## 1. Introduction

Diurnal slope and valley winds are an essential component of the fair-weather mountain atmosphere. They strongly influence the weather and climate in mountain valleys and, together with turbulent processes, control the land surface–atmosphere exchanges in mountainous regions. Also, the quantification of the associated fluxes of energy, momentum, moisture, and pollutants is important for many applications such as air-quality studies, numerical weather prediction, and climate modeling (e.g., Rotach et al. 2008; Weigel et al. 2007). Despite the importance of the diurnal mountain winds there is still some uncertainty regarding the driving mechanisms of the along-valley winds.

It is well established that the diurnal valley winds develop as a result of the larger amplitude of the diurnal temperature oscillation in the valley atmosphere in comparison to that of the atmosphere over an adjacent plain (e.g., Vergeiner and Dreiseitl 1987; Egger 1990; Whiteman 1990). The valley–plain temperature contrast produces hydrostatically an along-valley pressure gradient that accelerates the flow up the valley during the day and down the valley during the night. The relative importance of different valley heating and cooling mechanisms responsible for the larger diurnal temperature range in valleys is, however, less clear. Often the larger amplitude of the diurnal temperature oscillation in the valley is explained by means of the topographic amplification factor (TAF) concept (Wagner 1938; Steinacker 1984; McKee and O’Neal 1989). TAF is based on an argument stating that a given amount of daytime energy input (nighttime loss) applied to a valley heats (cools) a smaller volume of air than if applied over a plain, resulting in a larger heating (cooling) rate of the valley atmosphere. The main underlying assumption is that no heat is exchanged with the free atmosphere above the valley. Additional explanations include radiation effects (e.g., valley–plain difference in albedo or cloudiness) and differences in the conversion rate of net radiation to surface sensible heat flux—for example, valley–plain contrasts in vegetation type, soil moisture, or wind speed (Whiteman 2000).

Recently the role of the valley volume effect for the daytime valley wind evolution was questioned by Rampanelli et al. (2004). On the basis of idealized simulations, they observe that heat flows across the valley top and that therefore the valley volume heat budget is not closed as required by the TAF theory (p. 3104). Furthermore, they point to the importance of compensating subsidence within and above the valley induced by the upslope flows in warming the valley core. Subsidence heating is also used by Weigel et al. (2006, hereafter W06) to explain the warming in their high-resolution simulation of the flow in the Riviera Valley in southern Switzerland.

The closer examination provided herein reveals that the local subsidence heating mechanism is qualitatively consistent with the TAF argument. In fact, subsidence in the stable valley core has long been considered a key mechanism for heating the valley atmosphere (Vergeiner 1982; Whiteman and McKee 1982; Brehm 1986; Vergeiner and Dreiseitl 1987; Egger 1990). Since subsidence in the valley center is part of a circulation involving rising motion on the valley sides, the net effect of a purely thermally induced overshooting cross-valley circulation has been observed to export heat out of the valley, thus reducing the warming of the valley (Noppel and Fiedler 2002) and consequently the TAF can be significantly reduced. Conversely, convergence of downslope flows over the valley center and associated ascent are a key mechanism for cooling the valley center during the evening transition (Whiteman 1986).

Vertical motions associated with the divergence (convergence) of the mean along-valley flow may also contribute to a net heating (cooling) of the valley atmosphere. Indeed, Vergeiner et al. (1987) discuss the key role played by along-valley divergence for the evolution of the nighttime temperatures in Colorado’s Brush Creek Valley. The importance and origin of advection-induced daytime heating is, however, not well understood, as previous investigators have not made a clear distinction between subsidence heating induced by purely cross-valley circulations and along-valley divergence effects.

In this paper, we develop a new diagnostic framework for analyzing the evolution of the along-valley wind and apply it to numerical simulations of the valley wind system over an idealized three-dimensional valley–plain topography. The framework is used to investigate the relative importance of the different forcing mechanisms—in particular of the different heat exchange mechanisms (turbulent, along-valley, and cross-valley advection)—during the full diurnal cycle. In section 2, we derive a quantitative relation between the along-valley pressure gradient and the various forcing factors. The experimental setup and numerical model are introduced in section 3. The diurnal evolution is presented and analyzed in section 4 followed by a discussion in section 5 and conclusions in section 6.

