Hypercooling in the Nocturnal Boundary Layer: Broadband Emissivity Schemes

V. K. Ponnulakshmi Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India

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V. Mukund Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India

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D. K. Singh Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India

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K. R. Sreenivas Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India

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G. Subramanian Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India

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Abstract

Broadband flux emissivity schemes are often used to model infrared radiative exchanges in the atmosphere. In particular, such schemes help highlight the interaction of radiation with other transport processes, an aspect that is crucial to an understanding of phenomena relevant to the nocturnal boundary layer (NBL). Although the original schemes were restricted to radiatively black bounding surfaces, an extension of the same to nonblack surfaces has since been frequently used in NBL modeling. Herein, it is shown that the nonblack extension is erroneous and leads to a spurious yet intense near-surface cooling in the opaque bands. This paper presents the correct formulation that eliminates this cooling and discusses in some detail earlier NBL calculations affected by this error.

Corresponding author address: G. Subramanian, Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India. E-mail: sganesh@jncasr.ac.in

Abstract

Broadband flux emissivity schemes are often used to model infrared radiative exchanges in the atmosphere. In particular, such schemes help highlight the interaction of radiation with other transport processes, an aspect that is crucial to an understanding of phenomena relevant to the nocturnal boundary layer (NBL). Although the original schemes were restricted to radiatively black bounding surfaces, an extension of the same to nonblack surfaces has since been frequently used in NBL modeling. Herein, it is shown that the nonblack extension is erroneous and leads to a spurious yet intense near-surface cooling in the opaque bands. This paper presents the correct formulation that eliminates this cooling and discusses in some detail earlier NBL calculations affected by this error.

Corresponding author address: G. Subramanian, Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India. E-mail: sganesh@jncasr.ac.in

1. Introduction

Broadband flux emissivity schemes are computationally efficient and typically used in studies where the primary focus is not the radiation modeling alone, but also its interaction with turbulence, stratification, surface characteristics, and so on. (Penner et al. 2009; Pettijohn et al. 2009; Abraha and Savage 2008; Brubaker and Entekhabi 1996). Examples include micrometeorological phenomena that influence or are influenced by the thermal structure of the nocturnal boundary layer (NBL)—the formation and growth of inversion layers after sunset, the onset of radiation fog, etc. The broadband fluxes may be obtained from the corresponding spectral expressions by an integration over frequency, with a diffusivity factor used to model the angular dependence of the intensity (Goody 1964; Liou 2002); there are additional assumptions with regard to the thermodynamic scaling properties of the absorption paths. The central quantity in the resulting flux expressions is the broadband flux emissivity ϵf(u) defined as the ratio of the spectrally integrated emission from an isothermal column of participating medium of mass-absorption path length u to that of a blackbody; here, u is related to the actual column height z by , with δ being a thermodynamic scaling exponent, ρ(z′) the concentration of the participating component, and p(z′) the pressure at the level z′. Thus, an isothermal column emission at temperature T0 is . Here, ϵf(u) → 0 (1) for u → 0 (∞), so that short columns are nearly transparent while the emission of infinitely long columns is the same as that of a blackbody.

The broadband flux-emissivity in Fig. 1 is based on an empirical parameterization developed by Zdunkowski and Johnson (1965) and emphasizes the nonexponential variation of ϵf(u) with u for a nongray medium—in this case, a water vapor–laden atmosphere. An initial sharp increase, for small u, due to opaque band emissions is followed by a much slower increase due to weak emission in the transparent bands; the flux emissivities for other atmospheric gases exhibit a similar dependence on u. The deviation of ϵf(u) from the exponential increase characteristic of a gray medium is important. Although, by the very definition of ϵf(u), participating medium emission appears in a manner analogous to a solid surface, with ϵf(u) playing the role of the surface emissivity, the same is true for the attenuated reflected flux only for a gray medium wherein the spectral dependence of the intensity follows the Planck function. For a nongray medium with a multiplicity of photon path lengths, attenuation depends on the spectral contents of the incident radiation, and there can be no universal transmissivity; in particular, the attenuation corresponding to a broadband transmissivity defined by τf(u) = 1 − ϵf(u) applies only to a gray incident radiation. The prevailing scheme for nonblack surfaces proposed by Garratt and Brost (1981) fails to recognize this difference. The resultant erroneous transmissivity used to attenuate the reflected flux leads to an intense cooling within the opaque bands. We proceed to obtain the correct broadband reflected flux that eliminates this spurious cooling. The corrected emissivity scheme is then used to examine radiative flux divergence profiles within the NBL (modeled here as a simple inversion) as a function of the varying surface (ground) emissivity. Earlier applications of the erroneous scheme have exaggerated the role of the ground emissivity on the NBL thermal structure, and we discuss a few of these calculations to highlight the nature and magnitude of the error. The corrected scheme helps clarify the relative influences of the radiative and sensible flux divergences on NBL structure and evolution—an important issue in micrometeorology (Hoch 2005).

Fig. 1.
Fig. 1.

A schematic of ϵf(u) as a function of the mass absorber path length u; the expression used is given in section 2 [see (12)]. The flux emissivity for a gray medium, a simple exponential, is also shown with a photon mean free path of 200 m.

Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0269.1

Although the focus here is on the resolution of the aforementioned spurious cooling error within the framework of broadband emissivity schemes, the aforementioned spurious cooling error itself is generic in character and not merely confined to emissivity schemes. It is shown in an accompanying paper (Ponnulakshmi et al. 2012, manuscript submitted to J. Atmos. Sci.) that the error arises whenever the transmittance, averaged over the relevant frequency interval, deviates from an exponential. Owing to the extremely sensitive dependence of the absorption coefficient on frequency in the infrared, typical atmospheric gases behave as gray media only over spectral intervals comparable to or smaller than an elementary line width. An error is thus expected in any frequency-parameterized radiation scheme applied to the atmosphere over nonblack ground, including typical narrowband formulations, wherein the parameterization is over intervals significantly larger than an elementary line width. Correctly capturing the variability of the appropriate reflected flux on local, regional, and global scales is crucial to an accurate estimate of both the upwelling longwave fluxes and land surface temperatures via remote sensing in the thermal infrared (Wang et al. 2005), and elimination of the aforementioned error in frequency-parameterized schemes is thus of considerable significance. Determination of the spatial variations in the surface emissivity, via flux measurements, would also allow a sensible classification of surface types (Running et al. 1994).

