The Impact of Large-Scale Winds on Boundary Layer Structure, Thermally Driven Flows, and Exchange Processes over Mountainous Terrain

Jan Weinkaemmerer aInstitute for Atmospheric and Environmental Sciences, Goethe University, Frankfurt/Main, Germany

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Ivan Bašták Ďurán aInstitute for Atmospheric and Environmental Sciences, Goethe University, Frankfurt/Main, Germany
bEuropean Centre for Medium-Range Weather Forecasts, Reading, United Kingdom

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Jürg Schmidli aInstitute for Atmospheric and Environmental Sciences, Goethe University, Frankfurt/Main, Germany

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Abstract

The vertical heat and moisture exchange in the convective boundary layer over mountainous terrain is investigated using large-eddy simulation. Both turbulent and advective transport mechanisms are evaluated over the simple orography of a quasi-two-dimensional, periodic valley with prescribed surface fluxes. For the analysis, the flow is decomposed into a local turbulent part, a local mean circulation, and a large-scale part. It is found that thermal upslope winds are important for the moisture export from the valley to the mountain tops. Even a relatively shallow orography, possibly unresolved in existing numerical weather prediction models, modifies the domain-averaged moisture and temperature profiles. An analysis of the turbulent kinetic energy and turbulent heat and moisture flux budgets shows that the thermal circulation significantly contributes to the vertical transport. This transport depends on the horizontal heterogeneity of the transported variable. Therefore, the thermal circulation has a stronger impact on the moisture budget and a weaker impact on the temperature budget. If an upper-level wind is present, it interacts with the thermal circulation. This weakens the vertical transport of moisture and thus reduces its export out of the valley. The heat transport is less affected by the upper-level wind because of its weaker dependence on the thermal circulation. These findings were corroborated in a more realistic experiment simulating the full diurnal cycle using radiation forcing and an interactive land surface model.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jan Weinkaemmerer, weinkaemmerer@iau.uni-frankfurt.de

Abstract

The vertical heat and moisture exchange in the convective boundary layer over mountainous terrain is investigated using large-eddy simulation. Both turbulent and advective transport mechanisms are evaluated over the simple orography of a quasi-two-dimensional, periodic valley with prescribed surface fluxes. For the analysis, the flow is decomposed into a local turbulent part, a local mean circulation, and a large-scale part. It is found that thermal upslope winds are important for the moisture export from the valley to the mountain tops. Even a relatively shallow orography, possibly unresolved in existing numerical weather prediction models, modifies the domain-averaged moisture and temperature profiles. An analysis of the turbulent kinetic energy and turbulent heat and moisture flux budgets shows that the thermal circulation significantly contributes to the vertical transport. This transport depends on the horizontal heterogeneity of the transported variable. Therefore, the thermal circulation has a stronger impact on the moisture budget and a weaker impact on the temperature budget. If an upper-level wind is present, it interacts with the thermal circulation. This weakens the vertical transport of moisture and thus reduces its export out of the valley. The heat transport is less affected by the upper-level wind because of its weaker dependence on the thermal circulation. These findings were corroborated in a more realistic experiment simulating the full diurnal cycle using radiation forcing and an interactive land surface model.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jan Weinkaemmerer, weinkaemmerer@iau.uni-frankfurt.de

1. Introduction

Several publications point out the impact of thermally driven slope flows on the exchange of heat, mass, and momentum in the daytime convective boundary layer over mountainous terrain (Rotach et al. 2014; Serafin et al. 2018; Lehner and Rotach 2018; Gohm et al. 2009). Anabatic upslope winds are generated by horizontal temperature contrasts between the valley sidewalls and the valley center resulting in positive buoyancy over the slopes (Whiteman 2000). Apart from modifying the vertical profiles of temperature and moisture, thermally driven slope winds also control the transport and dispersion of pollutants over complex terrain (Gohm et al. 2009; Lehner and Gohm 2010; Giovannini et al. 2020). Together with other baroclinic circulations forming the mountain wind system, they are present especially in fair-weather conditions with weak synoptic forcing. In contrast to the larger-scale along-valley breezes, which are generated by hydrostatic pressure gradients resulting from temperature differences along the valley axis (Schmidli et al. 2009), the smaller-scale slope flows respond directly to the surface sensible heat flux and are generally turbulent (Serafin and Zardi 2010).

From a ridge-scale perspective, the slope–wind system appears as a cross-valley circulation consisting of an upslope branch, a partial recirculation to the valley center, and compensating subsidence in the stable core of the valley (Vergeiner and Dreiseitl 1987). Depending on the valley geometry, we encounter two or more vertically stacked circulation cells (Wagner et al. 2015). The resulting temperature profile often exhibits a three-layer thermal structure (Schmidli and Rotunno 2010; Schmidli 2013) with a valley mixed layer, a stable valley inversion, and an upper weakly stable layer followed by the free atmosphere.

Compared to the atmosphere over flat terrain, the valley atmosphere heats up more strongly with the same energy input due to the so-called valley-volume effect (Neininger 1982; Neininger and Liechti 1984; Schmidli and Rotunno 2010). This concept explains the higher heating rates by the reduced volume of air inside a valley. In flat terrain, a larger volume of air has to be heated up over the same area by the same solar energy input. The valley-volume effect can also be quantified in terms of the area-height distribution of a valley (Steinacker 1984). Using idealized large-eddy simulations (LESs), Schmidli (2013) has shown that the high diurnal temperature amplitudes in valleys can be explained to a large extent by the valley-volume effect. From a detailed heat-budget analysis, it is clear that advective cooling by the slope winds and heating by subsidence over the valley mixed layer combined with overall turbulent heating lead to a relatively homogeneous heating rate inside the valley. The total heating in the valley is reduced by a net export of heat out of the valley by overshooting thermals over the ridge tops.

Complementary to that work, other authors have concentrated on detailed analyses of tracer mass fluxes in idealized setups (Leukauf et al. 2016) or variable valley geometries with three-dimensional configurations of a valley with an adjacent plain (Wagner et al. 2015). Depending on stratification and surface heating, pollution can either be trapped inside a valley or released into the free atmosphere. This contrasting behavior is related to the internal energy necessary to break up the valley inversion. One notable result is that the tracer mass transport from the surface to the free atmosphere is 3–4 times more effective over valleys than over flat terrain (Wagner et al. 2015). In addition, vertical transport processes are found to be the strongest for deep and narrow valleys.

