class contributes the largest fraction for U = 20 m s−1, while the “Lee” class is the largest fraction for U = 5 and 10 m s−1. There is a very clear dominance of the “Lee” class in case of the sheared inflow wind profile independent of the wind speed (Fig. 12d).
Relative importance of the different pathways of air parcels entering the banner cloud for the (left) Pyramid, (center) Isolated Matterhorn, and (right) Reference Matterhorn configuration. (top) Constant wind profiles; (bottom)sheared wind profiles. The colored bars quantify the fraction of trajectories entering the banner cloud from the different directions. The numbers at the base of the color bars indicate the value of U (in m s−1). No backward trajectories were calculated for the Reference Matterhorn configuration with a constant inflow wind profile of U = 5 m s−1, because the maximum vertical displacement Δz was well below the selected threshold of 450 m.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0106.1
Turning next to the isolated Matterhorn configuration, we see a significant increase in the classes “South” and “North” for a constant wind profile (Fig. 12b). Although the “Windward” class remains the most important for a wind speed of U = 20 m s−1, the classes “South” and “North” together contribute the highest proportion for lower wind speeds, mostly at the expense of class “Lee.” In contrast, class “North” is not relevant for the sheared wind field, and class “South” alone represents the most important class (Fig. 12e). In comparison with the Pyramid configuration, the fraction of class “Lee” is significantly reduced.
Finally, the pathway analysis for Reference Matterhorn (right column in Fig. 12) results in a quite different behavior which contrasts the behavior in the previous two (more idealized) mountain configurations. The most striking difference is the fact that now the “South” class dominates independent of the wind speed and shear. Furthermore, the “Lee” class is even less important compared to the Isolated Matterhorn configuration. This result can directly be related to the unique feature of the large region of upwelling on the southern face of the Matterhorn in Figs. 3f and 3l. The underlying reason for this behavior must be attributed to the unique topographic conditions of the orography. In particular the ridges connected to the Matterhorn to the southwest and southeast serve to effectively channel the parcels along their trajectories. Presumably, this is also the reason why air currents are strongly separated from each other, as can be seen in Fig. 11. The “North” class that we found mainly for the Isolated Matterhorn configuration does not play a role here at all (see Figs. 12c,f).
5. Summary and conclusions
In this study we carried out numerical simulations to investigate the formation of banner cloud formation, i.e., clouds in the lee of a steep mountain. To this end we followed previous studies and used the vertical displacement as a proxy for the propensity of cloud formation. A flow field is associated with an increased likelihood of leeward cloud occurrence whenever there is a large windward–leeward asymmetry of vertical displacement with a plume of large values on the leeward side. We quantified this asymmetry by a single number P. In contrast with earlier studies that were exclusively based on idealized orography, we considered three model configurations that differed in the complexity of the underlying orography: 1) a quadratic Pyramid, 2) realistic orography from the topmost 1478 m of Mount Matterhorn protruding from a flat plain, and 3) fully realistic orography of the Matterhorn and its environment. In addition, we considered a range of inflow conditions that differed in the depth of the boundary layer. For all cases, we analyzed the structure and the properties of the time-averaged flow. Our main goal was to find out to what extent the results of earlier studies with idealized model configurations carry over to more realistic conditions.
As expected, more realistic orography is associated with more complex flow patterns. In particular, the spanwise symmetry from the idealized Pyramid configuration is broken on the leeward side. At the same time, the vortical structures appear to be much less coherent to an extent that those trajectories that reach the banner cloud were sometimes unrelated to these vortical features. Nevertheless, in all cases with banner cloud formation there are well-defined regions of uplift, although these are not necessarily confined any longer to the leeward side of the mountain.
For situations in which the ambient atmosphere is neutral up to the mountain summit, we found that the impact of the inflow wind profile is less pronounced for realistic orography in comparison with idealized orography. There is banner cloud formation for any wind speed, consistent with Wirth et al. (2020). We interpret this as a result of the upstream orography, which modifies the wind profiles such that the mountain “does not see” the inflow profile any longer. This is in stark contrast to earlier studies with flat terrain in the upstream region, in which we found rather strong sensitivity to the inflow wind profile.