The paper is organized as follows. In section 2a we write down the broadband flux emissivity formulation for black surfaces, as well as its prevailing extension to nonblack surfaces (ground) that includes an erroneous reflected flux component. After obtaining the correct reflected flux, the radiative flux divergence (cooling-rate) profiles are determined in section 2b, for model atmospheres, using both the prevailing (erroneous) and the correct formulations; these include a model inversion layer with an exponential increase in temperature with height. Tiny departures from a radiatively black surface lead to qualitative differences in the cooling-rate profiles obtained from the erroneous and correct schemes. We discuss the implications of these differences for existing NBL calculations in section 2c, where we also highlight a particularly serious instance of the error in the context of a theory for the Ramdas layer (Vasudeva Murthy et al. 1993). Section 3 presents the main conclusions. An appendix presents extensions of the correct emissivity scheme that allow for a nonisothermal atmosphere, for multiple reflections between a pair of reflective surfaces (a configuration commonly employed in laboratory experiments), and for an angular dependence of the radiant intensity; the final generalization avoids the use of a diffusivity factor and enables one to account for directional characteristics of surface emission and reflection.

2. The broadband flux emissivity formulation

a. The correct broadband reflected flux

The broadband fluxes in a participating medium, bounded below by a black surface, are given by Liou (2002):
e1
e2
where the dot denotes differentiation with respect to the argument, and ϵf(u) is the isothermal broadband flux emissivity defined as (Liou 2002)
e3
In the above equations, Tg is the ground temperature, T(u) denotes the atmospheric temperature profile, Bν(T) is the Planck function, and the broadband transmissivity is given by τf(u) = 1 − ϵf(u). In (3), the average of the transmission over all zenith angles has been replaced by the transmission along an effective zenith angle, , where β is the well-known diffusivity factor (β ≈ 1.66; see Liou 2002; Goody 1964), and τν(u, 1/β) is the monochromatic transmittance along this “effective” path. The emphasis here is on a fundamental error in the flux expressions; for purposes of simplicity alone, we will restrict ourselves to a water vapor–laden atmosphere, because water vapor (in the absence of clouds) is the dominant contributor to tropospheric radiative exchanges in the infrared (Liou 2002). In the absence of pressure–temperature scalings necessary for inhomogeneous paths, the vertical path length is given by , ρw being the water vapor density with ut in (1) denoting the top of the atmosphere (the water vapor scale height is around 2.7 km); note that the lack of thermodynamic scalings implies δ = 0 in the expression for u(z) given in section 1. The isothermality assumption in (3) neglects the temperature variation within a vertical layer, and the formulations (1) and (2) typically applies to each of many (nearly isothermal) layers (Liou 2002). This detail is not central to our arguments and we shall apply (1) and (2) to the entire water vapor–laden atmosphere. A nonisothermal generalization of the emissivity formulation by Ramanathan and Downey (1986) continues to be used in atmospheric general circulation models such as the Community Atmosphere Models, developed by the National Center for Atmospheric Research (NCAR) at Boulder, Colorado (Collins et al. 2002, 2006). An extension of the nonisothermal emissivity scheme to nonblack surfaces is given in the appendix.
The prevailing extension of (2) for an atmosphere bounded below by a nonblack surface with emissivity ϵg is given by Garratt and Brost (1981):
e4
The ground emission is weakened by ϵg, and there is an additional reflected flux component proportional to (1 − ϵg). The error in (4) arises from attenuating both the ground emission and the reflected flux with τf(u). As argued below, this is only correct for incident radiation whose spectral energy distribution follows the Planck function Bν(T), as is the case for the emission from a gray ground. The spectral energy distribution of the reflected flux at the ground [i.e., (1 − ϵg)F(0)], however, departs significantly from Bν(T).

The broadband transmissivity corresponding to the ϵf(u) in Fig. 1 is plotted in Fig. 2b. The use of τf(u) for gray-ground emission implies an initial rapid attenuation of incident energy due to opaque-band absorption, followed by a much weaker attenuation over longer distances in the transparent bands; for water vapor, in particular, there exists a region of the spectrum (8–14 μm), termed the atmospheric window, wherein the atmosphere remains largely transparent. Since the reflected flux at the ground represents the downward emission of the entire water vapor–laden air column, it is dominated by wavelength intervals corresponding to the principal band centers while being deficient within the window. Figure 2a highlights this difference between the spectra of and (1 − ϵg)F(0). The difference is evidently not accounted for when attenuating both fluxes with τf(u), leading to an error in the cooling rates. The reflected flux is already deficient in the transparent bands, and the initial rapid removal of opaque-band energy implies near-complete attenuation (see Fig. 2b). The use of τf(u) for (1 − ϵg)F(0) attributes a fraction of the incident opaque-band energy to the transparent bands, allowing it to escape to the upper atmosphere and beyond. The resulting deficiency over short length scales leads to a spurious near-surface cooling. This spurious cooling has been independently pointed out by Edwards (2009a) and Ponnulakshmi et al. (2009) [see also Mukund et al. (2010)], wherein the use of τf(u) is interpreted as an unphysical “band cross-talk.”

Fig. 2.
Fig. 2.

(a) The spectral energy distribution of (left) gray-ground emission , (middle) downwelling surface flux , and (right) reflected flux and (b) the broadband transmissivities for the emitted and reflected fluxes.

Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0269.1

Since an infinite water vapor–laden air column emits as a blackbody, obtaining the correct attenuated reflected flux involves accounting for the finite path length [O(ut)] of the reflected photons. Reflection only weakens the intensity by a factor 1 − ϵg, leaving the spectrum unchanged. Hence, rather than separately account for the downward emission from a column of height ut and subsequent attenuation of the upwelling reflected flux through a further distance u, it is convenient to consider the combined emission of a hypothetical air column of height u + ut extending from ut to −u. This emission is given by
e5
where a denotes air column emission, and the lower limit of the integral denotes the reversed reflected trajectory. The argument of the temperature field is , since the reflected photons traverse the region (0, u) twice in arriving at u. The expression (5) includes the (weakened) upward emission of the air column of height u, and the attenuated reflected flux alone is given by
e6
which further simplifies to
e7
an expression that may also be obtained directly from a frequency integration of the monochromatic reflected flux. The correct transmissivity , to be used in place of τf(u) to attenuate (1 − ϵ)F(0), is thus given by
e8
For an isothermal atmosphere, (8) may be written in terms of τf(u) as
e9
That (9) is appropriate may be seen by first noting that τf(u) is the attenuation, over a distance u, of incident radiation with a spectral energy distribution that follows the Planck function. Thus, τf(u + ut) may be interpreted as the additional attenuation, over a distance u, of incident radiation with band energies that survive after traversing an air column of height ut. Said differently, it denotes the additional attenuation of incident radiation with band energies that cannot be absorbed by an air column of height ut. Since what cannot be absorbed cannot be emitted (Kirchhoff’s law), τf(u + ut) is the attenuation, over a distance u, of incident radiation with band energies that are absent from the emission of an air column of height ut. Hence, must be proportional to τf(u) − τf(u + ut), the factor [1 − τf(ut)]−1 in (9) being needed for normalization.
With the incorporation of (7), the correct emissivity formulation for nonblack surfaces takes the form
e10
e11
The error in the prevailing scheme, given by (1) and (4), comes as a surprise since the correct reflected flux has appeared before in the literature, albeit not for a homogeneous atmosphere over reflective ground. In estimating the infrared forcing of cirrus layers overlying black ground, Liou and Ou (1981) had the correct reflected flux contribution (as part of the downwelling flux). Even earlier, Zdunkowski and coworkers (Zdunkowski et al. 1966) had the correct reflected flux as part of an emissivity scheme for an aerosol-laden atmosphere. The expression (7) neglects the anisotropy between the emissivities for upward and downward trajectories. There exist other emissivity parameterizations, used in atmospheric calculations (Siqueira and Katul 2010), that include such an anisotropy. The erroneous Garratt and Brost (1981) scheme, for instance, accounts for this anisotropy based on the Rodgers model (Rodgers 1967). It turns out, however, there is no rigorous manner in which it may be incorporated in a broadband formulation for nonblack surfaces. Indeed, neither of the aforementioned efforts, those of Liou and Ou (1981) and Zdunkowski et al. (1966), incorporates such an anisotropy.

