1. Introduction

Over flat, homogeneous terrain, nocturnal inversions are destroyed after sunrise predominantly by the upward growth of a convective boundary layer (CBL) from the ground through the action of penetrative convective plumes. In a valley, the upward heat flux also develops a CBL over valley surfaces but, in contrast, the heated sidewalls cause warmed air parcels to flow upslope. From the principle of mass continuity and the assumption of no mass convergence or divergence in the along-valley direction, as mass is removed from the base and sides of the stable core and carried away in the upslope flows, the elevated inversion must sink and warm adiabatically as a result of the subsiding motion over the valley center (Whiteman 1982).

To explain the gross features of the breakup of nocturnal temperature inversions in several deep mountain valleys of western Colorado, Whiteman and McKee (1982) formulated a bulk thermodynamic model in which sensible heat flux is the driving force. The model is extensively described by Whiteman and McKee (1982) and will not be presented in detail here. The primary inputs to the model are very simple, such as valley topography (i.e., valley floor width l and sidewall inclination angles i₁ and i₂), characteristics of the valley inversion at sunrise, and an estimate of sensible heat flux obtained from solar radiation calculations. Because of its simplicity, the bulk thermodynamic model provides a straightforward approach for obtaining quick but reliable preliminary estimates of inversion breakup in a wide range of valley topography (Whiteman and McKee 1982; Müller and Whiteman 1988; Zoumakis et al. 1992; Allwine et al. 1997; Savov et al. 2002; Whiteman et al. 2003; etc.). Moreover, Bader and McKee (1983, 1985) and Colette et al. (2003) concluded from dynamical model simulations of valley temperature inversion breakup that the bulk thermodynamic model accounts for the dominant mechanisms of inversion destruction.

Because two of the input parameters to the thermodynamic model (i.e., A₀ and k) are not available from routine meteorological observations (Whiteman et al. 2003, 2004), the general model equations cannot be applied directly. Instead, arbitrary values for the adjustable (free) parameters k and A₀ are chosen until the best simulation of the actual data is obtained with the model. The fitting of the model output to the observations is accomplished by first choosing the value of k so that the ascending CBL and descending inversion top meet at the proper observed height at the time of inversion destruction. The value of A₀ is then varied until the model outputs h(t) and H(t) fit the data well (Whiteman and McKee 1982, p. 300). Future theoretical and experimental work will help us to gain a better understanding of the factors that affect the partitioning of energy and thus affect the evolution of vertical temperature structure in actual valleys. In addition, because energy gain and loss from the valley volume occur primarily at the valley surfaces, further research is necessary to investigate simple schemes of radiation and surface energy budgets and partitioning of energy in the valley atmosphere, so as to eliminate the need for using arbitrary values for the adjustable model parameters.

In this contribution, a simple thermodynamic parameterization based on a modified version of the model described by Whiteman and McKee (1982) is presented to simulate the changes with time of the height of the inversion top and the depth of the convective boundary layer during the inversion breakup period in idealized valleys under fair-weather conditions. The proposed method adopts simplified semiempirical parameterizations of radiation and surface energy budgets at the valley floor and sidewalls and an empirical scheme for the partitioning of energy in the valley atmosphere, eliminating the need for using arbitrary values for the adjustable model parameters k and A₀. The model results are validated and compared with dynamical model predictions and actual measurements in a wide range of valley topography.

2. Radiation and surface energy budgets in idealized valleys

The thermodynamic inversion destruction model developed here has, as its basis, a simplified atmospheric energy budget approach in which sensible heat flux is the driving force. It is obvious that the energy available for sensible heat flux depends on solar heat flux and the partitioning of energy in the surface energy budget. Because the purpose of this study is to demonstrate a search approach rather than a model development, simplified semiempirical parameterizations of radiation and surface energy budgets at the major topographic surfaces (i.e., valley floor and sidewalls) and an empirical scheme for the partitioning of energy in the valley atmosphere were adopted in developing the proposed method.

a. Incoming solar radiation modeling

The valley atmosphere is driven by incoming solar radiation K↓, and so our primary interest in the radiation and surface energy budgets is the estimation of K↓ (Whiteman et al. 1989a, b). Incoming solar radiation is the sum of a directional component S and a diffuse component D, such that K↓ = S + D. As a first approximation, the solar radiation model “r.sun” was adopted to estimate the direct and diffuse solar radiation on an inclined (and horizontal) surface at level z (W m⁻²). The r.sun model was developed by the Department of Geography and Geoecology, University of Presov, and European Commission Joint Research Center, Institute for Environment and Sustainability, in Ispra Italy. The simulation code is extensively described by Hofierka and Suri (2002) and will not be presented in detail here. The solar radiation model requires only simple input parameters—elevation above sea level, slope inclination and aspect of the surface, latitude, day number, and a local solar time. In addition, the airmass 2 Linke atmospheric turbidity factor T_LK must be determined. The input parameter T_LK can be estimated in a variety of ways. For example, values of T_LK can be obtained from the European Solar Radiation Atlas (Rigollier et al. 2000). Also, the Linke turbidity factor can be calculated from actual measurements of direct solar radiation S using the formula

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where α is the solar altitude angle and the functional forms for the relative optical air mass m, Rayleigh optical thickness δ_R(m), and extraterrestrial irradiance I_Ext normal to the solar beam are taken from Hofierka and Suri (2002). Determination of the Linke turbidity factor from direct solar radiation data measured at two sites located on Colorado's Brush Creek Valley floor (Whiteman et al. 1989a) using Eq. (1) finally gives T_LK = 2.75, which is consistent with values obtained from the European Solar Radiation Atlas.

