## 1. Introduction

*u*above black ground (

_{t}*ε*

_{gj}= 1). Here,

*B*is the (approximately constant) Planck function in band

_{j}*j*and the overdot denotes differentiation with respect to the argument. The net flux is

*M*being the number of bands. The diffuse transmission function

Earlier studies have employed narrowband formulations to study the evolution of both diurnal (Savijärvi 2006) and nocturnal boundary layers (NBL; Duynkerke 1999; Savijärvi 2009); the formulation is sometimes used as a benchmark to test the validity of simpler models (Rodgers and Walshaw 1966; Ramanathan and Downey 1986). The purpose of this paper is to point out the error that arises in extending (1)–(2) to a nonblack surface (ground). The error is due to an incorrect reflected flux, and it arises from not discriminating between the spectral content of ground emission and the atmospheric column emission reflected from the ground. It manifests as an intense cooling close to ground, the intensity being a function of the vertical resolution used in a given calculation. We present in section 2 the correct reflected flux that removes this spurious cooling in a narrowband formulation. Thereafter, the discrepancy in the bandwise fluxes, calculated using the erroneous and correct formulations, is determined for a standard tropical atmosphere. Section 3 summarizes the main points of the analysis and highlights the inherent superiority of the correlated-*k* methods in this regard.

## 2. The narrowband formulation for reflective ground

*ε*

_{gj}in the

*j*th band is given by (Savijärvi 2006)

*ε*

_{gj}[

*πB*(

_{j}*T*)] and the reflected flux

_{g}*ε*

_{g}, and with

*τ*(

^{f}*u*) is the diffuse broadband transmissivity used (incorrectly) to attenuate both ground emission and the reflected flux. However, the transmissivity appropriate for the reflected flux is

*τ*(

^{f}*u*).

*G*(

*u*) being an arbitrary function of

*u*; using

*u*′ = 0 in (7), one concludes that

*G*is identically zero. Hence,

*τ*(

^{f}*u*) =

*e*

^{−αu}, where

*α*

^{−1}is the photon mean free path. It may be seen from (5) that

The water-vapor-laden atmosphere (water vapor is the principal participating component in a cloud-free atmosphere) is, however, pronouncedly nongray due to the enormous wavelength sensitivity of the water vapor absorption in the infrared and the resulting disparity in photon pathlengths even within small spectral intervals; thus, *τ ^{f}*(

*u*) for water vapor departs significantly from a decaying exponential [see Figs. 1 and 2 in Ponnulakshmi et al. (2012)]. One, therefore, expects the erroneous reflected flux to lead to a spurious cooling error in any atmospheric calculation with nonblack bounding surfaces and with radiation modeled using an emissivity scheme. An unlikely scenario may arise if (6) holds despite the broadband transmissivity not being an exponential; in which case,

*T*(

*u*′) would have to closely approximate the null eigenfunction of (6) (a Fredholm integral equation of the first kind). Rather than attempt to calculate this eigenfunction, it is easier to verify if such an exception occurs for typical atmospheric profiles. Figure 1 shows this not to be the case; the flux difference for a model tropical atmosphere remains comparable to the individual fluxes. In summary, a necessary and sufficient condition for the spurious cooling error to arise in a broadband emissivity scheme is for

*τ*(

^{f}*u*) to deviate from an exponential.

*j*th band in a narrowband formulation. Accounting for the non-Planckian energy distribution of the downwelling surface flux, in the same manner as in (4), one obtains the following expression for the upward bandwise flux to be used in a narrowband formulation for an atmosphere with reflective ground:

*δ*is the average line spacing,

*S*is the mean line intensity, and

*α*denotes the Lorentzian half-width. As is well known (Goody 1964), (11) exhibits three asymptotic regimes—the weak-line approximation when

_{L}*Su*≪

*πα*), the strong-line approximation with

_{L}*πα*≪

_{L}*Su*≪

*δ*

^{2}/

*πα*) when the line centers are strongly absorbed with additional absorption occurring in the wings, and subsequent saturation

_{L}*Su*≫

*δ*). An exponentially decaying transmittance is only realized in the heavily overlapping limit (

