## 1. Introduction

Atmospheric circulation in mountainous areas under weak synoptic conditions is mainly characterized by slope and valley winds (Simpson 1994). The slope wind system is driven by the horizontal temperature gradient between the air near the slope and that at some hundreds of meters from the slope at the same height, while the valley wind circulation results from the temperature difference between the air within the valley and that over an adjacent plain. These gradients are generated by the solar shortwave heating during the day and the radiative longwave cooling at the ground during the night. The resulting pressure gradient force drives airflow from the regions of low temperature toward those of high temperature near the ground and a compensating return motion in the upper part of the atmosphere. The overall circulation is then determined by baroclinicity and characterized by vorticity, as can be viewed applying the Bjerknes theorem. The same mechanism is common for all thermally driven local systems such as the sea and land breezes or flows due to the urban heat island effect (Martin 2006).

Typical speed values of these currents range from 0.5 m s^{−1} for low-intensity anabatic winds to more than 10 m s^{−1} for fast katabatic flows in regions with a significant snow or ice cover over long (tens of kilometers) slopes (Monti et al. 2002; Pettré and André 1991). While the depth of the anabatic current can easily exceed 500 m (Reuten et al. 2005), the katabatic wind is characterized by a vertical extension ranging from 5 to 30 m in midlatitude climates to more than 100 m over long slopes in polar regions. In the evening and in the early morning the temperature gradients tend to disappear and the resulting circulation intensity is consequently strongly reduced, creating a critical situation for the dispersion of pollutants. Therefore, these wind systems have a great influence on the air quality of inhabited mountain areas, since they can enhance ventilation within the valley atmosphere favoring the dilution of pollutants. Furthermore, they modify the stability profiles, which are crucial for the dispersion properties of the atmosphere. Such currents also play an important role in the formation of clouds, favoring the uplift of moist air masses above the lifting condensation level. Another important issue is that slope and valley winds, like other local circulation systems, play an important role in redistributing energy from the local to the regional scale (Noppel and Fiedler 2002). Since this effect can be of the same order as the parameterized surface turbulent exchange in global circulation models (GCM) but is not usually taken into account, the investigation of these wind systems is of great interest for the development of better parameterizations for the GCMs.

*w*_{*}= (*gQ*/_{s}h*θ*)_{s}^{1/3}is the convective velocity scale;*β*is slope steepness;*λ*is a calibration constant dependent on the surface roughness and the planetary boundary layer (PBL) depth;*Q*=_{s} is the kinematic surface heat flux;*w*′*θ*′_{s}*θ*is the surface temperature over the slope;_{s}*h*is the depth of the flow.

Many authors have measured the circulation over complex terrain during field studies. Brehm and Freytag (1982) analyzed the mass transport and the temperature structure in the Inn Valley (Austria) and pointed out the importance of subsidence motions and entrainment from the free atmosphere associated with the return flow. These phenomena, along with the reduced volume with respect to a plane geometry, are the main reasons for the enhanced diurnal warming of the air volume in a valley. Princevac et al. (2001) briefly reports measurements of the HU constant *λ* taken during the Vertical Transport and Mixing (VTMX) field campaign, conducted in Salt Lake City (Utah). Princevac and Fernando (2005), Princevac et al. (2008), Lee et al. (2006), and Monti et al. (2002) used data from the VTMX campaign to investigate the nighttime circulation, also computing the calibration parameters for the MS model. They pointed out the importance of the entrainment at low Richardson numbers and the development of along-slope periodic oscillations as the stability of flow increases during the night. Lee et al. (2006) used the VTMX measurements to implement and test a new PBL parameterization scheme based on a stability-dependent turbulent Prandtl number. The study by Monti et al. (2002) also includes an investigation of the different waves that can develop at the interface between the upslope flow and the return current, as a function of the stability of the flow, evaluated via the gradient Richardson number Ri* _{g}*. Reuten et al. (2005) investigated the flow pattern in the Fraser Valley (Canada), pointing out that the anabatic wind and the return current may exhibit equal velocity and vertical extension for valleys with steep slopes.

Laboratory-scale studies with simplified geometries were performed to validate theoretical and numerical models. Deardorff and Willis (1987) investigated the upslope flow by means of an inclined water tank heated from below, using oil parcels as a tracer. The complete daily cycle of heating and cooling was investigated by Chen et al. (1996). They used a salt-stratified water tank and two-dimensional ridge geometry. Their sensitivity analysis with respect to the background stratification confirmed a weak dependence on this parameter, as previously argued by Ye et al. (1987). Fernando et al. (2001) used a water tank with two different experimental setups, a simple variable slope and a two-dimensional sinusoidal basin, to reproduce the anabatic and katabatic currents in neutral and stratified background environment. Chan (2001), by means of experiments in a thermally controlled water tank, provided calibration values for the anabatic intensity relation derived by HU. Along with their analytical model, HU presented a laboratory study of the transition between the anabatic and the katabatic current above a two-dimensional valley showing the formation of a frontal structure over the slope. Cenedese et al. (2004) used a thermally controlled water tank with various background stratification values to reproduce the diurnal and nocturnal circulation over a simple slope. Calibration values for the MS and HU analytical models were also provided. A recent work by Princevac and Fernando (2007) focused on the steepness conditions leading to the onset of the anabatic flow. They used a variable simple slope and different water–glycerin solutions to obtain a relation for the critical slope angle at which the onset of the anabatic wind occurs, as a function of the Prandtl number.