## 2. Theory

To streamline the analysis of the along-valley wind dynamics and the derivation of the new diagnostic framework, we first review the processes contributing to the valley heat budget and introduce some useful notation.

### a. The valley heat budget

*R*= (

*θ*/

*c*)

_{p}T**∇**·

**R**. The left-hand side of this equation is the local net heating (cooling) rate. The terms on the right-hand side are, respectively, heating due to advection of potential temperature, turbulent heat flux divergence, and radiation flux divergence. To simplify the writing we henceforth refer to

*θ*simply as “temperature.”

*V*. For the sake of concreteness, let us take the control volume to be equal to the entire valley volume up to ridge height. The volume-averaged density-weighted heat budget equation iswhere

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

*H*

_{0}is the surface sensible heat flux normal to the surface. Equation (2) follows from the integration of (1) 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 words, the density-weighted volume-averaged temperature tendency (TND) is equal to the sum of the contributions due to the surface sensible heat flux (SHF), radiation flux divergence (RAD), total advection (ADV), and turbulent heat flux through

_{A}*A*(TRB). For future reference we derive a symbolic representation of (2) with the tendencies normalized by the area

_{A}*A*of the top surface of the control volume. Defining

*q*= (

^{X}*M*/

*A*)

*X*, where

*X*stands for any one of the tendency terms, (2) can be expressed aswith the density-weighted volume average defined by

*t*

_{0},

*t*

_{1}) iswhere

*Q*denotes the net heat input into the control volume

^{n}*V*normalized by the area of the upper control surface

*A*,

*Q*=

^{d}*Q*

^{SHF}+

*Q*

^{RAD}represents the normalized diabatic heat input,

*Q*= −

^{t}*Q*

^{ADV}−

*Q*

^{TRB}represents the normalized net heat loss due to advective and turbulent heat transport, and

### b. Dynamics of the along-valley wind

*h*with a horizontal floor, closed at the one end and open toward a plain at the other end (see Fig. 1). The Reynolds-averaged momentum equation for the along-valley wind at

*x*= 0,

*z*= 0, neglecting the Coriolis force, iswhere

*υ*is the along-valley velocity,

*ρ*is the density,

*p*is the pressure,

*y*is the Cartesian coordinate in the along-valley direction, and the vector

**T**

*is the turbulent flux of along-valley momentum. To distinguish between local and external influences on the valley flow, the valley-axis surface pressure is decomposed into a contribution from upper levels, denoted by*

_{υ}*p*=

_{h}*p*(

*x*= 0,

*y*,

*z*=

*h*,

*t*), and one from lower levels, denoted by

*p*=

_{l}*p*−

*p*. Then expanding the Lagrangian derivative yieldsAccording to (7) the local along-valley velocity can change because of the pressure gradient associated with low-level forcing (first term on the RHS), upper-level forcing (second term), and advective and turbulent transport of along-valley momentum (third and fourth term). Note that no approximations have been made so far.

_{h}*r*, and a deviation therefrom asFor

*hydrostatic*flow,

*p*′

*can be expressed aswith the buoyancy acceleration defined by*

_{l}*b*≡ −

*gρ*′/

*ρ*, the mass per unit area in the column (0,

_{r}*h*) by

*b̂*,

*b*=

*g*(

*θ*′/

*θ*), which implies that the low-level forcing is fully determined by the temperature distribution

_{r}*θ*′(

*x*= 0,

*y*,

*z*,

*t*). If we adopt a base state with a constant temperature

*θ*

_{0}, the (total) pressure gradient can be written asHere the first term on the right-hand side corresponds to the usual thermal wind term (Mahrt 1982).

*p*≡

*p*(0,

*y*, 0,

_{υ}*t*) −

*p*(0,

*y*, 0,

_{p}*t*) is given bywhere Δ

*p*is the upper-level valley–plain pressure difference and

_{h}*y*and

_{υ}*y*are points in the valley and on the plain, respectively. Factoring out

_{p}*τ*is defined asThe amplification factor is equal to the ratio of the vertically averaged temperature excess (day) or deficit (night) in the valley center (

*x*= 0) to that over the plain.

