## 1. Introduction

On fair-weather days, transport and mixing of heat, moisture, and other constituents over mountainous terrain is strongly influenced by the thermally forced mountain circulations, the slope, and valley winds. These mountain flows also play a key role in the formation of clouds and convection initiation (Banta 1990). Also, the quantification of the associated exchange processes is important for many applications such as air-quality studies, numerical weather prediction, and climate modeling (e.g., Rotach et al. 2004, 2008; Gohm et al. 2009). Nevertheless, there is still some uncertainty and debate regarding the basic role of the diurnal mountain flows and turbulent convection for exchange over complex terrain and in particular for the enhanced diurnal temperature amplitudes observed in valleys.

Why is the daily temperature range typically much larger in the interior of valleys than over the foreland? At the core of the current debate is the role of the slope-flow-induced cross-valley circulation for the heating of the valley atmosphere. It is clear that the cross-valley circulation acts to redistribute the heat within the valley atmosphere such as to equalize the temperature contrasts between the heated sidewalls and the valley core (e.g., Vergeiner 1982; Vergeiner and Dreiseitl 1987). But what is the net effect of the cross-valley circulation on the overall valley heat budget? Does it export heat out of the valley as suggested, for example, by Noppel and Fiedler (2002) and Schmidli and Rotunno (2010, hereafter SR10) or does it contribute to the daytime heating of the valley atmosphere as proposed by Rampanelli et al. (2004) and Serafin and Zardi (2010a, 2011)?

The major cause for the enhanced temperature range in valleys has long been attributed to the valley-volume effect, which can be quantified in terms of a topographic amplification factor (TAF; Wagner 1932a). The TAF concept is based on an argument stating that a given amount of energy input applied to a valley heats a smaller volume, and hence smaller mass, of air than if the same energy input is applied over the foreland, resulting in a larger heating rate of the valley atmosphere. Key to explaining the large observed differences between the valley and the foreland is a proper calculation of the volume effect by including the volume of side valleys and the effects of static stability (Steinacker 1984). Previous to Steinacker's seminal paper, the volume effect was not necessarily regarded as the key factor for explaining the enhanced valley heating; indeed, even Wagner regarded it as a secondary effect, in comparison to other effects associated with enhanced exchange within valleys and reduced exchange with the free atmosphere (Wagner 1932a, 1938).

Recently, the role of the valley-volume effect for the enhanced valley heating was questioned by Rampanelli et al. (2004) and Serafin and Zardi (2010a, 2011). Observing that the valley volume is not closed owing to overshooting slope flows, they suggest that the heating associated with the compensating subsidence in the valley center is an important additional contributor to the enhanced valley temperatures and that the cross-valley circulation brings down warm air from the stable free atmosphere (Serafin and Zardi 2011).

It appears that much of the recent debate and confusion regarding the valley-volume effect and subsidence heating arises from a failure to clearly distinguish between the local and bulk perspective of valley heating. In the local perspective, local temperature tendencies at any given point are explained in terms of local processes—in particular, processes leading to local advective and turbulent heat flux divergences. In the bulk perspective, the goal is to analyze the processes affecting the bulk temperature—that is, the valley-volume-averaged temperature. Clearly the TAF concept belongs to the second category.

As was pointed out by SR10, the local subsidence heating mechanism is not inconsistent with the volume effect argument. In fact, subsidence in the valley core has long been considered a key mechanism for heating the valley atmosphere (e.g., Vergeiner 1982; Vergeiner and Dreiseitl 1987). From the bulk perspective, however, the net effect of the thermally induced overshooting cross-valley circulation is to transport heat out of the valley, thus reducing the warming of the valley (Noppel and Fiedler 2002; SR10).

So far we have considered the role of the mean flow, the slope and valley winds, on the valley heat budget. Clearly, the heat transfer induced by turbulent convection is also important. Serafin and Zardi (2010a) show that the local heating tendency at any specific point in the valley atmosphere is the result of the interaction between top-down heating due to compensating subsidence in the stable valley core and bottom-up heating due to turbulent convection. This interaction leads to the well-known and frequently observed three-layer thermal structure of the valley atmosphere (e.g., Whiteman and McKee 1982; Brehm 1986). Also, strong mountain-top plumes lead to enhanced turbulent entrainment over the mountain ridges (Serafin and Zardi 2010a). This raises the question whether overall turbulent entrainment is increased over complex terrain in comparison to flat-terrain convective boundary layers (CBLs).

Although heat transfer over mountainous terrain is strongly influenced by turbulent convection and boundary layer (BL) structure, most modeling studies so far are based on mesoscale models using PBL turbulence schemes—that is, on solving the Reynolds-averaged Navier–Stokes equations. The crude representation of turbulence processes in typical PBL turbulence closures results in large uncertainties in BL structure (e.g., Schmidli et al. 2011) and the representation of turbulent processes such as entrainment.

The purpose of this paper is to clarify the role of the mean flow and turbulent convection in daytime heat transfer and exchange processes over mountainous terrain and to document the relevant structures of the daytime mountain boundary layer. Although the main focus is on the issue of enhanced valley heating, the broader aim is to clarify key principles of heat transfer in stratified fluids. Several three-dimensional large-eddy simulations (LESs) are performed for two- and three-dimensional topographies, corresponding to infinite and finite valleys, in order to evaluate the influence of the along-valley wind and the valley surroundings on the heat transfer processes and the cross-valley circulation. Key to our approach is the clear distinction between the local and bulk perspective in combination with the Reynolds decomposition of the turbulent flow into its mean and turbulent component.

The local and bulk perspectives are introduced in section 2. Section 3 describes the experimental setup and numerical model. The structure of the boundary layer and the associated cross-valley circulation is presented in section 4. The heat transfer processes are analyzed from the local and bulk perspective in section 5, followed by a sensitivity analysis, discussion, and conclusions in sections 6, 7, and 8.