The version of the solar radiation model r.sun used in this study considers only self shading (from the orientation of a surface with respect to the sun) and does not take the topographic shadowing effects (blocking of solar radiation by neighboring topography) into account. Also not included in this simple scheme is an additional contribution to K↓ from the proportion of reflected shortwave radiation received from surrounding terrain. On the other hand, the amount of diffuse radiation received on a surface in complex terrain depends on the fraction of the sky visible in the viewing hemisphere of the surface, called the sky-view factor υ. For a geometrically idealized valley cross section (with a horizontal valley floor and two linear sidewalls of different inclination angles), the smallest sky-view factors on the slopes occur at their bases. On the valley floor, sky-view factors increase as the valley center is approached (Whiteman 1990). For example, in the case of the Brush Creek Valley observations {made at five sites located on the valley's main topographic surfaces (Whiteman et al. 1989a, b): two on the valley floor (PNL and WPL sites), one on the ridgetop above the valley (RDG), and one on each of the opposing valley sidewalls [east (E) and west (W)]}, the sky-view factors were estimated as PNL: 0.74; WPL: 0.67; RDG: 0.92; E: 0.72; and W: 0.74. Observations of radiation components in the valley during September of 1984 (Whiteman et al. 1989a) have shown that 1) the orientation of the surface and shadows cast from surrounding topography have strong effects on instantaneous direct beam irradiance and 2) the diffuse radiation came predominantly from the sky but included single and multiple reflections from surrounding terrain within the viewing hemisphere of the radiometer. However, the instantaneous values of diffuse radiation were generally small and, although varying in time, did not vary significantly among the five radiation measurements sites at a given time.

Figures 1a and 1b show the comparisons of r.sun model simulations of K↓ and D with actual data for the Brush Creek Valley at the horizontal PNL site located on the valley floor (25 September 1984), and at the west sidewall site W#2 (29 September 1984), respectively. Notwithstanding the limitations discussed earlier (see also Whiteman et al. 1989a; Matzinger et al. 2003; Oliphant et al. 2003; Colette et al. 2003; etc.), the comparison plots show that the solar radiation model r.sun provides a good simulation of the actual data.

b. Radiation and surface energy budgets

Thermally developed valley wind systems are driven by heat transfer to and from the valley atmosphere, so that the sensible heat flux is the driving force in the energy budget equation. Also, the evolution of the wind systems and vertical temperature structure at individual sites is strongly dependent on sensible heat input into individual topographic elements within the valley. The energy available for sensible heat flux depends on solar heat flux (radiation budget) and the partitioning of energy in the surface energy budget (e.g., Whiteman 1982, 1990, 2000; Whiteman and McKee 1982).

Net all-wave radiation Q* provides the energy that is responsible for the turbulent exchange of sensible and latent heat between the earth's surface and the lower troposphere. It is often convenient to split the net radiation term into four components, and the radiation balance is thus expressed as (e.g., Holtslag and van Ulden 1983; van Ulden and Holtslag 1985; Moore et al. 1993; Bellasio et al. 2005)

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where the reduction factor (1 − r) is due to the reflection of incoming solar radiation by the surface, r is the albedo of the surface, K↓ is the incoming shortwave radiation, and L↓ and L↑ are incoming and outgoing longwave radiation, respectively. The surface energy budget relates the net radiation Q* to the various heat fluxes at the earth's surface. The energy balance of a homogeneous surface is given by the following equation (Oke 1978; Holtslag and van Ulden 1983):

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that is, the net all-wave radiation must be balanced by nonradiative fluxes, including surface sensible heat flux Q_H, surface latent heat flux Q_λE, and ground heat flux Q_G. Direct heat flux measurements are notoriously difficult to make on heterogeneous sloping surfaces (Whiteman 1990; Rotach et al. 2003), although several different measurement methods are available for estimating sensible, latent, and ground heat fluxes in complex terrain (Kimball et al. 1976; Whiteman et al. 1989b; etc.). Estimation of sensible heat flux often involves measurement of the individual terms in the radiation and surface energy budget equations in Eqs. (2) and (3).

To estimate the net radiation Q* at the surface, the components of the radiation budget must be determined. Several semiempirical parameterizations are available for estimating the components of the net radiation term (Whiteman et al. 1989a; Matzinger et al. 2003; Oliphant et al. 2003; etc.). The outgoing longwave radiation is given by (De Rooy and Holtslag 1999, p. 530; Oliphant et al. 2003, p. 118)

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where ε_s is the emissivity of the surface, σ is the Stefan–Boltzmann constant, and T_s is the radiation temperature of the surface. In complex terrain, the variability of topography and radiative properties of the surface combine to generate complicated spatial and temporal patterns of radiation and surface energy budgets (Oliphant et al. 2003). It is obvious then that the amount of incoming longwave radiation received on a surface in complex terrain depends on the sky-view factor. Therefore, the source of incoming longwave radiation is divided between surrounding terrain and the overlying atmosphere, depending on the sky-view factor. To account for the effect of υ, a very simple parameterization of the incoming longwave radiation L↓ is given by (Oliphant et al. 2003; Bellasio et al. 2005)

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with L_r = ε_rσT ⁴_r, where T_r is the air temperature at a reference height and ε_r is the atmospheric emissivity, which can be expressed as a function of air temperature, vapor pressure, and total cloud cover. Because the surface radiation temperature is not normally available, its fourth power can be substituted by the fourth power of the atmospheric temperature plus the correction term c₃Q*/σ, where c₃ is an empirical constant (Holtslag and van Ulden 1983). Because the empirical parameter c₃ may vary with surface moisture and air temperature, a better correction term is given by C_HQ*/σ, where C_H is an empirical heating coefficient that can be approximated as an analytical function of the Priestley–Taylor empirical parameter a_PT and air temperature (see van Ulden and Holtslag 1985). In the case of ε_s = 1 (i.e., the earth's surface is assumed to be a blackbody), combining the above equations finally yields