At the same time, we obtained important new results for ambient atmospheric profiles that have a shallow boundary layer with the top of the mountain protruding into the stably stratified free atmosphere. For these cases we found a strong dependence of banner cloud formation on the wind speed, which is in stark contrast with the results of Wirth et al. (2020) for a deep boundary layer. By analyzing profiles of potential temperature, TKE, and the shear production of TKE, we found that the underlying reason for this behavior is the fact that stronger turbulence in case of stronger wind tends to create a more neutral stratification. The latter is conducive to banner cloud formation according to Prestel and Wirth (2016). Interestingly, the wind speed needed to get a more neutral boundary layer turned out to be considerably lower in the case of a sheared ambient wind profile. The latter result is consistent with the fact that it is easier to create turbulence in shear flow than in a constant flow.
Finally, we investigated the different pathways along which parcels reach the banner cloud. Not surprisingly, more realistic orography breaks the spanwise symmetry and adds novel pathways that were not existent in the case of idealized orography used in earlier studies. In particular, there are new pathways on the spanwise faces of the mountain. In the case of the fully realistic Matterhorn orography, the fraction of the southern pathway dominates by a large margin. We argued that this is due to the existence of pronounced ridges connecting the Matterhorn summit to the surrounding terrain.
Overall, we conclude that even extremely complex orography allows for the formation of banner clouds. Similar to more idealized orography, the main mechanism involves the existence of a region with strong uplift, although the latter is not necessarily confined to the leeward side of the mountain any longer. In addition, weak stratification all the way up to the mountain summit is very important for banner cloud formation; the weak stratification must be either a property of the ambient atmosphere, or it can be self-created through turbulence arising from the interaction between the atmosphere and the orography in case of strong wind speeds.
Acknowledgments.
We are grateful to Zbigniew Piotrowski for his support during our transition from the anelastic to the compressible EULAG version and expert advice on numerous technical aspects. We also thank our colleagues Elmar Schömer and Ronja Schnur from the Institute of Computer Science for providing the Python package pygranite for GPU based trajectory calculation (Schnur 2022). All figures and animations were generated with the Python matplotlib package (Hunter 2007) and vedo package (Musy et al. 2020). This research was funded by the German Research Foundation through Grant WI-1685/13-1.
Data availability statement.
The simulation data necessary for reproducing figures in this study are available at https://zenodo.org/record/8171107.
REFERENCES
Glickman, T. S., Ed., 2000: Glossary of Meteorology. 2nd ed. Amer. Meteor. Soc., 855 pp., http://glossary.ametsoc.org/.
Goldstein, D., R. Handler, and L. Sirovich, 1993: Modeling a no-slip flow boundary with an external force field. J. Comput. Phys., 105, 354–366, https://doi.org/10.1006/jcph.1993.1081.
Haller, G., 2005: An objective definition of a vortex. J. Fluid Mech., 525, 1–26, https://doi.org/10.1017/S0022112004002526.
Hunter, J. D., 2007: Matplotlib: A 2D graphics environment. Comput. Sci. Eng., 9, 90–95, https://doi.org/10.1109/MCSE.2007.55.
Jeong, J., and F. Hussain, 1995: On the identification of a vortex. J. Fluid Mech., 285, 69–94, https://doi.org/10.1017/S0022112095000462.
Kurowski, M. J., W. W. Grabowski, and P. K. Smolarkiewicz, 2014: Anelastic and compressible simulation of moist deep convection. J. Atmos. Sci., 71, 3767–3787, https://doi.org/10.1175/JAS-D-14-0017.1.
Kurowski, M. J., W. W. Grabowski, and P. K. Smolarkiewicz, 2015: Anelastic and compressible simulation of moist dynamics at planetary scales. J. Atmos. Sci., 72, 3975–3995, https://doi.org/10.1175/JAS-D-15-0107.1.
