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  • View in gallery

    Schematic illustration of the model setup, implying an inflow–outflow configuration.

  • View in gallery

    Profiles specified at the inflow boundary: (a) zonal wind (m s−1), (b) potential temperature (K), and (c) moisture tracer (g kg−1).

  • View in gallery

    Time-mean variables in a vertical section through the center of the pyramid: (a) perturbation pressure (Pa), (b) vertical displacement (m), (c) streamwise component of the wind (m s−1), and (d) relative humidity (%). The arrows in all panels represent the (u, w) component of the time-mean wind.

  • View in gallery

    Hand-drawn schematic illustrating some key features of three-dimensional flow past a pyramid with a frictional boundary layer. The lines depict streamlines of the time-mean flow, and the shaded thick tube indicates the location and shape of the leeside bow vortex. The flow features shown here are believed to be generic as long as the atmosphere is close to neutrally stratified below the top of the pyramid and the inflow wind speed is nearly constant except in a shallow layer close to the surface.

  • View in gallery

    Schematic phase diagram for water substance illustrating the mixing-cloud mechanism. The solid line represents equilibrium vapor pressure as a function of temperature. Points A and B denote two unsaturated air parcels with different temperatures and vapor pressures. When the two parcels mix, the resulting mixture lies somewhere along the dashed straight line. The mixture in this case can become supersaturated (C), although the original two air parcels (A and B) were unsaturated. Points D and E and the associated mixing line represent a scenario that is consistent with our current model setup.

  • View in gallery

    Mixing diagnostics in a vertical section through the center of the pyramid. (a) Time-mean turbulent kinetic energy (m2 s−2), which is the sum of the parameterized subgrid-scale part and the explicitly resolved part. The arrows represent the time-mean (u, w) component of the flow. (b) Effective source (colors, m s−1) for the tracer χ, together with the xz component of the eddy flux (arrows).

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Mechanisms of Banner Cloud Formation

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Abstract

Banner clouds are clouds in the lee of steep mountains or sharp ridges. Their formation has previously been hypothesized as due to three different mechanisms: (i) vertical uplift in a lee vortex (which has a horizontal axis), (ii) adiabatic expansion along quasi-horizontal trajectories (the so-called Bernoulli effect), and (iii) a mixing cloud (i.e., condensation through mixing of two unsaturated air masses).

In the present work, these hypotheses are tested and quantitatively evaluated against each other by means of large-eddy simulation. The model setup is chosen such as to represent idealized but prototypical conditions for banner cloud formation. In this setup the lee-vortex mechanism is clearly the dominant mechanism for banner cloud formation. An essential aspect is the pronounced windward–leeward asymmetry in the Lagrangian vertical uplift with a plume of large positive values in the immediate lee of the mountain; this allows the region in the lee to tap moister air from closer to the surface. By comparison, the horizontal pressure perturbation is more than two orders of magnitude smaller than the pressure drop along a trajectory in the rising branch of the lee vortex; the “Bernoulli mechanism” is, therefore, very unlikely to be a primary mechanism. Banner clouds are unlikely to be “mixing clouds” in the strict sense of their definition. However, turbulent mixing may lead to small but nonnegligible moistening of parcels along time-mean trajectories; although not of primary importance, the latter may be considered as a modifying factor to existing banner clouds.

Corresponding author address: Dr. Volkmar Wirth, Institute for Atmospheric Physics, University of Mainz, Becherweg 21, 55099 Mainz, Germany. E-mail: vwirth@uni-mainz.de

Abstract

Banner clouds are clouds in the lee of steep mountains or sharp ridges. Their formation has previously been hypothesized as due to three different mechanisms: (i) vertical uplift in a lee vortex (which has a horizontal axis), (ii) adiabatic expansion along quasi-horizontal trajectories (the so-called Bernoulli effect), and (iii) a mixing cloud (i.e., condensation through mixing of two unsaturated air masses).

