## 1. Introduction

During daytime, temperature differences between the air heated immediately above a slope and the undisturbed adjacent atmosphere induce upslope currents. The basics of the phenomenon can be understood by considering the balance of forces acting on an air parcel near a heated infinite and plane slope. Heating creates positive temperature and negative pressure perturbations in the vicinity of the ground surface. Hence, the parcel is subject to a perturbation pressure gradient force directed approximately toward the slope and to a vertical buoyancy force (Haiden 2003). The horizontal component of the pressure gradient force is the primary driver of the upslope air motions. As the flow accelerates, retarding frictional forces will also grow (Fig. 1). If the temperature perturbation is uniform along the slope, the perturbation pressure gradient force is exactly normal to the slope itself, and the along-slope momentum budget becomes entirely determined by buoyancy and friction.

This ideal situation is considered in the first conceptual model of upslope flows, proposed by Prandtl (1952, see especially 422–425). At the steady state, heating at the slope surface is exactly compensated by the upslope advection of potentially cooler air. In this situation, a one-dimensional analytical model of the motion above a flat, uniform, and infinitely long slope can be derived: it was proved to fit data from field measurements reasonably well (Defant 1949), provided eddy diffusivity is properly chosen.

More recently, Schumann (1990) used large eddy simulations (LES) to analyze the same idealized case, but with a more realistic treatment of turbulent processes. Evidence was provided of the formation above gentle slopes of an upslope mixed layer, that is, a region with nearly constant vertical profiles of temperature and wind speed.

Although extremely useful to understand the basic features of up- and downslope motions, both Prandtl’s model and Schumann’s simulations suffer from intrinsic limitations, which make them hardly applicable to more realistic conditions.

First, real slopes can rarely be considered flat or uniformly heated owing to along-slope variations of the slope steepness and of the local surface energy balance (Vergeiner 1991). Second, in Prandtl’s and Schumann’s analyses the background atmospheric conditions are not affected by the onset of the slope flow system itself. Instead, it is widely accepted that upslope winds can significantly alter the thermal structure of the atmosphere, primarily by triggering subsiding motions that compensate flow divergence at the foot of slopes and advect potentially warm air toward the ground surface (Whiteman 2000; Rampanelli et al. 2004).

A study of the features of nonstationary anabatic winds in varying ambient conditions above nonuniform slopes is probably out of reach for simple analytical models. In fact, most of the insight on the physical processes occurring in a realistic context have been obtained by a careful analysis of physical models of the slope flow system, as described below (section 2). Results from numerical simulations have been comparatively less revealing, probably because of the challenges posed by simulations at a high spatial resolution in domains with inhomogeneous lower boundaries.

In this study we show that the variety of processes observed in water tank models can be reproduced numerically with a high degree of realism if an appropriate parameterization of turbulent processes is chosen. Therefore, we use a numerical model to evaluate the sensitivity of the slope wind system to changes in external conditioning factors (e.g., slope angle) that are not easily studied with physical models, whose configuration is fixed or difficult to change.

To highlight the basic mechanisms, our simulations intentionally consider a simplified idealized system in which the phenomenon of upslope flows can develop without being affected by perturbations deriving from large-scale atmospheric conditions, by pre-existing flow patterns (nocturnal katabatic flows or a mean ambient wind), or by small-scale spatial variations in topography or in the energy balance.

After reviewing the interaction between upslope flows and the convective boundary layer (CBL) in section 2 and summarizing the features of the numerical model in section 3, we describe simulation results in section 4. In section 5 we identify several representative regions in the domain: for each region we study the characteristic succession of phenomena that takes place if the atmosphere is heated and provide a simple conceptual model of each process. Technical details on the model setup and on the interpretation of simulation results are provided in the appendixes.

## 2. Interaction between upslope flows and the CBL

The mechanisms by which slope flows modulate heat exchange and turbulence development near the base of mountain ridges clearly have an effect on the dispersion of passive scalars and, as such, have relevant implications for air pollution meteorology in areas with a complex orography. This explains the attention recently devoted to the study of the interaction between slope flows and CBLs. The current body of knowledge about the issue can be traced back to the following points:

- The occurrence of slope flows gives the atmospheric boundary layer a number of features that are not normally observed in areas with flat topography. Extreme situations are, on one side, the suppression of the CBL growth at a slope base due to high-level subsidence (De Wekker 2008) and, on the other side, the development of “slope circulation wheels” entirely included within the CBL (Reuten et al. 2005).
- Topographic inhomogeneities in the slope surface help redistributing energy and passive scalars by means of processes other than turbulent convection. Turbulence within the CBL over a plain and thermal advection by slope winds actually modulate each other (Chen et al. 1996).
- The evolution of slope circulations gives the CBL a transient multilayered structure, uncommon over flat and uniform areas. The multiple-layer structure is preserved until sufficient heating eventually causes thermal convection to entrain all layers in a unique CBL (Reuten et al. 2007).
- The effect of these complex circulations on the dispersion of passive scalars is not univocal: both trapping and venting of pollutants can be observed from time to time (Reuten et al. 2005; De Wekker 2008). The volume of atmosphere where pollutants are dispersed may not be limited to the thermal boundary layer only but can also include elevated turbulent layers (Reuten et al. 2007).
- External conditions that modulate these processes include background stratification, intensity, and duration of thermal forcing (Chen et al. 1996; Reuten et al. 2007).