### c. Combining dynamics and thermodynamics

*τ*with the valley-volume argument, the fraction in (16) is expressed in terms of the temperature perturbation averaged over the valley control volume

*θ*′

^{υ}and that averaged over a plain control volume

*θ*′

^{p}, which results inwithThe factor

*τ*is the ratio of the vertical average at the valley center (

_{a}*x*= 0) to the valley volume-averaged temperature perturbation while

*τ*is the ratio of the volume-averaged temperature perturbation in the valley to that over the plain. For perturbations that are fairly uniform in the cross-valley direction (which will be shown to be the case below)

_{h}*τ*≈ 1, and therefore

_{a}*τ*can be considered a rough surrogate for

_{h}*τ*.

*τ*is expressed in terms of the valley and plain heat budgets. Substituting (5) into the definition of

_{h}*τ*, assuming

_{h}*M*/

_{p}*M*=

_{υ}*V*/

_{p}*V*, and writing

_{υ}*Q*=

^{n}*Q*(1 −

^{d}*f*) with

*f*=

*Q*/

^{t}*Q*yieldswhereHere

^{d}*τ*is the ratio of the plain to the valley volume, which is nothing other than the topographic amplification factor,

_{υ}*τ*is the ratio of the diabatic heat input into the valley to that into the plain volume, and

_{q}*τ*represents the advective and turbulent heat exchange through the boundaries of the control volumes. Taking

_{e}*A*=

_{υ}*A*, the factor

_{p}*τ*during the daytime is approximately equal to the ratio of the time-integrated valley-to-plain surface sensible heat flux

_{q}*τ*has been decomposed into four factorsThis new diagnostic relation enables us to evaluate quantitatively the valley volume argument discussed in the introduction. For the valley volume effect to explain the daytime up-valley acceleration (

*τ*> 1), with

*τ*≈ 1 and assuming

_{a}*τ*≈ 1, the valley volume effect (

_{q}*τ*> 1) must be large enough to offset the heat losses through the valley top (

_{υ}*τ*< 1). This will be shown to be the case in the analysis of numerical simulations presented below.

_{e}It should be noted that (23) and (24) are valid for very general valley shapes and sizes and choices of control volumes. By design the development here treats a flat valley floor and, based on the analysis of numerical simulations, the flow is justifiably taken to be hydrostatic. In summary (23) is a diagnostic equation designed to evaluate the major mechanisms giving rise to thermally induced along-valley winds; however, it must be kept in mind that the along-valley flow depends also on the momentum transport terms including surface friction and momentum exchange with the free atmosphere.

## 3. Numerical model simulations

### a. Experimental setup

To investigate the relative importance of the different forcing mechanisms of the thermally induced valley wind, as expressed by the *τ _{n}* in (24), we carried out numerical simulations for the idealized three-dimensional valley–plain topography shown in Fig. 1. This setup is similar to configurations used in previous studies (Li and Atkinson 1999; Rampanelli et al. 2004). The topography was chosen to satisfy the criteria listed in Rampanelli et al. (2004): 1) a horizontal valley floor, so that the up-valley wind has no upslope contribution, 2) a long valley, so that the along-valley flow can develop unaffected by numerical boundary conditions in the along-valley direction, and 3) moderate valley slopes that can be adequately represented by current mesoscale models. In addition to these conditions outlined in previous studies, a large computational domain was chosen in order to minimize the influence of the lateral boundaries on the simulated flow.

Note that the topography *h*(*x*, *y*) is the product of two simpler ones: a valley defined by two isolated ridges *h _{x}*(

*x*) and a slope connecting the plain with a plateau

*h*(

_{y}*y*). The origin of the coordinate system is located at the valley entrance where the valley first attains its full depth [

*h*(0) = 1].

_{y}We present simulations for a long, narrow, and moderately deep valley with the following dimensions: valley depth *h _{p}* = 1.5 km, valley length

*L*= 100 km, slope widths

_{y}*S*=

_{x}*S*= 9 km, and a crest-to-crest width of 20 km. Furthermore, the width of the valley floor and the mountain ridges is 1 km; thus,

_{y}*X*

_{1}= 0.5 km,

*X*

_{2}= 9.5 km,

*X*

_{3}= 10.5 km, and

*X*

_{4}= 19.5 km. The computational domain extends over

*x*= −60 to 60 km and

*y*= −200 to 100 km. The maximum inclination of the slopes, in both the

*x*and

*y*directions, is given by 0.5

*πh*/

_{p}*S*= 0.2618 (=14.7°).