Another aspect, which is still subject to discussion, is the impact of thermally induced winds on moist convection in complex terrain. Different to mechanically forced orographic convection, the process understanding is still poor and even the influence of basic terrain scales on thermally forced orographic convection remains uncertain (Kirshbaum et al. 2018). While upslope flows may trigger convection by transporting water vapor from the valley floor to the ridge tops inducing moisture convergence (Banta 1990), their cooling effect may also cause the contrary. Demko et al. (2009) argue that the advection of colder air along the slopes of a single mountain reduces the convective available potential energy (CAPE) thus inhibiting the initiation of convection. In contrast, Panosetti et al. (2016) come to the result that mass convergence and weaker convective inhibition (CIN) above mountains increase the probability of triggering convection. This is primarily due to the higher elevation of mountainous terrain in combination with water-vapor advection by upslope winds. Thus, convection initiation over mountain ridges may be enhanced or suppressed by thermal circulations depending on the given conditions (Schmidli 2013).

From the point of view of numerical weather prediction (NWP), thermally driven winds lie in a so-called gray zone where certain processes in the NWP models are neither completely subgrid nor fully resolved (Chow et al. 2019). Gray-zone effects are especially critical in complex terrain where turbulence and convection parameterizations need to consider the terrain influence even though the orography may not be accurately represented due to coarse grid resolutions or numerical limitations over steep terrain. Over a poorly resolved valley orography, neither the typical vertical temperature profile nor the slope–wind circulation can be reproduced properly (Wagner et al. 2014). Errors in the predicted boundary layer depth and the vertical exchange of heat and mass are the consequence. Apart from the gray-zone problem, already the processes in a convective boundary layer (CBL) in a fairly idealized setup over a quasi-two-dimensional valley-ridge orography involve three different scales (Lehner and Rotach 2018). First, small-scale turbulence dominating in the horizontally inhomogeneous mixed layer, second, the thermal cross-valley circulation, and third, an upper-level, geostrophic wind, which can be treated as the large-scale flow given by the synoptic conditions.

The overall goal of our study is to gain insight into the interaction between these scales, with a special focus on the role of an upper-level wind on the vertical transport and exchange processes. This aspect has not yet received much attention in the literature. Catalano and Moeng (2010) have shown second-order turbulent moments and turbulent kinetic energy (TKE) budgets for a similar case, but only locally at different positions along the slope and not in presence of a large-scale background wind. Panosetti et al. (2016) have found that the convective updraft is shifted downstream from the summit in presence of a background wind; however, no general correlation to the updraft strength or the formation of precipitation has been determined.

With regard to a possible future parameterization development, we are not only interested in the local effects of the involved processes, but also in their impact on averaged quantities. Therefore, LES is used over a simple orography to analyze the vertical heat and moisture transport in more detail. We decompose the flow into three components: A turbulent part, a thermal circulation, and a large-scale flow in order to study their interactions. The influence of the upper-level wind on the vertical transport is analyzed. Because the main goal of boundary layer models is the parameterization of heat and moisture fluxes, we take a close look at the budgets of these fluxes.

This paper is structured as follows: In the next section, the flow decomposition into turbulent part, thermal circulation, and large-scale flow, as well as the horizontal averaging method are introduced. Besides, the budget equations for TKE, turbulent sensible, and latent heat flux are presented. The experimental setup and the numerical model are presented in section 3. In sections 4 to 6, the simulation results are discussed. Conclusions are given in section 7.

2. Analysis methods

a. Flow decomposition

With the aim to distinguish between local turbulent fluctuations, the thermal circulation, and the large-scale background flow, the flow is decomposed into these three components with the help of corresponding averaging operators. A sinusoidal valley flanked by two ridges is oriented along the meridional or y direction of the domain. Thus, the slope–wind circulation is supposed to be bound to the xz plane forming a cross-valley circulation. Both in zonal and meridional direction, periodic boundary conditions are imposed, imitating an infinitely long valley. The symmetry of this quasi-two-dimensional setup used in this study simplifies the decomposition of the flow.

First, the turbulent fluctuations are separated from the mean flow by a Reynolds decomposition of the turbulent flow variable a(x, y, z, t). Note that z denotes the height above the valley floor. The required ensemble (Reynolds) average is approximated by an average in time and in the along-valley direction:
a¯(x,z,t)=1TLytTt0Lya(x,y,z,t)dydt,
where x, y, and z are the eastward, northward, and vertical directions, respectively; and t is time. An averaging time step of 1 min is used. The time-averaging interval T is set to 40 min. The same averaging period has been used in previous idealized studies of the convective boundary layer over orography (Catalano and Moeng 2010; Schmidli 2013) as a compromise between accuracy and stationarity. Ly equals the meridional domain length. Thus, the flow is decomposed into an ensemble mean part a¯ and a local turbulent part a′:
a(x,y,z,t)=a¯(x,z,t)+a(x,y,z,t).
For the sake of clarity, subgrid-scale fluctuations not resolved in the LES are not written out here and are treated as being included in the turbulent part.
The mean flow can be further decomposed into a local cross-valley circulation component and a large-scale background flow. This is achieved by averaging a¯ between two crests along the x axis on horizontal levels of constant height above the valley floor:
a¯(z,t)=1Lx0Lxa¯(x,z,t)dx,
where Lx represents the zonal domain length equal to the valley width. In case of terrain-following coordinates, this requires interpolation. The further decomposition results in
a¯(x,z,t)=a¯(z,t)+a¯c(x,z,t),
where the 〈 〉 operator denotes the horizontal domain average and a¯c is the mean local circulation, which is the deviation of the ensemble mean a¯ from the large-scale background flow a¯. For simplicity, this large-scale area mean a¯ is abbreviated with aB. The complete decomposition of the flow into the three components can be represented as follows:
a=a¯+a=aB+a¯c+a.
Following the Reynolds averaging rules, the ensemble mean of a product is given by
ab¯=(a¯+a)(b¯+b)¯=a¯b¯+ab¯.
The domain-averaged ensemble mean of a product is then
ab¯=(aB+a¯c)(bB+b¯c)+ab¯=aBbB+a¯cb¯c+ab¯.
Therefore, the total vertical transport of a quantity c can be expressed as follows:
wc¯total flux=wBcBlarge-scale flow+w¯cc¯cmean circulationmean advective flux+wc¯turbulent flux,
where w is the vertical velocity. In a periodic setup with no large-scale horizontal gradients and no large-scale advection, the turbulent and mean-circulation fluxes are the only contributions to the total (vertical) transport in the domain.
Note that the second-order turbulent fluxes consist of a resolved and a subgrid-scale part:
ab¯=ab¯|res+ab¯|sgs.
For the analysis, the subgrid parts are computed via a K-gradient ansatz analogous to Wagner et al. (2014). This reads, e.g., for a vertical scalar flux:
wc¯|sgs=Kcz,
where the vertical diffusion coefficient for scalars K is determined by the LES subgrid turbulence model.
Higher-order turbulent moments can be calculated in the same way with the help of second-order moments:
abc¯=abc¯a¯b¯c¯a¯bc¯b¯ac¯c¯ab¯.