In the present work, these hypotheses are tested and quantitatively evaluated against each other by means of large-eddy simulation. The model setup is chosen such as to represent idealized but prototypical conditions for banner cloud formation. In this setup the lee-vortex mechanism is clearly the dominant mechanism for banner cloud formation. An essential aspect is the pronounced windward–leeward asymmetry in the Lagrangian vertical uplift with a plume of large positive values in the immediate lee of the mountain; this allows the region in the lee to tap moister air from closer to the surface. By comparison, the horizontal pressure perturbation is more than two orders of magnitude smaller than the pressure drop along a trajectory in the rising branch of the lee vortex; the “Bernoulli mechanism” is, therefore, very unlikely to be a primary mechanism. Banner clouds are unlikely to be “mixing clouds” in the strict sense of their definition. However, turbulent mixing may lead to small but nonnegligible moistening of parcels along time-mean trajectories; although not of primary importance, the latter may be considered as a modifying factor to existing banner clouds.

Corresponding author address: Dr. Volkmar Wirth, Institute for Atmospheric Physics, University of Mainz, Becherweg 21, 55099 Mainz, Germany. E-mail: vwirth@uni-mainz.de

1. Introduction

Banner clouds are cloud plumes that extend downwind of steep mountains even on otherwise cloud-free days (Glickman 2000). Spectacular examples can be observed at Matterhorn in the Swiss Alps or Mount Zugspitze in the Bavarian Alps. Based on a set of time-lapse movies at Mount Zugspitze, Schween et al. (2007) provided a comprehensive definition of what should (and what should not) be considered a banner cloud. According to their analysis, a banner cloud must simultaneously satisfy four criteria: 1) the cloud should be in a fixed relation to the mountain and occur only on its leeward side, 2) the cloud should not be composed of snow crystals blown off the mountain by the wind, 3) the cloud should be persistent (i.e., live significantly longer than the time it takes for an air parcel to travel the horizontal extent of the cloud), and 4) the cloud should not be primarily of convective character. The time-lapse movies were later combined with additional in situ measurements, resulting in a comprehensive observational study of banner clouds at Mount Zugspitze (Wirth et al. 2012).

For many decades authors have speculated about the key physical ingredients of a banner cloud. Essentially, three hypotheses have emerged, explaining banner clouds as due to a lee vortex (Hann 1896; Douglas 1928; Hindman and Wick 1990; Geerts 1992a,b), as due to the Bernoulli effect (Humphreys 1920; Grant 1944; Huschke 1959; Beer 1974), or as a mixing cloud (Humphreys 1920). The first of the three hypotheses is based on the observation that often there seems to exist a vortex with a horizontal axis in the immediate lee of the mountain top, leading to upwelling close to the leeward face of the mountain. The studies of Schween et al. (2007) and Wirth et al. (2012) favor the lee-vortex hypothesis, but their conclusions are based entirely on observations. The numerical simulations of Reinert and Wirth (2009) indicate that a lee vortex on a pyramidal-shaped mountain can be considered as a viable mechanism of banner cloud formation, but they did not deal with the other two mechanisms in their paper.

It is the goal of the present study to systematically evaluate the three proposed mechanisms and quantitatively compare them with each other. We shall do this by means of numerical simulation in an idealized model setup (cf. Reinert and Wirth 2009). As we shall see, all three mechanisms may contribute toward banner cloud formation, but the lee-vortex mechanism turns out to be by far the most important.

The paper is organized as follows. The numerical model and the model setup are introduced in section 2. Details about model diagnostics are given in section 3. The results from the simulations are presented in section 4, and a short summary with our conclusions will be given in section 5.

2. Numerical model and model setup

We use the anelastic version of the Eulerian/semi-Lagrangian (EULAG) numerical model1 (Prusa et al. 2008) to simulate three-dimensional flow of dry air past idealized orography in a nonrotating atmosphere. The model is run in large-eddy simulation mode with a parameterization of subgrid-scale fluxes of momentum and heat through a turbulent kinetic energy (TKE) closure (Smolarkiewicz and Margolin 1997). Model variables are the Cartesian components of the three-dimensional wind u = (u, υ, w), perturbation pressure p*, and perturbation potential temperature θ*. Perturbations are defined as deviations from the inflow conditions, which are user specified (see below) and which are chosen to be hydrostatic and depend on altitude z only. The continuity equation · (ρ0u) = 0 involves a reference density ρ0(z). Orography is implemented with the help of the immersed boundaries (IB) method (Mittal and Iacarino 2005; Smolarkiewicz et al. 2007). The model code allows one to distribute the computation to multiple processors through domain decomposition. In the current work we used 256 processors.