_{x}*θ*= 280 K, Γ = 3.2 K km

_{s}^{−1}, the surface pressure

*p*= 1000 hPa, and a constant relative humidity of 40%. The initial temperature profile describes an atmosphere with a constant stratification corresponding to a Brunt–Väisälä frequency of about 0.011 s

_{s}^{−1}. The model is integrated for up to 40 h starting from sunrise which is at 0600 local time (LT).

### b. Numerical model

The Advanced Regional Prediction System (ARPS; Xue et al. 2000, 2001), which is a comprehensive regional to storm-scale prediction system, is used to simulate the thermally forced valley winds using full model physics. In our setup this includes a sophisticated radiation package (Chou 1990, 1992; Chou and Suarez 1994; Tao et al. 1996), a force–restore land surface soil–vegetation model (Noilhan and Planton 1989; Ren and Xue 2004), and a nonlocal turbulent kinetic energy (TKE)-based ensemble turbulence closure scheme after Sun and Chang (1986). Surface turbulent heat fluxes are based on the similarity theory of Monin and Obukhov (1954), while the stability functions for unstable conditions are based on Byun (1990). The minimum total wind speed for the calculation of surface fluxes is set to 1.0 m s^{−1}.

The computational domain for the simulations is 300 km in the along-valley direction and 120 km in the cross-valley direction. The horizontal grid spacing is 1 km. In the vertical, the domain extends to 12.2 km and the grid spacing varies from 20 m near the surface to a maximum of 200 m above 2 km. The time step is 12 s for the advection and physics terms and 1.2 s for the acoustic modes. Discretization of momentum and scalar advection is fourth-order accurate in the horizontal direction and second-order accurate in the vertical. The lateral boundary conditions are periodic in the cross-valley direction and free-slip rigid-wall conditions are imposed in the along-valley direction. These boundary conditions ensure that no mass 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 5 km to the top of the domain. The Coriolis force is turned off. Fourth-order computational mixing with a mixing coefficient equal to 10^{−3} s^{−1} is used.

The thermal forcing is determined by the incoming radiation and the land surface characteristics. To ensure a symmetric forcing of the valley wind system, the model domain is located at the equator and the time of the year is set to the spring equinox (21 March). The solar constant is reduced to 700 W m^{−2}, so that the resulting surface sensible-heat flux is of similar magnitude to that in previous studies (Rampanelli et al. 2004; W06). The radiative forcing is updated every 10 min. Uniform land surface characteristics are used: the soil type is sandy loam and the vegetation type is semidesert with a leaf area index (LAI) of 1.5, a vegetation fraction of 0.1, and an aerodynamic surface roughness of 0.1 m. The soil is initialized with a soil moisture saturation ratio of 20%, a near-surface soil temperature equal to the atmospheric surface temperature, and a 3-K warmer deep-soil temperature.

### c. Diurnal flow evolution

Before proceeding to the discussion of the flow evolution in terms of the theory developed in section 2, we briefly describe the diurnal flow evolution. Figure 2 shows the evolution of the flow in a cross-valley plane located 20 km up valley from the valley entrance (see the horizontal line in Fig. 1b). Three hours after sunrise, at 0900 LT, very shallow upslope-flow layers have developed on the valley sides. By 1200 LT the cross-valley circulation is well established with upslope flows on all sides, flow convergence over the mountain ridges, and weak subsiding motion over the center of the valley. By 1500 LT the core of the along-valley flow has attained a speed of over 4 m s^{−1}. One can also see that along-valley momentum is advected up the valley slopes by the slope flows. (The thermally induced along-valley pressure gradient is much too weak near ridge height to be able to produce the observed along-valley flow.) An interesting feature is visible at 1800 LT: a strong plain-to-mountain flow has entered the valley and is advecting the along-valley flow toward the valley center (cf. De Wekker et al. 1998). This plain-to-mountain flow is the result of the lower pressure in the valley in comparison to the surrounding plain. It brings air from heights of around 1 km over the plain and up to 10 km away from the mountain into the valley (not shown). Although thermally induced downslope flows start to develop soon after sunset, it takes several hours for the valley to cool sufficiently for the development of the nocturnal down-valley flow (see 2100 and 0300 LT). Because of its inertia, the down-valley flow persists for several hours after sunrise (see 0900 and 1200 LT), leading to a delayed evolution of the up-valley flow in comparison to the first day when the flow developed from an atmosphere at rest. Nevertheless, the upslope flows are established soon after sunrise, also on the second day (see 0900 LT).