## 2. Analysis methods

### a. Reynolds averaging and flow decomposition

### b. The local valley heat budget

^{r}indicates the filter operation and

**H**is the subfilter turbulent heat flux; see appendix B for further information on the LES and subfilter model and the diagnosis of the Reynolds-averaged fluxes. The lhs is the ensemble-mean local heating rate. The terms on the rhs are the heating (cooling) due to mean-flow advection, resolved-scale turbulent heat flux divergence, and subfilter-scale turbulent heat flux divergence. Under the usual Boussinesq approximation, the resolved turbulent term can be expressed as

### c. The bulk valley heat budget

*V*, typically the valley volume up to ridge height. The volume-averaged density-weighted heat budget equation is

*M*is the total mass of air in the control volume and

*H*

_{0}is the kinematic surface sensible heat flux normal to the surface. Equation (5) follows from the integration of (4) and by using Gauss' theorem to convert the resulting volume integral of the turbulent heat flux divergence into a surface integral and by decomposing the resulting surface integral into a land surface part

*A*and an atmospheric part

_{S}*A*. In other words, the net density-weighted volume-averaged temperature tendency (NET) is equal to the sum of the contributions due to the surface sensible heat flux (SHF), mean-flow advection (MEA), and turbulent heat exchange through

_{A}*A*(TRB_E).

_{A}### d. Description of PBL structure

The convective PBL is characterized by two layers: the mixed layer (ML), where profiles of potential temperature and other quantities are approximately height invariant, and an entrainment zone, which provides a transition from the turbulent ML to the (often stable) lower troposphere (Stull 1988). The top of the entrainment zone corresponds then to the top of the (turbulent) PBL. Over flat terrain the ML and PBL top are highly correlated; over complex terrain, that is not necessarily the case. Two different gradient methods are used to define the top of the ML and the PBL, respectively.

The ML height is defined as the height where the magnitude of the vertical gradient of along-valley-averaged potential temperature exceeds a predefined critical value of *γ _{c}* = 0.001 K m

^{−1}(i.e., at the base of the capping inversion). It is typically slightly below the height of the minimum average heat flux. The choice of the critical value follows Catalano and Moeng (2010).

The PBL height is defined as the vertical location of the largest increase in potential temperature—that is, the vertical position of the maximum *θ* gradient (following the gradient method of Sullivan et al. 1998). The PBL height is defined locally—that is, *z*_{PBL} = *z*_{PBL}(*x*, *y*, *t*). It is slightly below the height reached by the strongest thermals but above the height of the minimum average heat flux; it is well correlated with the entrainment interface—that is, with the top of the entrainment zone (Sullivan et al. 1998).

## 3. Numerical model simulations

### a. Experimental setup

To investigate the heat transfer processes over complex terrain, numerical simulations for idealized topographies are carried out. Different two- and three-dimensional topographies, corresponding to infinite and finite valleys, are used in order to investigate the influence of the along-valley wind and the valley surroundings on the heat transfer processes. The topographies are identical to those introduced in SR10 and Schmidli and Rotunno (2012, hereafter SR12) and are thus not shown. The experimental and numerical setup differs in several aspects from these previous studies and is therefore summarized below for the reference simulation. Further experiments are introduced in section 3c.

The physical setting of the reference case is that of an infinitely long, narrow, and moderately deep valley, periodic in the cross-valley direction, with the following dimensions: valley depth of 1.5 km, slope width of 9 km, and a crest-to-crest width of 20 km (see Fig. 2). The maximum inclination of the slope is 14.7°.

The simulations are started from an atmosphere at rest. The initial condition for the atmosphere is given by the potential temperature distribution *θ*(*z*) = *θ _{s}* + Γ

*z*, where

*θ*= 297 K, Γ = 3 K km

_{s}^{−1}, the surface pressure

*p*= 1000 hPa, and a constant relative humidity of 40%. The uniform land surface characteristics are summarized in Table 1. To simplify the analysis, the land surface properties and soil moisture are chosen such that the surface latent heat flux is virtually zero.

_{s}Summary of surface characteristics. The abbreviation *T _{a}* refers to the atmospheric surface temperature.

To obtain a near-steady state of the turbulent flow with quasi-steady turbulence statistics, a constant and horizontally uniform shortwave forcing is applied. This corresponds to a fixed position of the sun in the zenith and the neglect of the dependency of incoming radiation on the slope angle. A prescribed incoming shortwave radiation of 400 W m^{−2} is used, resulting in a domain-mean surface sensible heat flux of about 150 W m^{−2}, which roughly corresponds to typical daytime values in nonarid midlatitude valleys (Rotach et al. 2008). The current setup with prescribed radiative forcing allows for dynamic feedback of the evolving flow on the surface heat flux, which is in contrast to a setup with a prescribed surface heat flux.

The simulations are run for 6 h, with a focus on the situation after 4 h, representative of typical midday conditions in a deep midlatitude valley. As the depth of the CBL and the evolution of the along-valley wind is primarily determined by the time-integrated heating, simulations with constant and time-varying forcing will produce similar solutions, provided instances with the same amount of integrated heating are compared (Schmidli 2012). Regarding the spatial homogeneity assumption of the forcing, it is well known that surface heat fluxes are often very inhomogeneous in complex terrain (Rotach et al. 2008). While an inhomogeneous forcing may influence the symmetry of the evolving cross-valley circulation, it does not alter the basic heat exchange mechanisms between the valley and the free troposphere in complex terrain.

### b. Numerical model

As in SR10, the numerical simulations have been carried out using the Advanced Regional Prediction System (ARPS) model (Xue et al. 2000, 2001). In contrast to SR10, the model is run using variance-conserving fourth-order momentum advection and monotonic scalar advection, and, to ensure energy conservation, a flux-corrected transport scheme is used for potential temperature. The subfilter turbulence model is based on a 1.5-order turbulent kinetic energy (TKE) closure (Deardorff 1980). The standard ARPS implementation is modified in order to separate the implicit filter operation from the computational mesh, thus making the filter operation independent of the vertical grid stretching (see appendix B). Surface turbulent heat fluxes are based on similarity theory (Monin and Obukhov 1954; Byun 1990).