## 3. Numerical modeling tools

In this work, the features of the atmospheric convective boundary layer in a region of transition between a plain and a uniform slope are analyzed by means of a series of two-dimensional cross-sectional simulations. The ARPS model, developed by the Center for Analysis and Prediction of Storms of the University of Oklahoma (Xue et al. 2000, 2001), has been implemented at horizontal resolutions from about 1100 m down to about 70 m in order to study the variety of phenomena that can occur over different slopes ranging from very mild (2°) to moderately steep (30°).

The model code has been modified to output the values of the single terms of the prognostic equations for potential temperature and turbulent kinetic energy (TKE). In particular, model output includes the horizontal and vertical advection and turbulent mixing of potential temperature and the horizontal and vertical advection, shear production, buoyancy production or loss, dissipation, and turbulent mixing of TKE. Information about the magnitude of these quantities is used to identify the mechanisms that govern heat transfer from the immediate vicinity of a heated slope to the free atmosphere or to the bulk of the CBL.

Modifications to the ARPS model also include an adaptation of the surface heating parameterization, needed to obtain a realistic simulation of the CBL development, as briefly discussed in appendix A.

The idealized domains designed in this study represent either isolated slopes connecting two flat areas or symmetric mountain ridges. Both slopes and ridges have constant height (*D* = 1000 m) and variable steepness, *α*. An average vertical resolution *dz* = 60 m has been adopted in all simulations: the vertical grid spacing actually ranges from 20 m close to the surface to approximately 90 m at the model top. The horizontal resolution *dx* differs in each simulation and is related to the slope steepness and depth by 25*dx* = *D*/(tan*α*). This choice was made so as to resolve the area where the upslope flow develops with the same relative resolution in all model runs: the number of grid points in the slope region is constant in all simulations and equal to 26.

Since the slopes we consider are 1000 m deep, every surface grid point in the slope region lies at an altitude 40 m higher than its lower neighbor. This implies that the ratio between the height increment along the grid (40 m) and the near-surface vertical grid spacing (20 m) is 2 in all simulations, small enough to avoid large errors in the computation of horizontal pressure gradients near the slope. It was also verified that the main features of the flow field were well preserved in control simulations where the horizontal resolution was changed without altering the topography.

Variable horizontal resolutions impose particular choices on the parameterization schemes used to represent turbulent exchange. Horizontal resolutions coarser than a few tens of meters do not allow one to resolve explicitly the large eddies that mix the CBL. Surface wind gusts, important for their effect in modulating surface fluxes, are also subgrid-scale features at this resolution. Therefore, a mixing length closure accounting for the effect of large eddies (Sun and Chang 1986) and a surface gustiness parameterization (Beljaars 1994) were used in coarse-resolution runs. High-resolution simulations in which large eddies and surface gusts are explicitly resolved have, instead, been carried out in a two-dimensional LES arrangement using the Moeng (1984) closure. This has been used in simulations with grid sizes smaller than 200 m, comparable to those adopted in other LES studies of complex topography (e.g., Chow et al. 2006). More details about the parameterization of boundary layer processes and surface fluxes are provided in appendix A.

The time step used in the numerical integration is also affected by the varying horizontal resolution and becomes very small, 0.05 s for the highest-resolution runs. A summary of the features of different model simulations is reported in Table 1.

The depth of simulation domains is always 9000 m and their width is *nD*/(tan*α*), where *n* is equal to 9 or 10 in domains representing isolated slopes and symmetric ridges, respectively. With these settings, the slope or ridge region is always separated by the domain boundaries by two flat areas, each 4*D*/tan*α* wide. Accordingly, the total number of horizontal grid points in the domain is 25*n* + 1. It has been verified that with this setup the distance between the domain ends and the slope region is long enough to prevent spurious boundary effects from reaching the core of the domain during most of the simulation time.

All model runs are initialized with an atmosphere at rest and a stable potential temperature profile with constant gradient: ∂*θ*/∂*z* = 3 × 10^{−3} K m^{−1}. Simulations were forced using an idealized solar radiation cycle, where the sun path lies on a vertical plane, and the diurnal phase is exactly 12 h long. Surface forcing is then modeled using bulk relationships (see appendix A). Model runs start at 0930 LST in the morning: this initial time was chosen to coincide with the moment when the surface heat flux first becomes positive. Surface forcing is approximately uniform across the domain, although fluxes are slightly larger above the slope where the upslope wind is constantly blowing. The maximum surface sensible heat flux amounts to ∼0.4 K m s^{−1}, while the friction velocity *u*_{*} ranges from 0.2 m s^{−1} on the plains to 0.4 m s^{−1} on the slope. Under such forcing, which leads to a Monin–Obhukov length of approximately −10 m, it is expected that the flow regime approaches free convection, where turbulence production is dominated by buoyancy. All simulations last for 5 hours, a duration long enough to capture the physics of the phenomenon and short enough to prevent spurious boundary effects from contaminating the solutions.

## 4. Simulations of anabatic flow over isolated slopes

As recalled in section 1, anabatic flow originates near a sloping surface by heating from the ground. A downslope return current aloft is always associated with the upslope flow, as suggested by the Prandtl (1952) model. The upslope wind blows parallel to the slope up to a point where an abrupt change in inclination causes its detachment from the ground and the formation of a thermal plume.

