## 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 *ϵ*^{f}(*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 *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 *T*_{0} is *ϵ*^{f}(*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 *ϵ*^{f}(*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 *ϵ*^{f}(*u*) from the exponential increase characteristic of a gray medium is important. Although, by the very definition of *ϵ*^{f}(*u*), participating medium emission appears in a manner analogous to a solid surface, with *ϵ*^{f}(*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 *τ ^{f}*(

*u*) = 1 −

*ϵ*

^{f}(

*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).

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

*ϵ*

^{f}(

*u*) is the isothermal broadband flux emissivity defined as (Liou 2002)

*T*is the ground temperature,

_{g}*T*(

*u*) denotes the atmospheric temperature profile,

*B*(

_{ν}*T*) is the Planck function, and the broadband transmissivity is given by

*τ*(

^{f}*u*) = 1 −

*ϵ*

^{f}(

*u*). In (3), the average of the transmission over all zenith angles has been replaced by the transmission along an effective zenith angle,

*β*is the well-known diffusivity factor (

*β*≈ 1.66; see Liou 2002; Goody 1964), and

*τ*(

_{ν}*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

*ρ*being the water vapor density with

_{w}*u*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

_{t}*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.

*ϵ*

_{g}is given by Garratt and Brost (1981):

*ϵ*

_{g}, and there is an additional reflected flux component proportional to (1 −

*ϵ*

_{g}). The error in (4) arises from attenuating both the ground emission and the reflected flux with

*τ*(

^{f}*u*). As argued below, this is only correct for incident radiation whose spectral energy distribution follows the Planck function

*B*(

_{ν}*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 −

*ϵ*

_{g})

*F*

^{↓}(0)], however, departs significantly from

*B*(

_{ν}*T*).

The broadband transmissivity corresponding to the *ϵ*^{f}(*u*) in Fig. 1 is plotted in Fig. 2b. The use of *τ ^{f}*(

*u*) for gray-ground emission

*μ*m), 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

*ϵ*

_{g})

*F*

^{↓}(0). The difference is evidently not accounted for when attenuating both fluxes with

*τ*(

^{f}*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

*τ*(

^{f}*u*) for (1 −

*ϵ*

_{g})

*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

*τ*(

^{f}*u*) is interpreted as an unphysical “band cross-talk.”

(a) The spectral energy distribution of (left) gray-ground emission

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

(a) The spectral energy distribution of (left) gray-ground emission

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

(a) The spectral energy distribution of (left) gray-ground emission

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

*O*(

*u*)] of the reflected photons. Reflection only weakens the intensity by a factor 1 −

_{t}*ϵ*

_{g}, leaving the spectrum unchanged. Hence, rather than separately account for the downward emission from a column of height

*u*and subsequent attenuation of the upwelling reflected flux through a further distance

_{t}*u*, it is convenient to consider the combined emission of a hypothetical air column of height

*u*+

*u*extending from

_{t}*u*to −

_{t}*u*. This emission is given by

*a*denotes air column emission, and the lower limit of the integral denotes the reversed reflected trajectory. The argument of the temperature field is

*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

*τ*(

^{f}*u*) to attenuate (1 − ϵ)

*F*

^{↓}(0), is thus given by

*τ*(

^{f}*u*) as

*τ*(

^{f}*u*) is the attenuation, over a distance

*u*, of incident radiation with a spectral energy distribution that follows the Planck function. Thus,

*τ*(

^{f}*u*+

*u*) may be interpreted as the additional attenuation, over a distance

_{t}*u*, of incident radiation with band energies that survive after traversing an air column of height

*u*. Said differently, it denotes the additional attenuation of incident radiation with band energies that cannot be absorbed by an air column of height

_{t}*u*. Since what cannot be absorbed cannot be emitted (Kirchhoff’s law),

_{t}*τ*(

^{f}*u*+

*u*) is the attenuation, over a distance

_{t}*u*, of incident radiation with band energies that are absent from the emission of an air column of height

*u*. Hence,

_{t}*τ*(

^{f}*u*) −

*τ*(

^{f}*u*+

*u*), the factor [1 −

_{t}*τ*(

^{f}*u*)]

_{t}^{−1}in (9) being needed for normalization.

### b. Cooling-rate profiles

*ϵ*

^{f}(

*u*) used in these calculations (plotted in Fig. 1) is given by Zdunkowski and Johnson (1965):

*ϵ*

^{f}(

*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

*ϵ*

_{g}for an adiabatic lapse-rate atmosphere with

*T*(

*z*) =

*T*− Γ

_{g}*z*and

*T*= 300 K; here, Γ is the specified lapse rate, and

_{g}*T*=

_{g}*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

*ϵ*

_{g}= 1 the flux divergences calculated from the two formulations are coincident as they must be. For

*ϵ*

_{g}< 1 the deficient reflected flux in the erroneous formulation leads to a pronounced cooling near the ground. For

*ϵ*

_{g}= 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

*ϵ*

_{g}is absent in the correct formulation.