The computational domain for the reference simulation is 40 km in the cross-valley direction and 9.6 km in the along-valley direction. The horizontal grid spacing is 50 m. In the vertical, the domain extends to 5 km and the grid spacing varies from 8 m near the surface to a maximum of 50 m above 2.7 km. For most of the CBL depth, however, the grid has a constant vertical spacing of 20 m. The time step is 1 s. The lateral boundary conditions are periodic in both directions, thus ensuring that no mass and energy is transported into or out of the computational domain. At the upper domain boundary a rigid lid is employed with a Rayleigh sponge layer extending from 4 km to the top of the domain. To initiate turbulent motions the potential temperature on the five lowest model levels is disturbed with a random perturbation with a maximum amplitude of 0.5 K.

*α*is the solar surface albedo;

_{s}*ε*

_{a}and

*ε*

_{g}are the atmosphere and ground emissivities, with parameter values (

*α*,

_{s}*ε*

_{a},

*ε*

_{g}) = (0.27, 0.725, 0.995); and

*T*and

_{a}*T*are the surface air temperature and the ground temperature, respectively.

_{s}### c. Sensitivity experiments

The focus of the present study is on the detailed analysis of the reference simulation (REF2D) described above. Nevertheless, further simulations are undertaken in order to investigate the robustness of the findings derived from the reference simulation. These simulations are listed in Table 2 and discussed in section 6. They include simulations with three different valley-plain configurations and a cross-valley wind. Further simulations with different grid spacing and different lower boundary conditions are discussed in appendix C.

List of experiments.

The periodic configuration (VAL3D, identical to PERIODIC in SR12) consists of an infinite number of parallel valleys opening onto a common plain. The plain configuration (PLAIN, as in SR12) denotes an infinite valley formed by two isolated mountain ridges on a horizontal plain. The plateau configuration (PLATEAU, as in SR12) consists of an infinite valley cut into a large-scale plateau.

## 4. Flow evolution and structure

To provide some context for the following analysis of the heat transfer processes and valley heating, the evolution and structure of the mountain boundary is shown first, including the thermally forced circulations over the idealized valley. More detailed discussions of the PBL evolution over similar topographies can be found in the recent LES studies by Catalano and Cenedese (2010), Catalano and Moeng (2010), and Serafin and Zardi (2010b).

Since the flow is primarily driven by the surface heating, the time evolution of the *y*-averaged surface sensible heat flux is shown in Fig. 1a. The surface heat flux reaches a quasi-steady state after about 1.5–2 h, soon after the full development of the turbulence. The spatial pattern and magnitude of the surface friction velocity also attains a quasi-steady state after about 2 h (not shown). Despite a spatially uniform solar forcing, the surface heat flux varies strongly in the cross-valley direction, with the lowest values over the valley floor (≈120 W m^{−2}) and the highest values close to the mountain ridge (≈160 W m^{−2}). The cross-valley variation provides evidence of a strong coupling between the land surface and the slope flows. The time evolution of the *y*-averaged PBL depth is shown in Fig. 1b. In contrast to the surface heat flux, it generally does not reach a steady state, but shows a complex evolution with continued growth over the ridge but a peak and decline over the valley center. The latter is due to stronger downward motion over the valley center associated with the increasing strength of the upper circulation cell (see below).

### a. The mean flow

The mean cross-valley circulation and thermal structure including diagnostics of ML and PBL height are shown in Fig. 2 for *t* = 3 and *t* = 4 h. Differences between the two times are small. Apart from minor differences due to the slowly changing thermal structure, the flow is in a quasi-steady state. The mean cross-valley circulation is divided into two distinct circulation cells, separated by the valley inversion—the region of enhanced stability above the valley mixed layer. The lower cell is contained within the PBL and consists of the upslope winds and the compensating return flow over the valley center. The upper cell is associated with the strong thermal above the mountain ridge. The separation of the two circulation cells is clearly visible in the vertical velocity field; there is strong subsidence both below and above, but not in, the valley inversion.

Profiles of potential temperature at the valley center, over the slope, and at ridge top are shown in Fig. 3. The profile over the valley shows a three-layer structure, consisting of a lower-level mixed layer, a near-neutral layer at upper levels, and a stable layer in between, which is typical for many valleys (Brehm 1986). The profile at ridge top appears similar to that of the flat-terrain CBL. Over the slopes the shallow mixed layer is capped by an inversion layer and a deep weakly stratified layer. Further details on the mean-flow structure are provided in appendix C.

### b. Instantaneous flow patterns

The instantaneous patterns of the flow in terms of potential temperature and vertical velocity perturbations at *t* = 4 h, when the mean cross-valley flow is well developed, is shown in Fig. 4. In the horizontal sections an irregular cellular pattern, typical of a CBL, is observed. The typical size of the cells varies strongly in space, with larger cells over the valley center and the lower slopes and smaller cells over the upper slopes and over the ridge. The extent of the thermal plumes (the large energy-containing eddies) is clearly visible in the vertical cross section of the vertical velocity perturbations. It can be seen that they penetrate well into the weakly stable layer. The strongest temperature perturbations are found close to the surface and in the capping region above the mixed layer.

### c. Distribution of second-moment statistics

Figure 5a shows a vertical cross section of the total TKE at *t* = 4 h along with the mean wind vectors and the diagnosed ML and PBL height. Total TKE has three local maxima: one over the valley center, one over the lower slope, and the strongest maxima over the mountain ridge. To understand the cross-valley variation of the TKE, the individual components are presented in Figs. 5b–d. The color scale for the TKE components is chosen such that all four panels would look identical if the turbulence were isotropic. In most regions, the *w* component is the most important contribution to the TKE, as would be expected for a CBL with weak mean winds. Note that the ML and PBL heights correlate quite well with specific TKE contours: 0.5 m^{2} s^{−2} for the ML height and 0.1 m^{2} s^{−2} for the PBL height.

The turbulent fluxes and the potential temperature variance^{1} are shown in Figs. 5e–h. The vertical turbulent heat flux

## 5. Analysis of heat transfer processes and valley heating

As mentioned in the introduction, much of the recent debate regarding the valley-volume effect is likely due to the lack of a clear distinction between the local and bulk perspective of valley heating, where the former provides insight into the mechanisms leading to local temperature changes, while the latter is concerned with the dynamics of the valley-volume-averaged bulk temperature. Next, valley heating and heat transfer is analyzed from both perspectives with an emphasis on the separation of mean-flow and turbulent contributions.