*θ*) and TKE (

*E*) budgets:andTerms on the rhs of the

*θ*budget [Eq. (1)] represent advection and diffusion, respectively. Terms of the parameterized TKE budget [Eq. (2)] are advection, shear production, buoyancy production, dissipation, and diffusion. Coefficients

*K*and

_{m}*K*represent eddy diffusivities for momentum and heat, |def|

_{h}^{2}is the magnitude of the deformation tensor,

*C*is an empirical tuning coefficient, and

_{ϵ}*l*a vertical mixing length.

In the following analysis, we show that the onset of the upslope flow occurs at a greater speed on steeper slopes. Therefore, the early evolution of the phenomenon is studied by comparing the flow fields after two hours of simulation on slopes with different inclination (2°, 5°, and 10°; Figs. 2, 3, and 4a). The later stages are, instead, examined considering the further evolution of the flow on the steepest slope only (Figs. 4b,c) where the process evolves most rapidly.

### a. Model runs with parameterized turbulence: Flow over isolated slopes

Figure 2 represents the state of the TKE and thermal energy budgets 2 h after the onset of anabatic flow on an *α* = 2° slope. Advective effects in the potential temperature budgets are shown in Figs. 2a and 2b: while horizontal advection is relatively unimportant at this stage, vertical advection related to the anabatic flow causes potential cooling along the slope and warming in the region of the return flow. The isentropes in Fig. 2c show that the upslope flow region connects two well-developed mixed layers on the bottom plain and top plateau; the upslope flow layer is itself reasonably well mixed, particularly in its upper part (where isentropes are approximately vertical). Figure 2c shows that turbulent mixing effects on the *θ* budget are positive in the lower part of the CBL and negative in a limited entrainment region just below its top: subgrid-scale convection transports thermal energy away from the heated surface in the lower mixed layer (ML) and overshoots into the potentially warmer free atmosphere as the boundary layer grows.

The warming effect related to the return current appears to be larger above the bottom end of the slope (Fig. 2b); accordingly, isentropes are displaced downward there, causing a slight depression in the depth of the mixed layer (Fig. 2c). Conversely, a local increase in the mixed layer depth is observed at the upper end of the slope, in connection with a flow structure resembling a thermal plume where the upslope flow detaches from the surface (Fig. 2b). The plume advects potentially warm air up from the surface, causing local heating and the related upward displacement of isentropes close to the ground. Higher up, the plume reaches the free atmosphere where potential temperature is higher: therefore it causes potential cooling, which results again in an upward displacement of isentropes.

Figures 2d–i show how the upslope flow affects the components of the TKE budget. Advection effects are relatively small (Figs. 2d,e), and the balance is dominated by the buoyancy and shear production terms and by the mixing and dissipation terms. Buoyancy effects (Fig. 2h) generate turbulence close to the ground surface, where ∂*θ*/∂*z* is negative, and suppress it higher up where the gradient reverses. Shear production (Fig. 2i) is only relevant in the region of transition between upslope and return flow and is much smaller in magnitude than the buoyancy contribution. Turbulent mixing (Fig. 2f) typically diffuses TKE from regions where turbulence is high to regions where it is low; therefore, it contributes to reducing TKE close to the slope and increasing it at upper levels. Dissipation is negative everywhere within the boundary layer (Fig. 2g). Since dissipation is directly proportional to *E*^{3/2} in our model, the shading in Fig. 2g also allows one to evaluate how the intensity of turbulence varies in the domain.

At this stage (2 h after sunrise on an *α* = 2° slope) the TKE budget appears to be essentially one-dimensional: the preferential direction of turbulent exchange processes still appears to be the vertical since horizontal advection is irrelevant and none of the leading terms in the TKE budget exhibits significant horizontal variations in the domain. Therefore, the only visible effect of turbulent convection on the thermal budget is the development of a mixed layer close to the surface.

In contrast, horizontal variations exist in the potential temperature budget, mostly because of the vertical advection effects related to the return flow and to the thermal plume at the slope end: these processes locally induce respectively a depression (at the slope bottom) or an enhancement (at the slope top) of the turbulent region depth.

Therefore, although at this moment streamwise variations in the thermal structure of the atmosphere appear to have some influence on the development of turbulence, streamwise variations in the intensity of turbulence still do not visibly affect the potential temperature field.

The state of the *θ* and TKE budgets at the same time on a steeper slope (5°) is shown in Fig. 3. While the basic features are similar to the previous case, some new ones now become apparent. Compared to the 2° slope, the upslope flow is now stronger (approximately 5 m s^{−1} versus 3 m s^{−1}). The return current is also stronger and so is its potential warming effect (Fig. 3b). The boundary layer in the slope region is accordingly shallower (see the isentropes in Fig. 3c) and no longer well mixed: up- and downslope motions are now associated with two distinct regions of unstable and stable vertical *θ* gradients, respectively. While the thermal plume at the upper end of the slope is now slightly displaced toward the plateau, additional plumes have appeared in the downstream direction. New plumes have also appeared at the bottom end of the slope (Figs. 3a,b). The development of individual convective cells appears to be related to a spurious amplification of numerical errors, as discussed in appendix B.