b. Budgets for heat and mass

The aim of this study is to analyze the net vertical transport in the valley. For this, the evolution of temperature and tracer mass needs to be quantified with the help of their budget equations. Because the vertical transport over complex terrain involves different scales, we want to analyze transport contributions separated into turbulent motion, local circulation and large-scale flow. Thus, a decomposition of the variables needs to be used when deriving their budget equations. We start from the conservation equations for momentum, heat, and moisture written in the Boussinesq approximation (Wyngaard 2010):
uit+ujuixj=δi3gθ˜υθυ,01ρ0p˜xi,
θt+ujθxj=Sθ,
qt+ujqxj=Sq,
where ui = {u, υ, w} are the wind components in zonal, meridional, and vertical directions; θ is the potential temperature; q is the specific humidity; θ˜υ and p˜ are the deviations of the virtual potential temperature and the deviation of the pressure from a constant reference state; θυ,0 is the constant reference virtual temperature, ρ0 is the constant reference density; and S stands for any kind of source or sink term, e.g., originating from microphysical processes. Einstein summation convention is used. The flow is assumed to be incompressible and we have neglected rotation, viscosity, and molecular diffusivity. Inserting (2) into (13) and (14) and applying the averaging operators from (1) and (3) leads to the domain averaged ensemble-mean tendencies:
θ¯t=u¯jθ¯xjujθxj+Sθ¯,
q¯t=u¯jq¯xjCIRCujqxjTURB+Sq¯MP.
The first term on the rhs is the advective transport by the mean flow. In absence of large-scale gradients, it reduces to a local transport by the mean circulation (CIRC). The second and third term is the turbulent transport term (TURB) and a source or sink term by the microphysics (MP), respectively. In the LES analysis, the ensemble means and the sum of the above transport terms are computed directly from the model’s prognostic equations during the model run by a method described in section 3b. The cross-valley averages are calculated during postprocessing. No additional approximation is applied to the model equations.

c. Budgets for TKE and turbulent fluxes

Analogously, the budget equations for the turbulence kinetic energy (TKE) and both the turbulent sensible and latent heat flux can be derived from the conservation equations. Different from the traditional derivation (e.g., Stull 1988), additional terms emerge due to the three-component decomposition. Some budget terms are zero after horizontal averaging, thanks to the symmetry of the setup. The resulting budget equations resemble those for the boundary layer over flat terrain, except of additional terms representing the inter-scale transfer between turbulence and the local circulation:
12ui2¯t=gθυ,0wθυ¯Iuiw¯uB,izIIuiuj¯u¯c,ixj+u¯c,j12ui2¯xjIII12wui2¯zIV1ρ0wp¯zVεVI,
wθ¯t=gθυ,0θθυ¯Iww¯θBzIIujθ¯w¯cxj+wuj¯θ¯cxj+u¯c,jwθ¯xjIIIwwθ¯zIV+1ρ0pθz¯V2εwθVI,
where j = 1, 3 and u¯c,2=0, as there is no mean circulation in the along-valley direction. The scale interaction terms in (17) are marked as III. The two components in III represent the gradient production and advection due to the local circulation. The remaining terms on the rhs of (17) are the traditional TKE terms: Term I is the buoyant production/consumption term, II the mean-gradient production term, IV the vertical turbulent transport term, V the pressure correlation term, and VI the dissipation term. The situation in (18) is analogous with VI being the molecular destruction term. A budget equation for the domain-averaged turbulent moisture flux wq¯ looks similar to (18).

Technically, the dissipation term consists of the dissipation rate given by the subgrid turbulence model and the implicit diffusion arising from an odd-ordered advection scheme (Brown et al. 2000). In our case, the implicit-diffusion term can be received from the model’s diagnostics. While the total resolved transport term (TRT = II + III + IV) and the individual terms II and IV are calculated directly from the model data, the interaction term III is obtained as the difference: TRT − II − IV. It is not decomposed any further.

d. Boundary layer structure

To study the boundary layer structure, we start from the bulk perspective. Classically, the daytime atmospheric boundary layer over flat terrain is subdivided into a diabatic surface layer, followed by a nearly neutral, turbulent mixed layer (ML), topped by a stable entrainment zone representing the transition between the ML and the free atmosphere (Stull 1988). The top of the entrainment zone equals the height of the atmospheric boundary layer (ABL). We adopt this classification for our complex-terrain experiments and, following Schmidli (2013), define the ML height as the height where the vertical gradient of the along-valley averaged potential temperature first reaches a critical value of γc = 0.001 K m−1, marking the lower boundary of the capping inversion. The ABL height is diagnosed by a simple maximum-gradient method. It is computed from the local instantaneous height where the vertical θ gradient reaches its maximum. To obtain the x-dependent ABL height, the instantaneous local heights are averaged in time and along the valley axis according to (1).

3. Numerical model simulations

a. Experimental setup

To simplify our budget analysis, the flow is investigated over a two-dimensional sinusoidal valley (Schmidli 2013). The orography corresponds to an infinitely long, periodic valley and is described by
zs(x)=h2(1cos2πxW),
where zs is the surface height, h the ridge height from valley floor to crest, and W the width of the valley from ridge to ridge. The height h is set to 1500 m and the width W to 20.48 km which is a relatively narrow and moderately deep valley. The maximum inclination of the slope is 13.0°.

To study the influence of the upper-level wind on the flow within the valley, experiments with different upper-level wind velocities are performed. In the reference experiment (REF), the background atmosphere is at rest. The remaining setup of the experiments, unless stated otherwise, is as follows: The initial atmosphere is in hydrostatic balance, the pressure is 1000 hPa at z = 0, and a linear and stable potential-temperature profile is prescribed by θ(z) = θs + Γz with Γ = 0.003 K m−1 and θs = 297 K at the valley floor. A constant relative humidity of 40% is imposed leading to a water-vapor mixing ratio decreasing with height. The surface forcing is constant in space and time in order to come closer to a steady state during the simulation. The surface sensible heat and moisture flux is prescribed: (wθ¯)s=0.12kms1 and (wq¯)s=0.05gkg1ms1. In the reference case, no clouds are formed before 4 h 40 min and the ensemble-averaged cloud water mixing ratio does not exceed 9 × 10−6 g kg−1 during the rest of the simulation. Thus, water vapor can be approximately regarded as a passive tracer. Additionally, a simulation with exactly the same setup, but without orography is carried out for comparison (FLAT).