### a. Local perspective

The spatial distribution of the net heating rate and the local heat budget components, according to (4), is shown in Fig. 6. Strong, relatively uniform, net heating is observed within the valley below the inversion layer. The net heating rate decreases within the inversion layer, but remains positive across the valley section up to the altitude of the mixed layer over the mountain ridge (ridge thermal).

The net heating rate is spatially relatively uniform despite large gradients in the magnitude of the individual budget components. This provides evidence of strong coupling between the mean flow and the turbulence or in other terms of the effectiveness of the thermally induced flows in minimizing horizontal temperature gradients and ensuring a horizontally uniform heating of the valley atmosphere. The quantitative interplay between the two heating mechanisms is even more obvious in the profiles of the local heat budget terms shown in Fig. 7 for different locations. At all locations, the vertical variations of mean-flow and turbulent heating rates mostly compensate each other resulting in nearly uniform heating throughout the (turbulent) PBL.

Based on the heating patterns related to the mean flow and turbulence, five distinct zones can be distinguished: the slope wind layer, the ridge-top region, and the three layers over the valley center (cf. Fig. 3). In the mixed layer over the valley floor, the heating is dominated by the turbulent heat flux convergence, while at higher elevations mean-flow temperature advection is the dominant term. In the slope wind layer, there is a strong cooling tendency due to mean-flow advection of cooler air from below. This cooling is overcompensated by turbulent heating resulting in horizontally uniform warming of the valley atmosphere.

Regarding the question of potentially enhanced turbulent entrainment over complex terrain, it is found that the cross-valley-averaged entrainment heat flux is smaller than typically found over flat terrain. Although the entrainment heat flux attains 65% of the corresponding surface heat flux over the mountain ridge, the spatially averaged normalized entrainment heat flux is only 16%, owing to small entrainment fluxes over the slopes (minimum is 6%)—clearly smaller than the 25% obtained in a corresponding simulation with flat terrain (not shown). As with respect to the heat budget, it should be stressed that the entrainment heat flux merely redistributes energy from higher to lower levels and can thus only affect the heat budget if the area of upper-level cooling lies outside the control volume. Even in the hypothetical case of an “optimally” chosen control volume, the amount of lower-level warming over a 1-h period attributable to entrainment (i.e., upper-level cooling) is found to be only 2.9% for the valley case in comparison to 10.5% for the corresponding flat-terrain case (not shown).

The local heat budget is often discussed in terms of individual coordinate components—in particular, the vertical Cartesian component—although these components are of course not intrinsic physical quantities as they depend on the choice of the coordinate system. The spatial distribution of the individual Cartesian components is shown in Fig. 8. Focusing first on the three advection components, the decomposition shows that most of the mean-flow-induced warming over the valley is induced by subsidence (i.e., by pure vertical motion) but with significant contributions from the horizontal flow in the region of the upper circulation cell. Over the slopes, the individual Cartesian components are not really meaningful. They obscure what is really going on—namely, cooling by along-slope cold-air advection.

The decomposition of the turbulent component provides interesting insight. The individual components clearly demark the regions influenced by turbulent motions (and gravity waves generated by overshooting thermals), which is not always the case for the net component (e.g., in the valley inversion layer). Insight into the turbulent heat transfer process can also be gained. The vertical component carries the heat away from the heated surfaces. In the mixed layer it is the dominant component. Near the surface and in the inversion layer, horizontal motions induce cooling by transporting cooler air into the rising eddies near the surface and transporting cooler air away from overshooting thermals in the inversion layer.

In summary, the local heating is primarily due to mean-flow subsidence in the free valley atmosphere, turbulent heat transfer in the mixed layers, and a combination of both processes in the weakly stable regions above the mixed layers. The interaction of top-down advective warming and bottom-up turbulent warming has been pointed to also by Serafin and Zardi (2010a). Although these two processes might appear to be mutually exclusive (Serafin and Zardi 2010a, p. 3750), in fact there is a gradual transition from one to the other.

### b. Bulk perspective

Next, the processes that contribute to the evolution of the valley bulk temperature are examined. The time series of the bulk heat budget components (5) for the entire valley (see Fig. 9a) are shown in Fig. 9b for the reference simulation REF2D along with the corresponding heat budget for the valley wind simulation VAL3D (discussed in section 6). The net effect of both the advective and turbulent heat fluxes is to export heat out of the valley. Because of the stable stratification in the upper part of the valley, the total heat export is relatively small, amounting to about 15%–20% of the surface heat flux. The bulk heat budget is almost steady; only the turbulent export is increasing slowly throughout the simulation period.

How can the bulk heat budget be interpreted in terms of the heat fluxes in and out of the control volume? First, the advection term in (5) can also be converted into a surface integral, in analogy to the turbulence term. Second, owing to the symmetry of the setup, the only relevant fluxes are the advective and turbulent exchange fluxes through the upper surface of the control volume and of course the heat input by the surface sensible heat flux. Figure 10a shows the cross-valley variation of the mean advective and turbulent heat fluxes through the upper surface of the control volume (see Fig. 9a)—the integrands of the surface integrals. Indeed, there is a large downward heat flux over the valley center as implied by the subsidence heating argument, but it is overcompensated by an even larger upward heat flux over the ridges. Thus, the net effect of the mean-flow circulation is to export heat out of the valley as shown in Fig. 9b. The magnitude of the mean advective heat flux through the upper surface is rather arbitrary. Because of mass conservation (mass export due to the slope flows equals mass import due to subsiding motions), the two opposing heat fluxes largely compensate each other; the net effect of the circulation, however, is an export of heat out of the valley.

_{0}(

*z*) denotes the spatial mean at a given height

*z*.

^{2}This definition ensures that the perturbation temperature is small and that the compensation between upward and downward vertical heat fluxes is eliminated. The heat exchange through a horizontal control surface

*A*is then given by

The cross-valley variation of the perturbation advective heat flux

### c. Analysis of mean-flow vertical heat flux

*dz*and assuming stationary conditions, the divergence of the along-slope heat flux (UHT) is given by (Brehm 1986)

*H*

_{0}is the surface kinematic heat flux,

*ε*is a detrainment parameter,

*α*is the slope inclination, UH is the bulk along-slope mass flux, and

*θ*

_{0}is the valley core temperature. The first term on the rhs represents the net heat input into the slope wind layer by the surface heating, while the second term represents the advective cooling due to the along-slope flow for a given mass flux UH. If the surface heating term is larger than the advective cooling term, the along-slope heat flux increases with height.