The TKE budget exhibits some significant changes. Buoyancy effects now damp turbulence in a deeper layer above the slope. Conversely, shear production has become larger in the same region, as expected because the upslope flow has grown shallower and stronger. Most interestingly, advection effects are now important. Vertical advection related to the thermal plume on top of the slope is now strong enough to transport turbulence quickly from the surface to the top of the boundary layer (Fig. 3e). Moreover, detrainment of turbulent air from the plume now transports turbulence laterally, most notably to the left, toward the region above the slope (Fig. 3d).

Differently from the 2° case, in the 5° case the TKE budget exhibits some significant horizontal variability. In particular, the plume at the slope break vents turbulent air from the surface to the free atmosphere, increasing the intensity of turbulence in an elevated layer above the slope region. Furthermore, the horizontal *θ* gradient at the slope top has intensified (cf. Figs. 2h and 3c). This happened because the region left of the plume, above the slope, is subject to potential cooling due to the advection operated by the slope wind.

As the slope steepness is increased (to 10°), the features observed in the 5° run become even more pronounced: potentially cool advection by the upslope wind extends farther toward the upper plateau, where a significant thermal imbalance is created across the plume between the cooler slope flow region upstream and the warmer mixed layer downstream (not shown). Consequently, the plume at the slope top is located farther away from the slope (Fig. 4a). TKE advection operated by the plume above the slope top becomes stronger, and its horizontal component stretches farther above the upslope flow region, slowly creating an elevated turbulent layer (Figs. 4b,c). As the shading suggests, turbulence is weak at high levels: its occurrence here is merely a result of advection and not of local production. The TKE isolines that mark the top of the CBL in Fig. 4 present some bumps: they are related to the previously anticipated occurrence of spurious convective cells (appendix B).

The relationship between the evolution of these phenomena above slopes of different angles can be studied by comparing the variation in time of the along-slope surface wind field in the 2°, 5°, and 10° simulations (Fig. 5). In all of them, the gradual onset of a region of positive (upslope) wind speed is apparent in the middle of the domain. Two other processes are apparent in all simulations: first, convective cells (whose fingerprint in surface time series are alternating bands of positive and negative wind speeds) are progressively generated at growing distances from the slope, both upstream and downstream. Second, the anabatic wind region starts extending past the upper end of the slope after a shorter time interval as the slope steepness increases: the edge of the upslope flow region (i.e., the thermal plume that originates at the slope top) moves across the upper plateau at a slowly increasing pace. The plume motion speed on the plateau (estimated by considering the slope of the dark shade in Fig. 5) is between 1 and 1.5 m s^{−1} in all simulations, that is, lower than the wind speed at the surface. This suggests that, after a certain critical threshold is exceeded, the anabatic current starts flowing across the upper plateau in a structure similar to a cold breeze front, instead of being subject to a purely vertical acceleration at the slope break. Local breeze fronts are usually reported to advance with a speed lower than the wind behind the front (see Simpson 1994 for an example).

Apparently, the general features of the evolution of upslope flow systems are the same independently of the slope steepness, and what makes the difference between winds on shallow and steep slopes is the speed by which the upslope flow evolves in time. So, for instance, if we look at the same time at the position of thermal plumes at the top of slopes with different inclination, we find that they stand at different distances from the slope: the steeper the inclination, the longer the distance (Figs. 2, 3, and 4).

This finding is in accord with a result first found in the Schumann (1990) simulations. Above indefinitely long slopes, the early stages of the development of upslope flows are subject to a transient regime where the intensity of the flow varies regularly in time with a period of *τ* = 2*π*/(*N* sin*α*), where *N* is the Brunt–Väisälä frequency. For an average value of *N* ≈ 10^{−2} s^{−1}, *τ* corresponds to ∼5, ∼2, and ∼1 h, respectively, over slopes of 2°, 5°, and 10°. The proportion between these time lags is approximately the same that exists between the times of the first appearance of the thermal plume on the upper end of the slopes (∼2, ∼1, and ∼0.5 h, respectively).

Basically, both formation of the thermal plume and its motion across the plateau are related to a thermal imbalance at slope-top level, which occurs between the atmosphere above the slope and that above the plateau and is caused by the advective cooling effect of the slope wind. Both events are verified earlier above steep inclines because the onset of anabatic flow occurs on shorter time scales there since the slope-parallel component of buoyancy oscillations is larger.

After the thermal plume at the slope top has developed, the distinctive characters of the upslope flow are largely altered and soon diverge from those of a homogenous flow. This means that a steady-state upslope flow, like that studied by Schumann (1990), fails to be observed in a nonuniform framework, even if idealized: the flow would become horizontally inhomogeneous much before it reached a steady state, unless it developed on an extremely long slope.

Another relevant consequence of the processes highlighted above is the following: effects like a boundary layer depth depression at the base of a slope, or other localized enhancements or reductions of the turbulence intensity, are transient and only manifest in a particular phase of a continuous succession of processes—that is, upslope advection, return flow, venting by thermal plumes, and widespread thermal convection. Although the atmosphere might be locally subject to potential cooling or turbulence damping at a particular moment and in a particular place, the net global effect of this array of phenomena is heating and enhancement of the turbulence intensity.

A final remark about Fig. 5 is that it shows how spurious boundary effects begin to contaminate the model results. Shading indicates that an outward wind speed starts being manifest at the domain boundaries after a few hours (about 4 h in the 2° simulation and ∼2 h in the 10° simulation). The perturbation then extends toward the middle of the domain and affects the slope region as well. This is the reason why model simulations were not extended beyond the 5-h limit in the present work.