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or, equivalent (see also Bellasio et al. 2005),

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A very simple parameterization of the incoming longwave radiation L↓ in the absence of clouds, over flat homogeneous terrain, is given by L↓ = ε_rσT ⁴_r with ε_r = c₁T ²_r, where c₁ is an empirical coefficient (van Ulden and Holtslag 1985, p. 1200). Further, by setting υ = 1, it follows from Eq. (7) that (see also Holtslag and van Ulden 1983; van Ulden and Holtslag 1985)

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De Bruin and Holtslag (1982) show that the Penman–Monteith approach (Monteith 1981) can be simplified in such a way that it becomes more suitable for determining the partitioning of the surface sensible and latent heat fluxes. Then, a simple parameterization for estimating the sensible heat flux may be written (see Holtslag and van Ulden 1983):

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where s = ∂q_s/∂T (q_s is the saturation specific humidity), γ = C_p/Λ (Λ is the latent heat of water vaporization), the ratio γ/s can be expressed as a function of air temperature (van Ulden and Holtslag 1985), β (≈20 W m⁻²) is an empirical parameter for surface moisture for a grass-covered flat terrain, and Q_G = c_GQ* (c_G is an empirical coefficient for the ground heat flux).

Surface properties, including vegetation cover and soil moisture, are important in explaining differences between sites in reflected radiation, outgoing longwave radiation, and surface fluxes of heat and momentum. The Priestley–Taylor empirical parameter α_RT that affects the estimation of sensible heat flux through Eq. (9) has a seasonal variation depending on vegetation cover and soil moisture. For normal summer conditions in the Netherlands, preliminary estimates for a grass-covered surface (provided with enough water to evaporate) give α_RT ≈ 1. When there is lack of water, the value of α_RT decreases (De Bruin and Holtslag 1982). Also, we can determine values of α_RT for other surface and moisture conditions. For instance, for the Prairie Grass experiment, with rather dry vegetation, Holtslag and van Ulden (1983) found that typically α_RT ≈ 0.5. For bare soil, when there is no water to evaporate, α_RT vanishes (e.g., α_RT ≈ 0–0.2). It now becomes obvious, especially in complex terrain, that α_RT varies from site to site, depending on soil moisture and vegetation cover. Because α_RT is not generally evaluated in the field programs, it is usual to estimate the Priestley–Taylor empirical parameter from a visual terrain classification (Hanna and Chang 1992). However, as stated by Holtslag and van Ulden (1983) and Brutsaert (1982), the specific dependence of α_RT on the moisture conditions of the surface awaits further examination. In this study, typical values of 0.2–0.7 are chosen for the Priestley–Taylor empirical parameter that are probably consistent with the normally dry climates in deep mountain valleys of western Colorado.

Albedo r, defined as the absolute value of the ratio R↑/K↓ (where R↑ is the reflected shortwave radiation), depends on the reflective properties of bare soil and vegetation cover and on solar angles. Differing surface covers and characteristics generate complicated spatial and temporal patterns of albedo in complex terrain (Whiteman et al. 1989a; Matzinger et al. 2003; Oliphant et al. 2003). In the case of the Brush Creek radiation-measurement sites during the inversion breakup period, the average albedos were PNL: 0.15, WPL: 0.20, W: 0.12, and RDG: 0.14. The east sidewall (E) had a steady albedo increase during daytime, with an average albedo of about 0.21 (see Whiteman et al. 1989a, their Table 4, p. 419).

The empirical scheme used by Holtslag and van Ulden (1983) and van Ulden and Holtslag (1985) was designed for grass-covered surfaces, and its application is restricted to homogeneous level terrain. An alternative technique of calculating convective fluxes in complex terrain is the Bowen ratio energy budget method. This technique has been used in complex terrain experiments by several investigators and has the advantage of requiring small amounts of equipment and measurements. For a more detailed description of the Bowen ratio energy budget method of calculating convective fluxes in complex terrain see Whiteman et al. (1989b). Last, the ratio between ground heat flux and net radiation c_G = 0.1 was obtained for a grass-covered surface in the Netherlands (De Bruin and Holtslag 1982). The observations in the Brush Creek Valley (Whiteman et al. 1989b), however, showed that Q_G and c_G varied from site to site, depending mostly on soil moisture and vegetation cover (e.g., daytime ground heat flux totals were 0.06–0.15 of the daytime net all-wave radiation totals).

Figure 2 compares estimated values of Q* [from Eq. (7)] and Q_H [from Eq. (9)] with actual data for the Brush Creek Valley (25 September 1984) at the PNL site (Fig. 2a, reference simulation), at the W site (Fig. 2b), and at the E site [where an initial approximate value of air temperature T ≈ 286 K is tentatively adopted (from climatological data) and typical mean values α_RT = 0.3, 0.5, and 0.2 are chosen for the west sidewall, valley floor, and east sidewall, respectively (see Whiteman et al. 1989b); Fig. 2c]. The sensitivity of Q_H to the parameters α_RT and T is illustrated in Figs. 2d and 2e using the reference simulation above. For the normally dry climates in deep mountain valleys of western Colorado (e.g., assuming that α_RT ≈ 0–0.7) the estimated values of Q_H show less dependence on air temperature. During fair-weather conditions, in the dry Colorado valley atmosphere, the most sensitive parameter affecting Q_H is the incoming solar radiation. In contrast, when the surface moisture increases (e.g., α_RT ≈ 1), the available energy input is partitioned mainly into latent heat flux and the influence of the initial air temperature is much stronger on the estimated values of surface sensible heat flux. For comparison, the estimated Q_H from the surface energy budget equation used by Bader and McKee (1983, 1985) [see also Eq. (21)] for a horizontal site located on the valley floor in a typical Colorado valley is presented in Fig. 2d as a dotted line.