*ε*. The steady-state mass flux implied by the bulk model is compared with the LES results in Fig. 11a. Note the excellent agreement between the two in the region of the stable layer around 1 km AGL. This indicates that the slope-flow-induced mass flux is primarily determined by factors external to the slope wind layer and largely independent of turbulent processes in regions with sufficient stratification (

*ε*≈ 0).

Determination of the associated along-slope heat flux requires a parameterization of the mean temperature excess *θ*_{*} in the slope wind layer. Provided a good parameterization was available, the slope-flow-induced vertical heat flux could be also estimated a priori (see Fig. 11b). Here, the mean temperature excess has been determined by fitting to the LES results.

In summary, given the valley geometry and surface forcing, the vertical mass and heat fluxes at a given height are largely controlled by the atmospheric stability of the valley core at the same height. Thus, exchange between the land surface and the free atmosphere will be strongly controlled by the height regions with the strongest stratification (i.e., inversions, stable layers). Given the success of the bulk model, it might serve as a starting point for the development of a parameterization of thermally induced subgrid-scale exchange over complex terrain in coarse-resolution models.

### d. Horizontal advective heat transport

In addition to vertical exchange, the thermally forced circulations also induce horizontal exchange of heat and other properties. The horizontal heat exchange is important, for example, regarding the role of the thermally forced circulations for the initiation of convection. Figure 12 shows time series of the bulk heat budget components for a valley and a ridge volume. Both control volumes extend from midslope to midslope and from the surface to an altitude of 3000 m. The heating rate induced by the surface heat flux is much larger for the ridge volume than for the valley volume, mainly as the result of the volume effect, and to a lesser extent owing to the higher surface heat flux over the ridge region (see Fig. 1). Horizontal heat exchange between the two control volumes due to turbulence is negligible. Nevertheless, the net heating rate is larger for the valley volume. This is the result of the advective heat transport from the ridge to the valley volume. As the union of the two control volumes is approximately closed, the advective heat loss from the ridge volume has to be equal to the advective heat gain in the valley volume. Thus, the net effect of the slope-flow-induced circulation is to transport heat away from the ridge or from the mountain peak in the context of a mountain-plain setting. In other words, the mountain circulation cannot support convection initiation through the convergence of heat as hypothesized by Raymond and Wilkening (1980), but only through the convergence of moisture, or by providing a triggering mechanism (e.g., Banta 1990).

## 6. Influence of along-valley wind and other factors on the heat transfer processes

The reference simulation is characterized by a very idealized and symmetric setting. There is no along-valley wind, no larger-scale upper-level winds, and enforced symmetry over the ridge top owing to the chosen topography and lateral boundary conditions. In this section the generality of the results for the infinite valley is investigated by relaxing these assumptions (see Table 2 for a list of experiments). Further sensitivity tests related to the specification of the lower boundary conditions are described in appendix C.

### a. Along-valley wind

To investigate the influence of the along-valley wind on the BL evolution and the heat transfer processes, a large-eddy simulation with a three-dimensional valley-plain topography has been performed (VAL3D). The topographic configuration is identical to that for PERIODIC in SR12, allowing the development of an along-valley wind.

Figure 13 shows the mean cross-valley circulation and up-valley wind speed at two different locations at *t* = 4 h. Near to the valley entrance, the maximum up-valley wind already exceeds 3 m s^{−1}, while farther up valley it attains a value of about 1 m s^{−1}. Note that in contrast to the slope circulation, the along-valley wind has not yet reached a steady state and it will continue to increase. Despite the up-valley wind, the mean cross-valley circulation and boundary layer structure are very similar to that of the reference case, in particular at *y* = 50 km (cf. Fig. 2). The main difference is the weaker circulation and subsidence in the upper circulation cell in the section close to the valley entrance. The spatial distribution of the turbulence statistics is generally also very similar, but with higher TKE values, in particular near the valley entrance (not shown). The higher TKE values are to be expected, as the up-valley wind results in additional shear generation of TKE.

Although the influence of the up-valley wind on the local structure is relatively small, the bulk valley heat budget (see Fig. 9) is changed significantly. The up-valley wind brings cooler air into the valley resulting in strongly enhanced heat loss in the valley volume. In addition, the turbulent heat loss is also increased owing to the higher levels of turbulence—in particular, close to the valley entrance (see also Table 3). Finally, the SHF is significantly larger than for REF2D as a result of the increased surface–atmosphere exchange due to the larger wind speeds.

Bulk heat budget components at *t* = 4 h. The surface heat flux (SHF) is expressed in terms of the bulk temperature tendency of the corresponding control volume. All other components are expressed as a fraction of the SHF.

### b. Valley surroundings and upper-level cross-valley wind

Cross sections of the mean-flow structure for three further simulations are presented in Fig. 14; the corresponding bulk heat budget components are listed in Table 3. For PLAIN and PLATEAU (topographies as in SR12), the flow pattern and BL structure is qualitatively similar to that of REF2D. The main difference for PLAIN is the propagation of the ridge plume toward the valley center and for PLATEAU the propagation of the ridge plume away from the valley. These flow structures are similar to those found by SR12 in mesoscale simulations over three-dimensional topographies with identical cross-valley sections. Modification of the valley surrounding leads to an increased advective and turbulent heat export out of the valley control volume for PLAIN and a strongly reduced advective and turbulent heat export for PLATEAU (see Table 3).

With a weak upper-level wind (CROSS-U2), the symmetry of the cross-valley circulation and BL structure is broken, resulting in significantly different structures. The flow around the ridge-top plume becomes highly asymmetric and the two upper-level circulation cells merge into one cell spanning the entire valley cross section. Gravity waves are generated by the flow over and around the ridge-top plume. The circulation below the valley inversion is less affected, although the isentropic surfaces become slanted. Surprisingly, the influence of the upper-level wind on the bulk valley heat budget is relatively small (Table 3). While the turbulent heat loss is significantly enhanced, the heat loss due to the mean circulation is almost unchanged.