### b. Model runs with parameterized turbulence: Flow over symmetric mountain ridges

Upslope flow patterns above symmetric mountain ridges do not differ much from those that take place above isolated slopes. In these runs, the top of the mountain ridge is located exactly in the middle of the domain. Figure 6 shows cross sections referring to the 10° ridge simulation.

The flow field maintains a reasonable degree of symmetry during the whole simulation, although departures from symmetry become apparent when spurious boundary effects arise, that is, approximately after three hours of simulation. Numerical asymmetry is most likely related to the fact that boundary conditions are imposed on the basis of a locally diagnosed gravity wave phase speed, an approach particularly prone to numerical errors (Kantha et al. 1990). Better symmetry can indeed be achieved by converting to double precision all floating point variables in the model code (and therefore largely increasing computational time), but we believe that lack of it does not affect the physical plausibility of our results.

The onset of the anabatic flow above the two slopes of the ridge takes place at a rate similar to the one observed above isolated slopes, and maximum wind intensities at the surface reach a value of about 5 m s^{−1} (not shown). The features of the potential temperature and TKE budgets are analogous to those observed above isolated slopes (not shown). Accordingly, a similar evolution of the circulation can be observed.

A thermal plume develops at the top of the slope due to the convergence of equally intense anabatic winds from the opposite slopes of the ridge. In contrast to the isolated slope case, the plume remains stationary on the mountain top because the approximate symmetry of the flow field prevents the formation of temperature and pressure gradients across it (Fig. 6).

Notice that in this case the plume at the ridge top advects potentially cooler air, as opposed to plumes observed at the top of the isolated slopes, which are partially fed with air from the elevated plateau downwind of the slope. The latter is potentially warmer than that advected by the anabatic wind, merely because it originally lay at a higher altitude. Accordingly, in the case of a mountain ridge the plume overshoots to a lesser extent into the free atmosphere (see the height of the TKE isolines in Fig. 6).

Anyway, the horizontal detrainment of turbulent air toward the region above the plain is equally strong, if not larger, and elevated turbulent layers on both sides of the ridge quickly merge with the CBL growing below. After 4 h the CBL completely incorporates the mountain ridge, and its top slants with an angle about one-third of that of the slope below (Fig. 6c).

### c. Two-dimensional large-eddy simulation runs

The analysis carried out so far originates two questions: 1) What would the subsequent phases of the evolution of anabatic flows be like? 2) Would all of the features we found be visible also using a model with a more sophisticated treatment of turbulent exchange processes?

In the following, we try to answer both questions analyzing some model runs in which turbulence was modeled using a two-dimensional large eddy simulation (2D-LES) approach. Given the intrinsically three-dimensional nature of turbulence, fully 3D simulations in a domain uniform along one direction would have certainly provided physically sounder results. Nevertheless, we deem 2D simulations enough for the aim of the present study, as far as we restrict the analysis to potential temperature cross sections and instantaneous flow fields. Consistently, we do not perform any statistical evaluation of the turbulent fluxes of momentum or heat, or the energetics of turbulence (for which 3D simulations would be absolutely necessary). A comparison of the results of preliminary 2D and 3D simulations showed that the main features of the flow field were identical in the two settings, the only remarkable difference being that 2D simulations usually produced a CBL 5%–10% thicker compared to 3D simulations.

Figure 7 is analogous to Fig. 5 and portrays the evolution of the surface wind field above slopes and ridges with walls 15° and 30° steep. Simulation results (*dx* = 149 and 69 m, respectively) have been averaged over 10-min windows for the purpose. The characteristics of the early stages of the phenomenon resemble those observed on less inclined slopes. Upslope flow steadily develops as heating begins: the intensity of the anabatic wind is greater on 15° than on 30° slopes. The upslope breeze front, that is, the trace of the motion of the thermal plume away from the slope on the upper plateau, is clearly visible in the isolated slope domains.

Figure 8 shows consecutive cross sections of the flow fields above the 30° slopes and ridges. In the left panels, incipient convection above the plains is apparent: overshooting thermals cause undulations of the BL top surface. Thermal plumes at the slope and ridge top are visible, and, even in this case, the plume at the slope top penetrates higher into the free atmosphere, being fed by potentially warmer air from its right side. The plume on the slope has also moved away from the slope owing to the thermal imbalance across it (approximately 0.4 K in this case).

Two hours later (middle frames) the CBL has grown to a depth of approximately 1 km. In the slope domain, the thermal plume on the plateau has moved farther away from the slope top. The plume separates (right) a well-mixed CBL and (left) a moderately stable region extending toward the slope and surmounting the underlying growing CBL. The two layers left of the plume are separated by a rather strong inversion, around 1.5 K. Similar stable regions are also visible beside the walls of the symmetric ridge in the lower middle frame.

After two more hours convection has grown stronger. In the ridge domain the CBL has incorporated the whole mountain ridge, and large convective eddies, displaying an upslope branch close to the surface, are formed. In the slope domain, the elevated layer has grown in depth and has become on average closer to neutral. The stability of the thin layer separating it from the CBL below has also decreased. If further heating were provided, the two layers would merge into a single CBL, incorporating both upslope and downslope regions.