Mittal, R., and G. Iaccarino, 2005: Immersed boundary methods. Annu. Rev. Fluid Mech., 37, 239–261, https://doi.org/10.1146/annurev.fluid.37.061903.175743.
Musy, M., and Coauthors, 2020: Marcomusy/vedo: 2020.4.2. Zenodo, https://doi.org/10.5281/zenodo.4287635.
Prestel, I., and V. Wirth, 2016: What flow conditions are conducive to banner cloud formation? J. Atmos. Sci., 73, 2385–2402, https://doi.org/10.1175/JAS-D-15-0319.1.
Prusa, J. M., P. K. Smolarkiewicz, and A. A. Wyszogrodzki, 2008: EULAG, a computational model for multiscale flows. Comput. Fluids, 37, 1193–1207, https://doi.org/10.1016/j.compfluid.2007.12.001.
Reinert, D., and V. Wirth, 2009: A new large-eddy simulation model for simulating air flow and warm clouds above highly complex terrain. Part II: The moist model and its application to banner clouds. Bound.-Layer Meteor., 133, 113–136, https://doi.org/10.1007/s10546-009-9419-x.
Schappert, S., and V. Wirth, 2015: Origin and flow history of air parcels in orographic banner clouds. J. Atmos. Sci., 72, 3389–3403, https://doi.org/10.1175/JAS-D-14-0300.1.
Schnur, R., 2022: Pygranite version 1.5. GitHub, https://github.com/catheart97/pygranite.
Schumann, U., 1991: Subgrid length-scales for large-eddy simulation of stratified turbulence. Theor. Comput. Fluid Dyn., 2, 279–290, https://doi.org/10.1007/BF00271468.
Schween, J. H., J. Kuettner, D. Reinert, J. Reuder, and V. Wirth, 2007: Definition of “banner clouds” based on time lapse movies. Atmos. Chem. Phys., 7, 2047–2055, https://doi.org/10.5194/acp-7-2047-2007.
Smolarkiewicz, P. K., R. Sharman, J. Weil, S. G. Perry, D. Heist, and G. Bowker, 2007: Building resolving large-eddy simulations and comparison with wind tunnel experiments. J. Comput. Phys., 227, 633–653, https://doi.org/10.1016/j.jcp.2007.08.005.
Smolarkiewicz, P. K., C. Kühnlein, and N. P. Wedi, 2014: A consistent framework for discrete integrations of soundproof and compressible PDEs of atmospheric dynamics. J. Comput. Phys., 263, 185–205, https://doi.org/10.1016/j.jcp.2014.01.031.
Smolarkiewicz, P. K., W. Deconinck, M. Hamrud, C. Kühnlein, G. Mozdzynski, J. Szmelter, and N. P. Wedi, 2016: A finite-volume module for simulating global all-scale atmospheric flows. J. Comput. Phys., 314, 287–304, https://doi.org/10.1016/j.jcp.2016.03.015.
Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic, 666 pp.
Swisstopo, 2017: DHM25. Accessed 9 May 2017, https://www.swisstopo.admin.ch/de/geodata/height/dhm25.html.
Voigt, M., and V. Wirth, 2013: Mechanisms of banner cloud formation. J. Atmos. Sci., 70, 3631–3640, https://doi.org/10.1175/JAS-D-12-0353.1.
Wirth, V., M. Kristen, M. Leschner, J. Reuder, and J. H. Schween, 2012: Banner clouds observed at Mount Zugspitze. Atmos. Chem. Phys., 12, 3611–3625, https://doi.org/10.5194/acp-12-3611-2012.
Wirth, V., P. Bubel, J. Eichhorn, E. Schömer, T. Kremer, R. Erbes, S. Schappert, and I. Prestel, 2020: The role of wind speed and wind shear for banner cloud formation. J. Atmos. Sci., 77, 1199–1212, https://doi.org/10.1175/JAS-D-18-0354.1.