## 7. Discussion

Which processes are most important for the larger heating rates observed in the valley atmosphere in comparison to the atmosphere over the adjacent plain? To clarify this issue, the question is addressed in two steps:

How is the valley atmosphere heated?

Why is the heating rate in valleys larger?

### a. Local heating of the valley atmosphere

Neglecting radiation flux divergence and moist processes, as is done in the current and recent idealized studies, the local rate of temperature change within the valley atmosphere is simply the sum of the heating rates due to temperature advection (equal to the divergence of the advective heat flux) and the divergence of the turbulent heat flux (cf. Fig. 6). The balance between the two heating processes differs from region to region (cf. section 5a); the competition between the two mechanisms, top-down advective warming and bottom-up turbulent warming, is nicely illustrated in Figs. 6 and 7. The near exclusiveness of the two processes has been the basis for several conceptual models of the valley heating process (e.g., Whiteman and McKee 1982; Brehm 1986; Haiden 1990).

In the local perspective, the heating at any point within the stable valley core is the result of the local advection of warmer air from immediately above, as a result of the local stable potential temperature stratification. The subsidence velocity at any given height (within the stable core) is determined by the upslope mass flux within the adjacent slope wind layer. In a purely two-dimensional context, mass conservation requires that the upslope mass flux at any given height is compensated locally by an equal downward mass flux over the valley core. In the height range of the valley inversion, a simple steady-state estimate of the upslope mass flux agrees very well with the LES (cf. section 5c). Thus, for a given stratification and local slope angle, the strength of the upslope flow and hence of the heating in the valley core is determined by the magnitude of the surface sensible heat flux in the same height range.

The enhanced local heating of the atmosphere above the valley center, in comparison to flat terrain, is thus the result of the additional upper-level heat input to the stable valley core from the adjacent slopes, and not due to subsidence of warm air from the free atmosphere above the valley [as proposed by Serafin and Zardi (2010a)]. The magnitude of this additional heating term depends to a first approximation only on the surface sensible heat flux and the width of the valley in the corresponding height range (cf. section 5c; Brehm 1986; Haiden 1990).

### b. Bulk valley heating and the volume effect

*V*denotes the valley control volume, and

_{υ}Therefore, the enhanced heating rates observed in valleys can only be due to two factors: valley-plain differences in the area-averaged surface sensible heat flux and the valley-volume effect as quantified by the TAF, *V _{p}* denotes the control volume over the plain and

*h*is the depth of this control volume. While the former factor may be important under favorable conditions, the only systematic factor is the volume effect. Then, under the assumption of equal heat input per area, the TAF

_{p}*τ*quantifies the upper limit of the volume effect. The proponents of the TAF concept did not assume that the valley control volume is thermodynamically closed; rather, the question was whether the volume effect is sufficiently large to explain the observed valley-plain temperature differences despite the heat loss due to exchange with the free atmosphere and cold-air advection by the along-valley wind. Having only a valley tube in mind, Wagner (1932b, 1938) appears to have realized that the volume effect would not be sufficient to explain the diurnal temperature ranges observed in the Inn Valley. The question was finally solved by Steinacker (1984), who pointed out the importance of considering the effects of the side valleys and static stability.

What is the basis for the usual choice of taking the valley control volume up to ridge-top height? It is the observation that under real-world conditions with complex topography and larger-scale (synoptic) winds there is a rapid equilibration of local temperature and pressure contrasts above the ridge top. Above an effective ridge height, the daily pressure evolution becomes independent of the local conditions (Wagner 1932b). Thus, the evolution of the valley wind can be understood as the result of the combined effect of upper-level pressure gradients and the local thermal forcing as quantified by the valley-volume effect (SR10; SR12).

### c. Summary

The detailed analysis of heat transfer over complex terrain provides insight into key principles of heat transfer in stratified fluids in general. Thermally induced flows in stratified fluids develop as a result of horizontal temperature differences, as implied by the Bjerknes circulation theorem (Bjerknes 1898). The “goal” of these flows is to equilibrate horizontal temperature differences—between the slopes and the valley core in the case of the slope flows (lower circulation cell), between the ridge-top plume and the surroundings in the case of the upper circulation cell, between the valley and the foreland in the case of the along-valley wind, and between the mountains and the adjacent regions in case of the plain-to-mountain flow. This implies that the heat transport in stratified fluids is intrinsically coupled to the dynamics of the flow, in contrast to the transport of passive scalars. This intrinsic coupling may lead to results that at first may seem counterintuitive, such as, for example, the cooling of the mountain peaks by the thermally induced circulations.

## 8. Conclusions

The daytime heat transfer processes over mountainous terrain are investigated by means of LES over idealized valleys. Two key features of the analysis are the clear distinction between the local and bulk perspective of valley heating and the Reynolds decomposition of the turbulent flow variables into their mean and turbulent components.

The major findings can be summarized as follows:

The net effect of the (slope-flow induced) mean cross-valley circulation is to export heat out of the valley and away from the mountain ridge, as would be expected based on theoretical arguments. This confirms the results of previous studies, which were based on simple conceptual models or numerical model simulations with parameterized BL turbulence (e.g., Noppel and Fiedler 2002; SR10).

These results confirm that the volume effect (TAF) is the primary cause of the enhanced diurnal temperature amplitude in mountain valleys, as initially proposed by Steinacker (1984) and recently demonstrated by SR10. Under most conditions, the bulk valley heating is expected to be slightly less than that predicted by TAF, owing to advective and turbulent heat loss to the surroundings (SR12). Thus, the TAF can be considered an upper bound on the achievable enhancement of the valley temperature amplitude (assuming equal heat inputs).

In the local perspective, the slope-flow-induced subsidence leads to a warming of the air in the valley center. The energy for this warming, however, comes from the heated slopes and not from the free atmosphere above. In fact, the raison d'être of the thermally induced cross-valley circulation is to transport heat from the slopes to the valley center in order to equalize horizontal temperature differences.

While entrainment is strongly enhanced over the mountain ridge, it is reduced over the rest of the valley. Therefore, the warming that can be attributed to entrainment is smaller over the idealized valley than over flat terrain.