A further detail visible in Fig. 8 is that circulation cells within the CBL cause the excitement of internal gravity waves in the overlying atmosphere. The possible generation of gravity waves by convective updrafts hitting the bottom of a stable layer is well documented in the literature. Our simulations suggest that a mechanical oscillator effect (Clark et al. 1986) occurs at the interface between the CBL and the free atmosphere: when organized convective cells grow strong enough to overshoot into the stable atmosphere above the CBL, they create ripples in the upper surface of the CBL and excite internal waves.

However, the spatial organization of convection above flat regions in our simulations appears to be conditioned by spurious numerical effects. As a consequence, no reliable conclusions can be drawn as to the tilt and direction of propagation of internal gravity wave fronts in the vicinity of the heated slope. A description of the mechanism generating spurious convection is included in appendix B.

## 5. Discussion

The simulations described so far confirm that similar phenomena occur both near isolated slopes and in the vicinity of symmetric mountain ridges. The only relevant difference between the two cases pertains to the behavior of the thermal updraft at the top of the slopes, which moves forward like a breeze front in the isolated slope case, whereas it remains stationary on top of the mountain ridge. Moreover, the updraft is potentially cooler in the latter case and is therefore less efficient in transporting turbulence and thermal energy above the CBL. Apart from these minor discrepancies, the most important features of the interaction between slope flows and the adjacent CBL are the same in the isolated slope and symmetric ridge cases and can therefore be discussed together.

To explain the variety of processes highlighted by our numerical simulations, it is useful to identify some representative regions in the domain, considering the several locally generated forcings active in each of them. For instance, a heated parcel close to the slope surface would be shifted upward along the slope, dragging air aloft that has not been directly heated. As a result, a given isentropic surface would bend downward close to the slope and upward right above (as in Fig. 1). In turn, potentially cooler air shifted upward originates a weak negative buoyancy perturbation, which causes weak downward motion and warm advection farther up. Let us then choose a point (point 1) within the layer with positive *θ* perturbation close to the slope and another, point 2, in the layer with negative *θ* perturbation right above (Fig. 1). Furthermore, we consider point 3 at the top and point 4 at the bottom of the slope (Fig. 9).

As anticipated in the introduction, the balance of forces acting on a heated parcel at point 1 includes a buoyancy force and a pressure gradient force. A further relevant term is friction. The Coriolis force is of little relevance at the spatial scales typical of slope flows.

While the buoyancy force acts along the vertical, the perturbation pressure gradient is approximately normal to the slope. The usual analysis of upslope flows (e.g., the Prandtl model) considers a rotated reference frame, with one axis parallel and one axis normal to the slope: hence the conclusion that upslope flow is driven by the slope-parallel component of buoyancy. In a Cartesian reference frame with horizontal and vertical axes, acceleration toward the slope is provided by the horizontal component of the perturbation pressure gradient force. The intensity of the resulting anabatic wind will depend on the strength of the buoyancy force: if more buoyancy is available than what is necessary to sustain an exactly balanced slope-parallel wind, thermal convection will develop, as observed in our simulations.

Farther away from the slope, cooler parcels are subject to decreasing buoyancy acceleration, until at point 2 the net balance of forces is reversed: buoyancy is downward and the perturbation pressure gradient force has a horizontal component directed away from the slope. As mentioned earlier, cooling in this region is a result of frictional drag: it depends on the fact that air, which has not been directly heated, is carried along with the upslope wind, thereby creating a negative *θ* perturbation. This reversed balance consistently implies a downslope current. We emphasize that the downslope return current should not be explained by invoking the concept of mass conservation: downslope flow aloft is dynamically generated as the result of a local balance of forces acting on single air parcels, as already clearly suggested by Prandtl (1952).

When the upslope flow is still at an early stage of its development, the balance of forces at points 3 and 4, the top and bottom of the slope, respectively, is similar to that at points 1 and 2 except that the horizontal *θ* gradient (and therefore the horizontal component of the pressure gradient force) vanishes. This happens because the heating source (i.e., the ground) becomes horizontal there. So, for instance, when parcels embedded in the anabatic current reach point 3, they enter a region where the potential temperature field is still horizontally homogeneous and become subject to a purely vertical forcing. Consequently, a plume develops right above the slope break, point 3, where the leading end of the anabatic flow converges toward the CBL above the plateau. In contrast, downward vertical motions compensate for the flow divergence originated by the onset of the anabatic wind at the slope bottom at point 4 (Fig. 9, left).

As heating continues, this situation is progressively altered by the CBL growth above the flat areas left of point 4 and right of point 3. At point 3, the CBL above the plateau grows faster than above the slope, where part of the heat input is removed by advection from the upslope flow (Fig. 9, right). As the potential temperature imbalance grows, a horizontal pressure gradient force is established across the plume, which starts moving away from the slope across the plateau. Similarly, at point 4 the CBL above the plain grows faster than above the slope and cools the atmosphere at its top by entrainment. Thermal imbalance now occurs between the CBL and the compensating current of the upslope flow, where warm advection occurs. Consequently, the downward motion feeding the anabatic wind starts displaying a relevant component toward the slope, in analogy to what was observed by De Wekker (2008).

In the case of an isolated ridge, the rate of heating of the atmosphere above the sidewalls is approximately equal, and therefore no motion of the updraft at point 3 is observed. In any case, plumes vent air from the surface to the free atmosphere: vertical advection and mixing within the plumes tend to transport both heat and turbulence upward. Converging horizontal advection feeds heat and turbulence into the plumes at their bottom, while divergence at their top redistributes them into the free atmosphere above.