b. Cooling-rate profiles

We now examine the cooling-rate (flux divergence) profiles obtained from the erroneous formulation [(1) and (4)] and the correct formulation [(10) and (11)] for model atmospheres. The ϵf(u) used in these calculations (plotted in Fig. 1) is given by Zdunkowski and Johnson (1965):
e12
The original ϵf(u) had different representations for path lengths above and below 10−2 kg m−2. Since this would lead to an undesirable discontinuity in the cooling-rate profiles, we have chosen (12) over the entire range of path lengths. The focus here is on the near-surface flux divergence, and (12) is the representation close to ground. Figure 3 compares the cooling-rate profiles as a function of ϵg for an adiabatic lapse-rate atmosphere with T(z) = Tg − Γz and Tg = 300 K; here, Γ is the specified lapse rate, and Tg = T(0), so there is no temperature discontinuity at the ground. For the chosen value of Γ (−9.8 K km−1), the cooling rate decreases with height since the cooling-to-space contribution is overwhelmed by the upwelling flux from the ground and the underlying warmer air layers. This is consistent with earlier line-by-line (LBL) calculations for a water vapor–laden tropical atmosphere (Chou et al. 1993). For ϵg = 1 the flux divergences calculated from the two formulations are coincident as they must be. For ϵg < 1 the deficient reflected flux in the erroneous formulation leads to a pronounced cooling near the ground. For ϵg = 0.9, the erroneous surface cooling rate is already more than two orders of magnitude greater than its actual value, and there continues to be a substantial deviation even at 10 m. The heightened sensitivity to ϵg is absent in the correct formulation.
Fig. 3.
Fig. 3.

The flux-divergence profiles obtained from the erroneous formulation [(1) and (4)] and the correct formulation [(10) and (11)] for ϵg = (a) 1, (b) 0.9, and (c) 0.8, and for a model atmosphere with a constant lapse rate of −9.8 K km−1 (the ground temperature is assumed to be 300 K).

Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0269.1

Figure 4 shows the cooling-rate profiles for a standard inversion layer, characterized by , and with ϵg = 1; H here is specified a priori and ranges from 0.001 to 10 m in the calculations. In principle, H must be obtained from solving the energy equation and is then found to be a time-dependent quantity that increases through the night starting from very small values in the evening transitional layer (Edwards 2009a,b). The profiles in Fig. 4, for increasing H, may loosely be regarded as corresponding to successive instants of time after sunset. The nocturnal inversion layer provides a rather severe test for the vertical resolution used in radiation calculations (Räisänen 1996; Savijärvi 2006). The flux divergence profile in a homogeneous atmosphere includes a shallow warming zone just above ground, first predicted by Fleagle (1953); the warming arises because, for air layers sufficiently close to ground, the cooling-to-space contribution is dominated by opaque-band exchanges with warmer overlying air layers (Edwards 2009a). This warming, however, will not be observed in calculations resolved on a scale larger than H since the region of varying temperature now appears as a slip, T(0) − Tg, and leads to a near-surface cooling instead. The nature of the cooling-rate profile depends on the relative magnitudes of H and a representative photon mean-free-path in the opaque bands. The second plot in Fig. 4 shows that the vertical extent of the warming region decreases with decreasing , approaching the slip regime for .

Fig. 4.
Fig. 4.

(a) Inversion layer temperature profiles for different H. (b) Flux divergence profiles obtained as a function of (the characteristic optical depth τop of the inversion layer) for ϵg = 1.

Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0269.1

The spurious cooling error assumes particular significance for an inversion layer since the sign of the surface flux divergence, under stable nocturnal conditions, remains controversial (Ramdas and Atmanathan 1932; Funk 1960; Lieske and Stroschein 1967; Oke 1970; Nkemdirim 1978; Räisänen 1996; Hoch 2005; Edwards 2009a; Mukund et al. 2010, 2012, manuscript submitted to Quart. J. Roy. Meteor. Soc., hereafter MSP). Although the theoretical calculations (Edwards 2009a) predict a near-surface warming in the NBL for ϵg = 1 and a homogeneous atmosphere, the majority of observations support a radiative cooling in the lowest air layers (Funk 1960; Elliott 1964; Nkemdirim 1978; Sun et al. 2003). To date, there have been only two measurements, those of Lieske and Stroschein (1967) and, more recently, those of Hoch (2005), that support a near-surface warming due to an inversion. While this lack of agreement may arise from the sensitive dependence of the vertical extent of the warming region on the inversion stratification length scale, and result from a possible departure from homogeneity in the lowest air layers (Mukund et al. 2010; MSP), the focus here is on the theoretical prediction of a warming contribution being overwhelmed by the spurious cooling arising in the erroneous formulation for nonblack surfaces. The calculation below shows that this happens for a rather modest deviation of the surface emissivity from unity.