In the sensitivity experiments, the large-scale flow is initialized according to
u¯(z)={0,zh,zhΔhumax,h<zh+Δh,umax,z>h+Δh,
where u¯(z) is perpendicular to the valley axis, Δh = 400 m, and umax is the value of the upper-level wind. The sensitivity experiments are named U2, U4, and U8, having umax = 2, 4, and 8 m s−1, respectively. The zero wind below crest height avoids an immediate excitement of gravity waves and can be justified by interpreting (20) as a geostrophic wind independent of ABL processes. To facilitate the onset of convective turbulence, the initial atmosphere is altered by a random temperature perturbation with a maximum amplitude of 0.5 K.

The domain sizes for the basic experiments are listed in Table 1. The total duration of the simulations is 6 h. We lay a special focus on the situation after 4 h assuming it to be representative for typical midday conditions (Schmidli 2013). In section 6, the findings obtained with these idealized settings are compared against a case with spatially varying and time-dependent surface fluxes including radiation and an interactive land surface model. This extended case is integrated over one diurnal cycle.

Table 1

List of basic experiments where umax denotes the maximum background wind speed.

Table 1

b. Numerical model

The numerical simulations have been performed using CM1 (Cloud Model 1, Bryan and Fritsch 2002; Bryan 2016) in LES mode. CM1 is a nonhydrostatic, fully compressible numerical model with terrain-following σz coordinates. For this study, fifth-order horizontal and vertical advection using a WENO scheme for scalars (Jiang and Shu 1996) and the Klemp–Wilhelmson vertically implicit time splitting scheme is chosen. The subgrid-scale turbulence closure is a 1.5-order TKE scheme similar to Deardorff (1980). The horizontal grid resolution is set to 40 m. The lateral boundary conditions are periodic in both directions in order to mimic a vast mountainous terrain with a single terrain wavelength. Grid stretching is applied in the vertical starting from 8 m vertical grid spacing near the surface. Increasing with height, it reaches 40 m at 2.8 km height and is kept constant above. The domain top is a rigid lid and its height is chosen depending on the initial upper-level wind speed (see Table 1). To suppress spurious gravity wave reflections, a wave-damping sponge layer is employed in all cases. The setup of the wave-damping layer was adapted for each case accordingly because the vertical wavelength of the dominant gravity wave modes was found to be roughly proportional to the background wind speed. The damping time for the Rayleigh damping layer is 300 s for all cases. No radiation is included in the basic simulations. As sensible and latent heat fluxes are prescribed at the surface, no complex land surface model is required. Only the momentum fluxes are computed from the exchange coefficient for momentum corresponding to a neutral surface layer and a roughness length of z0 = 0.16 m. For the microphysics, a simple single-moment warm-rain scheme is selected (Kessler). The adaptive time step usually lies between 1 and 2 s.

The starting time of the simulation for the diurnal-cycle case (section 6) is set to 21 March at 0600 UTC. A longitude of 0° and a latitude of 55° is chosen for the NASA Goddard radiation scheme. Thus, the 12 h integration time covers the daytime period. The land use is set to grassland for the surface model (revised MM5 scheme, Jiménez et al. 2012). Although the surface characteristics do not change with the height of the orography, the surface heat fluxes are time and position dependent due to the movement of the sun. All other parameters are the same as for the simulations REF and U4.

An additional module has been implemented into CM1 allowing the online computation of time averages by recursion without the need to write any instant values to the disk (see appendix). Turbulence means are updated recursively during runtime over an averaging period. The result is then further averaged along the y axis before it is written to disk. This online calculation of the statistics reduces the required amount of disk space for the output files and replaces an otherwise expensive and time-intensive postprocessing.

4. Analysis of the reference case

a. Flow structure

Figures 1a–c shows the local circulation and key turbulence quantities for the reference case after 4 h, together with the diagnosed ML and ABL heights. Note that the ML and ABL are deepest at the valley center and over the ridges. This is where the turbulence is most intense (largest TKE values). The turbulent sensible heat flux is positive near the surface, decreases with height, and becomes negative in the upper part of the ML. The turbulent moisture flux is positive almost everywhere in the ABL, with the largest values inside the valley. The local circulation consists of a shallow upslope flow and an upper recirculation with subsidence over the valley center. Around halfway up the slope, an additional horizontal branch appears (horizontal intrusion, Leukauf et al. 2016).

Fig. 1.
Fig. 1.

Valley cross sections of the mean flow at t = 4 h for (a)–(c) REF and (d)–(f) U4, showing the cross-valley wind vectors (at every tenth grid point in x and z direction) and in shading the TKE, the turbulent sensible heat flux, and the turbulent moisture flux (with all quantities including the subgrid contribution). Also shown is (center) the potential temperature (contour interval: 0.5 K) and (right) the specific humidity (contour interval: 1 g kg−1). The dashed line marks the diagnosed ML height and the solid line the diagnosed ABL height.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

The thermally driven local circulation is partly induced by horizontal temperature gradients and consequently acts to weaken them. Cooling by ascent in the slope–wind layer and heating by subsidence in the valley core combined with overall turbulent heating leads to a rather homogeneous temperature distribution. The situation is different for moisture, since it does not have a significant impact on the flow dynamics. In the present context without cloud formation, the specific humidity is conserved under vertical motions and can thus be treated approximately as a passive tracer, except for its influence on the buoyancy term in the momentum equation, (12). In addition to that, the initial specific humidity profile decreasing with height favors a net transport of moisture from the lower part of the valley to the ridges. This leads to a drying in the valley center and an accumulation of moisture above the ridges (see also Kuwagata and Kimura 1997). This process increases the horizontal moisture gradient in and above the valley.

b. Temporal evolution

The temporal evolution of the TKE, the turbulent sensible heat flux, and the moisture flux at the valley center is shown in Fig. 2. Although the moisture-flux gradient leads to a moister valley ML compared to a rather dry free atmosphere, the valley center becomes gradually drier during the simulation. The ABL grows strongly and the temperature profile becomes nearly adiabatic near the end of the simulation. This corresponds to the breakup of the valley inversion.

Fig. 2.
Fig. 2.

As in Fig. 1, but for the temporal evolution of the quantities at the valley center (x = 0).

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

Figure 3 shows the ABL height above ground level at the valley center, the ridge top, and over the slope midway between the valley center and the ridge top. Because of the constant surface fluxes, the ABL continuously grows over the ridges. In contrast, the ABL height at the valley center is only slowly increasing and staying around 1000 m within the first 5 h. Over the slopes, the ABL height also stays almost constant at 500 m in the first 4 h indicating a nearly steady state. As the valley ABL continues to grow, we find a strong increase of the ABL height over the slope after 4 h of integration. With the breakdown of the valley inversion after around 5 h, the ABL height in the valley center also increases rapidly. At the end of the simulation, the ABL both over the valley center and over the slopes is deeper than over the ridge tops. The following analysis focuses on the situation after 4 h of integration, when the flow is in a quasi–steady state.

