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  • View in gallery

    Regime diagram for flow past a two-dimensional obstacle, adapted from Fig. 5.8 of Baines (1995). Depending on the combination of aspect ratio and the parameter , the flow belongs to one of three different regimes regarding flow separation from the surface. The dashed straight lines are isolines of /.

  • View in gallery

    Illustration of the model domain for the two-dimensional simulations.

  • View in gallery

    Illustration of the model domain for the three-dimensional simulations.

  • View in gallery

    Regime behavior in the two-dimensional simulations. The plots show streamlines of the time-average flow, with the overall flow direction from left to right. Simulations with different values for the parameters = h0/L (decreasing top–bottom) and = Nh0/U (increasing left–right), with the mountain steepness increasing logarithmically upward and the stratification parameter increasing logarithmically. The plots show only part of the model domain.

  • View in gallery

    Regime behavior in the two-dimensional simulations. The color fill represents the time-mean streamwise component of the wind 〈u〉 (m s−1), and the black contour is the zero contour of 〈u〉. Otherwise the figure conventions are as in Fig. 4.

  • View in gallery

    As in Fig. 5, but for cd = 0.001.

  • View in gallery

    Regime behavior in the three-dimensional simulations. The panels show streamlines of the time-averaged flow in a vertical cross section through the middle of the domain, with the overall flow direction from left to right. Simulations with different values for the parameters = h0/L (decreasing top–bottom) and = Nh0/U (increasing left–right), with the mountain steepness increasing logarithmically upward and the stability parameter increasing logarithmically. The plots show only part of the model domain.

  • View in gallery

    Regime behavior in the three-dimensional simulations. The color fill represents the time-mean streamwise component of the wind 〈u〉 (m s−1), and the black contour is the zero contour of 〈u〉. Otherwise, the figure conventions are as in Fig. 7.

  • View in gallery

    Regime behavior in the three-dimensional simulations. The panels show streamlines of the time-averaged flow in a horizontal cross section at z = 250 m. Otherwise the figure conventions are as in Fig. 7.

  • View in gallery

    Dependence of the recirculation area on aspect ratio and stratification in the three-dimensional simulations. The color fill quantifies the time-mean streamwise component of the wind 〈u〉 (m s−1) in the lowest model layer. The black contour delineates the horizontal extent of the mountain. Otherwise the figure conventions are as in Fig. 7.

  • View in gallery

    As in Fig. 8 except that surface drag has been set to zero.

  • View in gallery

    Regime behavior of vertical uplift in the three-dimensional simulations. The color fill shows the time-mean Lagrangian vertical displacement 〈Δz〉 (m) in a vertical section through the center of the domain. The black contour is the zero contour, and the white contour depicts 〈Δz〉 = +400 m—that is, an uplift almost as large as the mountain height, indicating significant potential for cloud formation. Otherwise, the figure conventions are as in Fig. 7.

  • View in gallery

    Regime behavior of the banner cloud formation potential P (see text for definition) for three different model configurations. The 12 colored tiles in each plot represent the value of P (m) for the same 12 combinations of and as considered in the previous figures. (a) Three-dimensional simulations with standard surface friction (cd = 0.01) corresponding to Fig. 12, (b) two-dimensional simulations with standard surface friction (cd = 0.01), and (c) three-dimensional simulations with zero surface friction (cd = 0).

  • View in gallery

    Vertical uplift for (, ) = (0.5, 0.3) in two sensitivity experiments, to be compared with the reference simulation in Fig. 12a. (a) Two-dimensional instead of three-dimensional flow; (b) surface friction set to zero (cd = 0). The color fill shows the time mean Lagrangian vertical displacement 〈Δz〉 (m) in a vertical section through the center of the domain. The black contour is the zero contour, and the white contour depicts 〈Δz〉 = +400 m.

  • View in gallery

    Results from an additional set of simulations for three-dimensional flow past two-dimensional orography with fixed = 0.25. The four columns represent the results for four different values of = Nh0/U. (a)–(d) The time-mean streamwise component 〈u〉 of the wind (m s−1, plot conventions as in Fig. 5); (e)–(h) the instantaneous wind υ in the spanwise direction (m s−1) at the end of the simulation in a horizontal section at z = 250 m; the region where this section is underground is indicated by gray shading.

  • View in gallery

    Results from three different simulations with different combinations of dimensional parameters N, L, U, and h0, but such that and remained fixed at = 0.5 and = 0.3. (a)–(c) The time-averaged streamwise component of the wind 〈u〉 (color, m s−1, zero contour black); (d)–(f) time-averaged vertical uplift 〈Δz〉 (color, m, zero contour black). (left) The original choice of parameters (shown before in Figs. 8a and 12a), (center) a simulation from set A, and (right) a simulation from set B (see text).

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What Flow Conditions are Conducive to Banner Cloud Formation?

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  • 1 Johannes Gutenberg University Mainz, Mainz, Germany
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Abstract

Banner clouds are clouds that are attached to the leeward slope of a steep mountain. Their formation is essentially due to strong Lagrangian uplift of air in the lee of the mountain. However, little is known about the flow regime in which banner clouds can be expected to occur. The present study addresses this question through numerical simulations of flow past idealized orography. Systematic sets of simulations are carried out exploring the parameter space spanned by two dimensionless numbers, which represent the aspect ratio of the mountain and the stratification of the flow. The simulations include both two-dimensional flow past two-dimensional orography and three-dimensional flow past three-dimensional orography.

Regarding flow separation from the surface, both the two- and the three-dimensional simulations show the characteristic regime behavior that has previously been found in laboratory experiments for two-dimensional orography. Flow separation is observed in two of the three regimes, namely in the “leeside separation regime,” which occurs preferably for steep mountains in weakly stratified flow, and in the “postwave separation regime,” which requires increased stratification. The physical mechanism for the former is boundary layer friction, while the latter may also occur for inviscid flow. However, flow separation is only a necessary, not sufficient condition for banner cloud formation. The vertical uplift and its leeward–windward asymmetry show that banner clouds cannot form in the two-dimensional simulations. In addition, even in the three-dimensional simulations they can only be expected in a small part of the parameter space corresponding to steep three-dimensional orography in weakly stratified flow.

Corresponding author address: Volkmar Wirth, Institute for Atmospheric Physics, Johannes Gutenberg University Mainz, Becherweg 21, 55128 Mainz, Germany. E-mail: vwirth@uni-mainz.de

Abstract

Banner clouds are clouds that are attached to the leeward slope of a steep mountain. Their formation is essentially due to strong Lagrangian uplift of air in the lee of the mountain. However, little is known about the flow regime in which banner clouds can be expected to occur. The present study addresses this question through numerical simulations of flow past idealized orography. Systematic sets of simulations are carried out exploring the parameter space spanned by two dimensionless numbers, which represent the aspect ratio of the mountain and the stratification of the flow. The simulations include both two-dimensional flow past two-dimensional orography and three-dimensional flow past three-dimensional orography.