Numerical modeling represents a way to extend theoretical and laboratory-scale insights toward more complex situations. The pioneering work by Orville (1964) on upslope winds was based on the integration of the vorticity equation in a two-dimensional domain with a simple slope of 45° and a plateau. The basic flow features, described at the beginning of this section, were captured despite the simplicity of his model. Bader and McKee (1983) analyzed the heat transfer mechanism in a valley, obtaining results in line with the observations made by Brehm and Freytag (1982). Ye et al. (1987) developed a numerical model to extend and validate their theoretical work. Segal et al. (1987) focused on the effects related to the slope orientation on anabatic flows, which becomes important especially for midlatitude winter conditions. The first large-eddy simulation (LES) study of the atmospheric boundary layer over a slope was performed by Schumann (1990) who assumed an unbounded homogeneous rough plane. The subgrid-scale (SGS) model was based on the turbulent kinetic energy (TKE) closure, periodic conditions were imposed for the lateral boundaries, and a constant and uniform heat flux was imposed at the bottom. The results showed the development of transverse convective rolls, which are an indicator of dynamical instability. Anquetin et al. (1998) investigated, by means of an LES model, the formation and breakup of the cold pool over an idealized two-dimensional valley with a resolution of 200 m along the horizontal directions and a stretched grid with 75 ≤ Δ*z* ≤ 100 m along the vertical direction. Skyllingstad (2003) performed the first LES of a katabatic wind, focusing also (Smith and Skyllingstad 2005) on the influence of along-slope steepness variations. He assumed an infinite slope with periodic lateral boundary conditions and open boundary conditions in the direction of the flow, finding a good agreement with the predictions of the MS hydraulic model. Rampanelli et al. (2004) used the Weather Research and Forecasting (WRF) model to investigate the diurnal circulation over a valley with simplified two- and three-dimensional geometries. The numerical resolution for this study was 1 km on the horizontal and 50 m on the vertical, which fall in the fine mesoscale range.

Most of the published LESs of the atmospheric circulation considered rather small domains with homogeneous and steady surface forcing; the complete diurnal and nocturnal cycle was not thoroughly investigated. The case of a valley, despite its relevance for environmental and wind-energy issues, has not been sufficiently studied. The aim of this paper is to describe an LES of the atmospheric circulation over an idealized periodic valley with time-varying surface forcing under weak synoptic conditions. The nonhydrostatic mesoscale model WRF will be modified and used in an LES mode to perform the numerical simulations.

When the fast katabatic current impacts the quiescent atmosphere of the valley, a hydraulic jump may form. This phenomenon is difficult to observe (Pettré and André 1991) and to reproduce with a numerical model because of its small characteristic length and time scales. Yu and Cai (2006) managed to reproduce in their idealized simulations the basic features of this structure, like the updraft and the correlation between the velocity and the temperature fields. According to these authors further insights can be derived from an LES study of the katabatic current, which is one of the issues we address in the present investigation.

The paper is arranged as follows: in section 2 a short description of the WRF model and its configuration to be used for our LES study is given. The daytime and nighttime results are presented in sections 3 and 4, respectively. Section 5 deals with some consideration about the complete heating–cooling cycle. Section 6 reports some comparisons with analytical models, field observations, and laboratory investigations from some of the literature works discussed above. Conclusions are given in section 7.

## 2. Model setup

The numerical simulations were performed using the three-dimensional meteorological model WRF (in LES mode) developed by the National Center for Atmospheric Research (NCAR) in cooperation with several universities and research groups. The model is based on the fully compressible nonhydrostatic Navier–Stokes equations with a terrain-following hydrostatic-pressure vertical coordinate. WRF employs a third-order Runge–Kutta (RK) scheme for the time integration, a fifth-order discretization scheme for the horizontal advection, and third-order for the vertical discretization. The integration of acoustic and gravity waves is performed with a smaller time step, using a time-splitting scheme incorporated into the RK loop, hence allowing for larger time steps in the RK integration. For a detailed description of the WRF modeling system see Skamarock et al. (2008).

Since the horizontal scales of the investigated phenomena are quite small we can neglect the Coriolis terms (Rossby number ≫ 1); in addition, we will limit the present study to a dry atmosphere. Under these simplifications the model equations are

- mass balance equation:
- momentum balance equations:
- thermal energy balance equation:
- equation of state:

*ρ*and*p*are the deviations from the background hydrostatic profiles of density*ρ*_{0}and pressure*p*_{0};*u*are the velocity components;_{i}*g*is the gravitational force per mass unit;*δ*is the Kronecker operator;_{ij}*θ*is the potential temperature;*γ*= 1.4 is the ratio of the specific heats at constant pressure and volume for dry air;*c*= 1004.5 J K_{p}^{−1}Kg^{−1}is the specific heat at constant pressure for dry air;is the gas constant for dry air; *τ*=_{ij} is the subgrid-scale stress tensor;*u*′′_{i}u_{j} is the subgrid-scale kinematic heat flux.*u*′′_{j}θ

^{3}and the prime is used for SGS variables. To simplify the notation, we have omitted the overbar for the resolved (RES) variables.

*K*and

_{m}*K*are modeled withwhere

_{h}*e*= ()/2 is the turbulent kinetic energy per mass unit; *C*= 0.15 is the Smagorinsky constant;_{k}- Pr
_{t}^{−1}= 1 + 2*l*/Λ is the inverse turbulent Prandtl number; *l*is the mixing length scale;- Λ is the LES filter width.