Elevated weakly turbulent layers, only slightly stable, slowly develop. Turbulence at this level is not produced locally but, rather, appears to be essentially a result of horizontal advection. Beyond heat and turbulence, advection from the plume at the slope top will also transport passive tracers, if present. Consequently, elevated layers would not only be slightly stable and weakly turbulent but would also contain a relevant concentration of tracers. These, originally present in the vicinity of the ground surface, would be vented out of the CBL through the slope circulation. Examples of passive tracers might be pollutants in the atmosphere or dyes in water tanks. A layering effect similar to that described above has, indeed, been identified by studying dye dispersion in laboratory experiments (Reuten et al. 2007).

Multiple layering persists until sufficient heating causes the erosion of the stable layer separating the CBL and the overlying elevated layers. The formation of a unique homogeneously mixed layer follows, and venting of tracers is replaced by trapping.

## 6. Conclusions

In the present study, we describe numerical simulations of anabatic flows over isolated slopes and mountain ridges, devoting particular attention to the relationships between the development of the slope wind system and the growth of an adjacent CBL. Although carried out in a completely idealized framework, the study provides evidence of a series of physical processes that are, indeed, likely to occur in nature.

- A characteristic sequence of events occurs above heated slopes. First, a local thermal imbalance triggers the onset of the anabatic wind, rather uniformly along the slope. Far from the slope ends, acceleration toward the slope is locally provided by the horizontal component of the perturbation pressure gradient force. At the slope break, the perturbation pressure gradient force gradually becomes vertical (i.e., its horizontal component vanishes as the upslope wind enters the uniformly heated plateau). Hence, air motions cease to be parallel to the surface and detach from it. Local patterns of flow convergence and divergence affect the momentum balance differently at the top and bottom of the slope, causing respectively the onset of a vertically rising plume or of sinking motions. While the latter are responsible for a transient depression in the depth of the CBL, as noted by De Wekker (2008), the former advect potentially cool and turbulent air toward the free atmosphere, causing enhanced mixing in the slope region. As a consequence, an elevated turbulent layer is produced above the slope flow. Then, if further heating is provided, a single boundary layer encompassing the plain, slope, and plateau regions is formed. In this respect, our numerical results agree well with experimental evidence by Reuten et al. (2007).
- Slope flow systems evolve with increasing speed on increasingly steep slopes. This suggests that physical processes, manifest in the early stages of the evolution of the upslope flow field (e.g., a boundary layer depression at the foot of the slope), might be observed preferentially near shallow slopes and be much more elusive in the vicinity of steep slopes.
- As already pointed out by Vergeiner (1991), under average conditions of atmospheric stability and over slopes of realistic steepness and length, it is hardly possible for slope winds to behave like the flows over homogeneous slopes studied by Schumann (1990). Variations in the slope steepness (ubiquitous in nature) rapidly induce inhomogeneities in the flow field in the along-slope direction. Streamwise variations in the
*θ*and TKE budgets result in a redistribution of thermal energy and turbulence above the anabatic flow layer, creating weakly elevated turbulent layers with a reduced static stability.

Our study confirms and completes previous results provided by Reuten et al. (2007) and De Wekker (2008). Still far from providing a comprehensive view on the possible interactions between a developing slope flow system and a CBL growing nearby, our work offers indications for some possible future development.

For instance, scaling arguments (Mahrt 1982) suggest that any sharp variation in the slope profile would favor the detachment of the upslope flow from the ground surface and, hence, the formation of thermal plumes. Our study highlights that such plumes give the CBL a layered structure, much more complicated than the rather simple profiles commonly observed over flat surfaces. The possible role of a changing atmospheric stability in modulating this layering effect is still poorly explored.

Further aspects deserving attention are 1) whether a similar evolution of the upslope wind system would take place, even starting from a different initial condition such as an atmosphere with different background stability, or previously subject to katabatic currents and intense low-level cooling or characterized by the presence of a residual turbulent layer above the night-time stable boundary layer, and 2) whether the presence of ambient wind and of the related shear turbulence production would alter the evolution of the slope flow field.

## Acknowledgments

The authors acknowledge support from CINECA, the Italian national supercomputing centre, where some of the computations for this study were performed. They also express their appreciation for the comments of two anonymous reviewers, who consistently helped to improve the original manuscript.

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## APPENDIX A

### ARPS Configuration

Turbulent eddy fluxes in the ARPS model equations are parameterized using a 1.5-order local closure. Accordingly, the flux of any quantity is proportional to its local gradient through a mixing coefficient, which in turn depends on the local turbulent kinetic energy and on an appropriate mixing length (Moeng 1984).

When coarse-resolution grids are used (*dx* > 200 m), an additional parameterization for the effect of large boundary layer eddies must be used: only in unstable conditions and below the boundary layer capping inversion, the vertical mixing length is imposed according to a predetermined semiempirical profile (Sun and Chang 1986).

Moist processes were deliberately not included in the present simulations so as to isolate the ideal features of thermally driven flows in a perfectly dry atmosphere. As a consequence, no parameterization scheme for microphysical processes was used. Accordingly, processes related to or conditioned by the presence of water vapor, liquid water, or ice were removed from the parameterization of the radiative energy transfer and of the surface energy budget.