In conclusion, notwithstanding the limitations mentioned previously (see also Whiteman et al. 1989a, b; Whiteman 1990, 2000; Matzinger et al. 2003; Oliphant et al. 2003; Colette et al. 2003; etc.), the comparison plots show that our highly simplified empirical scheme for radiation and surface energy budget daytime estimates (from routine weather observations) provides a reasonable simulation of the actual data.

c. An empirical scheme for the partitioning of energy in idealized valleys

Because energy gain and loss from the valley volume occur primarily at the valley surfaces, semiempirical parameterizations of radiation and surface energy budgets at the valley floor and sidewalls may be tentatively used to obtain an alternative empirical scheme for the partitioning of energy in the valley atmosphere. Following Whiteman's hypothesis, after sunrise a fraction of solar energy flux coming across the horizontal upper area L of the top of the inversion is converted to the sensible heat fluxes Q_Hf , Q_Hs1, and Q_Hs2 on the valley floor and opposing valley sidewalls, respectively. Because the surface sensible heat flux is the driving force that provides the energy input that warms the valley atmosphere, we assume that the sensible heat fluxes on the valley surfaces are the key variables that are relevant to the energy-partitioning phenomenon (note that advective terms and radiative flux convergences may be important in some circumstances). Following the above considerations, with the assumption of an idealized flat-bottomed valley and adoption of a highly simplified dimensional-analysis procedure, it is reasonable to make a tentative hypothesis that the contribution to the total rate of energy input from the fraction of solar radiation that is converted to sensible heat flux on the valley floor EI_VF may be proportional to the product of Q_Hf and an unknown characteristic length scale l_CBL, which is tentatively assumed to be analogous to the valley floor width l_CBL ∝ l, that is, EI_VF = w₁Q_Hfl_CBL, and the additional contribution to the total rate of energy input from the fraction of solar radiation that is converted to sensible heat flux on the valley sidewalls may be expressed by a similar relationship: EI_SW = w₂(Q_Hs1 + Q_Hs2)l_INV, in which the unknown characteristic length scale is tentatively assumed to be analogous to the difference between the inversion top width L and the valley floor width, that is, l_INV ∝ (L − l). The empirical proportionality factors w₁ and w₂ (nondimensional) may then be determined by regression analysis against the observed data (here, we assume them to be unity, in the absence of definitive evidence for alternative values). Thus, 1) the characteristic length scale for the total rate of energy input Q is assumed to be analogous to the top of the inversion L, 2) the characteristic length scale for the sensible heat flux from the valley floor is assumed to be analogous to the valley floor width l, and 3) the characteristic length scale for the remaining fraction of energy input, that is, for the total sensible heat flux from both sidewalls, is assumed to be analogous to the difference L − l.

Following Whiteman's hypothesis, it is reasonable to assume that the growth of a convective boundary layer at the valley floor, the sinking and warming of the stable layer induced by the slope winds, and their recirculation above the valley crests are all together responsible for the breakup of the stable layer (see also Whiteman 1982; Whiteman and McKee 1982; Colette et al. 2003; etc.). Therefore, a fraction of the total energy input, (KQ)_CBL, may be used to deepen and warm CBLs, and the remaining fraction, [(1 − K)Q]_INV, may be used to drive the slope flows that carry mass up the valley sidewalls, such that,

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where f_c and s_c are the fractions of EI_VF and EI_SW, respectively, that are used to drive the CBL growth [i.e., the remaining fractions (1 − f_c) and (1− s_c) result in the inversion descent] and the residual terms (e.g., advection terms) R_CBL and R_INV refer to a probable transferring of heat to and from the adjacent valleys (Hennemuth 1985). A highly simplified version of the empirical scheme for the partitioning of energy in the valley atmosphere may be obtained by assuming that s_c, R_CBL, and R_INV are nearly zero and f_c ≅ 1, such that

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In this special case, Eq. (12) is formulated so that the entire fraction of solar radiation that is converted to sensible heat flux on the valley floor is used to cause the CBL to grow, and Eq. (13) is formulated so that the entire fraction of solar radiation that is converted to sensible heat flux on the valley sidewalls goes solely to move mass up the valley slopes, causing the top of the inversion to descend.

Following Whiteman's hypothesis, the total energy requirement is composed of two parts: the energy increment Q₃ that causes a CBL to grow and the energy increment Q₂ that removes mass from the valley and allows the top of the inversion to sink. By differentiating the individual energies Q₂ and Q₃ with respect to time, an energy balance for the valley atmosphere is obtained by equating the fraction of the rate of energy input used to drive the growth of the CBL to dQ₃/dt and the remainder to dQ₂/dt, such that

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where the functional forms for dQ₃/dt and dQ₂/dt are taken from Whiteman and McKee (1982) and the fractions (KQ)_CBL and [(1 − K)Q]_INV of the total energy input may be tentatively determined by using the empirical scheme for the partitioning of energy in the valley atmosphere discussed earlier [i.e., Eqs. (10) and (11)]. Further, solving Eqs. (14) for the rate of ascent of the CBL dH/dt and for the rate of descent of the inversion top dh/dt results in a modified version of the Whiteman and McKee inversion destruction model:

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where C = (1/tani₁) + (1/tani₂) and c_P and ρ are the specific heat at constant pressure and density of air, respectively (see Whiteman and McKee 1982, p. 291). It is also assumed that mass is removed from the valley in such a way that the potential temperature gradient γ in the stable core is constant during the period of inversion breakup. The topographic amplification factor (Steinacker 1984; Müller and Whiteman 1988; Whiteman 1990; etc.) in Eqs. (15) and (16) was calculated on the basis of an idealized trapezoidal valley cross section having linear sidewalls and a flat-bottomed valley floor (this factor accounts for the reduced volume of air within the mountain valley relative to that over the plains for the same energy flux on a horizontal surface). When significant warming occurs in the neutral layer above the inversion top during the temperature inversion breakup period, extra energy is required to destroy the valley temperature inversion, because the inversion layer cannot be broken until the entire valley atmosphere is warmed to the temperature of the air above the valley. Therefore, the general model Eqs. (15) and (16) are modified to account for this extra energy requirement: Eq. (15) has the same form as before, but Eq. (16) can be approximated by (see Whiteman and McKee 1982, p. 297)

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where β_W (K s⁻¹) is the warm-air advection rate above the valley. It follows that Eq. (17) reduces to (16) when β_W is zero. An integration of the general model [Eqs. (15) and (16)] allows the simulation of the evolution of vertical temperature structure during the inversion breakup period in idealized valleys under fair-weather conditions. Last, the model is tested against specific datasets to test its performance in situations in which valley topography, initial inversion structure, and other external conditions vary.

3. Results

The bulk thermodynamic model of temperature inversion destruction developed here is based on the following fundamental assumption: simple schemes are available for estimating the fraction of solar radiation that is converted to sensible heat flux on the major topographic surfaces, that is, the valley floor and sidewalls, and the partitioning of energy in the valley atmosphere. As a working hypothesis, 1) the solar radiation model “r.sun” described in Hofierka and Suri (2002), 2) a modified version of the simple empirical scheme suggested by Holtslag and van Ulden (1983) and van Ulden and Holtslag (1985) [Eqs. (8) and (9)], and 3) a highly empirical scheme for the partitioning of energy in the valley atmosphere [see Eqs. (10) and (11)] were adopted in developing the numerical solution to the system of general model equations.

The inversion breakup phenomenon is driven primarily by solar heating of the ground in the morning. Therefore, accurate computation of the incoming solar radiation is critical for inversion breakup simulations. The solar radiation model r.sun is used in estimating the direct solar radiation and diffuse radiation on an inclined (and horizontal) surface at level z. The comparison plots in Figs. 1a and 1b show that r.sun provides a good simulation of the actual data.

A simple semiempirical scheme is presented that gives estimates of the net all-wave radiation Q* and surface sensible heat flux Q_H, avoiding the use of surface radiation temperature [Eqs. (6), (7), and (9)]. By setting υ = 1, this parameterization reduces to the same empirical scheme [Eqs. (8) and (9)] used by Holtslag and van Ulden (1983) and van Ulden and Holtslag (1985), which gives daytime estimates of Q* and Q_H from routine weather data. Therefore, in the absence of actual data for the sky-view factor and atmospheric (and surface) emissivity, as a working hypothesis this modified empirical scheme (because of its simplicity) was tentatively adopted for daytime estimates of sensible heat fluxes on the major valley surfaces [see Holtslag and van Ulden 1983, p. 521; van Ulden and Holtslag 1985, p. 1200] in developing the numerical solution to the system of general model equations. This scheme was designed for grass-covered surfaces, and the application of the suggested method is restricted to homogeneous level terrain. However, as stated in Holtslag and van Ulden (1983) and van Ulden and Holtslag (1985), the scheme contains parameters that take account of the surface properties in general. Here we must conclude that more work has to be done.

To complete the thermodynamic model, the partitioning of the energy input into CBL growth and mass transport must be estimated. Following sunrise, the CBL develops first over the illuminated sidewall or sidewalls and somewhat later over the valley floor when it is sunlit. The partitioning of energy approach used in the Whiteman and McKee inversion destruction model is controlled by an empirical function K defined as the fraction of sensible heat flux going to CBL growth. Furthermore, this empirical factor is assumed to be of the form

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where the remaining fraction 1 − K is assumed to be responsible for mass transport up the valley sidewalls, which results in inversion descent. Because the functional dependencies of K are not yet known, it is assumed that k is a constant and, by comparing model simulations with actual data, the values of k and A₀ that result in the best fit to data are determined (Whiteman and McKee 1982; see also Zoumakis and Efstathiou 2006, hereinafter Part I). The empirical scheme for the partitioning of energy in the valley atmosphere discussed in the previous section contains the unknown parameters f_c, s_c, R_CBL, and R_INV [Eqs. (10) and (11)]. Therefore, as a working hypothesis, a highly simplified version of this empirical scheme was adopted in developing the numerical solution to the system of general model equations. By adopting Eqs. (12) and (13), we tentatively hypothesize that most of the available sensible heat flux from the valley floor drives the CBL growth, and the remaining energy (i.e., the total sensible heat flux from both sidewalls) drives mass up the sidewalls, causing the top of the inversion to descend. Combining Eqs. (12) and (13) finally yields

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with

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where h_i is the depth of the initial inversion layer. As a consequence, it follows from Eqs. (19) and (20) that factor K depends on the topographic characteristics of the valley, that is, the (nondimensional) topographic parameter V_* = l/(h_iC), and sensible heat flux (see also Part I). Because the stable layer is destroyed from both the bottom (through surface heating) and the top (through the recirculation of warm air along the slopes into the valley from above, as well as the sinking and warming of the stable layer induced by the removal of air from the bottom of the valley) it is obvious that, in the actual valley atmosphere, the parameter f_c is nearly always less than 1 (i.e., a fraction of the solar radiation that is converted to sensible heat flux on the valley floor is lost in the upslope flows) and the parameter s_c is nearly always not equal to zero (i.e., a fraction of the solar radiation that is converted to sensible heat flux on the valley sidewalls goes to the growth of the valley-floor CBL). In any case, we have taken a first preliminary (crude) step toward an understanding of the energy-partitioning phenomenon. As is clear from the complex-terrain-meteorology experiments, valley flows in the atmosphere rarely occur under such idealized circumstances as assumed here. For example, there is nearly always mixing of momentum and energy on top of the valley, a process that will be difficult to describe. In contrast, at least for a narrow range of weak to moderate synoptic flow conditions, idealized deep valley topography produces more consistent inversion structures by protecting them from winds aloft (Whiteman 1982, 1990, 2000). However, in any case, the partitioning of energy in the actual valley atmosphere is an extremely complex phenomenon.