*≅ ⅓ in neutral and unstable conditions and has an upper bound of Pr*

_{t}*= 1 for stable regimes; this has proven to be successful in simulation of unstable to moderately stable boundary layers. However, some authors (Monti et al. 2002; Zilitinkevich et al. 2008) questioned its application for very stable regimes, since observations of nocturnal boundary layers have shown that, when Ri*

_{t}*is larger than ≅ 0.1, Pr*

_{g}_{t}can exceed unity. Axelsen and Van Dop (2009a,b) in their LES of the katabatic winds have employed an alternative formulation of the SGS Prandtl number, which takes into account its dependence on Ri

*, observing that an underestimation of Pr*

_{g}*generates lower surface buoyancy. Nevertheless, in the present work, the imposed thermal forcing does not generate very stable regimes and the results satisfy the limit Ri*

_{t}*≤ 0.1 (not shown), hence the current formulation for Pr*

_{g}*can be retained.*

_{t}*z*/(Δ

*x*, Δ

*y*) ≪ 1, the almost two-dimensional filtering dampens most of the fine scales in the z direction resulting in an underestimation of the constant

*C*. Scotti et al. (1993) proposed a modification of the filter width Λ based on a correction function

_{k}*f*(

*a*

_{1},

*a*

_{2}):where

*a*

_{1}and

*a*

_{2}are the aspect ratios of the filter and Δ

*, Δ*

_{i}*are the two smaller dimension of the cell. This modification of the length scale results, for the stretched grids, in an increase of the filter width both in the lower and in the upper part of the domain, where the aspect ratios are smaller.*

_{k}*l*differs for stable and unstable stratification:where the Brunt–Vaisälä frequency is defined by

The formulation (14) implies, for stable regimes, *l* < Λ near the surface, where the local potential temperature lapse rate (and hence *N*) is larger. This is consistent with the presence of smaller scales in the wall region (Sagaut 2006).

In a recent study (Catalano et al. 2007) the same geometry was investigated directly, imposing a sinusoidal time dependence for the turbulent heat flux at the bottom boundary to provide the thermal forcing at the surface; also, the anisotropy correction was not applied and the classical Deardorff’s (1970) definition for the filter width was assumed.

This previous formulation does not take into account the coupling between momentum and heat fluxes and the resulting differences of surface heat fluxes in zones of the domain where different flow conditions occur. This leads to deviations from the real conditions, especially during the nighttime period when in the valley the prevalent heat exchange occurs by radiative cooling and small negative values of surface heat flux are likely to occur, while over the slope high negative values are expected because of the momentum contribution on the cooling of the air.

The domain geometry is composed of a valley, symmetric with respect to *xz* and *yz*|_{x=Lx/2} planes, where *L _{x}* is the length of the domain along the

*x*direction (Fig. 1); the same plot also shows the four locations of the vertical profiles considered for the discussion.

Three simulations with different volumes of the valley and slope angles were performed to analyze the influence of these key parameters on the flow pattern and the distribution of the temperature. It is important to recall here that, since momentum and heat fluxes are coupled, a change in the geometry, which results in a modification of the circulation, will also result in a change of the computed surface heat fluxes.

*θ*with respect to the values

_{s}*θ*, corresponding to the initial stratification defined by the Brunt–Vaisälä frequency

_{s}*N*(Table 1):where Δ

*θ*

_{s,max}is the amplitude,

*T*= 24 h is the period, and

*t*is the time. This choice for the surface thermal forcing is supported by the observations of Monti et al. (2002), done during the VTMX campaign. For the beginning of October they reported a sinusoidal variation of the surface temperature with maximum amplitude of ≅5 K around the daily average.

The origin of time axis roughly corresponds to the sunrise. A preintegration of 24 h is made to obtain a capping inversion over the valley. Periodic lateral boundary conditions are assumed for the *x* and *y* directions (Table 1) to reproduce an infinite succession of valleys and ridges along the *x* direction. The horizontal grid resolution is uniform, Δ*x* = Δ*y* = 50 m; the vertical grid is defined with a parabolic stretching to have Δ*z* ≅ 2 m near the ground and Δ*z* ≅ 90 m at the top of the domain. Table 2 summarizes the main parameters for the three simulations.

Since both geometry and boundary conditions are independent with respect to the *y* direction, the mean flow can be considered two dimensional on a vertical plane and the variables are averaged along the *y* direction.

The simulations were run for 3 days; at the end of the first day a capping inversion forms over the valley (Fig. 2) as a result of the nighttime cooling. The solution is then almost cyclic after the first complete 24-h cycle. Hereinafter we will discuss only the second day of the simulation; the time scale will then be adjusted to have 0 ≤ *t* ≤ 24 h for the second daytime cycle.

In the following, case 2 will be used as the reference simulation for the analysis of the results. The stable layer at the end of the preintegration period extends up to about 500 m into the basin with a Brunt–Väisälä frequency *N* ^{2} = 9.8 × 10^{−4} s^{−2}.