*δ*≪

*πα*) when there is a direct transition from a linear decrease for small

_{L}*u*to an exponential one for large

*u*. Thus, the existence of a sensible strong-line regime is direct evidence of fine structure in the absorption spectrum and the related significance of “wing” contributions. The resulting departure of

*z*for a water-vapor-laden tropical atmosphere. The plots are for both an opaque (1550–1650 cm

^{−1}: the 6.3-

*μ*m vibration–rotation band) and a transparent band (720–800 cm

^{−1}) with

*u*+

*u*), and the cumulative emission in the

_{t}*j*th band is therefore given by

*u*, given by

Differences in the erroneous and correct reflected fluxes (normalized by the reflected flux at the surface in the particular band) for a tropical atmosphere and for the frequency ranges (a) 1550–1650 cm^{−1} (the 6.3-μm vibration–rotation band) and (b) 720–800 cm^{−1}; the band-averaged transmittance is given by (11) with the band parameters taken from Rodgers and Walshaw (1966).

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-095.1

Differences in the erroneous and correct reflected fluxes (normalized by the reflected flux at the surface in the particular band) for a tropical atmosphere and for the frequency ranges (a) 1550–1650 cm^{−1} (the 6.3-μm vibration–rotation band) and (b) 720–800 cm^{−1}; the band-averaged transmittance is given by (11) with the band parameters taken from Rodgers and Walshaw (1966).

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-095.1

Differences in the erroneous and correct reflected fluxes (normalized by the reflected flux at the surface in the particular band) for a tropical atmosphere and for the frequency ranges (a) 1550–1650 cm^{−1} (the 6.3-μm vibration–rotation band) and (b) 720–800 cm^{−1}; the band-averaged transmittance is given by (11) with the band parameters taken from Rodgers and Walshaw (1966).

Citation: Journal of the Atmospheric Sciences 70, 1; 10.1175/JAS-D-12-095.1

*f*(

*u*+

*u*) >

_{t}*f*(

*u*) +

*f*(

*u*). Now, with the equality sign, this relation is the Cauchy functional equation, satisfied by

_{t}*f*(

*u*) ∝

*u*, which corresponds, of course, to a gray atmosphere. The above inequality is satisfied when

*f*(

*u*) varies more rapidly than a linear function, say,

*f*(

*u*) ∝

*u*(with

^{x}*x*> 1); a transmittance of the form exp(−

*u*) would, therefore, lead to a spurious heating contribution. Although a mathematical possibility, such a functional form appears to be not relevant to tropospheric heat exchanges with pressure-broadened spectra. This is clearly evident from the form of the transmittance in the strong-line regime, in which case