Pure thermally forced mountain circulations cannot induce heat convergence over mountain ridges; they can, however, induce moisture convergence by transporting moisture from lower regions to the ridge top (e.g., Banta 1990). They may also provide a trigger. Thus convection initiation over mountain ridges may be enhanced or suppressed by the mountain circulations, depending on which effect of the circulation —the moistening/triggering or cooling effect—is stronger in any particular situation.

In the stable regions of the valley atmosphere, a simple steady-state bulk model provides an accurate a priori estimate of the upslope mass flux and also of the upslope heat flux, provided a reasonable parameterization of the mean temperature excess in the slope wind layer is available.

This study underlines the importance of the thermally forced mountain circulations and atmospheric stratification for heat and mass exchange over mountainous terrain. Although the simulations were limited to fairly simple setups, the qualitative conclusions are applicable to a broad range of real-world situations as they depend on fundamental physical principles. The effects of these mountain circulations need to be parameterized in coarse-resolution numerical models. Bulk models of the slope wind layer along the lines of Vergeiner (1982) might provide a useful starting point. With respect to an improved quantitative understanding, it would be desirable to extend the analysis to include further topographies; a time-dependent and inhomogeneous forcing such as, for example, real-world asymmetric radiation; the influence of moisture; and varying atmospheric stability profiles including the transition from stable to fully convective conditions throughout the valley atmosphere.

## Acknowledgments

The numerical simulations have been performed on the CRAY XE6 at the Swiss National Supercomputing Centre (CSCS) using the Advanced Regional Prediction System (ARPS) developed by the Center for Analysis and Prediction of Storms, University of Oklahoma. The figures have been produced using matplotlib (Hunter 2007). Comments by three anonymous reviewers and the editor are gratefully acknowledged.

## APPENDIX A

### Calculation of the Ensemble Average

*T*is the averaging period and

*L*the domain size in the along-valley direction. This average (and its numerical implementation in discretized model space) satisfies the usual Reynolds-averaging rules,

_{y}^{A1}in particular the product rule (2), exactly.

^{A2}

*a*. If

*a*=

*a*(

*t*), we have (Wyngaard 2010, his section 2.4)

*τ*the Eulerian integral time scale and

_{a}*T*the length of the averaging period. Combining time and space averaging (along the

*y*axis), the level of error

*l*is the integral scale in space. Assuming typical CBL values of

_{a}*l*≈ 1 km and

_{a}*τ*≈ 5 min, one obtains

_{a}*ε*= 0.11 for

*L*= 9.6 km and

_{y}*T*= 40 min (the values used for the reference case).

## APPENDIX B

### The Large-Eddy Model

Common subfilter models used in LESs, such as the Smagorinsky and Deardorff model, implicitly define and impose a filter operation (Mason and Callen 1986; Pope 2000). Here, the ARPS TKE-based subfilter model is modified in order to separate the implicit filter operation from the computational mesh, following the approach pioneered by Mason and Callen (1986) for the Smagorinsky model. In other words, the (implicit) LES filter-scale Δ_{f} is determined by an explicitly specified subfilter length scale *l _{n}* and not by the grid scale Δ as is usually done. Of course,

*l*is still chosen proportional to some measure of grid resolution. The motivation to follow this approach is twofold. First, an explicit specification of the length scale

_{n}*l*allows one to control the magnitude of numerical discretization errors (Mason and Brown 1999). Second, it results in a filter operation that is independent of vertical grid stretching, thus avoiding the introduction of nonphysical variations in the subfilter length scale.

_{n}For convenience, the large-eddy equations are summarized in section Ba. The subfilter turbulence model is introduced in section Bb, followed by the subfilter-TKE equation in section Bc, and the subfilter model coefficients in section Bd. The relation between the TKE and Smagorinsky subfilter model is briefly discussed in section Be, a turbulence Reynolds number is introduced in section Bf, and the calculation of the Reynolds-averaged turbulent fluxes is summarized in section Bg.

#### a. LES and subfilter fluxes

*ρ*

_{0}(

*z*) the basic-state density,

*g*the acceleration due to gravity,

*H*the subfilter heat flux vector, and

_{i}Conventional eddy-viscosity closures such as the Smagorinsky and Deardorff model only account for the Reynolds term. Thus the subfilter modeling error consists of three terms: neglect of the Leonard and cross term and the modeling error associated with the Reynolds term. In addition, the influence of gradients in subfilter TKE on the dynamics—that is, the difference between normal and modified pressure—is also often neglected, as for example in ARPS.

#### b. The subfilter turbulence model

*K*and

_{m}*K*are expressed as

_{h}*c*and

_{m}*c*are subfilter model coefficients, and the subfilter-TKE

_{h}*e*is computed using a prognostic equation following Deardorff (1980) (see below). The subfilter length scale

*l*is defined as

*l*is a neutral subfilter length scale.

_{n}Traditionally, the neutral subfilter length scale *l _{n}* is taken to be equivalent to the size of the grid mesh Δ (e.g., Lilly 1967), often defined as Δ ≡ (Δ

*x*Δ

*y*Δ

*z*)

^{1/3}. For a computational mesh with varying Δ

*z*this implies a vertical variation of the filter width, and thus introduces a (nonphysical) dependence of the filter operation on a particular choice of vertical grid stretching. Here the length scale

*l*is regarded as an externally specified parameter of the subfilter turbulence model. For the reference simulation we choose

_{n}*l*=

_{n}*β*Δ

*x*= 50 m; that is,

*β*= 1.

^{B1}

#### c. The subfilter-TKE equation

*w*

^{s}θ^{s})

^{r}=

*H*

_{3}, the third-moment terms are modeled diffusively (downgradient diffusion assumption),

*c*,

_{m}*c*,

_{h}*c*

*, and*

_{ε}*l*.