## 3. Daytime results

### a. Mean winds and temperature

During the daytime period, surface heating creates horizontal potential temperature gradients (Fig. 3), which, in turn, lead to the development of anabatic winds over the two slopes. At *t* = 5 h the thermal structure is characterized by the presence of a deep (≅800 m) mixed layer over the ridges and a shallow (≅200 m) convective boundary layer at the bottom of the basin. This is overlaid by a strongly stratified inversion layer with a vertical extension of about 200 m and a deep residual layer aloft. A significant along-slope potential temperature gradient, directed toward the ridge, is also observed. The symmetric geometry of the valley will force the circulation into a closed pattern, as can be seen from the streamfunction plot (Fig. 4). This representation was obtained after a decomposition of the wind field into spherical harmonics from which the streamfunction can be computed analytically by derivation (Adams and Swarztrauber 1997). Different features can be identified in the daily circulation:

- the anabatic current over the slopes, extending from the ground up to about 200 m;
- a horizontal breeze directed from the center of the valley toward the slopes, extending from
*z*≅ 450 m to*z*≅ 650 m; - a strong updraft and an associated intense roll vortex over the ridge;
- a region characterized by a free convection regime near the east and west lateral boundaries;
- a return current in the upper part of the domain directed from the ridges toward the center of the valley with a vertical extension ranging from 600 m in proximity of the top of the slope to 250 m over the valley;
- a deep (about 800 m) subsidence region from
*x*= 8000 m to*x*= 10000 m that closes the overall circulation.

The depth of the slope current *h* is determined, from the vertical profiles of *u*, by the height where the wind speed goes below 15% of its maximum value. The maximum intensity of the upslope wind is attained, for case 2, at *t* = 5 h of simulation. At that time, when the surface temperature is increased of ≅5 K, the depth of the current continues to increase because of the development of a horizontal breeze directed from the center of the valley toward the two ridges and located between 400 and 600 m. This current can be regarded as a secondary flow resulting from the layered thermal structure of the basin and can be explained considering the mutual influence of different factors. A first mechanism is related to the interaction of the subsidence current over the valley with the strong inversion layer below; the current is then forced to flow horizontally, interacting with the anabatic wind over the slope. An important aspect for the two-layer characteristic of the anabatic flow is the stronger stratification at low levels in the valley compared to that of the region above the inversion. A second mechanism is then related to the smaller mass flux associated with the lower slope region, which is characterized by a stronger potential temperature lapse rate. The large mass flux required by the upper circulation is then partially compensated by a horizontal flux at ridge height. A significant role in this phenomenon is also played by the higher values of the surface kinematic heat flux over the two ridges compared to the lower ones at the bottom of the valley at *t* = 5 h (Fig. 5), since *Q _{s}* drives and strengthens the return current and hence the subsidence flow.

A simulation run with the same geometry but with a constant prescribed surface heat flux (Catalano et al. 2007) did not revealed the two-layer feature of the anabatic flow. The daytime values of the surface kinematic heat flux over the slope (point B) are about 1.5 times larger than those observed on the valley floor (point D). Here *Q _{s}* shows a negative phase shift with respect to the imposed temperature on the slope and the valley base;

*Q*is in phase with the temperature on the ridge top. Also, the transition between positive and negative (or zero) heat flux occurs earlier on the valley ground than on the slope. The field measurements of the Riviera Project (Rotach et al. 2004) showed a similar daytime evolution of

_{s}*Q*for a steep symmetric Alpine valley. The authors attributed such behavior to the differences in exposure and slope angle. Instead, the similarity with the present results, which do not take into account the solar radiation, suggests that the surface heat flux evolution is mostly influenced by the local flow conditions (through the coupling with surface momentum fluxes). This complex distribution of the surface heat flux, which is in agreement with the numerical results of Chow et al. (2006), was not observed in previous studies, conducted imposing a surface heat flux constant in space (Catalano et al. 2007). The large negative values of the surface kinematic heat flux, associated with the nighttime period, are not comparable with those reported by Rotach et al. (2004) because of a sensible difference between the arbitrarily imposed surface temperature and the values measured during the night at that site.

_{s}The anabatic flow shows a distinctive layered structure, as appears in the two distinct maxima in the vertical profiles of horizontal velocity (Fig. 6). Vertical layering is a typical feature of thermally driven circulations over complex terrain (Fernando et al. 2001; Reuten et al. 2005; Reuten et al. 2007) and is a consequence of the interaction of multiple spatial and temporal scales. At *t* = 5 h the horizontal breeze is characterized by approximately the same intensity as the anabatic wind; as the upslope current decays at *t* = 9 h, the horizontal breeze is 3 times more intense than the anabatic wind. At the foot of the slope (Fig. 6b) the two-layer structure of the anabatic flow is even clearer and lasts for the entire daytime period; furthermore, a weak return current is observed between the anabatic wind and the horizontal breeze at ≅300 m. The intensity and depth of the upper return current are about the same in the middle of the slope and at its foot. The vertical profiles of *u* at *t* = 5 h taken at different *x* distances (Fig. 7) show that over the valley at *x* = 7750 m only the horizontal breeze is present; the anabatic current is very weak at *x* = 6500 m (about half of the strength of the horizontal breeze); at *x* = 5000 m the anabatic wind and the horizontal breeze have the same intensity and two different maxima can be observed; at the top of the slope (*x* = 3500 m) the two currents are completely merged and the wind is characterized by an intensity of 2 m s^{−1}. Because of the development of the horizontal breeze, the maximum vertical extension of the anabatic wind is reached later than its maximum intensity and is associated with a slower upslope current.