The transfer of radiant energy is treated with a simplified model, where radiative forcing is accounted for by semiempirical functions in the surface budget and land surface processes governing the energy exchange between soil and atmosphere are treated using a two-layer force–restore model (Xue et al. 2001). The values of the parameters of the model (and of the soil heat capacity in particular) are set so as to obtain reasonable diurnal cycles of the energy budget terms even if moist processes are not considered.

*H*

_{0}, is modeled using a bulk relationship:where

*U*= (

_{s}*u*

_{s}^{2}+

*υ*

_{s}^{2})

^{1/2},

*u*and

_{s}*υ*being the horizontal wind components at the surface;

_{s}*ρ*represents the density of air at the surface,

_{s}*c*its specific heat at constant pressure,

_{p}*T*its temperature at the lowermost model level;

_{a}*T*is the soil temperature, provided by the surface energy balance model. The drag coefficient

_{s}*C*is stability dependent.

_{d}In this study, when the horizontal resolution of simulations is fairly coarse [i.e., when the Sun and Chang (1986) mixing length parameterization is used], the wind intensity *U _{s}* in Eq. (A1) is redefined as

*U*= [

_{s}*u*

_{s}^{2}+

*υ*

_{s}^{2}+ (

*βw*

_{*})

^{2}]

^{1/2}, where

*w*

_{*}= [(

*g*/

*θ*)

_{s}*H*

_{0}

*z*]

_{i}^{1/3}is the Deardorff (1970) convective velocity scale,

*θ*being the potential temperature at the surface and

_{s}*z*the boundary layer depth. The additional term,

_{i}*βw*

_{*}(with

*β*= 1.2, following Beljaars 1994), represents a gustiness wind component, needed to parameterize the surface frictional effects of unresolved large turbulent eddies. If this additional component were not considered, Eq. (A1) would underestimate turbulent fluxes in free convection regimes or along convergence lines, that is, when or where no mean wind blows.

In all of our simulations, lateral boundary conditions are open, ideally allowing wave disturbances to exit the simulation domain with limited reflection. The top and bottom boundaries of the domain are treated as rigid walls with impermeable boundary conditions. Below the top boundary of the domain a 2000-m-deep Rayleigh damping layer is implemented to suppress spurious downward wave reflection.

## APPENDIX B

### Spurious Numerical Effects

In our simulations, convective cells are progressively generated at growing distances from the heated slopes, both upstream and downstream (see Fig. 5). All cells are 8–10 grid points wide, and their initiation appears to take place near the ground surface, as clearly visible from the vertical velocity contours in Fig. 3. The regular spacing between convective cells appears to be unphysical since it is well known that thermal updrafts in the CBL are generally stronger and narrower than downdrafts (Stull 1988).

Several control runs with different model setup suggest that the regular organization of convective activity is a spurious phenomenon resulting from imperfect numerics. It was verified that doubling or halving the horizontal model resolution doubles or halves the horizontal spatial scale of convective organization; hence, the forcing of convection is determined by the computational grid rather than by a physical triggering process.

While initial oscillatory perturbations in the temperature and velocity fields at the surface derive from numerical errors, the mechanism for their amplification is physical: owing to the statically unstable environment of the lower CBL, initial errors grow into a regular organization of symmetric up- and downdrafts (i.e., cooling and warming regions).

The occurrence of spurious regularly organized convective activity in numerical simulations has already been reported in literature, for example, in studies of the development of squall lines in the presence of a moist absolutely unstable layer (Takemi and Rotunno 2003; Bryan 2005). Rather than enhancing some parameters in the turbulence closure scheme in order to increase physical diffusion (as suggested by Takemi and Rotunno 2003), we chose to minimize the problem by including an artificial diffusional term in the model equations. Weak numerical diffusion along the horizontal direction (i.e., a fourth-order filter with a nondimensional damping factor of 5 × 10^{−4}) is also necessary to maintain numerical stability.

Theoretical considerations (Durran 1999) suggest that the value of the filtering constant determines the wavelength threshold below which damping is effective. The value chosen in the present simulations causes the removal of all disturbances with wavelength shorter than approximately 8*dx*, an exact match to the spatial scale of artificial convective cells. Increasing or decreasing the weight of the filter would simply shift the wavelength threshold. Thus, the spurious organization of thermals observed in simulations appears to be unavoidable unless the numerics of the ARPS model advection scheme are entirely revised.

Given the symmetry of updrafts and downdrafts in the organization of artificial convection cells, the bulk heating rate of the CBL is not expected to be conditioned by this problem. Consequently, we believe the physical plausibility of simulations is not invalidated.

Summary of the features of numerical simulations. In the first column, odd and even numbers identify simulations referred to isolated slopes and symmetric ridges respectively. The slope angle, horizontal resolution, and integration time step are reported in columns 2 to 4. Column 5 lists the turbulence closure scheme in use: (a) refers to the Moeng (1984) closure supplemented with the Sun and Chang (1986) mixing length parameterization and (b) refers to the Moeng (1984) closure only with a ±0.1 K random perturbation in the initial temperature field. Column 6 reports whether the wind gustiness parameterization has been used in Eq. (A1) or not. The vertical resolution *dz* is the same in all runs (60 m on average, 20 m in the vicinity of the surface).