An estimate for the warming flux divergence at the surface is obtained using (12) in (10) and (11). One finds
e13
where is the exponential integral (Abramowitz and Stegun 1990), and the constants a and b have been defined in (12). The expression shows that the warming flux divergence increases with decreasing ϵg. As argued by Zdunkowski et al. (1966), Lieske and Stroschein (1967), and more recently Edwards (2009a), this is because the reflected flux comprises downward flux contributions from warmer air layers. The spurious cooling contribution, obtained by using (12) in (4) and (10) for an isothermal atmosphere, is given by
e14
where . The cooling is evidently absent for ϵg = 1 but increases sharply with decreasing ϵg. As will be seen in section 2c, this steep increase has, in several earlier calculations, led to an unphysical exaggeration of the effect of ground emissivity on NBL flux-divergence profiles (Vasudeva Murthy et al. 1993; Varghese et al. 2003). One may now obtain ϵg−crossover, corresponding to a change in sign of the surface flux divergence due to spurious cooling, by equating (13) and (14), whence
e15
This yields ϵg−crossover ≈ 0.99 and 0.92, respectively, for optically thick and thin inversion layers. For ϵg < ϵg−crossover, the error in the prevailing formulation is large enough to change the sign of the surface flux divergence. The plots in Fig. 5 illustrate the competing effects of the near-surface warming and spurious cooling for values of ϵg on either side of ϵg−crossover, and for different values of .
Fig. 5.
Fig. 5.

The plots compare the flux divergence profiles, obtained from the erroneous formulation [(1) and (4)] and the correct formulation [(10) and (11)] for values of ϵg on either side of ϵg−crossover, with reference to (a),(b) optically thick and (c),(d) optically thin inversion layers.

Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0269.1

The assumed temperature profiles in the above calculations are devoid of any slip between the ground and the lowest air layers. The choice is deliberate since invoking a slip, as is done on an ad hoc basis in many calculations (Ha and Mahrt 2003; Varghese 2003), would trigger a net radiative exchange between the ground and the air layers in the opaque bands (Zdunkowski et al. 1966). The resulting flux divergence, however, masks the spurious cooling that arises even for zero slip in the above formulation. The primary focus here has been on the vertical distribution of radiant energy, and thence on the flux divergence profiles. The discrepancy in the fluxes, although not small, is not as significant. For instance, for an inversion layer with ΔT = 5 K and H = 10 m, the difference between the upward fluxes, obtained from the two formulations for ϵg = 0.8, increases from zero at the surface and asymptotically approaches about 20 W m−2 at larger heights.

c. The structure of the nocturnal boundary layer

The spurious cooling described in earlier subsections has arisen in a number of calculations beginning with Garratt and Brost’s attempt (Garratt and Brost 1981) to examine the roles played by radiation and turbulence in NBL evolution, as a function of ϵg, with the radiation being modeled using an emissivity scheme. Subsequent efforts that use the same model, or generalizations thereof, include André and Mahrt (1981), Vasudeva Murthy et al. (1993), Gopalakrishnan et al. (1998), Koračin et al. (1989), Rißmann (1998), Rama Krishna et al. (2003), Varghese (2003), Savijärvi (2006), and Siqueira and Katul (2010). In some instances, the model, although stated in its general (and therefore, erroneous) form, is used only for ϵg = 1. The error, when present, varies in magnitude because of differing vertical resolution, emissivity expressions, and varying levels of frequency parameterization (broadband flux emissivity schemes; narrowband formulations, etc.); it is shown in an accompanying note (Ponnulakshmi et al. (2012, manuscript submitted to J. Atmos. Sci.) that this error persists down to frequency intervals on the order of an elementary line width. Almost all of the above efforts examine NBL evolution due to a combination of radiation and turbulence within the plane-parallel formalism of a barotropic nondivergent atmosphere. Unlike the daytime convective layer, the NBL is significantly affected by radiation at low wind speeds (Stull 1988). Radiation increases the NBL height both via direct cooling near the top and in an indirect manner by differential warming–induced destabilization. Under calm conditions, there is a separation of scales with radiative cooling causing the height of the stratified inversion layer to significantly exceed that of the turbulent zone (Claude and Guedalia 1985; Rama Krishna et al. 2003). The structure of this turbulent zone is affected by the erroneous reflected flux.

Based on the relative magnitudes of the turbulent and radiative flux divergences for ϵg = 0.8 obtained from their simulations, Garratt and Brost (1981) proposed a three-layer structure for the NBL turbulent zone with the top and bottom of this zone being dominated by radiative cooling and the “bulk” dominated by turbulence. A similar structure has been found more recently by Gopalakrishnan et al. (1998) and Rama Krishna et al. (2003). The dominance of radiation near the ground in the original simulations is due to spurious cooling. In fact, the exaggerated cooling for the said ϵg leads to a spurious peak in the sensible heat flux—a feature absent in the ϵg = 1 simulations. For ϵg = 1, André and Mahrt (1981) highlight the two-layer NBL structure with a near-surface warming resulting from the inversion profile. This warming region is, however, absent in their calculations for ϵg = 0.965 (shown in their appendix), again because of the spurious cooling contribution. As pointed out earlier in section 2b, a reduced ϵg must lead to a relative warming, and the two-layer NBL structure must therefore remain qualitatively unaltered for a reduced ϵg.

For an inversion profile with a single-signed curvature (the exponential profile in section 2b being a specific example), the flux-divergence profile exhibits a single transition from cooling at greater heights to a near-surface warming. However, in presence of turbulence, the temperature profile exhibits regions of opposite curvature—an additional positive curvature region (expected from the Monin–Obukhov relations for the stably stratified regime; Turner 1973; Businger et al. 1971) below the original negative curvature zone (see Fig. 6a). This leads to a more complicated flux divergence profile. As shown by Edwards (2009b), the flux divergence, in certain instances, may change sign thrice! An initial change from a cooling contribution at larger heights leads to an intermediate warming zone due to the change in profile curvature. Since the lower part of the profile continues to resemble the original inversion with a single-signed curvature, the flux divergence changes sign again before finally approaching a near-surface warming. The intermediate warming region is present in earlier calculations including those of André and Mahrt (1981) (for the “mixed layer” temperature profile), Savijärvi (2006), and possibly even Steeneveld (2007) (the “radiation night” profile); in other instances, despite there not being an actual sign change, the cooling-rate profile does go through an intermediate minimum (Gopalakrishnan et al. 1998; Rama Krishna et al. 2003). The final transition to a near-surface warming is absent in all these studies because of either coarse resolution or spurious cooling. The profiles in Fig. 6, for the smaller surface emissivities (ϵg = 0.8, 0.9), show that the spurious cooling may even eliminate the second elevated region of warming. Since the latter is responsible, in part, for the prevailing notion of a radiative destabilization of the NBL, its disappearance will evidently affect the overall energy budget. Finally, in many instances, an exaggerated near-surface cooling arises even for a black surface because of an assumed temperature slip at the ground (Räisänen 1996; Duynkerke 1999; Ha and Mahrt 2003; Savijärvi 2006); the resulting changes in the NBL structure are the same as those due to spurious cooling. Although the assumption of a slip is physically motivated, being related to an unresolved inversion layer close to ground, it is nevertheless untenable in light of recent observations (Mukund et al. 2010; MSP).