In this section, a finite-difference form of the general model equations will be applied to simulate actual data collected in valleys on the western slope of the Rocky Mountains in western Colorado (Whiteman 1982). All valley surfaces are assumed to have the same uniform average albedo (r ≈ 0.16), and climatic mean temperatures are used as initial approximate values of air temperature T₀ for the iterative process. Based on Eqs. (15) and (16), Fig. 3 compares model simulations of H and h (solid lines) with actual data (dashed lines) for different values of α_RT: (a) for Eagle Valley on 16 October 1977 [note that in the reference simulation (thick solid line) the typical mean values α_RT = 0.7 and 0.3 are chosen for the valley floor and sidewalls, respectively], (b) for Eagle Valley on 9 July 1978, (c) for Gore Valley on 6 July 1978, (d) for South Fork White Valley on 29 August 1978, (e) for Yampa Valley on 9 August 1978, and (f) for Gore Valley on 19 October 1977. By using the reference simulation, the sensitivity of the inversion breakup to the parameters r, α_RT, and T₀ is illustrated in Fig. 3g for different values of r, and in Fig. 3h for different values of α_RT and T₀ [using the following model simulations: I (α_RT = 0.2 and T₀ = 273 K), II (α_RT = 0.2 and T₀ = 310 K), III (α_RT = 0.7 and T₀ = 273 K), and IV (α_RT = 0.7 and T₀ = 310 K)]. As far as idealized cases can be compared with real data, notwithstanding the limitations discussed in previous sections, the comparison plots for these simulations show rather good agreement with actual data. Also, the depths and lifetimes of the stable layer are reasonable. Last, it is concluded that the idealized patterns of inversion breakup described by Whiteman (1982) are successfully reproduced. This success is probably due to the predominant role of the direct solar beam on the overall radiation and surface energy budget in the normally dry Colorado valleys atmosphere during fair-weather conditions.

As a first approximation, following Whiteman's hypothesis, after sunrise the fraction of solar radiation that is converted to sensible heat flux on the valley's main topographic surfaces is assumed to be proportional to the extraterrestrial solar energy flux. It is important to note that the daytime values of sensible heat flux are nearly proportional to the net all-wave radiation flux at the ground surface and the net radiation typically becomes positive an hour or more after local sunrise in all deep valley sites (Whiteman et al. 1989b). In addition, the Whiteman and McKee (1982) inversion destruction model using extraterrestrial solar radiation as incoming energy does not differentiate between CBL growth over the major topographic surfaces, that is, between the valley floor and sidewalls. Thus, the initial overprediction of CBL growth over the valley floor is a characteristic feature of the bulk thermodynamic model equations (see Whiteman and McKee 1982, their Fig. 13, p. 300; see also the time of theoretical sunrise, the time of local sunrise, and the arbitrary values for the simulation starting time). As a consequence, by adopting simple parameterizations of radiation and surface energy budgets at the valley surfaces, it is obvious that the modified version of the Whiteman and McKee inversion destruction model [i.e., Eqs. (15) and (16)] provides a better simulation of the actual data (see Fig. 3).

The theoretical estimates from Eqs. (15) and (16) can be compared with dynamical model predictions. For example, Bader and McKee (1983, 1985) used a two-dimensional cross-valley dynamical model to study the morning inversion destruction cycle in a variety of deep mountain valley configurations. The surface sensible heat flux for each of the major valley surfaces was described by a sinusoidal function (Bader and McKee 1983, 1985)

View Expanded

where the day length T was set to 12 h to approximate the diurnal solar cycle, the amplitude (Q_H)_max based on observational data collected during clear, undisturbed synoptic conditions, and the heating factor F_i is an empirical coefficient for each of the major topographic surfaces. Also, Colette et al. (2003) used a three-dimensional, compressible, and nonhydrostatic large-eddy simulation model to investigate the effects of valley width and depth and topographic shading on inversion breakup in idealized mountain valleys. The surface characteristics were chosen so that the modeled valleys were similar to a typical Colorado valley (Bader and McKee 1983, p. 349; 1985, p. 826; Colette et al. 2003, p. 1258). Based on Eqs. (15), (16), and (21), Fig. 4 compares the bulk thermodynamic model simulations of H and h (solid lines) with the dynamical model predictions (dashed lines) in an idealized valley, for case 1 (Fig. 4a, reference case), case 2 (Fig. 4b), and case 3 (Fig. 4c) presented by Bader and McKee (1985). These simulations were run to compare the evolution of the boundary layer in valleys of differing widths (see Bader and McKee 1985, p. 827). Also, based on Eqs. (8), (9), (15), and (16) [assuming r ≈ 0.16 and a potential temperature gradient similar to that used by Bader and McKee (1983, 1985) to initialize their simulations], Fig. 4d compares the bulk thermodynamic model simulations of H and h (solid lines) for different values of α_RT with the dynamical model prediction (dashed line) in an idealized valley for the reference simulation presented by Colette et al. (2003). The comparison plots (see Fig. 4) show that the thermodynamic results agree quite favorably with the dynamical model predictions.