Regarding flow separation from the surface, both the two- and the three-dimensional simulations show the characteristic regime behavior that has previously been found in laboratory experiments for two-dimensional orography. Flow separation is observed in two of the three regimes, namely in the “leeside separation regime,” which occurs preferably for steep mountains in weakly stratified flow, and in the “postwave separation regime,” which requires increased stratification. The physical mechanism for the former is boundary layer friction, while the latter may also occur for inviscid flow. However, flow separation is only a necessary, not sufficient condition for banner cloud formation. The vertical uplift and its leeward–windward asymmetry show that banner clouds cannot form in the two-dimensional simulations. In addition, even in the three-dimensional simulations they can only be expected in a small part of the parameter space corresponding to steep three-dimensional orography in weakly stratified flow.

Corresponding author address: Volkmar Wirth, Institute for Atmospheric Physics, Johannes Gutenberg University Mainz, Becherweg 21, 55128 Mainz, Germany. E-mail: vwirth@uni-mainz.de

1. Introduction

Flow past mountains is almost inevitably associated with vertical air motion, giving rise to a whole range of meteorological phenomena (Whiteman 2000) including waves (Durran 2003), strong winds (Durran 1990), clouds (Banta 1990; Houze 1993), and precipitation (Roe 2005). From a theoretical point of view it is interesting to know whether and to what extent these phenomena are associated with specific flow conditions. In this paper we study flow past an isolated mountain in a nonrotating atmosphere and focus on so-called banner clouds (Glickman 2000). The question we will address is “Under which flow conditions can banner clouds be expected to occur?”

Banner clouds are clouds which appear to be attached to the leeward slope of a steep mountain. They have been observed, for instance, at Mount Matterhorn in the Swiss Alps or at Mount Zugspitze in the Bavarian Alps. A systematic definition was provided by Schween et al. (2007) based on time lapse movies taken at Mount Zugspitze. According to their definition a banner cloud has to simultaneously satisfy four criteria: 1) the cloud should exclusively appear in the lee of the mountain, where it is attached to the ground, 2) the cloud should not consist of snow crystals that are blown off the mountain, 3) the cloud should be persistent, and 4) the cloud should not be primarily of convective character. Observations of banner clouds were reported by Wirth et al. (2012), showing (amongst others) characteristic diurnal and seasonal cycles as well as the dependence of banner cloud occurrence on wind strength and direction.

Different hypotheses have been put forward to explain the formation of banner clouds. Voigt and Wirth (2013) tested these hypotheses through numerical simulations. They found that leeside lifting and the associated adiabatic cooling is the main formation mechanism, consistent with earlier suggestions by Schween et al. (2007), Reinert and Wirth (2009), and Wirth et al. (2012). The leeside upwelling was associated with a bow-shaped vortex right behind the mountain [see Fig. 4 in Voigt and Wirth (2013)], implying highly nonlinear flow. However, in all previous studies the model setup was more or less the same, judiciously chosen such as to provide large leeside uplift and, therefore, optimum conditions for banner cloud formation. This leaves open the following questions:

  1. Which aspect of the model setup is essential in order to produce the strong leeside uplift?
  2. What physical mechanism is responsible for the leeside vortex and, hence, for the strong leeside uplift?
To address these questions, we will systematically probe the relevant phase space and explore the different flow regimes in the present paper. This will clarify the flow conditions and the physical mechanisms that are conducive to banner cloud formation.

As mentioned above, a key ingredient of banner cloud formation is upwelling on the leeward face of the mountain all the way up to its summit. In combination with upward flow on the windward side, this implies the separation of the flow from the ground right at the top of the mountain (Scorer 1955) and, hence, the formation of a lee vortex. In the past there have been two distinct physical mechanisms associated with the formation of vortices in the lee of a mountain. The first one is surface friction, and this mechanism does not require the atmosphere to be stably stratified (Scorer 1955). It is well known in the engineering literature and goes back to the seminal work of Ludwig Prandtl [see, e.g., Anderson (2005)]. The second mechanism relies on the baroclinicity of the flow past a three-dimensional mountain (Smolarkiewicz and Rotunno 1989; Smith 1989; Rotunno and Smolarkiewicz 1991; Rotunno et al. 1999; Epifanio and Rotunno 2005). While this mechanism only operates in a stably stratified atmosphere, it does not require viscosity and surface friction to be present.

Generally, in a nonrotating atmosphere the response of flow past a mountain depends on the stability of the approaching air, the wind profile, and the characteristics of the topography (Whiteman 2000). These properties can be combined into two dimensionless parameters: namely, the aspect ratio and the parameter ,
e1
e2
where h0 is the mountain height, L is the horizontal extent of the mountain at altitude h0/2, U is the wind speed of the approaching flow, and N is the Brunt Väisälä frequency. The aspect ratio represents the steepness of the obstacle, while the parameter is proportional to the stratification of the incoming flow. In the past, has been referred to as nonlinearity parameter, nondimensional obstacle height, or inverse Froude number, although there are reasons not to use the latter terminology for stratified flows [see Baines (1995), chapter 1.4]. Based on laboratory experiments with flow past a two-dimensional obstacle, it has been found that these two parameters allow one to characterize the flow response (Baines and Hoinka 1985; Baines 1995). The resulting behavior can succinctly be displayed in a regime diagram (Fig. 1). For steep mountains ( ≫ 1), the flow separates from the surface right at the top of the mountain, while for shallow mountains and weak stratification ( ≪ 1, ≪ 1) there is no flow separation at all. In addition, there is a third regime called “postwave separation,” which occurs for flow with strong stratification ( ≫ 1). In this regime, the flow separation from the surface is associated with gravity waves; it does not occur at the mountain top, but some distance downstream. Thus, there is downwelling flow between the summit and the point of separation, which may be associated with a downslope storm (Klemp and Lilly 1975; Durran 1986, 1990). Based on vorticity arguments, Ambaum and Marshall (2005) developed an elegant theory to explain the occurrence of these regimes in the framework of two-dimensional flow.
Fig. 1.
Fig. 1.

Regime diagram for flow past a two-dimensional obstacle, adapted from Fig. 5.8 of Baines (1995). Depending on the combination of aspect ratio and the parameter , the flow belongs to one of three different regimes regarding flow separation from the surface. The dashed straight lines are isolines of /.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Although these developments provide some guidance regarding flow separation, they are not directly applicable to our problem. The reason is that the simulations of Schappert and Wirth (2015) suggest that banner cloud formation requires genuinely three-dimensional orography. On the other hand, both the experiments of Baines and Hoinka (1985) and the theory of Ambaum and Marshall (2005) are based on flow past two-dimensional obstacles. Fortunately, it seems that flow past three-dimensional obstacles is similar to flow past two-dimensional obstacles in many aspects. For instance, Hunt and Snyder (1980) carried out towing tank experiments with a three-dimensional hill of moderate slope for different Froude numbers. These experiments indicate overall similar behavior regarding flow separation, although they do not cover the complete parameter space of Baines (1995). We are, therefore, motivated to revisit the regime diagram of Baines (1995) for three-dimensional instead of two-dimensional orography.