Fig. 6.
Fig. 6.

(a) A typical temperature profile obtained from the interaction of radiation and turbulence. The corresponding flux-divergence profiles, obtained from the erroneous and correct formulations, for ϵg = (b) 1, (c) 0.9, and (d) 0.8.

Citation: Journal of the Atmospheric Sciences 69, 9; 10.1175/JAS-D-11-0269.1

An exaggerated instance of the spurious cooling error has occurred in the context of a well-known micrometeorological paradox known as the Ramdas layer (Ramdas and Atmanathan 1932; Lake 1956; Geiger 1965; Oke 1970; Lettau 1979). The layer refers to a region very close to ground that supports a nonmonotonic variation of temperature with height under calm cloudless conditions. The occurrence of such a lifted temperature minimum (LTM) remains unexplained (Mukund et al. 2010; MSP) and highlights the subtle role played by radiative processes in the NBL. The prevailing explanation put forward by Vasudeva Murthy et al. (1993), based on a flux emissivity scheme, unfortunately interprets the spurious cooling as a physical process leading to the LTM. In contrast to the other calculations cited earlier (Garratt and Brost 1981; André and Mahrt 1981; Gopalakrishnan et al. 1998), the spurious cooling is the central feature of the theoretical explanation with the cold air layer arising from a balance of this cooling and conduction. A comparison of the Vasudeva Murthy et al. (1993) and Garratt and Brost (1981) predictions also reveals the singular nature of the energy density that drives the spurious cooling. The former differs only in that the flux emissivity is determined from an experimental fit down to much smaller scales relevant to the LTM. As a result, for a decrease in ϵg from unity to 0.8, the near-surface cooling rate predicted by Garratt and Brost (1981) increases from about 17 to 77 K day−1, while that predicted by the Vasudeva Murthy et al. (1993) model shoots up from a mere 0.3 to 1728 K day−1! This scale-dependent sensitivity to ϵg becomes obvious on examining the flux divergence for an isothermal atmosphere using (1) and (4):
e16
The factor 1 − ϵf(ut) denotes the overall medium transparency and is insensitive to an increased resolution of the opaque bands. In contrast, scales as the inverse of the smallest photon path length resolved, and is directly responsible for the enhanced cooling in the Vasudeva Murthy et al. (1993) model. With the addition of increasingly opaque bands, the portion of the reflected energy that is (incorrectly) attributed to the transparent bands corresponds to shorter photon path lengths; the resulting deficiency in warming flux is felt closer to the surface, leading to an increasingly large cooling flux density. Thus, an attempt to refine a given prediction by a careful synthesis of the flux-emissivity function leads one further away from the correct answer (close to the surface). It is worth noting that although the Vasudeva Murthy et al. (1993) surface cooling-rate prediction for ϵg = 0.8 is exaggerated, the cooling rate predicted by the model at 1 m is about 125 K day−1 (Varghese et al. 2003), which is comparable to the Garratt and Brost prediction at 2 m (the lowest level used in the calculation), again emphasizing the resolution-dependent magnitude of the spurious cooling error.

3. Conclusions

In this paper, we have highlighted a fundamental error in the prevailing emissivity scheme for reflective ground together with its consequences for atmospheric cooling-rate profiles. The error results from using the same transmissivity for both the surface emission and the reflected fluxes, in spite of their differing spectral content, and leads to a spurious cooling near the surface. We present a consistent extension of the original scheme, given by (10) and (11) in section 2a and repeated here for convenience,
e17
e18
which is to be used for an atmosphere bounded below by ground with emissivity ϵg. Despite the correct broadband reflected flux [as part of (18)] having appeared in earlier literature, the error in the formulation of Garratt and Brost (1981) has gone unnoticed. Their flux expressions have since been used by several research groups, primarily for investigations of the NBL, suggesting that the physical implications of the error have not been appreciated. Recently, Edwards (2009a) and Mukund et al. (2010) have interpreted the error in terms of a band cross-talk but only in the context of a spectral scheme. Herein, we have explained the error within the framework of a broadband emissivity scheme, allowing us to compare the erroneous and correct schemes on the same footing. Our study shows that, in contrast to earlier investigations, the cooling-rate profiles are largely insensitive to a departure of the surface from radiatively black behavior.

It is worth emphasizing that the emissivity scheme proposed by Garratt and Brost (1981) for nonblack surfaces [and narrowband formulations such as that of Savijärvi (2006); see Ponnulakshmi et al. (2012)] suffer from two sources of error. The first is the parameterization error, inherent in any averaged scheme, due to the approximate modeling of the unresolved finescale structure (Goody 1964). The second source of error is related to spurious cooling and, crucially, arises only for nonblack surfaces. The validation of an averaged scheme by comparison with an LBL calculation for ϵg = 1, as is the norm (Fels et al. 1991; Ellingson et al. 1991; Savijärvi 2006; Collins et al. 2006), does not prevent the emergence of spurious cooling when the scheme is applied to nonblack surfaces. The correct reflected flux, presented above and derived in section 2a for a broadband emissivity scheme, eliminates the second error component. This ensures that, even with a varying ϵg, the discrepancy between an LBL computation and a broadband emissivity scheme is solely due to the limitations of parameterization.

APPENDIX

Extensions of the Flux Emissivity Scheme

Herein, we present extensions of the basic emissivity scheme [(10) and (11)] that account for departures from the simplest scenario of an isothermal emissivity formulation for a cloud-free atmosphere above reflective ground.