In addition to the standard model variables we consider two quasi-conserved tracers χ and q. They have a specified profile at the inflow boundary and evolve according to /Dt = Mχ and Dq/Dt = Mq, respectively, where D/Dt denotes the material derivative and Mχ and Mq represent the sources due to mixing from the TKE closure. The use of these tracers will be explained below. Temperature is related to potential temperature through T = θ(p/p0)κ, where p is 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 model setup is illustrated in Fig. 1. Any point in space is represented by its Cartesian coordinates x = (x, y, z), where x is taken for the streamwise direction, y for the spanwise direction, and z for the vertical direction. The size of the model domain is 8 km in the x direction, 4.8 km in the y direction, and approximately 3 km in the z direction. The grid spacing in both horizontal directions is Δx = Δy = 25 m. In the vertical we use a stretched coordinate with the grid spacing smoothly increasing from Δz ≈ 10 m at the bottom of the domain to Δz ≈ 100 m at the top of the domain. Our orography is a square-based pyramid of height H = 997 m and width L = 930 m at the surface, corresponding to a slope angle α = 65°; the pyramid is located 3 km downstream of the inflow boundary and centered in the y direction. Note that our plots below show only the inner part of the model domain.

Fig. 1.
Fig. 1.

Schematic illustration of the model setup, implying an inflow–outflow configuration.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-12-0353.1

Periodic conditions are specified at the spanwise boundaries, open conditions at the outflow boundary, and a rigid lid at the upper boundary. There is a sponge layer in the upper part of the domain (z ≥ 2 km); in that layer the wind is being relaxed toward the specified inflow profile. In the flat portion of the lower boundary we use a drag law , where τ is the surface stress, ρ is the air density, and CD = 0.01 is a drag parameter.

At the inflow boundary (x = −3000 m) we specify profiles of wind, temperature, and tracers. The wind is prescribed as u = (u, 0, 0) with
e1
and k = 0.4, u* = 1.01 m s−1, z0 = 5.1 m, and u0 = 9 m s−1 (see Fig. 2a). This profile is close to what one would obtain if an initially constant wind profile with u0 = 9 m s−1 blows over a flat surface with CD = 0.01 for a few hours. Potential temperature is specified such that the stratification is neutral (∂θ/∂z = 0) up to a height of z = 1 km, and stably stratified above (with ∂θ/∂z = 4 K km−1; see Fig. 2b). The profiles of χ and q are specified as
e2
e3
where q0 = 7 g kg−1. The tracer χ is a heightlike tracer used to estimate the altitude of air parcels at the time when they entered the model domain. The tracer q (Fig. 2c) is meant to represent an idealized profile of specific humidity [see Wirth et al. (2012), their Figs. 10a and 12a, for observed profiles of specific humidity]. Note that, in contrast with potential temperature, specific humidity is not constant with altitude. It implies that our inflow conditions do not represent a well-mixed boundary layer. This is consistent with the work of Reinert (2009) and Reinert and Wirth (2009), arguing that deviations from a well-mixed boundary layer are essential for the occurrence of banner clouds.
Fig. 2.
Fig. 2.

Profiles specified at the inflow boundary: (a) zonal wind (m s−1), (b) potential temperature (K), and (c) moisture tracer (g kg−1).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-12-0353.1

Although the tracer q provides a method to diagnose humidity throughout the domain, we do not simulate the flow of moist air. In particular, we do not represent the effect of latent heat release from condensation of water vapor. It was shown by Reinert and Wirth (2009) that the latter affects the size of the banner cloud, but the associated changes can be considered as small without consequences for the overall flow geometry and the basic mechanism. For the benefit of straightforward interpretation we, therefore, stick to flow of dry air in this study.

Our simulations were carried through for a duration of 80 min. The first 20 min are needed to let the transients decay and the flow reach a statistically stationary state; during the remaining 60 min we accumulated flow statistics. In particular, time averages for all model variables ψ at times tn > ts = 20 min were computed recursively as follows (e.g., Fröhlich 2006):
e4
where the overbar denotes the time average, the index (n and n + 1, respectively) denotes the time step, α = Δt/(tnts), and Δt is the time increment.