Our investigation is based on both two- and three-dimensional simulations of the turbulent flow past an idealized mountain. We will carry out several sets of simulations, analyzing regime behavior and associated flow patterns. It will be shown that many aspects of the two-dimensional regime diagram of Baines (1995) apply also to the three-dimensional simulations. Nevertheless, banner clouds turn out to occur only in a small subspace of the parameter space explored. This means that flow separation is a necessary, but not a sufficient condition for banner cloud formation. In addition, we will elucidate through sensitivity experiments the physical mechanisms which are responsible for flow separation in the different regimes.

The remainder of the paper is organized as follows. Section 2 describes the numerical model, the model setup, as well as the diagnostics used to detect flow separation and banner cloud occurrence. The results are presented in section 3, where we discuss regime behavior and vertical uplift in both two- and three-dimensional simulations and compare them with each other. Finally, section 4 provides some discussion and our conclusions.

2. Model and diagnostics

a. Numerical model and model setup

We use the nonhydrostatic anelastic version of the Eulerian/semi-Lagrangian fluid solver (EULAG; Prusa et al. 2008) in order to simulate the flow of dry air past an idealized mountain in a nonrotating atmosphere. The equations of motion are formulated in a terrain-following coordinate system. The model is run in large-eddy simulation (LES) mode, and the subgrid-scale fluxes of momentum are parameterized through a turbulent kinetic energy (TKE) closure (Sorbjan 1996). The advection scheme is MPDATA (Smolarkiewicz 1983), which contains just enough numerical diffusion so as to keep the numerics stable (Smolarkiewicz 1983; Smolarkiewicz et al. 2007; Beaudoin et al. 2013). For further model details, see also Voigt and Wirth (2013) and Schappert and Wirth (2015).

The orography for the three-dimensional simulations has a cosine shape according to
e3
where h0 = 500 m is a constant mountain height, with the horizontal coordinates x (streamwise direction) and y (spanwise direction), and where (x0, y0) denotes the center of the mountain. The orography for the two-dimensional simulations is defined through h(x, y = y0) with h given by (3). In both cases, the orography has deliberately been chosen to be very smooth, which is in distinct contrast to the pyramid-shaped orography in our earlier investigations (Reinert and Wirth 2009; Voigt and Wirth 2013; Schappert and Wirth 2015). In the latter, we were primarily interested in a model setup that produces a realistic banner cloud. A pyramid is favorable to banner cloud formation, because the viscous flow separates at its salient edges almost by necessity (Scorer 1955). On the other hand, in the present study we want to give the flow a fair chance not to separate from the obstacle, because this allows us to better distinguish those conditions that favor flow separation from those conditions that do not.

The model domain for our two-dimensional simulations consists of two parts: an outer part in which we implemented two sponge layers (one at the top and one at the inflow boundary) and an inner part that is free of any wave absorbers (see Fig. 2). In the x direction (streamwise direction), the extent of the inner part of the domain is specified as a function of mountain steepness to be 9.6−1 km, while in the vertical it extends to a fixed height of 3.0 km. The equations are discretized on a uniform mesh with nx × nz = 768 × 120 grid points (corresponding to a fixed vertical grid spacing of δz = 25 m in the flat portion of the domain). The sponge layer above the inner part of the model domain has a thickness of Δzsponge = 1.5 km. Inside this sponge layer, the wind is relaxed toward the inflow profile. This layer is meant to minimize the reflection of upward propagating gravity waves. The second sponge layer is located at the inflow boundary and has a thickness of Δxsponge = 1.6−1 km. It turned out necessary to avoid reflections at the inflow boundary which otherwise occurred for flows with > 0.3. Several tests with a larger model domains revealed that the flow close to the mountain and, especially, the flow separation behavior are only marginally sensitive to the exact size of our model domain.

Fig. 2.
Fig. 2.

Illustration of the model domain for the two-dimensional simulations.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

The requirements on the size of the model domain are less severe for the three-dimensional simulations, because in this case the gravity waves are significantly weaker. Here, the inner part of the model domain extends 4.0−1 km in the x direction (streamwise direction), 2.4−1 km in the y direction (spanwise direction), and a fixed amount of 1.5 km in the vertical (see Fig. 3). The equations of motion are discretized on a uniform mesh with nx × ny × nz = 320 × 192 × 60 grid points (corresponding, again, to a vertical grid spacing of δz = 25 m in the flat portion of the domain). Outside of the inner part of the model domain we added the same sponge layers as for the two-dimensional simulations. The mountain is centered in the spanwise direction, and its streamwise position x0 is located 1.5−1 km downstream of the inflow boundary of the inner part of the model domain. Again, several tests indicated that the key flow features changed only marginally for larger domains.

Fig. 3.
Fig. 3.

Illustration of the model domain for the three-dimensional simulations.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

At the inflow boundary, we specify constant wind and constant Brunt Väisälä frequency N. This is achieved through the following inflow profiles for wind u = (u, υ, w) and potential temperature θ:
e4
e5
where U = 9 m s−1, g = 9.81 m s−2 is the acceleration due to gravity, and θ0 = 283 K is a constant. As a consequence, the flow approaching the mountain has a well-defined stability parameter .
The boundary condition at the outflow boundary ideally should simulate a nonreflecting boundary. We effectively achieved this by specifying the inflow profiles at the outflow boundary and pushing this boundary far enough downstream. Indeed, as stated before, we verified that the flow close to the mountain depends only marginally on the size of the model domain. At the boundaries in the spanwise direction in the three-dimensional simulations we used periodic conditions. At the lower boundary a drag law is implemented according to
e6
where τ denotes the surface stress, v0 = (u0, υ0) is the surface wind, ρ0 = 1.23 kg m−3 is a constant density, and cd is a drag coefficient. Unless noted otherwise we used cd = 0.01.

The equations are integrated forward in time using a time step Δt = 0.5 s. Each model run consists of two parts: the spinup run and the ensuing main run. The spinup run allows the model variables to reach a statistically stationary state. Several tests revealed that this was reached to a satisfying approximation after a duration of ts = 10−1 minutes. The duration of the main run is 60 min.

b. Diagnostics used for the analysis

Although our primary interest is a specific type of orographic clouds, we only simulate flow of dry air past a mountain. At first sight this appears strange, but the approach was chosen quite deliberately. For one thing, earlier simulations have indicated that the release of latent heat, albeit not completely negligible, is of minor importance for banner cloud formation (Reinert and Wirth 2009). It would, therefore, be sufficient to advect a passive tracer q representing specific humidity and to diagnose relative humidity RH from this tracer; a cloud could be said to exist wherever RH > 100% (Voigt and Wirth 2013). However, the need to initialize the tracer q introduces some arbitrariness, since obviously cloud formation depends on the amount of moisture in the air. Instead, it turned out useful to only diagnose the Lagrangian vertical uplift of air parcels, because this can be taken as a proxy for the potential of banner cloud formation: to the extent that uplift is larger on the leeward side than on the windward side, one may expect a banner cloud to occur given suitable moisture conditions.

