## 1. Introduction

Infrared radiation (IR) is a physical process that plays a prominent role in atmospheric physics—especially through interaction with clouds. It is the most important physical phenomenon that drives radiation fog formation (Davis 1994). To study atmospheric radiation, the question arises whether to adopt a 1D, 2D, or 3D approach to compute radiative transfer (RT). Many sophisticated treatments of the radiative transfer equation (RTE)—Monte Carlo method (MCM) (Fleck 1961) for angular integration and line-by-line (LBL) calculations for the integration over the spectrum (Rothman et al. 1992)—aim to simulate accurately radiative processes. However, as is the case in many atmospheric simulations, radiative processes are only one aspect of the whole modeling problem. Once RT is coupled with fluid dynamics, the energy balance equation has to be solved within the computation of the heating or cooling rate. This procedure is still time consuming even if RT calculations are activated periodically (RT values are assumed unchanged during a period of time).

Therefore, a compromise between accuracy and computing cost should be found to compute radiative quantities in 3D simulations. The Spherical Harmonics Discrete Ordinate Method (SHDOM; Evans 1998) proposes a good compromise to compute 3D RT by combining a directional resolution with a multispectral method. In many meteorological applications, band models are commonly employed to compute the spectral integration. In these schemes, the averaged transmissions are computed across spectral bands. For statistical band models, hypothesis on the line spacing are invoked to perform the integration over the spectrum and, thereby, derive a band-averaged transmissivity (Goody 1952; Malkmus 1967). Another widely used approach consists in the integration over the values of the absorption coefficient [the correlated-*k* distribution (CKD) method] to compute the transmission (Lacis and Oinas 1991; Fu and Liou 1992). Mathematically, it consists in taking Lebesgue’s definition of the integral for the transmission calculation. However, most atmospheric cases, such as studies of the nocturnal boundary layer (NBL) use emissivity schemes to treat the spectral dependence of the RTE (Viskanta et al. 1977; Savijärvi 2006; Edwards 2009; Siqueira and Katul 2010). For its computational efficiency, broadband emissivity schemes are a useful approach to highlight the interaction of radiation with other transport processes. The concept of broadband emissivities was introduced by Elsasser (1942), and Yamamoto (1952) produced a radiation chart on which Sasamori (1968) fitted analytical expression of emissivities.

The radiative model developed by Musson-Genon (1987) in the computational fluid dynamics (CFD) software Code_Saturne (Archambeau et al. 2004) with this parameterization in infrared wavelength shows good agreement with measurements made during fog events such as ParisFog in 2006 (Zhang et al. 2014). Although a case study with a 3D mesh—for CFD—has been done, radiative exchanges were assumed unidimensional. The presence of buildings and vegetation changes at least the surface emission and makes radiative transfer three dimensional. Moreover, even if we may reasonably assume that atmospheric gas concentrations are horizontally homogeneous, horizontal radiative effects induced by the heterogeneous liquid water field or complex terrain with buildings and vegetation should be taken into account with a three-dimensional solver (Hoch et al. 2011). In local-scale models with a fine spatial discretization, an MCM is unusable for its very high computational cost when several radiative outputs are needed like the heating rate field (Evans 1998). MCMs—like the LBL method to calculate the integration over the spectrum—are used as a reference solution for benchmark (Modest 2003). The discrete ordinates method (DOM) is a good alternative to MCM to reduce the computational time. First proposed by Chandrasekhar (1950), DOM is based on the directional variation of the radiance. The RTE is solved for a set of discrete directions spanning the total solid angle of 4*π*. The unknown variables are now a discrete set of intensities along the chosen angular directions, and the original integral over the solid angle is replaced by a discrete weighted sum (quadrature).

The purpose of this paper is to give a gray formulation of radiative properties for 3D applications using the DOM of gray RTE (equivalent absorption coefficient), based on the emissivity formula of Sasamori (1968). The authors derive expressions for the equivalent absorption coefficient that appear in the formulation of the gray RTE, using the analytical expressions of upward and downward fluxes from Lenoble (1993). The expressions for these equivalent absorption coefficients are obtained by matching the upward and downward fluxes obtained from a broadband emissivity scheme for the restricted case of isothermal atmosphere. Thus, equivalent absorption coefficients can directly be linked with emissivities. By separately treating upward and downward fluxes, we propose a formulation for upward and downward equivalent absorption coefficients that respectively depend on upward and downward emissivities. This expression mathematically ensures the positivity of equivalent absorption.