Figure 3 shows four snapshots of the flow in an along-valley plane located at the valley center (*x* = 0 km). The asymmetry between the evolution of the daytime up-valley flow and the nighttime down-valley flow is apparent. During the daytime, the strong up-valley flow extends almost to the valley head (100 km from the valley entrance). The down-valley flow, on the other hand, is restricted to heights well below the ridge height and high velocities are found only near the valley entrance and over the adjacent plain. It should be noted that in comparison to the simulation reported in Rampanelli et al. (2004), the upper-level return flow above the valley center is weak because in the present setup—two isolated mountain ridges on a plain—it is not restricted to the region above the valley center.

## 4. Results

### a. Formation of the along-valley wind

The evolution of the along-valley wind averaged over the valley cross section and the valley–plain pressure difference is shown in Fig. 4a. The two curves are almost perfectly anticorrelated. Note the almost linear increase in the strength of the up-valley wind between 1000 and 1400 LT and the long transition from up-valley flow to a well-developed down-valley flow after midnight. In contrast to the daytime, when the along-valley flow speed is constantly changing, a quasi-steady-state down-valley flow develops after midnight from about 0100 to 0800 LT. The evolution of net radiation and the surface heat fluxes is shown in Fig. 4b. The surface sensible heat flux reaches a maximum of about 180 W m^{−2} at noon and a minimum of about −40 W m^{−2} during the night. Note that significant heating only starts after 0900 LT. Because of the very dry soil conditions the latent heat flux is very small.

It is clear that for the current setup—a start from an atmosphere at rest, with no externally imposed pressure gradients, a horizontal valley floor, and a relatively large valley—the along-valley flow develops as a result of a thermally induced along-valley pressure gradient. Analysis of comparable numerical simulations confirms that the flow occurs under hydrostatic balance (not shown; see Rampanelli et al. 2004) and that the forcing is thus fully determined by the temperature distribution. Moreover, the contributions to the surface pressure difference from the pressure difference at the valley top, Δ*p*_{ref}, is negligible, at least in this case of a relatively deep valley. This point will be further addressed in the conclusions.

Figure 5 shows vertical profiles of temperature on the valley axis (*x* = 0 km) over the plain (*y* = −40 km) and within the valley (*y* = 80 km). There are significant differences in the structure and average temperature of the plain and valley profiles. The daytime profiles (1200 LT) show the typical structure of a well-mixed convective boundary layer over the plain, and a three-layer structure in the valley. The three-layer structure, consisting of a shallow mixed layer near the ground, a near-neutral layer at upper levels, and a stable layer in between, is typical for many valleys (Brehm 1986). On average the temperature is about 1 K higher in the valley than over the plain at 1200 LT. This temperature excess is the proximate cause of the along-valley pressure gradient that produces the daytime up-valley winds, as shown for example in Figs. 11 and 12 of Rampanelli et al. (2004). During the nighttime the situation is reversed. The valley atmosphere is significantly cooler than the atmosphere over the plain. In contrast to the daytime situation, the maximum temperature difference occurs at lower levels.

As was shown in section 2, the valley–plain pressure difference on the valley axis is determined by *τ*, the ratio of the vertically averaged temperature at the center line of a valley to that at a location over the plain. Rampanelli et al. (2004) analyzed the local temperature tendency Eq. (1) to show that local subsidence creates a warm anomaly in the valley center with respect to the plain and hence (in the present terminology) *τ* > 1. In section 2 a method was introduced to relate the latter local evaluation to traditional arguments based on valley volume-averaged temperature perturbations; we apply that method next to the present simulations.