Following our earlier approach (Reinert and Wirth 2009; Voigt and Wirth 2013; Schappert and Wirth 2015), we diagnose Lagrangian uplift through a Eulerian tracer χ(x, t), which evolves according to
e7
Here, D/Dt denotes the material rate of change following the flow and Mχ represents material nonconservation owing to the parameterized subgrid-scale turbulence. As discussed in our earlier work, Mχ is small except in regions of strong shear. The tracer is initialized with χ(x) = z, and at the inflow boundary it is set to χ(z) = z. As argued in Schappert and Wirth (2015), for all practical purposes during the main run the tracer χ represents the altitude of the parcel at the time when it entered the domain. As a consequence, the Lagrangian vertical displacement Δz throughout the model domain can be computed from
e8
The field Δz represents upward (Δz > 0) or downward (Δz < 0) displacement of air parcels owing to the presence of the mountain compared to their unperturbed altitude. Positive values of Δz will be referred to as “uplift” in the following. A banner cloud can be said to exist if the uplift has a significant windward–leeward asymmetry with a plume of large values in the lee. Note that uplift (Δz > 0) has to be clearly distinguished from “upwelling” (w > 0), because the latter turns out not to be a good proxy for banner cloud formation (Voigt and Wirth 2013).

The flow in our simulations is highly nonstationary, and for some simulations there is quasi-periodic shedding of vortices in the lee of the mountain (Schär and Durran 1997). However, in this paper we are only interested in the major characteristics of the flow that show up in the time-averaged fields. The time average used in our analyses extends over the entire main model run and will be denoted by angle brackets 〈⋅〉.

To identify flow separation from the surface, we use the time-averaged streamwise wind component 〈u〉. The separation point is characterized by a change in 〈u〉 from positive to negative values in the streamwise direction at the surface. Additionally, we use streamlines of the time-averaged wind in order to show the overall behavior of the flow and to delineate the spatial extent of lee vortices (if present).

3. Results

a. Two-dimensional simulations

We start with the two-dimensional simulations. Both and will be varied systematically by varying the values for L and N while keeping U and h0 constant. This allows us to probe the parameter space corresponding to the regime diagram of Baines (1995). Although this regime diagram is firmly established, we consider this exercise as important, because it provides evidence that our model code is well suited for our purposes. This cannot be taken for granted a priori, because flow separation is sensitive to the numerical treatment close to the bottom boundary, which is a nontrivial challenge for any large-eddy simulation model (Moeng and Sullivan 2003; Fröhlich 2006). In addition, we will later compare the two- and the three-dimensional simulations in terms of their potential for banner cloud formation, which has not been done before. All simulations are restricted to < 0.5, because larger aspect ratios led to numerical instabilities.

Streamlines of the time-averaged flow for 12 different combinations of and are shown in Fig. 4, and the corresponding time-averaged streamwise component of the wind 〈u〉 is shown in Fig. 5. Overall, the figures indicate gravity waves of different sorts in the lee of the mountain as soon as the stratification is large enough (second, third, and fourth columns of Fig. 4). For large aspect ratios and low stratification the flow separates right at the top of the mountain (Fig. 4a), and this is associated flow reversal in the lee (Fig. 5a). Following Baines (1995), we call this behavior “leeside separation.” Increasing the stratification renders the recirculation area on the leeward side of the mountain smaller (Figs. 4b,c and 5b,c) until it vanishes (Figs. 4d and 5d) and the separation of the flow from the surface is delayed further into the lee. The latter is referred to as “postwave separation.” The delayed separation is due to the fact that parcels that are lifted on the windward side of the mountain in a stably stratified atmosphere acquire negative buoyancy, and this helps to avoid flow separation right behind the summit. Expressed in more technical terms, the leeside separation vanishes by the time the horizontal length scale of the gravity wave triggered by a (smooth) mountain becomes smaller than the horizontal length scale of the mountain. Note that the flow in Figs. 4c and 5c cannot be uniquely associated with a specific regime, as it contains elements of both leeside separation and postwave separation. Nevertheless, we would argue that in our simulations the transition from leeside separation to postwave separation for = 0.5 occurs between = 1.2 and 2.4. For obstacles with a smaller aspect ratio of = 0.25 (second row in Figs. 4 and 5), the transition from leeside separation to postwave separation occurs at lower stratification (approximately between = 0.3 and 0.6).

Fig. 4.
Fig. 4.

Regime behavior in the two-dimensional simulations. The plots show streamlines of the time-average flow, with the overall flow direction from left to right. Simulations with different values for the parameters = h0/L (decreasing top–bottom) and = Nh0/U (increasing left–right), with the mountain steepness increasing logarithmically upward and the stratification parameter increasing logarithmically. The plots show only part of the model domain.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Fig. 5.
Fig. 5.

Regime behavior in the two-dimensional simulations. The color fill represents the time-mean streamwise component of the wind 〈u〉 (m s−1), and the black contour is the zero contour of 〈u〉. Otherwise the figure conventions are as in Fig. 4.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

For shallow obstacles in weakly stratified flow there is no flow separation at all (Figs. 4i and 5i). This behavior is referred to as “no separation.” Keeping the stability parameter fixed at = 0.3 and decreasing the aspect ratio (going downward in the left column in Figs. 4 and 5) leads to a shrinking of the recirculation area on the leeward side until it vanishes. Outside the frictional boundary layer the flow in Figs. 4i and 5i is very close to potential flow, as can be shown by reference to vorticity (no figure shown). The transition from leeside separation to no separation occurs in our simulations between = 0.25 and 0.125. Keeping the aspect ratio low ( = 0.125) and increasing the stratification (bottom rows in Figs. 4 and 5) yields a transition from no separation to postwave separation, with the transition occurring between = 0.3 and 0.6.

Overall, our results are in a good agreement with the diagram of Baines (1995), which makes us confident that our numerical model is suitable for our purposes. Note that the amount of surface friction can slightly modify the exact location of the transitions between the regimes (cf. Doyle and Durran 2002). To quantify the related sensitivity we carried out the same set of simulations with cd = 0.001 instead of 0.01. The result is shown in Fig. 6. Overall, the regime behavior is unchanged with leeside separation, no separation, and postwave separation all being present quite like in the previous simulations. At the same time there are subtle differences, and the transitions between the regimes have shifted to different values of and . For instance, the recirculation tends to be much stronger in Fig. 6 than in Fig. 5, which presumably is a direct consequence of the reduced surface friction. More importantly in our present context, the reduction of the drag coefficient implies less vertical shear close to the ground, which makes it harder for the boundary layer to separate. As a consequence, the transition from leeside separation to no separation shifts toward larger aspect ratios (cf. the left column of Fig. 6 with the left column of Fig. 5). With cd = 0.001, the leeside recirculation area almost disappears for = 0.25 (Fig. 6e), while for the same mountain the recirculation area is well developed with cd = 0.01 (Fig. 5e). Similarly, with reduced surface friction the postwave separation is delayed further into the lee (cf. Fig. 6g with Fig. 5g), and the strength and downward penetration of the downslope winds is larger (right column in Fig. 6). The somewhat ambiguous behavior in Fig. 5c, which we commented on earlier, vanishes for the reduced surface friction (see Fig. 6c), which can now uniquely be attributed to the postwave separation regime.

Fig. 6.
Fig. 6.