Fig. 3.
Fig. 3.

Temporal evolution of the ABL height above ground level at the valley center (x = 0), at the slope (x = 5.12 km), and at the ridge top (x = 10.24 km).

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

c. Area-mean profiles

1) Deep valley (1500 m)

To see the impact of the turbulent exchange and the local circulation on the large-scale quantities, the area-mean profiles of wind, potential temperature, and specific humidity are computed by horizontal averaging over the whole domain. These profiles are depicted in Figs. 4a–c.

Fig. 4.
Fig. 4.

Area-mean profiles (height above valley floor) at t = 4 h of the zonal wind u, the potential temperature θ, and the specific humidity q for (a)–(c) the deep valley and (d)–(f) the shallow valley.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

The temperature and moisture profiles in all cases with orography are more structured compared to the FLAT case. This is caused by the cellular character of the flow in the valley (cf. Schmidli 2013). Due to the valley volume effect, the temperature profile is warmer for all cases with orography. The moisture profile is dryer in the ML and moister above the ML because moisture is transported out of the valley by the slope winds and accumulates over the mountain tops.

The difference in the transport of heat and moisture can also be seen from the decomposed vertical heat and moisture fluxes (Figs. 5g–i and 5j–l) and from the budgets of heat and moisture [see section 4d(1)]. Heat is transported mainly by turbulent mixing. The local circulation has a smaller impact on the heat transport, since the temperature distribution evolves to be relatively horizontally homogeneous. In contrast, the local circulation dominates the transport of moisture, as it exports moisture from the valley center to the ridges. For the momentum flux (Figs. 5d–f), however, both the turbulent and the circulation part is zero in the REF case due to the symmetry of the flow in a symmetric valley.

Fig. 5.
Fig. 5.

Area-mean profiles (height above valley floor) at t = 4 h of (left) the turbulent components, (center) the local circulation components, and (right) their sum for (a)–(c) the kinetic energy, (d)–(f) vertical momentum flux, (g)–(i) sensible heat flux, and (j)–(l) moisture flux. The turbulent part includes the subgrid contribution.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

The intensity of turbulent mixing represented by the TKE (Fig. 5a) shows an additional secondary maximum compared to the FLAT case caused by increased turbulence in strong convective updrafts over the mountain ridges (cf. Wagner et al. 2014). Secondary maxima can also be observed in the turbulent heat and moisture flux profiles.

Compared to the FLAT case, the kinetic energy in the valley is contained not only in the large-scale flow and in the turbulent motions, but also in the local circulation (CKE). The distribution of the kinetic energy between the turbulence and the local circulation explains the lower values of TKE in the cases with orography. The CKE (Fig. 5b) is largest at altitudes where the circulation has mainly horizontal components. Thus, the two peaks in the CKE profile are related to the upper horizontal branches of the local circulation.

2) Shallow valley (500 m)

So far, we have only looked at the processes in a CBL over a deep valley where the ridge height is similar to or larger than the ABL height. However, shallow to moderately deep valleys can be expected to be even more frequent around the world. We repeat the REF and U4 simulations for a valley of only 500 m depth which is completely embedded in the ABL after a few hours of simulation. The result is shown in Figs. 4d–f and Fig. 6 after 4 h of simulation.

Fig. 6.
Fig. 6.

As in Fig. 5, but for the shallow-valley variants (h = 500 m) of the REF and the U4 case; without the kinetic energy and vertical momentum-flux plots.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

One can see in Figs. 4d–f that the heating and drying effect extends over the whole ML and is not limited to the valley atmosphere. Moistening occurs in a shallow zone above 1.5 km. Below crest height, the specific humidity decreases almost linearly down to the valley bottom. The profile of the turbulent sensible heat flux in Fig. 6a resembles that over flat terrain, but shows slightly higher values above the mountain tops. At 500 m, a second heat-flux maximum becomes evident. The circulation-induced heat flux in Fig. 6b is weak. All in all, the mean CBL structure over complex-terrain with low ridge heights is qualitatively similar to that over flat terrain. Still, the impact of the valley-volume effect and the drying of the valley atmosphere by the thermal circulation is obvious even for shallow valleys.

d. Budget analysis

1) Budgets for heat and moisture

Next, we take a closer look at the heat and moisture budgets according to (15) and (16). Figure 7 shows valley cross sections of the local budget terms and Fig. 8 their horizontally averaged profiles, respectively. The contribution from the microphysics is zero, since cloud processes are negligible in our simulations at t = 4 h (not shown). For the FLAT case, Fig. 8 reveals homogeneous heating in the ML and cooling in the entrainment zone. At the same time, there is a slight drying of the ML, while the entrainment zone is strongly moistened. Over orography, the heating rates in the valley ML are approximately 1.5 times higher than in the FLAT case. Both from Figs. 7a–c and Fig. 8 it can be seen that the heating of the lower valley atmosphere is mainly caused by turbulence, while the slope winds mainly have a cooling effect. At higher levels, the air is recirculated to the center of the valley leading to a local warming. In contrast, Figs. 7g–i shows that the moisture is transported out of the lower part of the valley (z0.5km) mainly by the slope winds to a height of 2 km where it accumulates and contributes to possible cloud formation over the ridges. The turbulent moistening in the entrainment zone and the drying effect of the subsiding air almost cancel out in the total tendency.

Fig. 7.
Fig. 7.

Valley cross sections as in Fig. 1, but showing the local budget terms for (a)–(f) heat and (g)–(l) moisture for REF and U4 at t = 4 h. The budgets terms include the total tendency (TOTAL), mean-flow advection (CIRC), and turbulence (TURB), corresponding to the terms in (15) and (16), respectively. The contribution from the microphysics (MP) is zero.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

Fig. 8.
Fig. 8.

Vertical profiles of the heat and moisture budget for FLAT, REF, and U4 at t = 4 h (h = 1500 m). The budgets terms include turbulence (TURB) and mean-flow advection (CIRC) corresponding to the terms in (15) and (16) as well as the total tendency (TOTAL). The contribution from the microphysics (MP) is zero.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

2) Budgets for TKE and turbulent fluxes

Next, the turbulence budgets are investigated, especially in terms of the impact of the local circulation on the turbulent processes. Budgets of TKE, wθ¯, and wq¯ according to (17) and (18) at t = 4 h are shown in Fig. 9. Only the resolved parts are presented because a subgrid contribution was not available for every single budget term. Tests with the TKE budgets have shown that the differences caused by the subgrid-scale terms are negligible above the surface layer (not shown). The net tendency of the budgets is close to zero almost everywhere, this means that the flow is close to a steady state.