### b. Time evolution of the forcing factors

Figure 6 shows the evolution of the vertically and volume-averaged temperature perturbations and the factors *τ _{n}* appearing in (24) during the day and night. The reference time for the calculation of the temperature perturbations is 0600 LT for the daytime plot and 1900 LT (at which time

*p*≈ 0) for the nighttime plot. As already seen in Fig. 5, the valley atmosphere at

*x*= 0 is warmer than the plain atmosphere during the day and cooler during the night. Figure 6a shows that while the vertically averaged temperature perturbation on the valley axis is notably warmer than the valley volume-average temperature, the two different averages over the plain are of course indistinguishable. The difference between the vertical and volume average temperature perturbation in the valley is the result of the different weighting with height. While for the vertical average the temperatures at all heights within the valley are weighted equally, for the volume average the temperatures at higher levels receive more weight.

Figures 6c and 6d show that the total forcing factor *τ* in (17) is far from constant in time as it changes significantly throughout the day and night. If the valley volume effect were the only driver of the along-valley wind then *τ* should be equal to *τ _{υ}* (=2 for the current topography). It can be seen in Fig. 6c that

*τ*starts off being larger than

*τ*during the morning hours, but then it rapidly drops to values below

_{υ}*τ*, primarily because of an increase of the heat export out of the valley atmosphere (

_{υ}*τ*< 1). During the night, Fig. 6d shows that

_{e}*τ*is larger than

*τ*because of the heat exchange with the free atmosphere (

_{υ}*τ*> 1). The factor

_{e}*τ*, which is the ratio of vertically to volume-averaged temperature, remains more or less constant (≈1.2) throughout the entire simulation. The factor

_{a}*τ*, which is the ratio of the valley to plain surface sensible heat flux, is slightly larger than unity during the day and early night and slightly smaller than one after midnight. As the land surface characteristics are constant throughout the model domain, this valley–plain difference must be the result of a feedback between the flow dynamics and the surface fluxes. For the present setup, however, the effect is quite small. From this analysis one may infer that during the day there is heat lost from the valley but that the specific heat that remains in the valley volume is still in excess of that over the adjacent plain and that therefore the valley volume argument (which can include local subsidence heating within the valley) can qualitatively explain the along-valley flow.

_{q}### c. Time evolution of the valley heat budget

Next, we examine in more detail the processes that contribute to the heating and cooling of the valley atmosphere as a whole. Time series of the heat budget components (section 2) for the entire valley are shown in Fig. 7. It can be seen that the dominant source of daytime heating is the surface sensible heat flux, which follows the pattern of incoming solar radiation, while the contribution from the radiation flux divergence is almost negligible. During the night, however, the cooling tendency due to radiation flux divergence is almost as large as the contribution from the surface sensible heat flux. Total advection is a large source of cooling during the day and early night and a large source of heating after midnight and during the morning transition of the second day. Thus, total advection opposes the driving forcing of the valley wind most of the time except during the transition periods. During the evening transition, total advection helps to rapidly cool the valley, and during the morning transition, it helps to warm the valley for several hours after sunrise (second day). Turbulent heat transport is significant during the day but negligible during the night.

Results obtained for two smaller control volumes near the valley entrance and the valley head qualitatively follow those of the entire valley (not shown). The main difference is the larger daytime heat export for the control volume near the valley entrance due to stronger along-valley cold-air advection and the smaller heat export for the control volume near the valley end due to a negligible tendency from along-valley advection, except during the evening transition.

### d. Prescribed forcing simulation with free-slip lower boundary

*ω*= 2

*π*/(24 h),

*S*

_{max}= 200 W m

^{−2}, and

*S*

_{min}= −50 W m

^{−2}. With respect to velocity, free-slip lower boundary conditions are used (i.e., zero surface momentum flux).

As can be seen in Fig. 8, the qualitative evolution of the heat budget components is similar to that for the full-physics simulation. The main difference is the earlier onset of heating in the morning for the prescribed-forcing simulation. Clearly, the qualitative findings are robust with respect to the details of the forcing.