Over the ridge the stable layer developed during the night of the preintegration period is very shallow, hence the evolution of the convective boundary layer promptly interacts with the residual layer, generating a well-defined mixing layer (Fig. 8a); over the slope and in the middle of the valley the strong stable layer is gradually eroded. Over the slope the warm air, advected by the return current of the anabatic winds enhances the erosion of the inversion layer and the residual layer above is reached at 7 h (Fig. 8b). In the middle of the valley the extension of the stable layer is progressively reduced by the combined effects of the subsidence above and the growing of a mixing layer below (Fig. 8c). Princevac and Fernando (2008, hereafter PF) proposed a cold pool destruction mechanism based on the relative importance of the heat flux and the stratification. According to their conceptual model, supported by laboratory measurements in a V-shaped valley, when the stratification prevails (type I), the erosion of the nocturnal inversion is mainly driven by horizontal intrusion at middepth of the basin. Otherwise, if the heat flux is significant compared to the stratification (type CI), the destruction of the stable core is driven by upslope advection and subsidence over the valley center. The variables that determine the prevailing inversion breakup mechanism are the buoyancy parameter *B* = *N* ^{3}*h*^{2}/(*Q _{s}g*/

*θ*

_{0}) and the slope angle

*β*; the intrusion dominates for values of

*B*larger than

*B*=

_{c}*Cβ*

^{2/3}. With the value of the empirical constant (

*C*= 1750) suggested by PF, our results are expected to follow the type I mechanism for all the simulations, since

*B*∼ 10

^{3}and

*B*∼ 10

_{c}^{2}. However, the return current in the present cases is located higher up the valley top and we did not observe any intrusion at middepth of the basin and the characteristics of the cold pool breakup are ascribable to type CI. This can be explained by the different geometry of PF, in particular their negligible bottom width, as mentioned by the authors themselves. This phenomenon can affect the air quality of cities in similar environments, since the pollutant will be likely to stagnate for a long time under such conditions.

The subsidence over the valley induces warming, in agreement with the measurements made by Brehm and Freytag (1982). The warming effect is more important for smaller valleys, as can be seen from the comparison of the potential temperature profiles for cases 2 and 3 (Figs. 8c, 9).

It is interesting to note that in case 1, which has a gentler slope, the vertical profiles at mid depth of the valley (Fig. 10a) show that the anabatic current is completely merged with the horizontal breeze until 7 h, while at the foot of the slope (Fig. 10b) the horizontal breeze prevails. This demonstrates that the slope steepness plays a much more important role during the onset of the anabatic, in particular at the foot of the slope.

The high resolution of this LES study makes it possible to identify some characteristic microscale features of the anabatic current, like the presence of waves at the interface between the upslope current and the horizontal breeze (Fig. 11), resulting from the entrainment of warmer air into the anabatic current. A similar wave activity was observed by Schumann (1990) between the anabatic flow and the return current for slope angles of about 10°. In the current case, the presence of the horizontal breeze restricts larger overturning motions to the lower part of the slope (6000 ≤ *x* ≤ 6500 m), where a weak eastward current is observed at ≅300 m (Fig. 6b).

### b. Turbulence

Most of the past LES investigations assumed horizontal homogeneity and computed the statistics as spatial averages over horizontal planes or time averages on the steady-state solution. The present investigation considers time-varying surface forcing and the only homogeneous direction is *y*, hence we will compute the averages for every (*x*, *z*) location in space. To improve the data sample, the statistics are next averaged over a 40-min time interval. This choice for the averaging period allows including a significant number of the characteristic time scales, as suggested by Sakai et al. (2001). As an example, at *t* = 5 h, the PBL depth *z _{i}* ≅ 200 m,

*w*

_{*}≅ 0.81 m s

^{−1}, and the convective time scale

*t*

_{*}=

*z*/

_{i}*w*

_{*}= 247 s, hence the averaging interval is ≅10

*t*

_{*}.

The vertical profiles of TKE (total, resolved, and subgrid scale) at *t* = 5 h show (Fig. 12) that the maximum values (≅0.75 m^{2} s^{−2}) are attained over the ridge at *z* ≅ 700 m because of the instabilities associated with a strong roll vortex associated with the reverse of the anabatic current and the presence of deep thermals. In the middle of the slope (Fig. 12b), the maximum values of TKE (≅0.25 m^{2} s^{−2}) are located close to the surface; a second maximum is observed at *z* ≅ 1200 m, perhaps because of the shear generation at the interface between the return current and the free atmosphere. Over the valley (Fig. 12c), the maximum values, located ≅100 m above the ground, are about 5 times lower than over the ridge, whereas the very low turbulence activity in the upper layer can be explained by the weak intensity of the subsidence motions. The resolved part of TKE is the most important contribution, apart from the first levels close to the ground where the grid spacing, even with a vertically stretched grid, is still not enough to fully resolve the very fine structure of that part of the boundary layer. The horizontal breeze determines the presence of a relative maximum of TKE in the upper part of the domain over the slope (Fig. 12b). This mechanism can be of great importance in the computation of the exchange coefficient in the PBL parameterizations for mesoscale or GCM models, which usually neglect orographic effects, as long as simple one-dimensional models are used.

The level where the vertical profiles of TKE go below 15% of their maximum value is used to estimate the vertical extension of the PBL. It is important to recall here that the PBL depth differs from the height of the total anabatic flow (defined in section 3a), which includes the upper horizontal breeze. The maximum velocity of the anabatic wind component is located within 10%–20% of the boundary layer depth, while the maximum of the horizontal breeze is associated with the upper maximum in the TKE profiles.