As in Fig. 5, but for cd = 0.001.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

It is interesting to note that the theory of Ambaum and Marshall (2005) does not explicitly contain any treatment of surface friction; rather, it only stipulates that there must be some surface friction such as to produce a boundary layer. This is consistent with the idea that the regimes of boundary layer separation in viscous two-dimensional flow are universal, transcending the exact formulation of surface friction. We take the same view in the present paper and focus mostly on the qualitative behavior with the existence of three distinct regimes and the transitions between them.

b. Three-dimensional simulations

We now extend our simulations to three-dimensional flow past three-dimensional orography, allowing us to systematically explore whether and to what extent the above regime behavior remains valid. To cover the regime diagram of Fig. 1 we varied, again, and by varying N and L while keeping U and h0 constant. The results are shown in Figs. 7 and 8 in the same format as before. Generally, the leeside vortex in the upper-left part of the figure (Fig. 7a) has a smaller leeward extension than in the two-dimensional simulations (see Fig. 4a). Nevertheless, for weak stratification the transition between leeside separation and no separation (left column in Fig. 8) looks qualitatively similar as in the corresponding two-dimensional simulations (left column in Fig. 5). For strong stratification, the three-dimensional simulations (right column in Fig. 8) show overall weaker gravity wave activity and a more extended leeside recirculation area than the two-dimensional simulations (right column in Fig. 5). Both features are presumably related to the fact that the air can go around (instead of over) the mountain, which is the preferred pathway especially for stable stratification.

Fig. 7.
Fig. 7.

Regime behavior in the three-dimensional simulations. The panels show streamlines of the time-averaged flow in a vertical cross section through the middle of the domain, with the overall flow direction from left to right. Simulations with different values for the parameters = h0/L (decreasing top–bottom) and = Nh0/U (increasing left–right), with the mountain steepness increasing logarithmically upward and the stability parameter increasing logarithmically. The plots show only part of the model domain.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Fig. 8.
Fig. 8.

Regime behavior in the three-dimensional simulations. The color fill represents the time-mean streamwise component of the wind 〈u〉 (m s−1), and the black contour is the zero contour of 〈u〉. Otherwise, the figure conventions are as in Fig. 7.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Streamlines in a horizontal section are shown in Fig. 9, and the horizontal extent of the recirculation area close to the ground can be gleaned from Fig. 10. For a steep mountain with weak stratification (Fig. 9a), there are two counterrotating vortices with a vertical axis in the immediate lee of the mountain; combining this information with the leeside vortex pattern in the vertical section (Fig. 7a), one recognizes that these features are signatures of a three-dimensional bow-shaped vortex, which has been mentioned and sketched before by Voigt and Wirth (2013, see their Fig. 4). For weak stratification (left column in Fig. 10) the leeside recirculation area decreases with decreasing aspect ratio, and the transition from leeside separation to no separation occurs between = 0.25 and 0.125. This is consistent with the corresponding result from the two-dimensional simulations (see previous section). In addition, there is recirculation on the windward side in most of the plots shown in Fig. 10, and this represents orographic upstream blocking of air (Baines 1987; Baines and Smith 1993; Epifanio and Rotunno 2005). It is only for weakly stratified flow over a shallow mountain that the air is not blocked on the windward side (Figs. 10e,i,j). It is also interesting to note how the streamwise wind component for steep mountains changes as the stratification is increased (top row in Fig. 10): there are two bands of enhanced wind on either side of the mountain extending into the lee, and these bands get stronger and extend further into the lee as stratification is increased. We interpret this as essentially a transition from “flow over the mountain” to “flow around the mountain.”

Fig. 9.
Fig. 9.

Regime behavior in the three-dimensional simulations. The panels show streamlines of the time-averaged flow in a horizontal cross section at z = 250 m. Otherwise the figure conventions are as in Fig. 7.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Fig. 10.
Fig. 10.

Dependence of the recirculation area on aspect ratio and stratification in the three-dimensional simulations. The color fill quantifies the time-mean streamwise component of the wind 〈u〉 (m s−1) in the lowest model layer. The black contour delineates the horizontal extent of the mountain. Otherwise the figure conventions are as in Fig. 7.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Apparently, the best combination of parameters for leeside separation right at the mountain top is a steep mountain in weakly stratified flow (Figs. 7a and 8a). Increasing the stratification for a large aspect ratio ( = 0.5, top row in Fig. 8) renders the recirculation area in the lee smaller until no separation occurs any longer right at the obstacle but, instead, is shifted further into the lee (Fig. 8c). The transition from leeside separation to postwave separation in our simulations occurs between = 0.6 and 1.2. For a flat mountain (bottom row in Figs. 8 and 10) the flow separates on the leeward slope of the mountain if the stratification is strong enough (Figs. 8k,l); however, this part of the parameter space belongs to the regime of postwave separation, which means that the separation is not right at the mountain top but shifted into the lee by some distance. The transition from no separation to postwave separation for = 0.125 is between = 0.6 and 1.2. The middle rows of Figs. 8 and 10 are particularly interesting: here we can observe all three regimes, starting with leeside separation (Figs. 8e and 10e), turning into no separation (Figs. 8f and 10f), and, finally, to postwave separation (Figs. 8g,h and 10g,h). Note that this behavior is, indeed, foreseen for intermediate values of in the regime diagram of Baines (Fig. 1).

For all simulations with strong stratification ( = 2.4, right columns in Figs. 8 and 10) the separation point is close to the top of the obstacle, even though not right at the top. The leeside recirculation area is stronger and farther extended into the lee compared to the leeside separation case (Fig. 8a). However, as mentioned above, the strong stratification favors flow around rather than over the mountain, and the flow separation occurs preferentially at the sidewise edges of the obstacle rather than at its top. This can be seen more clearly from the vertical wind (not shown), which indicates overall significantly smaller values in the three-dimensional simulations than in the two-dimensional simulations. This indicates the completely different asymptotic behavior between two- and three-dimensional simulations in the limit of strong stratification: while the parcels simply have to go over the mountain in the two-dimensional simulations, they follow an increasing tendency to go around the mountain in the three-dimensional simulations. This is consistent with the theory of Drazin (1961), which stipulates purely two-dimensional horizontal flow in this limit.

As mentioned earlier, there are two distinct physical mechanisms that can generate flow separation and the formation of lee vortices: one based on surface friction and another one based on the baroclinicity of the flow. Since our simulations account for both surface drag and stable stratification, they contain both mechanisms at the same time, and it is not clear which one is primarily responsible for flow separation and vortex formation in the different parts of the parameter space. To separate the two mechanisms, we switched off surface friction by setting cd = 0 and repeated all simulations. The result is shown in Fig. 11. Apparently, for weak stratification (first and second column on the left) the flow does not separate from the surface any longer even for the steep mountains, in distinct contrast with the simulations from Fig. 8. This indicates that leeside separation is essentially due to a viscous mechanism requiring nonzero surface friction. On the other hand, for the runs with strong stratification the flow separates from the surface even in the absence of surface friction (right column in Fig. 11). This is consistent with earlier results where lee vortex formation was found in simulations with a free-slip boundary condition (Smolarkiewicz and Rotunno 1989; Schär and Durran 1997). We also carried out a set of simulations with neutral stratification (i.e., with = 0), which is equivalent to switching off baroclinicity. This renders our parameter space one dimensional, with the aspect ratio being the only remaining parameter, and the postwave separation regime gets lost. The corresponding results (not shown) look very similar to the left columns in Figs. 7, 8, and 12, with no sign of postwave separation whatsoever. It follows that flow separation and vortex occurrence in the postwave separation regime must be associated with the baroclinic mechanism.