_{n}#### d. Subfilter model coefficients

Assuming that the neutral subfilter length scale equals the filter scale, *l _{n}* = Δ

_{f}, and in the absence of stability corrections—that is, in the homogeneous isotropic limit—values for the subfilter model coefficients can be derived for steady-state solutions to the subfilter-TKE equation, the exact values depending on the assumed filter and the spectral distribution of the subfilter energy (Lilly 1967; Schumann 1991). For a Gaussian filter and the Kolmogorov inertial range spectra one obtains (

*c*,

_{m}*c*

*) = (0.081, 0.66), which implies a Smagorinsky constant of*

_{ε}*c*= 0.168 (Sullivan et al. 2003). The default values used in ARPS follow Deardorff (1980); hence, (

_{s}*c*,

_{m}*c*,

_{h}*c*

*) = (0.1, 0.3, 0.7). In local equilibrium, these values correspond to a Smagorinsky constant of*

_{ε}*c*= 0.1944.

_{s}*c*

*is increased by a wall-effect factor to*

_{ε}*c*

*= 1.4, similar to Deardorff (1980).*

_{ε}#### e. Relation between the TKE and Smagorinsky subfilter model

*c*. In local equilibrium dissipation is locally balanced by the conversion of mean-flow energy into subfilter energy as the time-rate-of-change and transport terms are neglected. In this limit, (B12) takes the form

_{s}*l*and the subfilter coefficients), (B16) can be solved for

*e*,

*N*

^{2}/

*S*

^{2}is the local gradient (or deformation) Richardson number. The definition of the subfilter eddy viscosity implies that

*c*/

_{m}*c*is the turbulent Prandtl number, Rf is the flux Richardson number, and

_{h}#### f. The effective turbulence Reynolds number

*R*is proposed:

_{t}*λ*

_{0}is the Smagorinsky mixing length (Mason and Brown 1999) of the subfilter model. The derivation of this estimate follows Bryan et al. (2003), but the LES filter length is replaced by the Smagorinsky mixing length

*λ*

_{0}, leading to a much better agreement with the a posteriori calculated value of

*R*.

_{t}#### g. Reynolds-averaged turbulent fluxes

^{r}+ ϕ

^{s}, the turbulent heat flux, for example, can be expressed as

## APPENDIX C

### Mean-Flow Structure and Convergence

Here some additional quantitative information on the mean-flow structure, numerical convergence, and sensitivity to lower boundary conditions is given. The setup of the sensitivity experiments is summarized in Table C1.

List of sensitivity experiments. The Smagorinsky length scale is *λ*_{0} = *c _{s}β*Δ

*x*with

*c*= 0.1944. The effective turbulence Reynolds number is defined as

_{s}*R*= (

_{t}*l*/

_{e}*λ*

_{0})

^{4/3}, where

*l*= 1 km is a large-eddy length scale.

_{e}#### a. Mean-flow structure

Profiles of mean wind, potential temperature, and total and subfilter TKE in the slope wind layer at three different locations are shown in Fig. C1. Although there are large along-slope differences in the depth of the mixed layer and the overall wind profile, in the lowest 200 m AGL the wind profile is qualitatively similar at the three locations, with a maximum at 20–25 m above ground. At upper levels the differences are significant, with a return flow above the ML at *x* = 3 and *x* = 5 km and a second upper-level maximum related to the upper circulation cell at *x* = 7 km. In general, total TKE is relatively uniform throughout the mixed layer.

As an example, profiles of the heat flux and velocity and temperature variances taken in the middle of the slope are shown in Fig. C2. The turbulent heat flux decreases approximately linearly with height with negative values in the capping inversion, as is typical of the CBL. The profile of the vertical velocity variance is similar to that of a typical CBL. The two horizontal velocity variances, on the other hand, show a secondary peak in the upper part of the mixed layer, related to the cross-valley circulation. The temperature variance peaks at the surface and in the inversion layer. Both the velocity and temperature variances are nonzero above the top of the inversion layer, possibly owing to gravity wave activity. Note that the subfilter contribution to the fluxes and variances is small, except near the surface.

#### b. Convergence of the large-eddy simulation

At a horizontal grid spacing of 50 m the inertial subrange of the CBL turbulence is just starting to be resolved (e.g., Sullivan and Patton 2011). How accurate are the flow statistics calculated at 50-m grid spacing? The convergence of four key statistics is illustrated in Fig. C3. Differences between DX25 and DX50 are generally very small—in particular for the mean wind, potential temperature, and vertical turbulent heat flux. Differences are somewhat larger for total TKE in the lowest 200 m above ground level. In summary, increasing grid spacing from 50 to 25 m has only a minor influence on the slope wind layer profiles, indicating near convergence at *dx* = 50 m for the quantities considered. Note that the reference simulation (*l _{n}* = 50 m;

*β*= 1) is even closer to DX25. DX100 deviates more strongly from the higher-resolution simulations.

#### c. Sensitivity to choice of lower boundary condition

The sensitivity of the simulations to the specification of the lower boundary condition was investigated by conducting three additional simulations (see Table C1). The reference simulation is driven by a constant prescribed incoming shortwave radiation of 400 W m^{−2} (cf. section 3). The additional experiments include a simulation driven by a constant prescribed surface heat flux of 150 W m^{−2} (SH150), a fixed soil temperature of ΔΘ_{s} = 7 *K* with respect to the initial surface temperature (TSOIL7), and a simulation with the thermal roughness length set equal to the momentum roughness length (Z0T). The prescribed soil temperature was chosen such that the resulting surface heat flux is about 150 W m^{−2}.

The main effect of the different lower boundary conditions is to modify the strength and cross-valley variation of the surface sensible heat flux (not shown). Any direct influence on the local turbulence structure of the simulated flow seems to be negligible in comparison to the influence resulting from differences in the surface heat flux forcing. Apart from slight quantitative differences due to the different forcing, the flow structure and evolution, including the local temperature tendencies, are very similar for all simulations.

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^{1}

The subfilter temperature variance is diagnosed from the subfilter heat flux and the subfilter TKE as in Nieuwstadt et al. (1993).

^{2}

To simplify notation, *φ* in this section.

^{A1}

Except for the commutative property with the time derivative.

^{A2}

Provided that the weakly time-dependent ensemble-mean

^{B1}

Alternatively, one can regard the Smagorinsky mixing length scale *λ _{n}* ≡

*c*=

_{s}l_{n}*C*Δ

_{f}_{f}as the externally specified parameter (Mason and Brown 1999). For the reference simulation, one obtains

*λ*=

_{n}*c*= 9.7 m, where

_{s}l_{n}