The total rate of energy input into the valley atmosphere used in the thermodynamic valley inversion destruction model of Whiteman and McKee (1982) is tentatively assumed to be a constant fraction A₀ of the solar energy flux coming across the horizontal upper area of the top of the inversion that is converted to sensible heat flux on the valley surfaces [see Eq. (3) in Part I]. The fraction k [see Eqs. (18) and (14)] uniquely determines the parameters Λ and μ that affect the partitioning of E_TOTAL in the valley atmosphere [see Eqs. (24) and (25) in Part I]. In the normally dry Colorado valleys the most important factors affecting the available energy are surface moisture (latent heat flux) and albedo (see Whiteman and McKee 1982, p. 294). Based on Eqs. (15) and (16) herein and Eqs. (17) and (29) in Part I, the solid lines in Fig. 5a represent the dependence of A₀ on α_PT (for different values of r) and the solid lines in Fig. 5b represent the dependence of the energy-partitioning parameter k on the Priestley–Taylor empirical parameter (α_PT)_VF for surface moisture at the valley floor, for different values of the empirical parameter (α_PT)_SW for surface moisture at the valley sidewalls, with r = 0.1 (solid lines) and r = 0.3 (dashed lines). It follows from Fig. 5a (for a given r) that A₀ decreases with increasing α_PT (i.e., when the surface moisture increases, the available energy input is partitioned mainly into latent heat flux). With the assumption of a constant α_PT, Fig. 5a illustrates the expected decrease in A₀ with increasing r (i.e., when the albedo increases, the amount of energy input available to destroy the stable layer decreases). In the case of a constant (α_PT)_SW, Fig. 5b illustrates the expected decrease in k with increasing (α_PT)_VF (i.e., as the surface moisture increases at the valley floor the inversion destruction more nearly follows pattern 2). In the case of a constant (α_PT)_VF, Fig. 5b illustrates the expected increase in k with increasing (α_PT)_SW (i.e., partitioning of much of the available energy into latent heat flux at the valley sidewalls reduces the sensible heat flux there). Last, with the assumption of uniform surface characteristics over the valley (i.e., using a homogeneity assumption), it follows from Fig. 5b that the effect of α_PT and r on the energy-partitioning parameter k is relatively small. It is obvious that the topographic characteristics of the valley may affect the mode of inversion destruction because they control, to a certain extent, the divergence of mass in the CBL and thus determine whether inversion destruction more nearly follows pattern 1 or pattern 2 (see also Whiteman and McKee 1982, p. 294).

4. Discussion

It would be wise to note some of the obvious limitations of the proposed method discussed in previous sections: actual valleys tend to have a much more complicated geometry than that assumed here, and thus the roughness elements (Zoumakis 1994) influence the circulation in a manner that is considerably different from what has been described here. Also, in complex terrain, the variability of topography and surface characteristics, including vegetation cover, soil moisture, and radiative properties of the surface combine to generate complicated spatial and temporal patterns of radiation and surface energy budgets. Therefore, among the limitations, the most noteworthy fact is that the traditional view of the surface radiation and energy budgets as described by Eqs. (2) and (3) is significantly more complicated in highly complex terrain, in the following four ways. 1) Direct-beam shortwave radiation varies spatially, because the orientation of the surface (with respect to the sun) and the shading effects (resulting from the blocking of solar radiation by neighboring topography) have strong effects on the directional component of the incoming solar radiation. Diffuse-beam shortwave radiation also varies over space because of its partial anisotropy, the predominant role of sky-view factor on the isotropic portion, and the multiple reflections from surrounding terrain. The source of incoming longwave radiation is also divided between surrounding terrain and the overlying atmosphere, depending on the sky-view factor (Whiteman et al. 1989a; Whiteman 1990; Oliphant et al. 2003; Matzinger et al. 2003; etc.). 2) Significant spatial differences in instantaneous and daily total sensible and latent heat fluxes occur within complex terrain areas. For example, in the dry, high-altitude Brush Creek Valley, observations showed that strong contrasts in instantaneous latent and sensible heat fluxes occurred between the opposing northeast- and southwest-facing valley sidewalls as insolation varied through the course of the day and as shadows propagated across the valley. Whiteman et al. (1989a, 1989b) and Matzinger et al. (2003) emphasized the importance of the topographic shading effects, which, by delaying the local sunrise (e.g., see Fig. 1), strongly affect the net radiation and surface energy balance in the actual valley atmosphere. The delay of about 1 h of the actual data from the estimated values of Q* and Q_H in Fig. 2 is probably due to the local topographic shading effects (see Whiteman et al. 1989a, p. 426). In addition, the differential heating and moistening of the air above the opposing valley slopes produced cross-valley circulations that resulted in redistribution of heat and moisture throughout the valley volume. Therefore, the field of latent and sensible heat fluxes in the actual valley atmosphere often contains a mosaic of microclimatic regimes, with pronounced differences occurring over very small spatial scales (Whiteman et al. 1989a, 1989b; Whiteman 1982, 1990, 2000; etc). Given the large spatial range of surface (turbulent) heat flux densities expected for each component of the surface radiation and energy budgets, the homogeneity assumption that surface characteristics are uniform over the valley is generally incorrect. 3) Estimation of sensible heat flux often involves measurement of the individual terms in the radiation and surface energy budget equations. However, actual observations at a particular location on a valley slope may be unrepresentative of the mean conditions over the slope because of the known inhomogeneity and nonstationarity of the upslope flows on valley sidewalls in real topography (Whiteman 1990). As mentioned in the previous section, direct surface heat flux measurements are notoriously difficult to make on heterogeneous sloping surfaces in highly complex terrain because of instrumental restrictions or difficulties. On the other hand, in many studies over nonideal terrain (and often even over near-ideal terrain) the surface energy balance is found to be unclosed (Rotach et al. 2003). Furthermore, a key question regarding the measurements of individual terms in the radiation and surface energy budget equations is the representativeness of the point measurements for estimating convective fluxes over the valley as a whole. 4) Overall, the bulk thermodynamic model predictions agree very favorably with real data but lack some of the detail necessary to describe fully the inversion breakup in the actual valley atmosphere. It must be stressed that the results presented here are preliminary in the sense that not all cases of interest have yet been considered (see also Whiteman and McKee 1982). An alternative approach is to solve the coupled conservation equations of mass, momentum, and energy with appropriate initial and boundary conditions. Therefore, an adequate incorporation of the combined action of these conservation equations represents a fundamental step in the development of a realistic dynamical model to simulate the evolution of vertical temperature structure during the inversion breakup period in actual valleys. However, although the physical processes affecting the valley local circulations may be identified, the relative importance of the different processes varies from valley to valley, from time to time in the same valley, and even from segment to segment along a valley's length. Moreover, the valley atmosphere consists of many interrelated layers, and the coupling of the conservation equations for the different layers (to simulate potential temperature changes in the valley atmosphere as a whole) would be extremely difficult because of topographic complexity, geometrical considerations, and the lack of detailed information on physical characteristics of the various interrelated layers (Whiteman and McKee 1982; Whiteman 1990; Rotach et al. 2003; etc). Therefore, in this study a highly simplified approach is taken in which a bulk thermodynamic model is developed to simulate the inversion breakup phenomenon in idealized valleys.