Fig. 11.
Fig. 11.

As in Fig. 8 except that surface drag has been set to zero.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Fig. 12.
Fig. 12.

Regime behavior of vertical uplift in the three-dimensional simulations. The color fill shows the time-mean Lagrangian vertical displacement 〈Δz〉 (m) in a vertical section through the center of the domain. The black contour is the zero contour, and the white contour depicts 〈Δz〉 = +400 m—that is, an uplift almost as large as the mountain height, indicating significant potential for cloud formation. Otherwise, the figure conventions are as in Fig. 7.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Smolarkiewicz and Rotunno (1989) considered a rather shallow mountain corresponding to = 0.12 in our notation. They found lee vortices to occur (even without surface friction) when their Froude number was less than 0.5, corresponding to > 2 in our notation. This is consistent with our results displayed in the bottom row in Fig. 11, where strong recirculation associated with two counterrotating lee vortices only occurs in the rightmost panel (i.e., Fig. 11l), where > 2.

We conclude this section by reemphasizing that the regime behavior in terms of flow separation is quite similar for our two- and three-dimensional simulations, with the same three regimes appearing in the same parts of the phase space.

c. Banner cloud formation

As mentioned before, the occurrence of banner clouds is accompanied by the occurrence of flow separation and lee vortices, but the latter two are only necessary, not sufficient conditions for banner cloud formation. The key diagnostic is vertical uplift Δz, which has to feature a plume of large values in the lee of the mountain. This is what we want to analyze now.

The regime behavior of 〈Δz〉 for our three-dimensional simulations is shown in Fig. 12. Steep mountains in combination with weak stratification (Fig. 12a) show a pronounced windward–leeward asymmetry with a plume of large values in the lee. Of all the panels shown in Fig. 12, this is the situation which is most conducive to banner cloud formation. Keeping stratification fixed and decreasing the aspect ratio leads to a more symmetric distribution of 〈Δz〉 (left column in Fig. 12); the leeward–windward asymmetry and, hence, the potential for banner cloud formation vanishes for shallow mountains. Instead, Fig. 12i suggests that the most likely location of a cloud is right above the mountain, and this cloud would be symmetric with respect to the streamwise direction. These features are usually observed in connection with cap clouds (Glickman 2000). Figure 12 also shows that for intermediate stratification the vertical displacement is downward (rather than upward) in the lee, which is associated with adiabatic warming and the dissolution of clouds, akin to foehnlike conditions (Steinacker 2006; Richner and Hächler 2013). It also shows that optimum conditions for this foehnlike behavior change with changing aspect ratio. Finally, for strong stratification (right column in Fig. 12) the magnitude of vertical displacement in the lee is overall rather small. Of course this is intuitively right, since strong stratification suppresses vertical parcel displacement, and it is consistent with theoretical considerations that predict purely two-dimensional horizontal flow in the limit of very strong stratification (Drazin 1961). In our current context this means that although postwave separation is associated with upward flow in the lee of the mountain, the magnitude of the uplift in this part of the parameter space is rather small such that one would not expect banner clouds to occur.

We noted before that for steep mountains there is a transition from leeside separation to postwave separation as stratification is increased. The top row in Fig. 12 indicates that this transition is accompanied by a gradual change from Lagrangian upward displacement in the lee (i.e., banner cloud conditions; Fig. 12a) to Lagrangian downward displacement producing foehnlike conditions with a tendency for clouds to dissolve in the lee (Fig. 12c). This demonstrates, again, that the existence of flow separation and lee vortex formation in the postwave separation regime is not sufficient to provide conditions conducive to banner cloud formation.

It is also instructive to consider the dependence on stratification for shallow mountains (bottom row in Fig. 12). For weak stratification the displacement is symmetric about the mountain top with its maximum right above the mountain, as one would expect for potential flow. Increasing the stratification renders the uplift asymmetric about the mountain with larger values on the windward side and smaller values on the leeward side (Fig. 12j). Again, this is exactly the opposite of what is required for banner cloud formation. Increasing stratification even further (Figs. 12k,l) continues the tendency for the region of uplift to shift to the windward side, while the pattern on the leeward side gets more complex. Although this more complex behavior is associated with vortexlike flow features (see the bottom-right panels of Figs. 7 and 9) including some uplift, the magnitude of this uplift is smaller than on the windward side. This makes the formation of banner clouds unlikely in this part of the parameter space. It transpires that gently sloping mountains cannot be expected to form banner clouds no matter how small or large the stratification is.

The results in Fig. 12 can also viewed from a scale analysis perspective (e.g., Epifanio and Rotunno 2005). Vertical displacement in stratified flow is generally limited to roughly ΔzU/N, implying
e9
It follows that for large typical vertical displacements are much smaller than the height of the mountain. This is consistent with Fig. 12, where displacements Δz on the order of the mountain height h0 only occur in the left columns, while in the right columns the magnitude of Δz is significantly smaller than h0.
We condense our findings about vertical uplift into a diagnostic P that is an integrated measure of the potential for a specific flow situation to give rise to banner cloud occurrence. As stated before, this requires a large asymmetry in 〈Δz〉 between the leeward and the windward side with larger values in the lee. Correspondingly, we define
e10
where the integral is taken on the surface y = y0, “lee” refers to the area given by x0xx0 + L, zsfczh0, and “ww” refers to the area given by x0Lxx0, zsfczh0, with zsfc denoting the altitude of the bottom surface. Positive values of P indicate conditions that are conducive to banner cloud formation, P ≈ 0 means that one can expect clouds to occur on either side of the mountain with approximately the same likelihood, and negative values of P indicate a situation with clouds to be more likely on the windward than on the leeward side. We will refer to P as “banner cloud formation potential.” A plot of P as a function of and is shown in Fig. 13a. Only three tiles in the top-left corner of this plot show red colors, indicating potential for banner cloud formation only in this part of the parameter space. Obviously, this is consistent with the above results.
Fig. 13.
Fig. 13.

Regime behavior of the banner cloud formation potential P (see text for definition) for three different model configurations. The 12 colored tiles in each plot represent the value of P (m) for the same 12 combinations of and as considered in the previous figures. (a) Three-dimensional simulations with standard surface friction (cd = 0.01) corresponding to Fig. 12, (b) two-dimensional simulations with standard surface friction (cd = 0.01), and (c) three-dimensional simulations with zero surface friction (cd = 0).