The flux-divergence profiles obtained from the erroneous formulation [(1) and (4)] and the correct formulation [(10) and (11)] for *ϵ*_{g} = (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

The flux-divergence profiles obtained from the erroneous formulation [(1) and (4)] and the correct formulation [(10) and (11)] for *ϵ*_{g} = (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

The flux-divergence profiles obtained from the erroneous formulation [(1) and (4)] and the correct formulation [(10) and (11)] for *ϵ*_{g} = (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 *ϵ*_{g} = 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) − *T _{g}*, 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

(a) Inversion layer temperature profiles for different *H*. (b) Flux divergence profiles obtained as a function of *τ*_{op} of the inversion layer) for *ϵ*_{g} = 1.

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

(a) Inversion layer temperature profiles for different *H*. (b) Flux divergence profiles obtained as a function of *τ*_{op} of the inversion layer) for *ϵ*_{g} = 1.

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

(a) Inversion layer temperature profiles for different *H*. (b) Flux divergence profiles obtained as a function of *τ*_{op} of the inversion layer) for *ϵ*_{g} = 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 *ϵ*_{g} = 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.

*a*and

*b*have been defined in (12). The expression shows that the warming flux divergence increases with decreasing

*ϵ*

_{g}. 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

*ϵ*

_{g}= 1 but increases sharply with decreasing

*ϵ*

_{g}. 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

*ϵ*

_{g−crossover}, corresponding to a change in sign of the surface flux divergence due to spurious cooling, by equating (13) and (14), whence

*ϵ*

_{g−crossover}≈ 0.99 and 0.92, respectively, for optically thick

*ϵ*

_{g}<

*ϵ*

_{g−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

*ϵ*

_{g}on either side of

*ϵ*

_{g−crossover}, and for different values of

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 *ϵ*_{g} on either side of *ϵ*_{g−crossover}, with reference to (a),(b) optically thick

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

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 *ϵ*_{g} on either side of *ϵ*_{g−crossover}, with reference to (a),(b) optically thick

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

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 *ϵ*_{g} on either side of *ϵ*_{g−crossover}, with reference to (a),(b) optically thick

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 Δ*T* = 5 K and *H* = 10 m, the difference between the upward fluxes, obtained from the two formulations for *ϵ*_{g} = 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 *ϵ*_{g}, 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 *ϵ*_{g} = 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 *ϵ*_{g} = 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 *ϵ*_{g} leads to a spurious peak in the sensible heat flux—a feature absent in the *ϵ*_{g} = 1 simulations. For *ϵ*_{g} = 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 *ϵ*_{g} = 0.965 (shown in their appendix), again because of the spurious cooling contribution. As pointed out earlier in section 2b, a reduced *ϵ*_{g} must lead to a relative warming, and the two-layer NBL structure must therefore remain qualitatively unaltered for a reduced *ϵ*_{g}.

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 (*ϵ*_{g} = 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).

(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 *ϵ*_{g} = (b) 1, (c) 0.9, and (d) 0.8.

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

(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 *ϵ*_{g} = (b) 1, (c) 0.9, and (d) 0.8.

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

(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 *ϵ*_{g} = (b) 1, (c) 0.9, and (d) 0.8.

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

*ϵ*

_{g}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

*ϵ*

_{g}becomes obvious on examining the flux divergence for an isothermal atmosphere using (1) and (4):

*ϵ*

^{f}(

*u*) denotes the overall medium transparency and is insensitive to an increased resolution of the opaque bands. In contrast,

_{t}*ϵ*

_{g}= 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

*ϵ*

_{g}. 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 *ϵ*_{g} = 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 *ϵ*_{g}, 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

*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

*ϵ*

_{g}= 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

*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

*ϵ*

_{g}, 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

*ϵ*

_{g}< 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

*ϵ*

_{g}and

*ϵ*

_{c}and a nonzero upper surface transmissivity

*τ*[keeping in mind the atmospheric context; see Liou and Ou (1981)], the flux expressions are

_{c}*τ*= 0, appears in Gille and Goody (1964).

_{c}#### c. Directional flux emissivity formulation

*θ*,

*ϕ*) denotes the direction on the unit sphere with

*μ*= cos

*θ*and

*d*Ω = −

*dμdϕ*;

*ϵ*

_{gΩ}and

*ρ*

_{Ω′→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,

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