To analyze the effect of the new formulation for the filter width it is convenient to examine the behavior of the simulated turbulent flow in wavenumber space. Since the vertical direction is not homogeneous we will limit the analysis to the horizontal wavenumber spectra. The one-dimensional spectra of the horizontal component of the velocity along the longitudinal wavenumber direction *k _{x}* are obtained by performing a two-dimensional Fourier transformation of the

*u*field for a fixed vertical level and then integrating the two-dimensional periodogram along the transverse wavenumber direction

*k*. Even if the flow described in this paper is characterized by a marked anisotropy and inhomogeneity along the

_{y}*x*direction, it is still possible to isolate a portion of the domain where homogeneous and isotropic conditions are approximately satisfied, like in the middle of the convective boundary layer over the valley. Furthermore, to reduce the inhomogeneities of the flow field being analyzed, a detrending operation is performed following Errico (1985).

*k*≡ |

**k**| =

*α*= 1.5. The

*u*component of the spectrum tensor along the

*k*wavenumber has the expression

_{x}*k*:where

_{x}*k*=

_{N}*E*

_{ux}(

*k*) differs from the log–log line with slope

_{x}Figure 13 shows the spectrum obtained for the valley subdomain (6500 m ≤ *x* ≤ 11 500 m) for level 8 of the grid [(*z*–*z _{s}*) ≈ 60 m] at

*t*= 5 h taken from case 2 and from the same simulation recomputed without the anisotropy correction; both the curves are compared with the filtered truncated Kolmogorov spectrum (K41). This level has been chosen because it is located in a well-mixed zone of the PBL, thus not being too close to the ground where the flow tends to be less resolved, and because the important aspect ratio of the grid at that level (∼5) allows us to evaluate the influence of grid anisotropy. The spectrum obtained with the corrected length scale clearly lies closer to the theoretical curve, apart from the high wavenumber region where data seems to be slightly affected by aliasing.

## 4. Nighttime results

### a. Mean winds and temperature

During the nighttime period, as the ground temperature goes down, a stable layer develops in the valley, thus eroding the residual layer just above (Fig. 14). The surface cooling is responsible for the development of downslope currents, flowing from the top of the two ridges. The maximum depth of the katabatic current (∼20 m), determined by the sign of the vertical profiles of *u*, is several times lower than the vertical extent of the anabatic flow. Instead, the intensity of the downslope wind is about 2–2.5 times greater than the anabatic one (Fig. 15). Another important difference with respect to the daytime circulation is the absence of a return current of significant intensity, in agreement with the flow measurements of Princevac et al. (2008) and the LES results of Skyllingstad (2003). The katabatic wind, in fact, can be interpreted as a low-level jet in a stratified flow.

The downslope wind is a gravity current, so the length and the angle of the slope are the most important control parameters. The vertical profiles of the wind speed show that, during the onset and the early stage of the katabatic current, the longer slope of case 1 (Fig. 15a) is characterized by intensity and depth similar to those of the shorter but steeper slopes (Figs. 15b,c). Next, the interaction with the growing stable boundary layer inside the basin will make the late evolution of the flow essentially governed by the volume of the valley. In fact, in the smaller basin the growth of the stable boundary layer is faster, causing an earlier damping of the katabatic wind.

In the valley system a stable stratification develops. Over the slope (Fig. 16), negative values of surface kinematic heat flux (Fig. 5) are associated with the katabatic wind. Over the valley (Fig. 17), *Q _{s}* is almost zero and the cooling is driven by the advection of cold air by the downslope currents. Three different zones can be distinguished in the vertical direction:

- A shallow surface layer with a strong lapse rate can be observed. Over the slope this layer corresponds to the vertical extension of the katabatic current (Fig. 16). The lapse rate in this zone reaches its maximum of 0.3 K m
^{−1}at 17 h, at the same time where the maximum wind speed is attained. - There is a stably stratified layer above with a smaller lapse rate, corresponding to the evolution of the cold pool in the basin; the intensity of the stratification increases during the night.
- There also is a residual layer above 500 m and extending up to the maximum daytime PBL depth.

Up to 17 h the atmosphere above the katabatic layer has a very small lapse rate of 0.007 K m^{−1}. After this time the depth and intensity of the katabatic currents begin to decay and the stratification extends to the higher levels of the atmosphere. This phenomenon can be interpreted considering that the katabatic wind produces an increase in the negative turbulent heat fluxes near the ground, which enhances surface cooling. At the same time, warmer air from the ridge is advected in the upper level of the downslope current; this contributes to maintain a shallow stable layer during the first hours of the night.

It is important to recall here that, during the night, over the ridge and the valley the heat exchange is almost zero, while over the slope it reaches absolute values even bigger than the daytime ones (Fig. 5) because of the intense momentum fluxes generated by the katabatic wind. Such large negative surface heat flux reflects a limitation of the imposed thermal forcing; in fact, prescribing the soil temperature has the drawback that the feedback with the uppermost soil layer, which would reduce the surface temperature anomaly, is neglected. Nevertheless, our choice for the thermal forcing stresses the importance of considering the coupling between heat and momentum fluxes instead of directly imposing the surface heat flux.

The main parameter that controls the evolution of the nocturnal PBL under this topographical configuration is found to be the volume of the valley; in fact, the smaller valley exhibits a faster cooling (Fig. 16). Slope steepness also plays an important role, since the faster current flowing over the steepest slope slightly reduces and delays cooling with the mechanism discussed above.