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

Earlier we saw that the two- and three-dimensional simulations are qualitatively similar regarding their regime behavior (cf. Figs. 4 and 7). Does this also mean that the potential for banner cloud formation has a similar parameter dependence for the two- and the three-dimensional simulations? To address this question, we compare the vertical uplift in the three-dimensional simulations (Fig. 12) with the corresponding results in the two-dimensional simulations shown earlier. Here, we focus on the steep mountain in weak stratification (Fig. 12a), which in the case of the three-dimensional simulations gives rise to the characteristic windward–leeward asymmetry conducive to banner cloud formation. For the corresponding two-dimensional simulation (Fig. 14a), there is also some windward–leeward asymmetry. However, below the mountain top the uplift is now larger on the windward rather than the leeward side; above the mountain top there is a (somewhat asymmetric) plume of large values of uplift, but this plume extends to high altitudes way beyond the mountain top. Both features are inconsistent with the observed character of a banner cloud (Schween et al. 2007). Summarizing these results in terms of our metric P gives Fig. 13b. Apparently, all tiles are blue, indicating that for the two-dimensional simulations there is no potential for banner cloud formation in any part of the parameter space.

Fig. 14.
Fig. 14.

Vertical uplift for (, ) = (0.5, 0.3) in two sensitivity experiments, to be compared with the reference simulation in Fig. 12a. (a) Two-dimensional instead of three-dimensional flow; (b) surface friction set to zero (cd = 0). The color fill shows the time mean Lagrangian vertical displacement 〈Δz〉 (m) in a vertical section through the center of the domain. The black contour is the zero contour, and the white contour depicts 〈Δz〉 = +400 m.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

We return to the three-dimensional simulations and further investigate the mechanisms of lee vortex formation. As mentioned earlier, banner cloud formation is associated with a bow-shaped lee vortex, and this vortex owes its existence to surface friction. This raises the question: how does the field of vertical uplift look like in a run without any surface friction (i.e., for cd = 0). The resulting regime behavior in terms of Δz turns out to be qualitatively similar to that in Fig. 12 except for steep mountains with weak stratification (see Fig. 14b, to be compared with Fig. 12a). In this part of the parameter space the windward–leeward asymmetry is completely lost when surface friction is switched off. This demonstrates, again, that the bow-shaped vortex associated with banner cloud formation arises from flow separation due to surface friction. Condensing this result in terms of our metric P, we obtain Fig. 13c. This plot looks similar as Fig. 13a except in the top-left corner, which indicates that surface friction has a major impact on P in the case of steep mountains and weak stratification; it is only through the inclusion of surface friction that the top-left tiles in the figure turn red. This is but another manifestation of the result that banner cloud formation relies on surface friction.

d. Three-dimensional flow past two-dimensional orography

We considered a third model configuration (not mentioned so far), in which we simulated three-dimensional flow past two-dimensional orography. This setup appears interesting, because two- and three-dimensional flows are known to be fundamentally different regarding their turbulence behavior (e.g., Vallis 2006). To the extent that inertial range arguments apply, kinetic energy is expected to cascade toward larger scales in two-dimensional turbulence but toward smaller scales in three-dimensional turbulence. This can have profound implications for the occurrence of coherent structures (McWilliams 1984). On the other hand, in the present context we do not expect huge differences between two- and three-dimensional flow past two-dimensional orography. This is because the laboratory experiments of Baines and Hoinka (1985) and the theory of Ambaum and Marshall (2005)—both using two-dimensional orography—indicate overall very similar regime behavior despite the fact that the former are based on three-dimensional flow, while the latter is based on two-dimensional flow.

The domain size for the simulations in the present section is like for the two-dimensional simulations in both the streamwise and the vertical direction, while its extent in the spanwise direction is half that from the three-dimensional simulations. The orography is a ridge with translational symmetry in the y direction and with its profile in the streamwise section given by h(x, y = y0) with h taken from (3). We carried out 12 simulations for the same 12 combinations of and as previously. It turns out that the overall regime behavior and, in particular, the behavior regarding flow separation at the surface is very similar as in the strictly two-dimensional simulations. For illustration, we show results for = 0.25 in Fig. 15. The top row of this figure should be compared with the middle row of Fig. 5. The only noticeable differences occur in the lee of the mountain: in the leftmost panel, the strictly two-dimensional simulation shows a somewhat smaller recirculation area, and in the rightmost panel the strictly two-dimensional simulations show significantly larger wave amplitudes. However, these differences have little impact on the regime behavior regarding flow separation on the mountain. Not surprisingly, the difference in terms of vertical uplift between the two sets of simulations (not shown) is very small, too.

Fig. 15.
Fig. 15.

Results from an additional set of simulations for three-dimensional flow past two-dimensional orography with fixed = 0.25. The four columns represent the results for four different values of = Nh0/U. (a)–(d) The time-mean streamwise component 〈u〉 of the wind (m s−1, plot conventions as in Fig. 5); (e)–(h) the instantaneous wind υ in the spanwise direction (m s−1) at the end of the simulation in a horizontal section at z = 250 m; the region where this section is underground is indicated by gray shading.

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

We conclude that the dimensionality of the flow is of minor importance for the flow topology in the neighborhood of the mountain and, therefore, for the question which we address in this paper. Arguably this is due to the fact that steep orography represents a major external forcing such that inertial range arguments do not straightforwardly apply in the neighborhood of the orography. Consistent with this interpretation, the only region where dimensionality of the flow seems to make a difference is in the lee of the orography, where the turbulence (which was generated by the flow over the mountain) is allowed to develop freely. We tested this interpretation by ploting the instantaneous spanwise component υ of the wind in the bottom row of Fig. 15. Apparently, significant amplitudes of υ only occur in the lee of the mountain, and they have a clear maximum in Fig. 15h corresponding to that part of the parameter space in which 〈u〉 shows the largest differences owing to the dimensionality of the flow (Fig. 15d versus Fig. 5h).

e. Relevance of the dimensionless parameters and 

So far we have explored the two-dimensional parameter space spanned by and through varying L and N while keeping U and h0 constant. This allowed us to organize the different simulations in a two-dimensional regime diagram. We now want to extend these results by showing that these two dimensionless parameters are, indeed, the key parameters in the sense that changing the dimensional parameters U, N, L, and h0 leaves our regime diagram unchanged as long as and remain unchanged.

For this purpose, we carried out two additional sets of 12 three-dimensional simulations each. In set A we replaced both N and U by half of the value used previously, while in set B we replaced h0, L, and U by half of the value used previously. Both modifications leave the dimensionless numbers and unchanged. In addition, we adjusted the grid spacings such that the Courant number remained unchanged—that is, Δt → 2Δt in set A and (Δx, Δy, Δz) → 0.5(Δx, Δy, Δz) in set B. Here we only show the result of one simulation from each set (Fig. 16), corresponding to that part of the regime diagram that is relevant for banner clouds. Note that in these plots the contour intervals have been adjusted to reflect the corresponding changes in U and h0. Apparently, the two new simulations (second and third column) are very similar to the original simulation (left column) both in terms of the flow (top row) and in terms of the vertical uplift (bottom row). It turns out that the differences between the three sets of simulations are very small not only for this particular combination of dimensionless parameters but also for all other combinations that we considered earlier (but did not show in Fig. 16). As a consequence, the regime behavior is practically identical in all three sets of simulations. We conclude that and are, indeed, the relevant dimensionless parameters in the present context.

Fig. 16.
Fig. 16.