3. Model diagnostics

a. Perturbation pressure

As mentioned earlier, perturbation pressure is defined as p* = ppe, where pe(z) is the pressure profile specified at the inflow boundary. The time-mean p* is shown in Fig. 3a. Although we use the anelastic version of the EULAG model, the diagnosed values of p* can be expected to be a good approximation to the actual pressure perturbation, because we are dealing here with low–Mach number flows (e.g., Smolarkiewicz and Szmelter 2009).

Fig. 3.
Fig. 3.

Time-mean variables in a vertical section through the center of the pyramid: (a) perturbation pressure (Pa), (b) vertical displacement (m), (c) streamwise component of the wind (m s−1), and (d) relative humidity (%). The arrows in all panels represent the (u, w) component of the time-mean wind.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-12-0353.1

b. Vertical displacement

A key variable in our analysis is the Lagrangian vertical displacement Δz that a parcel has experienced since the time it entered the model domain. Following Reinert and Wirth (2009) we estimate Δz by computing
e5
Figure 3b shows the time-mean Δz for our model setup. As noted by Reinert and Wirth (2009), there is a pronounced windward–leeward asymmetry with a plume of large uplift in the immediate lee of the mountain.

c. Relative humidity

Relative humidity is diagnosed from the specific humidity tracer q as follows. First, saturation vapor pressure es is computed using Bolton's approximation
e6
where e0 = 6.112 hPa and is temperature (°C) (Bolton 1980; Emanuel 1994). Relative humidity (RH) is then obtained from
e7
where e is the actual vapor pressure, p denotes pressure, and ε = 0.622 is the ratio between the molecular mass of water and dry air. Grid boxes with RH > 100% are considered cloudy; all other grid boxes are considered cloud free. Figure 3d shows time-mean relative humidity in our model setup.

d. Turbulent fluxes

The evolution of a materially conserved tracer ψ follows
e8
In flux form this can be rewritten as
e9
We partition the model variables according to
e10
where Φ is either u or ψ, the overbar denotes the time average, and the prime denotes the deviation therefrom. Inserting this into (9) and forming the time average, we obtain
e11
or, equivalently,
e12
where . Comparison of (12) with (8) shows that the time mean tracer follows a similar equation regarding the time-mean flow as the original variable ψ regarding the full flow u except that there is a source term Seff.

Our tracers χ and q are not strictly conserved owing to the parameterized subgrid turbulence. This gives rise to an additional term on the right-hand sides of (8) and (12). However, in our simulation the TKE from (parameterized) subgrid-scale motions is less than 10% of the TKE from explicitly resolved motions throughout the domain except very close to the surface (no figure shown)—consistent with the basic idea of a large-eddy simulation. We, therefore, consider the term Seff to quantify the effect of mixing regarding our two tracers. It is computed explicitly from model output for ψ′ and u′ using (4).

4. Results from the analysis

The three mechanisms of banner cloud formation mentioned in the introduction can be classified into two categories, corresponding to two distinct causes for the condensation of water vapor: (i) mixing between two different air masses and (ii) adiabatic expansion of air parcels and the associated temperature decrease. In the former case, condensation essentially results from the nonlinearity of the increase of saturation water vapor with temperature [e.g., section 6.8 in Bohren and Albrecht (1998)], which can occur isobarically. In the latter case, condensation goes back to the fact that for adiabatic expansion the decrease in es(T) exceeds the decrease in p such that RH in the expression (7) increases [e.g., section 6.9 in Bohren and Albrecht (1998)].

In the following, we analyze all three mechanisms and address the question: to what extent may one expect the formation of a banner cloud from each of them?

a. Uplift in a lee vortex

It has been shown that flow past a pyramid is associated with complex flow geometry involving, among others, a bow-shaped vortex in the lee of the pyramid (Martinuzzi and AbuOmar 2003; Reinert and Wirth 2009). Figure 4 serves for illustration. Close to the bottom boundary, the air is forced to go around the pyramid, resulting in two counterrotating vortices with vertical axis in the immediate lee of the obstacle. On the other hand, in an xz section through the center of the pyramid, one obtains a vortex with horizontal axis close to the summit of the pyramid; this vortex is associated with pronounced upwelling close to the leeward face of the mountain. As indicated in Fig. 4, the three vortices mentioned above should actually be considered as part of a single three-dimensional vortex being bent in a bow shape. This vortex is referred to as a “bow vortex” in the following (cf. Larousse et al. 1993). Note that Fig. 4 must be taken with a grain of salt, since there are at least two important aspects not represented in the schematic: first, the flow is turbulent and second, the bow vortex need not be stationary (i.e., there may be vortex shedding with a certain quasi periodicity).