## 5. Discussion

### a. TAF formulations

*A*=

_{υ}*A*=

_{p}*A*, as in section 3.1 in Steinacker (1984), then the ratio of the temperature change in the valley to that over the plain iswhich corresponds to Eq. (4) in Steinacker (1984). Here

*A*

_{h}is the (map) area averaged over the height of the valley, or the height interval under consideration, respectively. If one assumes only a uniform surface sensible heat flux, and not

*A*=

_{υ}*A*, then the ratio of the temperature change in the valley to that over the plain isas in Whiteman (1990).

_{p}*θ̂*(

*y*

_{0}+

*y*) =

*θ̂*

_{0}

*τ*(

*y*) with

*τ*(

*y*)=

*θ̂*(

*y*

_{0}+

*y*)/

*θ̂*

_{0}and

*θ̂*

_{0}=

*θ̂*(

*y*

_{0}), it can be shown thatandwhere

*τ*= (

_{g}*V*

_{0}

*A*)/(

*VA*

_{0}) =

*h*

*h*

_{0}and

*τ*

_{q′}=

*Q*/

^{d}*Q*

_{0}

*. Neglecting*

^{d}*p*and the

_{h}*τ*term, the pressure gradient can be written aswhich corresponds to (10) in McKee and O’Neal (1989).

_{e}### b. Heat exchange with the free atmosphere

Further insight into the heat exchange of the valley with its surroundings can be obtained by decomposing the temperature tendency due to total advection into different components. Perhaps the simplest is to decompose it into a component flux through the valley top and one through the valley entrance. Another decomposition that has been used in the literature is to partition the heat flux into contributions from the along-valley flow and the cross-valley circulation.

**∇**(

*ρ*

**v**) = 0—the volume-averaged advective temperature tendency term can be written in terms of a surface integral,where

**n**is the outward-directed surface normal. For simplicity, consider a straight valley and a control volume that does not exceed the height of the surrounding mountain ridges and with the along-valley faces oriented perpendicular to the valley’s axis: then the advective tendency can be expressed aswhere

*A*denotes the top surface of the control volume and

_{T}*A*and

_{S}*A*denote the two along-valley faces of the control volume. To avoid large compensating heat fluxes, the heat fluxes can be redefined in terms of a potential temperature perturbation

_{N}*θ̃*=

*θ*−

*θ*

*θ*

Figure 9 is computed for a control volume limited by the valley end wall (at which *υ* = 0) and hence there are only two contributions to the heat-transport terms—flux through the valley top and flux through the valley entrance. The net heat flux through the valley top (*A _{T}*) is out of the valley during the first day and the evening transition (until midnight) and into the valley during the second half of the night and the morning transition of the second day (until about 1100 LT). The net heat flux through the valley entrance (

*A*) shows a similar behavior, producing a cooling tendency during the day and a heating tendency during the night. Its magnitude, however, is generally smaller and the nighttime positive phase is significantly longer. As seen in section 4c, the total effect of the advective heat transport is to cool the valley during periods of up-valley flow and to heat the valley during periods of down-valley flow.

_{S}This finding is confirmed by the real-case simulations of W06. The temperature tendency due to net advection is positive during the early morning hours (Figs. 15–17 in W06) and becomes negative more or less simultaneously with the transition from down-valley to up-valley flow (Fig. 2 in W06).

**v**

*= (*

^{c}*u*, 0,

*w*) and

**v**

*= (0,*

^{a}*υ*, 0). In general, the decompositions (37) and (38) lead to different results. Applying Gauss’ theorem to the volume integral of

**∇**· (

*ρ*

**v**

*), it can be shown that the heat flux through the valley top is given byIt is thus generally not possible to deduce the magnitude or the sign of the heat exchange with the free atmosphere using only the cross-valley advection term (as attempted, for example, in W06). The second term on the right-hand side involving the divergence of the cross-valley circulation is in general not zero because of the strong coupling between the along-valley and cross-valley flows (via the continuity equation). The difference between the heat flux through the valley top and the cross-valley advection term is particularly large during the period of down-valley flow between midnight and late morning (Fig. 9).*

^{c}θ̃In summary, the net advective temperature tendency opposes the driving diabatic forcing most of the time, but it is in line with the diabatic forcing during the transition periods leading to additional warming during the morning transition and to additional cooling during the evening transition. In general, the net advective temperature tendency or the heat flux through the valley top cannot be simply attributed to either the cross-valley or the along-valley flow component since these flows are closely coupled through the continuity equation.