Results from three different simulations with different combinations of dimensional parameters N, L, U, and h0, but such that and remained fixed at = 0.5 and = 0.3. (a)–(c) The time-averaged streamwise component of the wind 〈u〉 (color, m s−1, zero contour black); (d)–(f) time-averaged vertical uplift 〈Δz〉 (color, m, zero contour black). (left) The original choice of parameters (shown before in Figs. 8a and 12a), (center) a simulation from set A, and (right) a simulation from set B (see text).

Citation: Journal of the Atmospheric Sciences 73, 6; 10.1175/JAS-D-15-0319.1

4. Discussion and conclusions

Previously, the formation of banner clouds has been investigated through numerical simulations, but in these studies the model setup and flow configuration have always been such as to provide conditions that are favorable to banner cloud formation. This left open the question as to which flow conditions actually are conducive to banner cloud formation. It is the goal of the present paper to answer this question through sets of large-eddy simulations, systematically exploring the parameter space spanned by two dimensionless numbers, namely the mountain aspect ratio and the stability parameter . The focus of our analysis is on the occurrence of flow separation from the surface, lee vortices, and large Lagrangian uplift in the lee of the mountain.

In a first step we carried out two-dimensional simulations and were able to reproduce the regime behavior regarding flow separation, which was established some time ago through laboratory experiments (Baines 1995). The three regimes correspond to 1) leeside separation, 2) no separation, and 3) postwave separation. We went on to show that this regime behavior remains qualitatively valid for three-dimensional flow past three-dimensional orography. On the other hand, regarding Lagrangian uplift we found substantial differences between the two- and the three-dimensional simulations. We showed explicitly that these differences are due to the different dimensionality of the orography, not the different dimensionality of the flow: banner clouds require genuinely three-dimensional orography—they cannot occur behind two-dimensional ridges. This conclusion is consistent with the study of Schappert and Wirth (2015), who found that a large fraction of those parcels which end up in a banner cloud have a rather complex and fully three-dimensional path history, flowing around the mountain at rather low altitudes. We also showed that banner clouds require the mountain to be steep and the stratification to be weak such that = Nh0/U is significantly smaller than 1. Hence, banner clouds are restricted to a fairly small part of the parameter space lying within the regime of leeside separation. It also implies that flow separation from the surface is only a necessary but not a sufficient condition for banner cloud formation. We also showed through additional sets of simulations that and are, indeed, the relevant nondimensional parameters in the present context.

It is informative to distinguish hydrostatic from nonhydrostatic flow. Previously, nonhydrostaticity has been diagnosed as NL/U/ ≪ 1 (e.g., Baines 1995). The dashed straight lines in Fig. 1 are contours of constant /. In our simulations, the condition / ≲ 1 is satisfied only in that part of the parameter space that is represented in the top-left panels in the respective figures. Since this part of the parameter space coincides with significant potential for banner cloud formation (see Fig. 13a), we conclude that banner clouds are likely to be associated with nonhydrostatic flow.

We discussed two different physical mechanisms that have previously been invoked to explain flow separation from the surface and lee vortex formation: a viscous one associated with surface friction and an inviscid one based on baroclinicity. Our simulations indicate that banner clouds are inevitably associated with boundary layer separation due to surface friction. By contrast, the baroclinic mechanism requires strong stratification, which suppresses vertical uplift and is, therefore, not conducive to banner cloud formation. The fact that friction is required to produce banner-cloud-like conditions was clearly recognized earlier in a conference contribution by Geerts (1992); however, this work seems not to have led to a journal publication that would have allowed one to obtain a more detailed understanding for the line of argument, including the important distinction between upwelling (w > 0) and Lagrangian uplift (Δz > 0) in the lee of the mountain.

Considerations of numerical stability restricted our simulations to aspect ratios ≤ 0.5. This is somewhat less than the value for truly steep, but relevant mountains like, for example, Mount Matterhorn, which we estimate to be somewhat larger than 1. However, we expect that our results extend well into the region ≥ 1. Using a different model configuration, where the mountain was represented through immersed boundaries (Mittal and Iacarino 2005; Smolarkiewicz et al. 2007) instead of terrain-following coordinates, we were able to extend well beyond the value 1. These simulations showed a very similar windward–leeward asymmetry of vertical uplift as in our Fig. 12a. Yet, for the present work we preferred the terrain-following coordinates, because this allows better control over the amount of surface friction, which was essential for our analysis.

How well are our results supported by available observations? First, we have shown that steep mountains are favorable for banner cloud formation. This is consistent with the general experience that banner clouds typically occur at steep mountains such as Matterhorn, Mount Zugspitze, Fitz Roy, or Mount Everest. Second, we have shown that weak stratification is favorable for banner cloud formation. This is consistent with the observations of Wirth et al. (2012); their measurements indicate that during two specific events at Mount Zugspitze the stratification was weak below the summit of the mountain. Note that for these two events the stratification turned out to be rather strong in the free atmosphere above the mountain. However, this seems to be of little relevance to the phenomenon of banner clouds, as we could show through additional simulations (not presented in this paper).

Quite deliberately we use in our simulations an idealized model setup with a smooth mountain and with inflow profiles characterized by constant wind and constant N. This allows us to explore generic regime behavior, but at the same time it excludes more subtle phenomena that require vertical variations of wind and stratification (e.g., Sheridan and Vosper 2006). We are currently investigating different mountain shapes and inflow profiles and find some nonnegligible sensitivities. For instance, a mountain with salient edges is almost inevitably associated with flow separation (Scorer 1955), but the flow may quickly reattach and the magnitude of the uplift is small—except when is large and is small. We also note that the pattern of the uplift in our Fig. 12a does not agree very well with prime samples of observed banner clouds. In the latter, the leeside cloud extends only up to the summit of the mountain and the windward side is completely cloud free. We believe that the somewhat unrealistic shape of our plume of large 〈Δz〉 is due to the smoothness of our idealized orography, which contrasts with the salient ridges of banner cloud–prone mountains like the Matterhorn or Mount Zugspitze. Ongoing simulations suggest that by adjusting the mountain shape and the inflow profile it is easy to produce a pattern of uplift that is in much better agreement with a real banner cloud. However, it was not the goal of the present study to investigate a scenario that is as realistic as possible.

In summary, our results (Figs. 12 and 13a) clearly indicate that there is a general tendency for uplift to be larger on the windward side than on the leeward side except for steep mountains and weak stratification. This is consistent with the intuitive notion that generally a mountain forces the air to rise during the approach, leading to positive vertical parcel displacements and, possibly, cloud formation on the windward side. In this sense the formation of a banner cloud on the leeward side is the exception rather than the rule, and this is possibly why it almost invariably captures the attention of the observer.

Acknowledgments

We sincerely thank the anonymous reviewers for their insightful comments and suggestions, which led to significant enhancements during revision. We thank P. Smolarkiewiocz for his expert advice on numerical modeling and S. Serafin for fruitful discussions on regime behavior. The second author gratefully acknowledges discussions with R. Rotunno about the subtleties of flow past mountains. Computational resources were made available by Deutsches Klimarechenzentrum through support from the Bundesministerium für Bildung und Forschung, as well as by the Zentrum für Datenverarbeitung at the Johannes Gutenberg University Mainz. This work was supported by the German Research Foundation Grant WI 1685/11-1 as well as the Center for Computational Sciences Mainz.

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