Fig. 4.
Fig. 4.

Hand-drawn schematic illustrating some key features of three-dimensional flow past a pyramid with a frictional boundary layer. The lines depict streamlines of the time-mean flow, and the shaded thick tube indicates the location and shape of the leeside bow vortex. The flow features shown here are believed to be generic as long as the atmosphere is close to neutrally stratified below the top of the pyramid and the inflow wind speed is nearly constant except in a shallow layer close to the surface.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-12-0353.1

Vertical motion in the neighborhood of the pyramid is associated with a pronounced windward–leeward asymmetry of vertical uplift, with a plume of large values of Δz (maximum values in excess of 500 m) attached to the immediate lee of the mountain (Fig. 3b). At the same time, Fig. 3d indicates a “cloud” (i.e., a region with RH > 100%) in the lee of the mountain, which is practically coincident with the plume of large values of Δz.

The close spatial correlation between Δz and RH in Fig. 3 suggests that vertical uplift and its leeward–windward asymmetry is responsible for banner cloud formation—at least indirectly. For a more sound interpretation, we refer to the expression (7) for relative humidity. Rising parcels experience both a pressure reduction and an associated adiabatic temperature decrease; as already mentioned, the latter dominates such that relative humidity increases. The key question in our context is how this effect leads to supersaturation in the lee, while at the same time the windward side remains unsaturated.

Let us, to this end, compare two sites close to the mountain at the same altitude level, one on the windward side and one on the leeward side, respectively. As will be shown below, the pressure is almost equal at both sites. In addition, given that our model atmosphere is neutrally stratified below the summit, vertical motion does not affect local temperatures such that the temperature is approximately the same at both sides. Inspecting (7), it transpires that the only factor that can be significantly different is the value of q: the parcel on the leeward side originated at a lower level and carries, therefore, a larger amount of specific humidity, because in our inflow profile, q increases downward. In other words, the larger vertical uplift on the leeward side allows the flow to tap moister air from farther below.

b. Bernoulli effect

In the immediate neighborhood of a mountain peak, one typically observes wind speeds larger than elsewhere, because the presence of the mountain decreases the cross section available for the air to pass. This seems to suggest that pressure reduction through the so-called Bernoulli effect may play a role for banner cloud formation.

To clearly separate the discussion of the Bernoulli effect from the discussion of the lee-vortex mechanism, we start with Bernoulli's theorem for inviscid adiabatic flow of an ideal gas in a gravitational field, stating that the quantity
e13
is constant along streamlines for stationary flow (g denotes the acceleration due to gravity). Note that θ is constant along a streamline, because the flow is assumed to be adiabatic. It follows that changes in pressure must be associated with changes in either altitude or wind speed squared (or both) in order for B to remain constant. In the former case one obtains (for small changes), which is nothing but the hydrostatic relation, and the related pressure drop from vertical uplift has been discussed in the previous subsection. However, this is not what is usually meant in this context. Instead, previous authors (quoted in the introduction) implicitly alluded to the pressure reduction associated with the wind speed increase due to the presence of the mountain. In terms of (13) this corresponds to a compensation between the first and the third term on the right-hand side or, equivalently, assuming strictly horizontal flow.

We, therefore, assume (somewhat hypothetically) that the flow is strictly horizontal, such that along a trajectory. This allows us to estimate the order of magnitude of the effect. Assuming that the presence of the mountain increases the wind speed from 9 to 11 m s−1 (cf. Fig. 3c), and using ρ = 1 kg m−3, we obtain Δp = −ρΔ|u2|/2 = 20 Pa.