Fig. 9.
Fig. 9.

Vertical profiles of the area-mean budgets for TKE and the turbulent fluxes of heat and moisture according to (17) and (18) at t = 4 h for the three cases FLAT, REF, and U4. Note that subgrid contributions are not included.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

Generally, the budgets for our FLAT case agree with the expectations from literature (e.g., Stull 1988). For the TKE budget, the buoyancy term and the dissipation term are the dominant source and sink terms. The turbulent transport profile indicates an export of TKE out of the lower ABL to higher levels. For the turbulent fluxes of heat and moisture, the gradient production is nonzero due to significant vertical temperature and moisture gradients. Turbulent transport and molecular destruction are less important compared to the remaining terms. They reach their highest values near the surface and at the inversion height.

In the REF case, secondary maxima appear in the profiles of the buoyancy and the dissipation term in the TKE budget. Different to the FLAT case, the buoyancy term is positive throughout the ABL. The magnitude of the turbulent transport term is reduced compared to the FLAT case. Instead, the interaction term representing the advection and gradient production by the local circulation acts mainly as a consumption term inside the valley. This indicates that TKE is exported out of the valley by the slope winds resulting in generally lower TKE values inside the valley compared to the FLAT case.

For the turbulent fluxes, the interaction term is mainly positive and opposite to the gradient term, but they are not completely balanced. This means that the local circulation generally transports heat and moisture in the direction of the gradients. Turbulent transport is even less important than in the FLAT case. The turbulent sensible heat flux is essentially characterized by the balance between the buoyancy and the pressure term, the two dominant terms. For the turbulent moisture flux, the buoyancy and gradient terms are dominant, but the interaction term also reaches high values close to the surface and near the ridge height. The shapes of the curves are more complicated.

5. Impact of an upper-level wind

a. Flow structure and gravity waves

In the following, we investigate the impact of an upper-level wind on the flow structure, the exchange processes, and the temperature and moisture distribution. The structure of the local flow is displayed for the reference case and two cases with an upper-level wind in Fig. 10. Two vertically stacked circulation cells can be found over each slope in the REF case. This structure is modified in the U2 and U4 cases by gravity waves which form distinct wave patterns over the valley. It is remarkable that gravity waves are generated even though the initial wind profile has been chosen to avoid orographic waves. This means that the waves emerge from the deflection of the large-scale wind by the local circulation as a sort of convectively generated gravity waves (Clark et al. 1986). Thus, although not directly of orographic origin, the resulting wave pattern shows typical characteristics of a stationary mountain wave. As expected for mountain waves, the vertical wavelength increases with the large-scale wind velocity.

Fig. 10.
Fig. 10.

Valley cross sections showing the local circulation wind vectors as well as the zonal and vertical parts of the cross-valley circulation (u¯c and w¯c, shading) at t = 4 h for REF, U2, and U4. Also showing the isolines of the potential temperature in K. The dashed line marks the diagnosed ML height and the solid line the diagnosed ABL height.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

Regarding the area-mean profiles in Fig. 4a, the large-scale wind profile u¯ is clearly influenced by the local processes in and above the valley. Below about 2 km, the wind speed has decelerated in the U4 and U8 case. In addition, u¯ is even negative at a height around 1 km due to the deformation of the cross-valley circulation by the large-scale wind. The profile of the turbulent vertical momentum flux uw¯ explains this deceleration, showing that momentum from around 1.5 km is transported into the valley via the turbulence. Turbulent mixing dominates the vertical momentum transport between the valley and the free atmosphere, while the circulation momentum flux u¯cw¯c is nonzero mainly in the free atmosphere. However, this is caused by gravity waves which are accounted for as local circulation, as they have the same spatial scales as the thermal circulation. For U8, that gravity waves’ momentum flux is smaller than for U2 and U4 at 4 h, but exceeds them at later times (not shown). The lower vertical group speed of the gravity waves for higher ambient wind speeds could be the cause for this effect. The deceleration of the large-scale flow can also be observed over the shallow valley (Fig. 4d).

The CKE profiles in Fig. 5b are oscillating with height due to the occurrence of gravity waves, especially above crest height. Although the local circulation is modified in structure, Fig. 10 shows that the circulation wind velocities are not generally reduced by the upper-level wind. Thus, the impact of the upper-level wind on the vertical heat and mass transport cannot be explained by a generally weaker thermal circulation.

Looking at the budgets for the second-order moments in Fig. 9, we find that the upper-level wind does not strongly change the character of the TKE budget. The gradient production by the large-scale flow contributes only little to the TKE budget.

b. Impact on temperature and moisture distribution

It is remarkable that the upper-level wind has only a minor influence on the θ¯ profile (Figs. 4b and 4e). Although the local circulation is modified by the large-scale flow, the temperature distribution is largely unaffected by this change. In contrast, there is a significant impact of the upper-level wind on the area-mean profile of the specific humidity q¯. For stronger winds, less moisture seems to be exported out of the valley. The q¯ values near the valley bottom are slightly higher which is most obvious for U8. The cross sections in Figs. 1d–f reveal that the large moisture accumulations above the ridges disappear for the U4 case and the moisture distribution becomes more horizontally uniform, at least above crest level. The turbulent heat flux does not change strongly, while the turbulent moisture flux becomes asymmetric with respect to the valley center and reaches higher values around crest level compared to the REF case.

Correspondingly, the profiles for the sensible heat flux in Figs. 5g–i do not change strongly for different upper-level wind velocities, especially within the valley. In fact, the change in the total sensible heat flux is only visible around a height of 1.5 km. The circulation part of the heat flux w¯cθ¯c shows slightly stronger deviations from the REF case, especially in its upper parts. However, due to its smaller magnitude, it has only a small impact on the resulting temperature profile. In contrast, the vertical moisture flux is very sensitive to the upper-level wind (Figs. 5j–l). For U8, the peak in w¯cq¯c is diminished almost to half of its REF value. Also the profile of the turbulent component is strongly altered. While w¯cq¯c plays the dominant role in the REF case, both parts attain similar importance for stronger background winds.

For the shallow valley, the impact of the upper-level wind on the mean moisture profile is limited to a reduced drying of the valley atmosphere (Fig. 4f). However, the turbulent and the circulation part of the moisture flux are almost interchanged between REF and U4 (Figs. 6d–f). Regardless, the total moisture flux remains relatively unchanged.