Intersecting the vertical profiles of potential temperature over the valley at different times with the neutral profile at the end of the diurnal period, it is possible to analyze the evolution of the depth *h _{p}* of the cold pool into the valley (Fig. 18). The life cycle of the nocturnal inversion starts when the lapse rate becomes positive near the valley floor and ends when the stability regime turns back to be unstable driven by the positive surface heat flux (morning of third day of simulation). In case 3 the cold pool evolves quickly during the first hours of the night, reaching a depth of ≅150 m at

*t*= 11 h, while for the other cases the stable layer has grown only a few tens of meters. A rapid increase of the cold pool is observed when the base of the former inversion layer is reached; this occurs at

*t*≅ 10.5 h for case 3, at

*t*≅ 11.5 h for case 1, and at

*t*≅ 12.5 h for case 2. At

*t*≅ 15 h the depth of the stable core is roughly the same for the three cases; from this point on the growth rate decreases for cases 1 and 2 and a difference in depth of ≅ 50 m with respect to case 3 is kept. As already mentioned, one can see the role played by the volume of the valley and the effect of the slope steepness in delaying the formation of the stable boundary layer.

### b. Turbulence

The maximum values of turbulent kinetic energy (≅0.5 m^{2} s^{−2}) over the slope are observed near the surface (Fig. 19). At this point the depth of the PBL (≅20 m) roughly equals the vertical extension of the katabatic current. Despite the increased vertical resolution near the ground, the horizontal grid spacing is still too coarse to fully capture all the fine turbulent structures in the lowest model layer, hence the modeled components of TKE are the most important contribution during the nocturnal period. In this sense, the results obtained for the nighttime period can be regarded as the output of a Reynolds-averaged Navier–Stokes (RANS) model with a second-order closure for the turbulence kinetic energy and a first-order closure for the temperature fluxes. The difference in the model behavior with respect to the diurnal period is due to the fact that the characteristic length scales of turbulence are strongly reduced in presence of stratification. It should be recalled here that the aim of this paper is to simulate the entire diurnal cycle over a quite large periodic ridge–valley domain. The choice of the spatial resolution is then a compromise between the need to resolve the main features of the circulation and the computational demand.

*t*= 17.5 h it is located between

*x*= 4500 m and

*x*= 5000 m, with an extension of 500 m in the horizontal and 100 m in the vertical. If we consider the atmosphere in the framework of a two-layer bulk hydraulic model (Ball 1956), the densimetric Froude numbercan be assumed to characterize the instability region. Hereis the reduced gravity,

*ρ*is the density of the katabatic layer, and

_{k}*ρ*is the density of the atmosphere above the current.

_{a}Although there is a dramatic decrease in Fr immediately after the recirculation zone, its values upwind are less than unity (Fig. 21); hence, the phenomenon cannot be strictly termed a jump. In any case, it must be said that the uncertainties on the determination of the height of the current make the estimation of Fr prone to some error, since *h* is the most sensitive parameter for its computation. It must also be observed that the values of *h* downwind of the jump refer to the small katabatic layer flowing below the instability zone, rather than the vertical extent of the zone itself. Despite the uncertainties associated with the computation of Fr, the structure has most of the characteristics of a katabatic jump: a significant positive vertical velocity can be observed in the middle of the jump (Fig. 22) as well as stronger values of turbulent kinetic energy (Fig. 20). Further evidence of the presence of the jump can be inferred from the surface kinematic heat flux plot in B (Fig. 5), which shows a strong discontinuity in correspondence to *t* = 17.5 h.

In agreement with Yu and Cai (2006), waves are observed downstream of the jump, as shown in the contour plot of the vertical velocity (Fig. 23); the wavelength is about 250 m. These authors attributed the generation of this kind of oscillations to the deformation induced in the wind field by the presence of an updraft region. It is worth noting that Yu and Cai (2006) considered intense (∼10 m s^{−1}) katabatic winds over strongly cooled and long (∼70 km) slopes in Antarctica. The present study confirms the development of such disturbances on shorter slopes in a valley; with this geometry, the growth of the cold pool assumes an important role in triggering mechanism for the formation of the jump.

## 5. Considerations on the complete cycle

The time evolution over a complete diurnal cycle of the maximum wind speed *U*_{max}, along with the height of the flow over the slope and the imposed surface temperature *θ _{s}* in the middle of the left slope (Fig. 24), gives a sketch of the response of the atmosphere to the variable thermal forcing in the presence of complex geometry. This plot also summarizes some of the main characteristics of the upslope and downslope winds, like the differences in intensity and vertical extension. Comparison of the three curves demonstrates that the maximum intensity of the anabatic current occurs about at the same time as the maximum surface forcing. During the day the maximum height of the flow takes place about 2 h later (

*t*= 8 h) with respect to the imposed temperature maximum because of the horizontal breeze. In the night the maximum depth of the katabatic wind is promptly reached 1 h after the inversion of the surface forcing, then it remains almost constant until the cold pool reaches the middle of the slope at

*t*= 19 h; from this time on the current begins to decay both in intensity and in depth. One can also observe an important drop (≅2 m s

^{−1}) in the wind speed and flow depth (≅15 m) at 17.5 h, when the weak jump is taking place upstream. It is worth noting that the weak jump is an isolated and almost instantaneous phenomenon in our case.

## 6. Comparisons

In the absence of an accurate dataset of measurements taken in conditions similar to those described in this work, we will compare the results with the prediction of two well-known and verified theoretical models. The differences between the hypotheses of the theoretical models and our geometry and boundary conditions pose some limitations that are discussed below.