^{x}*f*(

*u*+

*u*) <

_{t}*f*(

*u*) +

*f*(

*u*), implying that

_{t}The arguments above show that the deviation of the band-averaged transmittance from an exponential is directly linked to the existence of a strong-line regime, and that the nature of this deviation is such as to lead to a spurious cooling contribution. It is therefore worth emphasizing that typical infrared spectra of atmospheric gases have pressure-broadened line widths [*O*(0.01–0.1 cm^{−1})] smaller than the smallest interline spacing [*O*(1 cm^{−1}) due to rotation transitions]; in other words, atmospheric radiative exchanges correspond largely to the strong-line regime (Goody 1964). Indeed, the importance of the atmospheric window implies that radiative cooling in the lower troposphere is dominated by wing contributions; the weak window attenuation, modeled as the water vapor continuum, is thought to arise from cumulative far-wing contributions (Bignell 1970; Clough et al. 1989); although, the additional role of water vapor dimers continues to be debated (Ptashnik et al. 2011). The significance of the strong-line regime is also evident in emissivity parameterizations used for NBL modeling (Siqueira and Katul 2010; Garratt and Brost 1981; Rodgers 1967); the emissivity, for small *u*, being expressed in terms of *u*^{1/2}, rather than *u*. Scaling approaches for an inhomogeneous atmosphere have again been based on the strong-line approximation (Cess 1974; Ramanathan 1976; Chou and Arking 1980). Clearly, one expects a spurious cooling contribution in typical narrowband formulations. To the extent that the frequency interval used in a narrowband formulation is arbitrary (Ramanathan and Downey 1986), a spurious cooling error is expected in any frequency-parameterized scheme with the parameterization applied to intervals larger than an elementary line width; the error, of course, arises only when such a scheme is applied to reflective ground. The calculations of André and Mahrt (1982), Schaller (1977), and more recently, Savijärvi (2006), are examples in this regard. Because of the spurious cooling error, the cooling rate profiles obtained by Savijärvi (2006), using a narrowband formulation, are very sensitive to a departure of *ε*_{g} from unity. Reducing *ε*_{g} from 1 to 0.8, for a midlatitude summer (MLS) atmosphere led to perceptible cooling rate differences at heights of up to a kilometer; the cooling rate at 0.1 m, in particular, changed from 3.8 K day^{−1} for *ε*_{g} = 1 to 9.5 K day^{−1} for *ε*_{g} = 0.8. In contrast, the actual cooling rate profiles for the (dry) lapse rate atmosphere have been shown to be fairly insensitive to *ε*_{g} (Ponnulakshmi et al. 2012). Morcrette’s original narrowband calculation (Morcrette 1977), extended by André and Mahrt (1982) to model NBL over a nonblack ground, highlights the incorrect effect of *ε*_{g} on the cooling rates. In contrast to the expected enhancement with decreasing *ε*_{g}, expected for a nocturnal inversion (Lieske and A. Stroschein 1967; Edwards 2009), the warming layer is already lost for *ε*_{g} = 0.965.

## 3. Conclusions

In this paper, we have highlighted a fundamental error in the prevailing narrowband formulation for reflective ground together with its consequences for atmospheric cooling rate profiles. A necessary and sufficient condition for the error, in the form of an intense near-surface cooling, to occur is the deviation of the appropriate frequency-averaged transmissivity function from a simple exponential decay; the deviation results from the multiplicity of photon pathlengths in the relevant frequency interval. The latter is almost always true for atmospheric radiative exchanges. The infrared spectra of most atmospheric gases are dominated by vibration–rotation bands, and the specificity of the underlying (discrete) transitions renders the photon mean free path an extremely sensitive function of frequency. Thus, any frequency-parameterized radiation scheme that does not resolve intervals comparable to or smaller than an elementary line width will suffer from a spurious cooling error. We have presented the corrected scheme that removes the error.

The spurious cooling error inherent in a frequency-parameterized radiation scheme highlights the superiority of the *k*-distribution method and its extension (the correlated-*k* method) to an inhomogeneous atmosphere (Liou 2002). The method is based on the grouping of absorption coefficient (*k _{ν}*) values. Thus, the frequency integrals in the expressions for the monochromatic fluxes are replaced by integrals over

*k*weighted by the cumulative probability distribution of photon pathlengths

*g*(

*k*). The smooth variation of

*g*with

*k*, in sharp contrast to the rapid variation of

*k*with

*ν*, leads to an immediate computational advantage. In the present context, forming contiguous intervals based on

*k*values naturally negates the error in the reflected flux, since the error arises due to the disparity in photon pathlengths over short frequency intervals. Finally, it is worth mentioning that several calculations have been based on an exponential-sum fitting of the narrowband transmittances (Liou and Sasamori 1975; Stephens 1978), this being a discrete representation of the

_{ν}*k*-space integral with each decaying exponential corresponding to a gray subband; importantly, the flux divergence must be calculated as a sum of subband contributions. Not conforming to this procedure will lead to an error due to the huge disparity in the subband photon pathlengths, within a single band, as is the case in Varghese et al. (2003). Similar to Savijärvi (2006), the predicted flux-divergence profiles (for an MLS atmosphere) remain sensitive to a small deviation of

*ε*

_{g}from unity even at heights on the order of a kilometer; the surface cooling rate changes from 4.5 K day

^{−1}for

*ε*

_{g}= 1 to 37.5 K day

^{−1}for

*ε*

_{g}= 0.8.

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