5. Conclusions

It is important to note that a full investigation of radiation and energy budget schemes at valley surfaces and the partitioning of energy in the valley atmosphere was beyond the scope of this study. Therefore, because the purpose of this study was to demonstrate a search approach rather than a model development, highly simplified semiempirical schemes [i.e., Eqs. (6)–(9) and (10)–(14)] were adopted to develop the method. Nevertheless, the major conclusion is that there is clear evidence that improvement of the inputs to the model (e.g., α_RT, β, c_G, and r) can also improve the quality of thermodynamic model simulations of the height h of the inversion top and the depth H of the convective boundary layer during the inversion breakup period in idealized valleys. Here again we must conclude that more work has to be done. Moreover, instead of using semiempirical parameterizations of radiation and surface energy budgets designed for grass-covered homogeneous level terrain [i.e., Eqs. (8) and (9)], there is also clear evidence that the input parameters to the thermodynamic model can be improved by the use of an alternative technique of calculating radiation and convective fluxes in highly heterogeneous complex terrain. An example thereof is the use of surface radiation models that are particularly suited to complex topographical sites (Oliphant et al. 2003; Bellasio et al. 2005; etc.) based on a topographically driven surface radiation scheme, which determines the radiation fluxes over a surface. Using this model, combined with theoretical equations for the sun's movement through the sky on the date of interest (Whiteman and Allwine 1986; Hofierka and Suri 2002; etc.), self-shading and topographic shading algorithms may be used to simulate the propagation of shadows throughout the valley (Colette et al. 2003). An initial approach to estimating the variability of energy budget terms in complex terrain has been to consider that the surface energy budget is driven primarily by net radiative gains during daytime and to estimate the spatial variability of such gains by investigating the contributions to net radiation from individual terms in the radiative energy budget (Whiteman 1990). Therefore, instead of using a homogeneity assumption for each of the major valley surfaces [e.g., see the characteristic length scales in Eqs. (12) and (13)], the energy input available for sensible heat flux that warms the valley atmosphere may be properly determined by using the radiation and surface energy budget [Eqs. (2) and (3)] on individual topographic elements within the valley [i.e., enhancement of domain resolution (i.e., smaller topographic elements are assumed to have uniform surface characteristics) may be possible to improve model results]. As more is learned about the individual components of the overall valley system (including the various interrelated layers and stable core) the bulk thermodynamic model may be refined to simulate more appropriately the inversion breakup phenomenon in idealized valleys (see also Whiteman and McKee 1982). On the other hand, it may not be necessary to know the detailed structure of the valley wind systems, the variability of valley topography and surface characteristics, and the radiation and surface energy budgets on individual topographic elements within the valley to determine bulk valley warming or to simulate potential temperature changes in the valley atmosphere as a whole. Such changes are caused by the total energy input to the valley volume and its distribution through the valley atmosphere by the integrated slope flows (Whiteman 1990). It would be wise to note that chaotic components of valley flows may be important in the actual atmosphere, so that as the model becomes more complicated its behavior becomes more unpredictable. However, it is realized that this idea needs further verification. In the future, additional research will be necessary to compare the general model equations with meteorological observations from different representative valley geometries to determine the validity of the new approaches.

In summary, a simple thermodynamic parameterization based on a modified version of the Whiteman and McKee (1982) inversion destruction model is presented to simulate the changes with time of the height of the inversion top and the depth of the convective boundary layer during the inversion breakup period in idealized valleys under fair-weather conditions. The proposed new method tentatively adopts simplified semiempirical parameterizations of radiation and surface energy budgets at the valley floor and sidewalls, and a highly empirical scheme for the partitioning of energy in the valley atmosphere, eliminating the need for arbitrary values for the adjustable model parameters. In addition, the bulk thermodynamic model is simple to use, because it depends on routinely available data. Because of its simplicity and its fair agreement with observations, the proposed method may be used in applications in boundary layer, air pollution, and complex-terrain meteorology.