The HU bulk theoretical model was developed for the anabatic flow over a finite slope and a ridge. Since this model assumes a steady-state flow, in order to make a comparison with measurements or results from prognostic numerical models, quantities must be considered averaged over the diurnal period (positive surface heat flux). The calibration parameter *λ* for the HU model, defined in (1), is obtained from the current simulations averaging over the depth of the anabatic layer the values of the wind speed taken in point B. The values of *λ* are compared with the field measurements of Princevac et al. (2001), the laboratory investigation of Chan (2001) and Cenedese et al. (2004), and the simulations of Catalano et al. (2007). It can be seen (Table 3) that the development of the horizontal breeze, which is a peculiarity of our investigation, causes lower values of *λ* because of the resulting higher values of the flow depth.

The MS model predicts the depth of the katabatic wind as a function of the distance from the beginning of the slope; they considered a semi-infinite slope. Our results for cases 1 and 2 (Figs. 25a,b) show a fairly good agreement for the upper part of the slope, then they deviate from the theoretical predictions as the wind impacts with the quiescent air of the cold pool. For case 3 (Fig. 25c), at this time (17.5 h) the current has been significantly weakened by the cold pool development and hence the comparison is quite poor. An interesting feature is that the depth of the katabatic wind inside of the cold pool layer is reduced proportionally to the length of the slope, with the longer slope (case 1) showing a deeper residual flow. Another difference with respect to the theoretical predictions can be seen near the upper end of the slope because the MS model was developed for a slope without a ridge; here it appears that the current tends to propagate downward on the ridge for a certain extent with a depth of ≅5 m.

The plot for case 2 gives further evidence of the weak jump, which appear as a discontinuity in the flow depth with a slight increase upwind and a strong decrease downwind.

## 7. Summary and conclusions

The three-dimensional nonhydrostatic meteorological model WRF was modified to perform large-eddy simulations of the complete diurnal cycle of the thermally driven circulation over a valley under calm geostrophic wind conditions.

To correctly reproduce the fine structures of the anabatic and katabatic winds, a vertically stretched grid was introduced with a finer mesh close to the ground. With highly anisotropic grids, problems arise with the classical definition of the filter width, so the length scale of the SGS model was modified according to the theoretical considerations of Scotti et al. (1993).

Most of the past LES studies in the literature introduced the forcing at the bottom boundary of the domain by directly imposing the surface kinematic heat flux as the right-hand side term in (4). The most important drawback of this formulation is that it does not take into account the coupling between heat and momentum fluxes in the surface layer and thus cannot reproduce the differential heating of zones with different flow characteristics. In the present study the LES model is coupled with a surface layer scheme; the surface heat flux as well as the friction velocity were computed according to the Monin–Obukhov similarity theory. Differences with respect to a previous study without coupling of Catalano et al. (2007) were emphasized. It should be recalled here that prescribing the surface temperature anomaly (as in this study) still has the drawback that the feedback with the uppermost soil layer cannot be simulated.

The evolution of the diurnal boundary layer was investigated, pointing out the great importance of the subsidence over the valley and its related warming. A horizontal breeze develops as a consequence of the layered thermal structure of the basin. This flow can be explained considering the mutual influence of two principal factors: 1) the interaction between the subsidence current over the valley and the strongly stratified layer below; 2) the smaller mass flux associated with the lower slope region, which is characterized by a stronger stratification. The large mass flux required by the upper circulation is then partially recovered horizontally. The higher values of the surface kinematic heat flux over the ridge compared to those at the bottom of the valley also cover an important role in this phenomenon.

The waves at the interface between the anabatic wind and the horizontal breeze were correctly reproduced and shown by the streamlines. The parameter *λ* from the HU theoretical model was computed and compared with previous investigations; the most relevant effect of the valley geometry is a lowering of this parameter due to the higher values of the flow depth caused by the interaction of the anabatic wind with the horizontal breeze.

The nocturnal PBL was investigated along with the evolution of the cold pool into the valley. It was found that the volume of the valley is the most important parameter for the evolution of the katabatic winds in such geometry, since it controls the rate of the growth of the stable boundary layer into the basin. The resolution of this study, even if not large enough to fully resolve the small surface structures of the stable boundary layer, allowed the capture of the flow separation that occurs as the fast katabatic wind impacts with the quiescent air of the cold pool. Comparisons with the MS hydraulic model show an agreement outside of the cold pool layer; in the lower part of the basin the depth of the katabatic current is reduced proportionally to the volume of the basin and the length of the slope. The formation of a hydraulic jump induces an increase of the flow depth upwind and a sensible decrease downwind of the discontinuity with respect to the theoretical predictions.

During the nighttime period the SGS contribution to total TKE is significantly larger than the resolved one, thus the results should be interpreted like those of a RANS model with a second-order closure for the turbulence kinetic energy and a first-order closure for the temperature fluxes. The compromise on the spatial resolution is motivated by the aim of this work to reproduce both diurnal heating and nocturnal cooling over a quite large periodic ridge–valley domain.

We thank Dr. Stefania Espa for her helpful discussions about the spectral analysis. We express our gratitude to three anonymous reviewers whose comments led to significant improvement of the manuscript.

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Domain configurations: geometry is given in Fig. 1; *V*_{valley} is the volume of the basin.

Parameters of the simulations: Δ*θ*_{s,max} is the amplitude of the imposed thermal forcing, *N* ^{2} is the initial Brunt–Väisälä frequency, and *z*_{0} is the surface roughness. LBC denotes lateral boundary conditions.