Obviously, this estimate involves a number of assumptions and approximations. To obtain a more realistic estimate we draw on our numerical simulation and analyze p* from the model output (Fig. 3a). Apparently, there are positive values of p* on the windward side and negative values of p* on the leeward side. This general behavior is similar to what is being observed for flow around surface mounted bluff bodies (e.g., Castro and Robins 1977). There is no clear correlation between large wind speed and low pressure perturbation as one would expect from a naive application of (13); this is mostly due to the fact that on both sides of the pyramid, p* has a substantial component perpendicular to the trajectory such that the pressure force changes direction rather than speed. Interestingly, the clear windward–leeward asymmetry in p* with a distinct negative anomaly in the lee favors banner cloud formation. Assuming, again, that the parcel goes around the mountain along a strictly horizontal trajectory, its value of q is the same on both sides of the mountain. Thus, the adiabatic pressure increase and concomitant temperature increase when approaching the windward side of the pyramid decreases relative humidity and, therefore, favors cloud-free conditions. The opposite (pressure decrease, temperature decrease, and increase of relative humidity) happens as the parcel reaches the leeward side of the pyramid, thus favoring cloud formation there.

The magnitude of the pressure reduction in the lee in Fig. 3a is on the order of 15 Pa, which is consistent with our previous order of magnitude estimate. This value must be put into perspective by comparing it with a typical pressure reduction experienced by a parcel in the rising branch of the lee vortex. As shown above, Δz in the lee can readily exceed 500 m. Using the hydrostatic equation, a pressure drop of Δp = 15 Pa can be related to a corresponding uplift Δz through , where we used g ≈ 10 m s−2 and ρ ≈ 1 kg m−3. It follows that the adiabatic expansion due to the so-called Bernoulli effect is more than two orders of magnitude smaller than that due to the lee-vortex effect.

c. Mixing cloud

A strict definition of “mixing cloud” is given in Glickman (2000). As a prerequisite there must be two air parcels with different trajectories which come into close proximity to each other. In particular, they arrive at the same pressure level, but they are both still unsaturated before they mix. Then the mixing process sets in, which is assumed to be isobaric and adiabatic. Owing to the nonlinearity of the Clausius–Clapeyron equation the mixed air can become supersaturated; that is, a cloud can form. The process is illustrated in Fig. 5.

Fig. 5.
Fig. 5.

Schematic phase diagram for water substance illustrating the mixing-cloud mechanism. The solid line represents equilibrium vapor pressure as a function of temperature. Points A and B denote two unsaturated air parcels with different temperatures and vapor pressures. When the two parcels mix, the resulting mixture lies somewhere along the dashed straight line. The mixture in this case can become supersaturated (C), although the original two air parcels (A and B) were unsaturated. Points D and E and the associated mixing line represent a scenario that is consistent with our current model setup.

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-12-0353.1

Apparently, a mixing cloud cannot occur unless the warmer air mass is moister (in terms of specific humidity or vapor pressure) than the colder air mass (line A–C–B in Fig. 5). However, warmer air can never be moister in our model setup (cf. line D–E in Fig. 5). This can be seen as follows. Since both θ and q are materially conserved, the correlation between both throughout the domain is given by their correlation at the inflow boundary. There, we specified the profiles of θ and q such that air with higher values of θ cannot be associated with higher values of q (see Figs. 2b and 2c). Since mixing occurs isobarically, potential temperature essentially reflects temperature. It follows that mixing of two distinct unsaturated air parcels can never lead to supersaturation in our model setup. To the extent that our inflow profiles are generic, the foregoing argument renders it unlikely that banner clouds are mixing clouds in the strict sense of the definition.

d. The role of mixing

Yet, there is another point of view on mixing and its role for cloud formation. If we define the mean or resolved flow through the time-averaged flow (cf. section 3d), anything that is associated with the turbulent (transient) eddies can be considered as mixing. It is conceivable that a hypothetical parcel following the mean flow could remain unsaturated everywhere, while the instantaneous flow field (i.e., including the turbulent components) might become saturated at some points in space. In this case one might legitimately say that turbulent mixing leads to cloud formation. Note that this process is distinct from the mixing-cloud mechanism discussed earlier, because here cloud formation occurs through adiabatic cooling in the rising branch of a turbulent eddy rather than through mixing of two unsaturated air masses.

As a quantitative measure for turbulent mixing we show turbulent kinetic energy in Fig. 6a. There is strong mixing in the immediate lee of the summit owing to the strong shear and boundary layer separation in this region. Interestingly, the mixing is significantly weaker in the lee below the summit, which is where the mean trajectories experience their mean uplift.

Fig. 6.
Fig. 6.