a. Nonisothermal flux emissivity formulation

An expression for the reflected flux [as part of F(u)] may also be obtained within the nonisothermal emissivity formulation of Ramanathan and Downey (1986) (henceforth, the RD formulation). The flux expressions in this formulation, obtained from an integration by parts of the original broadband fluxes [see (1) and (2) in section 2a], are given by
ea1
ea2
for ϵg = 1, where ϵ and A are referred to as the nonisothermal emissivity and absorptivity, respectively. The said authors developed a parameterization for ϵ and A, based on a comparison with narrowband calculations, in terms of a dependence on both the emitting level temperature and the temperature of the equivalent homogeneous path (Ramanathan and Downey 1986). For nonblack surfaces, the upward flux takes the form
ea3
where
ea4
ea5
Here, the up and down arrows denote the direction of the trajectory after and before reflection at the ground, respectively. Thus, although the nominal emitting and absorbing levels remain the same, the actual trajectories involved in (A4) and (A5) are bidirectional, having undergone a reflection at the ground; hence, the equivalent path temperature will differ from that used to originally parameterize ϵ and A even for the same z and z′. One therefore needs to define new temperature-dependent functions—in the sense of requiring an independent parameterization. An erroneous generalization of the Ramanathan and Downey (1986) formulation, for an arbitrary ϵg, appears in Garand (1983). Successive versions of the Community Atmosphere Models, originally developed at the National Center for Atmospheric Research (NCAR) as the Community Climate Model (CCM), use the Ramanathan and Downey (1986) formulation and differ only in the parameterization for particular atmospheric gases (Collins et al. 2002). The spurious cooling error assumes importance since, for ϵg < 1, such a contribution will likely negate any marginal improvement from one model version to the next resulting from a more accurate parameterization.

b. Flux emissivity formulation for a pair of reflective surfaces

The basic emissivity scheme may also be generalized to the case where the participating medium is bounded on either side by reflective surfaces, and one needs to account for the infinite hierarchies of reflections that result. In the atmospheric context, the flux expressions would pertain, for instance, to a reflective ground in presence of an overlying cirrus layer. The analogous scenario is common in the laboratory where polished metallic surfaces are often used as part of apparatuses that examine the role of radiation on the onset of convection in the classical Rayleigh–Benard configuration (Gille and Goody 1964; Schimmel et al. 1970; Hutchison and Richards 1999). With surface emissivities ϵg and ϵc and a nonzero upper surface transmissivity τc [keeping in mind the atmospheric context; see Liou and Ou (1981)], the flux expressions are
ea6
ea7
for the portion of the medium between the two surfaces, and
ea8
ea9
above the upper surface. A truncated version of (A6) and (A7), with τc = 0, appears in Gille and Goody (1964).

c. Directional flux emissivity formulation

One may also write down the broadband fluxes without the aid of a diffusivity factor, taking explicit account of both the differences in path lengths of the reflected photons arising from the differing inclinations of their trajectories to the vertical and any additional anisotropy induced by interaction with a non-Lambertian surface. The fluxes in such a directional emissivity scheme are given by
ea10
ea11
where Ω ≡ (θ, ϕ) denotes the direction on the unit sphere with μ = cosθ and dΩ = −dμdϕ; ϵgΩ and ρΩ′→Ω denote the directional surface emissivity and bidirectional surface reflectivity, respectively (Siegel and Howell 2002). For simplicity, the flux expressions are restricted to the case of a single reflective surface (ground). In (A10) and (A11), the directional flux emissivity is given by an expression similar to (3), but without the use of a diffusivity factor to approximate the integral over zenith angles. Thus,
ea12
and the empirical expression given in section 2b [see (12)] may still be used with an appropriate change in argument .

REFERENCES

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    • Export Citation
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    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Lieske, B. J., and L. A. Stroschein, 1967: Measurements of radiative flux divergence in the arctic. Theor. Appl. Climatol., 15, 67–81.

    • Search Google Scholar
    • Export Citation
  • Liou, K. N., 2002: An Introduction to Atmospheric Radiation. Academic Press, 392 pp.

  • Liou, K. N., and S. C. Ou, 1981: Parameterization of infrared radiative transfer in cloudy atmospheres. J. Atmos. Sci.,38, 2707–2716.

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  • Oke, T. R., 1970: The temperature profile near the ground on calm clear nights. Quart. J. Roy. Meteor. Soc., 96, 1423.

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    • Search Google Scholar
    • Export Citation
  • Pettijohn, J. C., G. D. Salvucci, N. G. Phillips, and M. J. Daley, 2009: Mechanisms of moisture stress in a mid-latitude temperate forest: Implications for feedforward and feedback controls from an irrigation experiment. Ecol. Modell., 220, 968978.

    • Search Google Scholar
    • Export Citation
  • Ponnulakshmi, V. K., G. Subramanian, V. Mukund, and K. R. Sreenivas, 2009: The Ramdas layer remains a micro-meteorological puzzle. Jawaharlal Nehru Centre for Advanced Scientific Research Rep. JNCASR/EMU/2009-1, 41 pp.

  • Räisänen, P., 1996: The effect of vertical resolution on clear-sky radiation calculations: Tests with two schemes. Tellus,48, 403–423.

  • Rama Krishna, T. V. B. P. S., M. Sharan, S. G. Gopalakrishnan, and Aditi, 2003: Mean structure of the nocturnal boundary layer under strong and weak wind conditions: EPRI case study. J. Appl. Meteor., 42, 952969.

    • Search Google Scholar
    • Export Citation
  • Ramanathan, V., and P. Downey, 1986: A nonisothermal emissivity and absorptivity formulation for water vapor. J. Geophys. Res.,91 (D8), 8649–8666.

  • Ramdas, L. A., and S. Atmanathan, 1932: The vertical distribution of air temperature near the ground during night. Beitr. Geophys., 37, 116117.

    • Search Google Scholar
    • Export Citation
  • Rißmann, J., 1998: Der einflußlangwelliger strahlungsprozesse auf das bodennahe temperaturprofil. Ph.D. thesis, Institut für Meteorologie der Universita ät Leipzig, 146 pp.

  • Rodgers, C. D., 1967: The use of emissivity in atmospheric radiation calculations. Quart. J. Roy. Meteor. Soc.,93, 43–54.

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    • Search Google Scholar
    • Export Citation
  • Savijärvi, H., 2006: Radiative and turbulent heating rates in the clear-air boundary layer. Quart. J. Roy. Meteor. Soc., 132, 147161.

    • Search Google Scholar
    • Export Citation
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  • Siegel, R., and J. R. Howell, 2002: Thermal Radiation Heat Transfer. Taylor and Francis, 840 pp.

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  • Steeneveld, G. J., 2007: Understanding and prediction of stable atmospheric boundary layers over land. Ph.D. thesis, Wageningen University, 199 pp.

  • Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Springer, 670 pp.

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  • Varghese, S., 2003: Band model computation of near-surface longwave fluxes. Ph.D. thesis, Jawaharlal Nehru Centre for Advanced Scientific Research, 178 pp.

  • Varghese, S., A. S. Vasudeva Murthy, and R. Narasimha, 2003: A fast, accurate method of computing near-surface longwave fluxes and cooling rates in the atmosphere. J. Atmos. Sci., 60, 28692886.

    • Search Google Scholar
    • Export Citation
  • Vasudeva Murthy, A. S., J. Srinvasan, and R. Narasimha, 1993: A theory of the lifted temperature minimum on calm clear nights. Philos. Trans. Roy. Soc. London, 344, 183206.