## 6. Conclusions

The physical mechanisms governing the evolution of diurnal along-valley winds in mountain valleys have been examined. A diagnostic framework for the along-valley wind has been developed and applied to numerical simulations of thermally driven flow over an idealized valley–plain topography. The major findings can be summarized as follows:

- A realistic diurnal evolution of the valley wind system is simulated with a strong and deep up-valley wind during the day and strong and more shallow down-valley wind during the night extending far out onto the plain (Fig. 3).
- The analysis within the new diagnostic framework confirms the importance of the valley volume effect (TAF concept) as a qualitative explanation of the thermally induced along-valley wind, despite the fact that the basic assumption of the TAF argument of negligible heat exchange with the free atmosphere rarely holds. While advective and turbulent transport contribute significantly to heat exchange between the valley and the free atmosphere, most of the time these transport processes
*oppose*the driving diabatic forcing. They produce a cooling tendency during the day and a warming tendency during the night. During the transition periods, however, advective transport supports the driving diabatic forcing, leading to additional warming during the morning transition and additional cooling during the evening transition (Figs. 6 and 7). - While slope-flow-induced local subsidence in the valley center is one of the main mechanisms for heating the valley core during the morning transition (Rampanelli et al. 2004), the
*net*effect of the circulation is to cool the valley atmosphere. As a rule, the net vertical heat transport of a pure (divergence free) slope-wind-induced cross-valley circulation is always positive (see also Noppel and Fiedler 2002). A cross-valley circulation extending above the valley top exports heat out of the valley atmosphere and cools the valley. Therefore, the cross-valley circulation cannot be an additional driving mechanism of the valley wind; rather, it results in less heating of the valley and thus weaker valley winds (Figs. 7 and 9). - Subsidence induced by the divergence of the along-valley flow is an important contributor to the heating of the valley atmosphere during the nighttime and the morning transition (cross in Fig. 9). This along-valley flow-induced subsidence is an intrinsic feature of the three-dimensional valley wind system.
- Dynamically induced valley–plain contrasts in the surface sensible heat flux were found to enhance the daytime up-valley forcing and reduce nighttime down-valley forcing for the present idealized case (Fig. 6).

Since the desire here was to deepen our understanding of the basic dynamics and thermodynamics of the valley wind system, we intentionally limited the simulations to a fairly simple setup. The diagnostic framework developed here and the qualitative conclusions of our study are, however, applicable to a broader range of valley shapes and sizes, as the derivation does not depend fundamentally on the simplified shape chosen for study here.

Nevertheless, it should be stressed that the link between the valley’s heat budget and the thermally induced along-valley pressure gradient does not represent the entire story with respect to the evolution of the along-valley wind. The evolution of the along-valley wind depends also on the upper-level pressure gradient Δ*p _{h}* and the advective and turbulent momentum exchange with the free atmosphere (7). These terms can become important because of external influences such as larger-scale pressure gradients and downward transport of upper-level high-momentum air (Whiteman and Doran 1993; Schmidli et al. 2009) but also in cases of pure thermal forcing without external disturbances. For shallow valleys, for example, the thermal perturbation can extend well above ridge height and induce a significant upper-level pressure gradient Δ

*p*.

_{h}With respect to an improved quantitative understanding, such as under which conditions Δ*p _{h}* is negligible, it would be desirable to carry out a systematic and more comprehensive series of numerical simulations varying the topographical parameters, such as valley size and shape, and the atmospheric stability profiles over the range of observed values.

Finally we list a number of potentially important effects that for simplicity have been neglected, such as an upward-sloping valley floor, the Coriolis effect, nonhydrostatic effects, and the effect of a curved valley. Although we simulated only a simple valley with no side valleys, the diagnostic framework is valid for more complicated valleys. Perhaps the most fundamental limitation with respect to the simulations is the accuracy of the simulated local structures such as slope wind layer profiles because of the reliance on parameterized turbulent convective heat transfer.

## Acknowledgments

M. Sprenger for comments on an earlier version of this manuscript and the support of the Swiss National Science Foundation (Grant PA002-111427) for the first author are gratefully acknowledged.

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* The National Center for Atmospheric Research is sponsored by the National Science Foundation.