Mixing diagnostics in a vertical section through the center of the pyramid. (a) Time-mean turbulent kinetic energy (m2 s−2), which is the sum of the parameterized subgrid-scale part and the explicitly resolved part. The arrows represent the time-mean (u, w) component of the flow. (b) Effective source (colors, m s−1) for the tracer χ, together with the xz component of the eddy flux (arrows).

Citation: Journal of the Atmospheric Sciences 70, 11; 10.1175/JAS-D-12-0353.1

According to (7), relative humidity on a given pressure level depends on specific humidity and temperature. At the inflow boundary, both are functions of altitude only. Therefore, the effect of mixing can be estimated through the “effective source” Seff associated with the heightlike tracer χ. The advantage of this approach is that this renders the analysis independent of the moisture profile specified at the inflow boundary. The result is shown in Fig. 6b. There is a distinct dipole-shaped pattern in the lee of the mountain at a downwind distance of a few hundred meters. The upper lobe of the dipole has negative values, which means that mixing reduces the values of ; in other words, the mean parcel acquires tracer values from farther below. Thus, to the extent that air below is moister, the negative values of Seff indicate moistening through turbulent mixing.

The units of Seff are the same as those of /Dt (m s−1), suggesting the following interpretation: in areas of negative Seff the values of are being modified by turbulent mixing to an extent which is equivalent to a mean rising motion with vertical velocity Seff. This allows us to quantify the mixing effect and put it into perspective with the effect from uplift in the mean flow. According to Fig. 6b, a typical scale of Seff is 1 m s−1. Assuming that the parcel travels for a streamwise distance of 1 km through the mixing region with a speed of 10 m s−1 (cf. Fig. 3c), this corresponds to a time scale τ = 100 s and, thus, to an effective net vertical displacement of τ × Seff ~ 100 s × 1 m s ~ 100 m. This needs to be compared with the uplift in the lee vortex discussed earlier, which is on the order of 500 m. It follows that the lifting in the lee vortex is substantially (factor of 5) larger than the “effective lifting” through turbulent mixing. It is, therefore, unlikely that turbulent mixing is the primary process of cloud formation, although in particular cases it may just tip the scale and lead to cloud formation for an otherwise cloud-free mean flow.

There is another argument indicating that turbulent mixing alone is unlikely to be the primary mechanism for banner cloud formation. The shape of the blue region in Fig. 6b is qualitatively different from the shape of a typical banner cloud. A banner cloud usually has its deepest vertical extent right at the mountain, while it gets thinner as one moves downwind away from the mountain [see our Fig. 3d and the discussion of Schween et al. (2007)]. In distinct contrast, the shape of the upper lobe of the dipole pattern of Seff is quite different: it is not directly attached to the mountain but starts only about 100 m downstream, reaching maximum values some 500 m away from the mountain. In addition, as a parcel travels downwind through the effective moisture source, the effect of mixing accumulates with time such that its impact should be felt stronger farther downstream than closer to the mountain. The corresponding cloud would be strongest with deepest vertical extent farther downstream rather than closer to the mountain. This is just the opposite of what we consider to be a typical banner cloud.

Nevertheless, our quantitative analysis shows that, unlike the Bernoulli effect, mixing is not completely negligible. One may, thus, expect that turbulent mixing affects an existing banner cloud (that has formed through the lee-vortex mechanism). In particular, comparing the location of the dipole in Fig. 6b with the expected cloud location from Fig. 3d indicates that turbulent mixing moistens the upper part of the cloud and dries the lower part of the cloud. One may expect a moderate modification of the shape of the cloud from this mechanism.

It is also interesting to note that both diagnostics in Fig. 6 indicate relative minor impact of mixing on the leeward side close to the mountain, where the rising branch of the lee vortex is located (cf. the arrows in Fig. 3). This implies that the large moisture values of the air close to the surface, which flows around the mountain and is subsequently lifted in its immediate lee, are hardly reduced by mixing. Thus, the lack of mixing in this part of the flow actually favors banner cloud formation.

5. Summary and conclusions

In the past, different mechanisms have been suggested to explain the formation of orographic banner clouds. This paper reports on numerical simulations with a large-eddy simulation model with the aim to compare and quantitatively evaluate the different mechanisms. Our model setup considers flow of dry air past an idealized pyramid-shaped mountain. The inflowing air is neutrally stratified up to the summit of the pyramid, stably stratified above, and has decreasing specific humidity with altitude. Using various types of diagnostics we obtained the following results.