    • Search Google Scholar
    • Export Citation
  • Wang, K., Z. Wan, P. Wang, M. Sparrow, J. Liu, X. Zhou, and S. Haginoya, 2005: Estimation of surface longwave radiation and broadband emissivity using Moderate Resolution Imaging Spectroradiometer (MODIS) land surface temperature/emissivity products. J. Geophys. Res.,110, D11109, doi:10.1029/2004JD005566.

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Save
  • Abraha, M. G., and M. J. Savage, 2008: Comparison of estimates of daily solar radiation from air temperature range for application in crop simulations. Agric. For. Meteor., 148, 401–416.

  • Abramowitz, M., and I. Stegun, 1990: Handbook of Mathematical Functions. Dover, 1046 pp.

  • André, J. C., and L. Mahrt, 1981: The nocturnal surface inversion and influence of clear-air radiative cooling. J. Atmos. Sci., 39, 864878.

    • Search Google Scholar
    • Export Citation
  • Brubaker, K. L., and D. Entekhabi, 1996: Analysis of feedback mechanisms in land-atmosphere interaction. Water Resour. Res., 32, 13431357.

    • Search Google Scholar
    • Export Citation
  • Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181–189.

  • Chou, M. D., W. L. Ridgway, and M. M. Yan, 1993: One-parameter scaling and exponential-sum fitting for water vapor and CO2 infrared transmission functions. J. Atmos. Sci., 50, 2294–2303.

  • Claude, E., and D. Guedalia, 1985: Influence of geostrophic wind on atmospheric nocturnal cooling. J. Atmos. Sci., 42, 26952698.

  • Collins, W. D., J. K. Hackney, and D. P. Edwards, 2002: An updated parameterization for infrared emission and absorption by water vapor in the National Center for Atmospheric Research Community Atmosphere Model. J. Geophys. Res., 107, 4664, doi:10.1029/2001JD001365.

  • Collins, W. D., and Coauthors, 2006: The formulation and atmospheric simulation of the Community Atmosphere Model version 3 (CAM3). J. Climate, 19, 21442161.

    • Search Google Scholar
    • Export Citation
  • Duynkerke, P. G., 1999: Turbulence, radiation and fog in Dutch stable boundary layers. Bound.-Layer Meteor., 90, 447477.

  • Edwards, J. M., 2009a: Radiative processes in the stable boundary layer: Part I. Radiative aspects. Bound.-Layer Meteor., 131, 105126.

    • Search Google Scholar
    • Export Citation
  • Edwards, J. M., 2009b: Radiative processes in the stable boundary layer: Part II. The development of the nocturnal boundary layer. Bound.-Layer Meteor., 131, 127146.

    • Search Google Scholar
    • Export Citation
  • Ellingson, R. G., J. Ellis, and S. B. Fels, 1991: The intercomparison of radiation codes used in climate models: Longwave results. J. Geophys. Res.,96 (D5), 8929–8953.

  • Elliott, W. P., 1964: The height variation of vertical heat flux near the ground. Quart. J. Roy. Meteor. Soc.,90, 260–265.

  • Fels, S. B., J. T. Kiehl, A. A. Lacis, and M. D. Schwarzkopf, 1991: Infrared cooling rate calculations in operational general circulation models: Comparisons with benchmark computations. J. Geophys. Res.,96 (D5), 9105–9120.

  • Fleagle, R. G., 1953: A theory of fog formation. J. Mar. Res., 12, 4350.

  • Funk, J. P., 1960: Measured radiative flux divergence near the ground at night. Quart. J. Roy. Meteor. Soc., 86, 382389.

  • Garand, L., 1983: Some improvements and complements to the infrared emissivity algorithm including a parameterization of the absorption in the continuum region. J. Atmos. Sci., 40, 230243.

    • Search Google Scholar
    • Export Citation
  • Garratt, J. R., and R. A. Brost, 1981: Radiative cooling effects within and above the nocturnal boundary layer. J. Atmos. Sci., 38, 27302746.

    • Search Google Scholar
    • Export Citation
  • Geiger, R., 1965: The Climate near the Ground. Harvard University Press, 600 pp.

  • Gille, J., and R. M. Goody, 1964: Convection in a radiating gas. J. Fluid Mech., 20, 4779.

  • Goody, R. M., 1964: Atmospheric Radiation. Clarendon Press, 600 pp.

  • Gopalakrishnan, S. G., M. Sharan, R. T. McNider, and M. P. Singh, 1998: Study of radiative and turbulent processes in the stable boundary layer under weak wind conditions. J. Atmos. Sci., 55, 954960.

    • Search Google Scholar
    • Export Citation
  • Ha, K. J., and L. Mahrt, 2003: Radiative and turbulent fluxes in the nocturnal boundary layer. Tellus, 55A, 317327.

  • Hoch, S. W., 2005: Radiative flux divergence in the surface boundary layer, a study based on observations at Summit, Greenland. Ph.D. thesis, Swiss Federal Institute of Technology ETH, 180 pp.

  • Hutchison, J. E., and R. F. Richards, 1999: Effect of nongray gas radiation on thermal stability in carbon dioxide. J. Thermophys. Heat. Transfer,13, 25–32.

  • Koračin, D., B. Grisogono, and N. Subanovic, 1989: A model of radiative heat transfer effects in the atmospheric boundary layer. Geofizika, 6, 7586.

    • Search Google Scholar
    • Export Citation
  • Lake, J. V., 1956: The temperature profile above bare soil on clear nights. Quart. J. Roy. Meteor. Soc., 82, 187197.

  • Lettau, H. H., 1979: Wind and temperature profile prediction for diabatic surface layers including strong inversion cases. Bound.-Layer Meteor., 17, 443464.

    • Search Google Scholar
    • Export Citation
  • Lieske, B. J., and L. A. Stroschein, 1967: Measurements of radiative flux divergence in the arctic. Theor. Appl. Climatol., 15, 67–81.

    • Search Google Scholar
    • Export Citation
  • Liou, K. N., 2002: An Introduction to Atmospheric Radiation. Academic Press, 392 pp.

  • Liou, K. N., and S. C. Ou, 1981: Parameterization of infrared radiative transfer in cloudy atmospheres. J. Atmos. Sci.,38, 2707–2716.

  • Mukund, V., V. K. Ponnulakshmi, D. K. Singh, G. Subramanian, and K. R. Sreenivas, 2010: Hyper-cooling in the nocturnal boundary layer: The Ramdas paradox. Phys. Scripta,T142, 014041, doi:10.1088/0031-8949/2010/T142/014041.