The relatively small impact of the upper-level wind on the heat transport is also seen in the budgets (Figs. 7a–f) where the differences between REF and U4 are rather small. Also the total heating rate in the profiles of Fig. 8 remains almost unchanged. In contrast, the moisture tendencies in Figs. 7g–l change very distinctively with an upper-level wind. With a horizontally more uniform moisture distribution, the local circulation leads to positive moisture tendencies in the slope–wind layer around crest height and negative tendencies above. Although the turbulent exchange counteracts the circulation tendencies, the total amount of exported moisture is significantly reduced, at least in the case of the deep valley. This is seen also in Fig. 8 where there is no longer a negative peak between 1 and 1.5 km in the circulation term for moisture. The circulation term even becomes positive there. As a result, also the total moisture tendency in the upper part of the valley between 1 and 1.5 km is now positive. Compared to the REF case, the positive peak above 2 km is now mainly caused by turbulent transport. It is, however, significantly smaller.

For the second-order moments (Fig. 9), the biggest differences between REF and U4 are, as expected, in the moisture-flux budget, specifically at crest height where the gradient production becomes positive. Generally, the buoyancy and the gradient production term turn out to be smaller than in the REF case.

In summary, an upper-level wind affects mostly the vertical moisture transport, while the resulting temperature profile remains almost unchanged. In this setup, this is facilitated by the initial moisture profile decreasing with height. Moisture export out of the valley is mostly achieved by the upslope winds and the stationary updrafts above the ridge tops. The turbulent moisture flux leads to a moistening of the slope–wind layer below crest height. Consequently, the slope–wind layer remains rather moist compared to the relatively dry valley core. A horizontal upper-level wind changes the transport patterns by making the moisture distribution more horizontally uniform. In the end, this reduces the moisture export out of the valley. In contrast, the thermal circulation acts to weaken horizontal temperature gradients. The resulting, horizontally more homogeneous temperature distribution is much less affected by an upper-level wind.

6. Bulk heat and mass transport

To compare the impact of the upper-level wind on the vertical exchange of heat and mass with other factors, more simulations were performed. Specifically, sensitivity studies were conducted for the initial temperature lapse rate γ and the valley width W. The present comparison focuses on the total sensible heat and moisture fluxes out of the valley atmosphere, that is the time-integrated total flux across a horizontal surface at z = 1.5 km. The fluxes, normalized by the total time-integrated surface fluxes, are shown in Fig. 11. The intention is to depict the vertical net flux out of the valley (ratio of export). However, the accumulated fluxes are also influenced by a small amount of direct heating by the surface fluxes at the ridge tops. Generally, there is a much smaller net export of heat than of moisture. While the upslope winds have a local cooling effect and the valley is heated quite homogeneously, the moisture is exported out of the valley by the slope–wind circulation. After 6 h, the fraction of moisture exported is 50%–150%, while for heat the fraction is only 5%–15%.

Fig. 11.
Fig. 11.

Time series of the time-integrated total sensible heat and moisture flux at crest height (1500 m, sum of turbulent and local-circulation part) normalized by the total time-integrated surface fluxes for the different cases where either the initial lapse rate γ (green), the valley width W (blue), or the upper-level wind (orange) has been varied.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

For both the heat and the moisture flux, the background stratification seems to have the strongest impact, especially when the initial lapse rate is below the value of the REF simulation (0.003 K m−1). Even a slight reduction to 0.002 K m−1 leads to the onset of cloud formation at t = 4 h and a considerable increase of the heat export related to an early break-up of the valley inversion (Leukauf et al. 2016). In contrast, when the initial lapse rate is larger than the REF value, there is almost no decrease of the heat export ratio before 5 h. For the moisture export, the effect of a weaker stratification is similar, while a stronger stratification results in a significant decrease of the export ratio.

As varying only the valley width does not affect the area–height distribution of the terrain, the valley atmosphere receives the same heat input per surface area. Thus, the heating of the valley should not be affected. However, wider valleys seem to facilitate the sensible-heat export, especially after the first hours of simulation. The result for moisture is contrary to that for sensible heat, although there seems to be a compensating effect at the end of the simulation. Here, it is difficult to go into more detail without more simulations and longer integration times.

In comparison to a variation of the background stratification, the upper-level wind has a smaller impact. Nevertheless, even a weak wind of 2 m s−1 leads to a 30% reduction of total moisture transport. The export of sensible heat seems to be enhanced at least for the cases U2 and U4, questioning the nearly unmodified θ¯ profiles (Fig. 4b). However, this observation, as well as the effect of varying the valley width, is put into perspective by the generally small magnitude of the heat export ratio. It has only a slight impact on the temperature profile.

Finally, the results concerning the upper-level wind are validated in a more realistic simulation forced by a diurnal cycle of solar radiation. The results for the accumulated fluxes at crest height shown in Fig. 12 agree with the above findings: An upper-level wind can lead to a significant reduction of the total moisture export, while the heat transport from valleys to the free atmosphere is less affected.

Fig. 12.
Fig. 12.

As in Fig. 11, but during a diurnal cycle induced by a homogeneous radiative forcing.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-21-0195.1

7. Conclusions

In this study, the impact of a large-scale upper-level wind on the daytime vertical heat and mass transfer over mountainous terrain has been investigated using idealized large-eddy simulations. A key feature of this work is the decomposition of the flow into a small-scale turbulent part, a local mean circulation, and a large-scale background flow as well as the distinction between turbulent and advective transport by means of heat and moisture budget analysis. The test cases include simulations over a periodic, quasi-two-dimensional valley of different heights and widths and with different upper-level wind velocities.

The major findings are summarized as follows:

  • The thermally driven upslope wind and the horizontal and subsiding recirculation form a mean cross-valley circulation which can be clearly separated from small-scale turbulence and the large-scale flow. Its main effect is to export moisture out of the valley through the slope–wind layer and to accumulate moisture over the mountain tops. This builds up a horizontal moisture gradient between the slopes and the valley center. In contrast, the temperature distribution is more or less horizontally homogeneous, as the local circulation is directly driven by temperature gradients and acts to reduce them. The temperature distribution is characterized by enhanced heating rates inside the valley due to the valley-volume effect. Heating by the mean circulation is mainly provided in the elevated branch of the circulation.

  • From a horizontally averaged perspective, moisture is mainly transported by the slope flows and turbulent exchange plays a minor role for the vertical transfer of moisture. In contrast, turbulent mixing is the dominant process for the heating of the mixed layer. The thermal circulation is mainly balancing horizontal temperature gradients and contributes less to vertical heat exchange. A comparison of the turbulent-flux budgets between a flat and a mountainous-terrain boundary layer shows that the gradient-production terms are reduced because of additional advection terms arising due to the thermal circulation. In case of the turbulent kinetic energy budget, these additional terms act as a sink of TKE in the lower parts of the valley.