  • As was previously known, the three-dimensional geometry of the flow in the neighborhood of the pyramid is rather complex, with a bow-shaped vortex in the lee of the pyramid. This is associated with a striking windward–leeward asymmetry in the vertical uplift with particularly large values in the immediate lee just beneath the summit of the mountain. A cloud may thus form as a consequence of expansion and adiabatic cooling in the rising part of the lee vortex. By contrast, the chance for cloud formation on the windward side is smaller because of the smaller vertical uplift; to the extent that specific humidity increases downward, banner cloud formation is due to the fact that the larger leeward uplift is able to tap moister air from farther below. This mechanism—the so-called lee-vortex mechanism—has been examined before in some detail by Reinert and Wirth (2009).
  • There is a pronounced windward–leeward asymmetry in pressure perturbation with a positive anomaly on the windward side and a negative anomaly on the leeward side of the mountain. At first sight this suggests a role for banner cloud formation, because anomalously low pressure on the leeward side favors cloud formation (and vice versa on the windward side). However, both a rough estimate using Bernoulli's theorem and the examination of the perturbation pressure in the model output reveal that this mechanism is quantitatively much too weak: the pressure drop along a quasi-horizontal trajectory would be more than two orders of magnitude smaller than the pressure drop along a trajectory in the rising branch of a lee vortex (see previous item). Hence, for all practical purposes the Bernoulli mechanism cannot be considered to be relevant for banner cloud formation.This conclusion is consistent with the observations of Wirth et al. (2012), who found that banner cloud occurrence at Mount Zugspitze is independent of wind speed. If the Bernoulli mechanisms were of primary importance, one would expect banner clouds to occur preferentially for strong wind conditions, but apparently this is contrary to observational evidence.
  • The mixing-cloud mechanism implies that adiabatic isobaric mixing of two subsaturated air masses may produce supersaturation owing to the nonlinear dependence of saturation vapor pressure on temperature. In our model setup, potential temperature increases upward and specific humidity increases downward, rendering the above mechanism impossible. Our inflow profiles were motivated by the observations of Wirth et al. (2012); to the extent that they are generic, one can argue that the mixing-cloud mechanism is not a primary mechanism for banner cloud formation. Of course this does not exclude the viability of this mechanism in specific rare situations.
  • We also investigated the role of mixing in the sense that we quantified the effect of turbulent eddies on the time-mean flow. Moistening from turbulent mixing is quantitatively less important (by almost an order of magnitude) than moistening from uplift in the time-mean lee vortex. In addition, the shape of the mixing region is not compatible with the shape of a typical banner cloud. This suggests that turbulent mixing alone is unlikely to be the primary mechanism for banner clouds, but it may well modify an existing banner cloud. Relative lack of mixing in the rising branch of the lee vortex is somewhat favorable for banner cloud formation.
Our model setup, albeit idealized, is believed to be prototypical for the occurrence of banner clouds. A preliminary sensitivity analysis (Reinert 2009) indicates that the overall character of the flow does not sensitively depend on the inflow wind and temperature profiles, as long as the atmosphere is close to neutrally stratified below the mountain summit and the wind profile is nearly constant except in a shallow layer close to the bottom boundary. A more systematic analysis is planned for future work.

In our analysis we tried to separate the different mechanisms as clearly as possible. Of course, in reality all mechanisms can be expected to coexist. However, our results strongly suggest that the lee-vortex mechanism dominates in most cases and can, hence, be considered to be the basic mechanism for banner cloud formation.

Acknowledgments

We gratefully acknowledge funding from the German Research Foundation (Grant WI 1685/5-4) as well as the Center of Computational Sciences at the University of Mainz. Björn Brötz helped to implement the EULAG model setup. Most of this paper was written while one of the authors, V. Wirth, visited the University of Bergen (Norway) and the National Center for Atmospheric Research (Boulder, Colorado); he wishes to thank the wonderful hospitality of his hosts, Jochen Reuder (Bergen) and Pjotr Smolarkiewicz (Boulder), as well as the NCAR visiting scientist program. We thank the reviewers for numerous constructive comments, which led to a significant improvement of the presentation.

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We switched from the model of Reinert and Wirth (2009) to the EULAG model because the former was designed to run on a single processor only, while the latter scales well on parallel computer architectures.

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