  • Nkemdirim, L. C., 1978: A comparison of radiative and actual nocturnal cooling rates over grass and snow. J. Appl. Meteor.,17, 1643–1646.

  • Oke, T. R., 1970: The temperature profile near the ground on calm clear nights. Quart. J. Roy. Meteor. Soc., 96, 1423.

  • Penner, J. E., Y. Chen, M. Wang, and X. Liu, 2009: Possible influence of anthropogenic aerosols on cirrus clouds and anthropogenic forcing. Atmos. Chem. Phys., 9, 879896.

    • Search Google Scholar
    • Export Citation
  • Pettijohn, J. C., G. D. Salvucci, N. G. Phillips, and M. J. Daley, 2009: Mechanisms of moisture stress in a mid-latitude temperate forest: Implications for feedforward and feedback controls from an irrigation experiment. Ecol. Modell., 220, 968978.

    • Search Google Scholar
    • Export Citation
  • Ponnulakshmi, V. K., G. Subramanian, V. Mukund, and K. R. Sreenivas, 2009: The Ramdas layer remains a micro-meteorological puzzle. Jawaharlal Nehru Centre for Advanced Scientific Research Rep. JNCASR/EMU/2009-1, 41 pp.

  • Räisänen, P., 1996: The effect of vertical resolution on clear-sky radiation calculations: Tests with two schemes. Tellus,48, 403–423.

  • Rama Krishna, T. V. B. P. S., M. Sharan, S. G. Gopalakrishnan, and Aditi, 2003: Mean structure of the nocturnal boundary layer under strong and weak wind conditions: EPRI case study. J. Appl. Meteor., 42, 952969.

    • Search Google Scholar
    • Export Citation
  • Ramanathan, V., and P. Downey, 1986: A nonisothermal emissivity and absorptivity formulation for water vapor. J. Geophys. Res.,91 (D8), 8649–8666.

  • Ramdas, L. A., and S. Atmanathan, 1932: The vertical distribution of air temperature near the ground during night. Beitr. Geophys., 37, 116117.

    • Search Google Scholar
    • Export Citation
  • Rißmann, J., 1998: Der einflußlangwelliger strahlungsprozesse auf das bodennahe temperaturprofil. Ph.D. thesis, Institut für Meteorologie der Universita ät Leipzig, 146 pp.

  • Rodgers, C. D., 1967: The use of emissivity in atmospheric radiation calculations. Quart. J. Roy. Meteor. Soc.,93, 43–54.

  • Running, S. W., and Coauthors, 1994: Terrestrial remote sensing science and algorithms planned for EOS/MODIS. Int. J. Remote Sens., 15, 35873620.

    • Search Google Scholar
    • Export Citation
  • Savijärvi, H., 2006: Radiative and turbulent heating rates in the clear-air boundary layer. Quart. J. Roy. Meteor. Soc., 132, 147161.

    • Search Google Scholar
    • Export Citation
  • Schimmel, W. P., J. L. Novotny, and F. A. Olsofka, 1970: Interferometric study of radiation–conduction interaction. Proceedings of the Fourth International Heat Transfer Conference, Elsevier, R2.1.

  • Siegel, R., and J. R. Howell, 2002: Thermal Radiation Heat Transfer. Taylor and Francis, 840 pp.

  • Siqueira, M. B., and G. G. Katul, 2010: A sensitivity analysis of the nocturnal boundary-layer properties to atmospheric emissivity formulations. Bound.-Layer. Meteor.,134, 223–242.

  • Steeneveld, G. J., 2007: Understanding and prediction of stable atmospheric boundary layers over land. Ph.D. thesis, Wageningen University, 199 pp.

  • Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Springer, 670 pp.

  • Sun, J., S. P. Burns, A. C. Delany, S. P. Oncley, T. W. Horst, and D. H. Lenschow, 2003: Heat balance in the nocturnal boundary layer during CASES-99. J. Appl. Meteor.,42, 1649–1666.

  • Turner, J. S., 1973: Buoyancy Effects in Fluids. Cambridge University Press, 382 pp.

  • Varghese, S., 2003: Band model computation of near-surface longwave fluxes. Ph.D. thesis, Jawaharlal Nehru Centre for Advanced Scientific Research, 178 pp.

  • Varghese, S., A. S. Vasudeva Murthy, and R. Narasimha, 2003: A fast, accurate method of computing near-surface longwave fluxes and cooling rates in the atmosphere. J. Atmos. Sci., 60, 28692886.

    • Search Google Scholar
    • Export Citation
  • Vasudeva Murthy, A. S., J. Srinvasan, and R. Narasimha, 1993: A theory of the lifted temperature minimum on calm clear nights. Philos. Trans. Roy. Soc. London, 344, 183206.

    • Search Google Scholar
    • Export Citation
  • Wang, K., Z. Wan, P. Wang, M. Sparrow, J. Liu, X. Zhou, and S. Haginoya, 2005: Estimation of surface longwave radiation and broadband emissivity using Moderate Resolution Imaging Spectroradiometer (MODIS) land surface temperature/emissivity products. J. Geophys. Res.,110, D11109, doi:10.1029/2004JD005566.

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  • Fig. 1.

    A schematic of ϵf(u) as a function of the mass absorber path length u; the expression used is given in section 2 [see (12)]. The flux emissivity for a gray medium, a simple exponential, is also shown with a photon mean free path of 200 m.

  • Fig. 2.

    (a) The spectral energy distribution of (left) gray-ground emission , (middle) downwelling surface flux , and (right) reflected flux and (b) the broadband transmissivities for the emitted and reflected fluxes.

  • Fig. 3.

    The flux-divergence profiles obtained from the erroneous formulation [(1) and (4)] and the correct formulation [(10) and (11)] for ϵg = (a) 1, (b) 0.9, and (c) 0.8, and for a model atmosphere with a constant lapse rate of −9.8 K km−1 (the ground temperature is assumed to be 300 K).

  • Fig. 4.

    (a) Inversion layer temperature profiles for different H. (b) Flux divergence profiles obtained as a function of (the characteristic optical depth τop of the inversion layer) for ϵg = 1.

  • Fig. 5.

    The plots compare the flux divergence profiles, obtained from the erroneous formulation [(1) and (4)] and the correct formulation [(10) and (11)] for values of ϵg on either side of ϵg−crossover, with reference to (a),(b) optically thick and (c),(d) optically thin inversion layers.

  • Fig. 6.

    (a) A typical temperature profile obtained from the interaction of radiation and turbulence. The corresponding flux-divergence profiles, obtained from the erroneous and correct formulations, for ϵg = (b) 1, (c) 0.9, and (d) 0.8.

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