  • An upper-level wind has a minor impact on the temperature profile, as the temperature distribution is already fairly horizontally homogeneous. The additional shear generated by the large-scale flow does not significantly increase turbulent mixing, at least in the current setup. Instead, it strongly affects the moisture distribution with its strong horizontal gradient between the rather dry valley center and the moister slope region. In consequence, an upper-level wind leads to a reduced vertical transfer of moisture even though the strength of the slope flow is not significantly diminished. Thus, it can strongly reduce the export of moisture out of a valley and into the free atmosphere. Similar results were found in a more realistic setup with variable surface fluxes determined by an interactive land surface model driven by solar radiation.

In summary, this study contributes to an improved process understanding of heat and mass exchange over mountainous terrain. The situation is quite different from the widely described flat convective boundary layer where heat and mass are mainly transported by turbulent motions. Over mountainous terrain, thermally induced winds interact with turbulence and the large-scale flow. This situation has been studied by comparing the evolution and distribution of temperature and moisture under the influence of an upper-level wind. The strong impact of the upper-level wind on the vertical mass transport can be an important factor for cloud formation and tracer dispersion over mountains. The results may not only be relevant for a better insight into the physical processes, but also for future parameterization development. Although the simulations are fairly idealized, we expect that the main findings are relevant also in more complex situations.

It is clear that it would be more difficult to apply the analysis in a more complex three-dimensional orography. Without a symmetry axis of the terrain, the ensemble mean would have to be calculated either by temporal averaging only or by temporal averaging combined with box- or section-wise spatial averaging only over parts of the valley. It has to be kept in mind that other symmetry effects, like the disappearing turbulent momentum flux in the cross-valley average, will not arise with a more realistic orography. Furthermore, along-valley winds occurring in three-dimensional terrain would interact with the slope–wind circulation. Another factor would be cloud formation over the ridges which may result in precipitation. That would change the heat and moisture balance between the valley and the free atmosphere significantly. Additionally, three-dimensional radiative effects, like the shading of a valley by a ridge or a mountain peak and different land surface characteristics, could also be considered. The investigation of these more complex aspects of thermally driven flows over mountainous terrain is left for future research.

Acknowledgments.

This research was funded by Hans Ertel Centre for Weather Research of DWD (3rd phase, The Atmospheric Boundary Layer in Numerical Weather Prediction), Grant 4818DWDP4. We thank the anonymous reviewers for their valuable suggestions that improved the manuscript considerably.

Data availability statement.

The configuration files, outputs and visualization scripts for the simulations are openly available in Zenodo at https://doi.org/10.5281/zenodo.7097422. All figures were generated with the Python matplotlib package (Hunter 2007).

APPENDIX

Online Calculation of LES Statistics

To compute the LES statistics in a postprocessing step, it would be necessary to write out the complete 3D data fields in 1-min intervals. To reduce the demand of disk space for model output, an online time averaging has been implemented into CM1, meaning that the time averaging is done during the model run. The temporal average atav [a part of the ensemble mean in (1)] is calculated by recursion at the end of each 1-min interval as follows:
atavn=atavn1+1n(aatavn1).
Here, n is the current summation step. An output is only necessary at the end of every time period T (in our case T = 40 min).

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  • Whiteman, C. D., 2000: Mountain Meteorology: Fundamentals and Applications. Oxford University Press, 376 pp.

  • Wyngaard, J. C., 2010: Turbulence in the Atmosphere. 1st ed. Cambridge University Press, 406 pp.

  • Fig. 1.

    Valley cross sections of the mean flow at t = 4 h for (a)–(c) REF and (d)–(f) U4, showing the cross-valley wind vectors (at every tenth grid point in x and z direction) and in shading the TKE, the turbulent sensible heat flux, and the turbulent moisture flux (with all quantities including the subgrid contribution). Also shown is (center) the potential temperature (contour interval: 0.5 K) and (right) the specific humidity (contour interval: 1 g kg−1). The dashed line marks the diagnosed ML height and the solid line the diagnosed ABL height.

  • Fig. 2.

    As in Fig. 1, but for the temporal evolution of the quantities at the valley center (x = 0).

  • Fig. 3.

    Temporal evolution of the ABL height above ground level at the valley center (x = 0), at the slope (x = 5.12 km), and at the ridge top (x = 10.24 km).

  • Fig. 4.

    Area-mean profiles (height above valley floor) at t = 4 h of the zonal wind u, the potential temperature θ, and the specific humidity q for (a)–(c) the deep valley and (d)–(f) the shallow valley.

  • Fig. 5.

    Area-mean profiles (height above valley floor) at t = 4 h of (left) the turbulent components, (center) the local circulation components, and (right) their sum for (a)–(c) the kinetic energy, (d)–(f) vertical momentum flux, (g)–(i) sensible heat flux, and (j)–(l) moisture flux. The turbulent part includes the subgrid contribution.

  • Fig. 6.

    As in Fig. 5, but for the shallow-valley variants (h = 500 m) of the REF and the U4 case; without the kinetic energy and vertical momentum-flux plots.

  • Fig. 7.

    Valley cross sections as in Fig. 1, but showing the local budget terms for (a)–(f) heat and (g)–(l) moisture for REF and U4 at t = 4 h. The budgets terms include the total tendency (TOTAL), mean-flow advection (CIRC), and turbulence (TURB), corresponding to the terms in (15) and (16), respectively. The contribution from the microphysics (MP) is zero.

  • Fig. 8.

    Vertical profiles of the heat and moisture budget for FLAT, REF, and U4 at t = 4 h (h = 1500 m). The budgets terms include turbulence (TURB) and mean-flow advection (CIRC) corresponding to the terms in (15) and (16) as well as the total tendency (TOTAL). The contribution from the microphysics (MP) is zero.

  • Fig. 9.

    Vertical profiles of the area-mean budgets for TKE and the turbulent fluxes of heat and moisture according to (17) and (18) at t = 4 h for the three cases FLAT, REF, and U4. Note that subgrid contributions are not included.

  • Fig. 10.

    Valley cross sections showing the local circulation wind vectors as well as the zonal and vertical parts of the cross-valley circulation (u¯c and w¯c, shading) at t = 4 h for REF, U2, and U4. Also showing the isolines of the potential temperature in K. The dashed line marks the diagnosed ML height and the solid line the diagnosed ABL height.

  • Fig. 11.

    Time series of the time-integrated total sensible heat and moisture flux at crest height (1500 m, sum of turbulent and local-circulation part) normalized by the total time-integrated surface fluxes for the different cases where either the initial lapse rate γ (green), the valley width W (blue), or the upper-level wind (orange) has been varied.

  • Fig. 12.

    As in Fig. 11, but during a diurnal cycle induced by a homogeneous radiative forcing.

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