This parameterization allows us to separate gaseous and liquid water absorption. The equivalent absorption coefficients suited to DOM have been developed to give results similar than the 1D broadband emissivity scheme. Three-dimensional infrared radiation appears with water droplet formation. Then, a local gray absorption coefficient is computed to take into account absorption by a 3D distribution of water droplets. Thus, although gaseous absorption is treated in 1D, the solution of the gray RTE is actually 3D. This paper will compare the new parameterization for the 3D solver with the existing 1D broadband emissivity scheme on plane-parallel atmospheres and show its ability to catch 3D infrared radiative effects.

This paper is organized according to the following outline:

- Analytical calculations are performed in section 2 to provide analytical 1D solutions of the RTE within broadband emissivity definition in the case of isothermal atmosphere.
- By using the two formulations of the RTE (the differential and the integral), a relation between the equivalent absorption coefficient and emissivity is found in section 3 for the 3D radiation code.
- In section 4, numerical results are shown for clear-sky and cloudy conditions. A comparison between the 1D (two-stream approach using broadband emissivity) and the 3D (gray RTE solved by the DOM using the new radiative properties parameterization) radiation code of Code_Saturne is performed. For cloudy and clear atmospheres, we used atmospheric profiles from the ParisFog field experiment from the intensive observation period 13, simulated by Zhang et al. (2014).
- The section 5 is dedicated to an exploratory simulation to show 3D radiative effects captured by this new parameterization.

## 2. The broadband emissivity for 1D modeling

### a. The two-stream approximation

**S**is the propagation direction of the radiance

^{−1}) for the

*λ*wavelength, and the source term is given by the Planck function

*h*is the Planck constant,

*c*is the velocity of light in vacuum, and

*T*is temperature.

*μ*is the cosine of the zenith angle

*θ*. Separating the RTE according to upward or downward directions leads us to define the upward and the downward radiances:to obtain from (2.2) the system

*z*, expressions for the fluxes are obtained:where

### b. The broadband emissivity

*k*distributions (Fu and Liou 1992). But in operational models the computation cost is the decisive factor. An alternative method is to use a broadband emissivity scheme, which parameterizes the total emissivity of an isothermal layer over the whole spectrum. The isothermal broadband emissivity (Liou 2002) along

*μ*m band and rotation band, respectively, and

_{2}O)

_{2}(the self-broadened continuum).

*z*to 0 remains transparent along 0 to

*z*after reflection—and should be rewritten as in Ponnulakshmi et al. (2012):The term

## 3. Analytical expression of gray absorption coefficient for 3D modeling

**x**is the position in space and withare the radiative quantities spectrally integrated.

*z*coordinates, becomerespectively. By using the analytical expressions of fluxes for an isothermal atmosphere, (2.16) and (2.17), the above lead torespectively. The upward and downward absorption coefficients depend on upward and downward emissivities; that is, they are both functions of pressure and temperature, but also of optical path. The difference with the monochromatic absorption coefficient is the nonlocal nature of

- The two-stream approximation leads to two independent equations for fluxes or radiances. The heating rate is the contribution of the divergence of upward and downward fluxes. The absorption coefficient has the following expression:
- The dependence of
and on optical paths is due to the definition of equivalent absorption. They have been defined as functions that satisfy both (3.6) and (3.7). Analytical solutions of fluxes given for an isothermal atmosphere are spatially integrated. They contain the optical information for the layers between 0 and *z*and betweenand *z.*This gives upward and downward coefficients their nonlocal nature. - The expressions of
and derive from mathematical considerations under the isothermal atmosphere assumption. Treating separately upward and downward directions for fluxes but also for extinction coefficients is not a new idea (Edwards 1996). Generally, , but in a GCM the equivalent extinction is based on net fluxes. Assuming leads to the following expression:Numerical problems may appear and we may have negative extinction. Separating upward and downward direction provides the positivity of the equivalent absorption coefficients. - The spectral profiles of downward and upward fluxes are totally different (Ponnulakshmi et al. 2012), which justifies separating upward and downward directions for treating the spectral integration.

## 4. Numerical experiments

### a. Fluid dynamics and radiative transfer solvers

To compute the radiative source term, including the radiance distribution, both an integration over the solid angle and a spatial integration are needed. Code_Saturne is a 3D computational fluid dynamics code using a collocated finite-volume scheme (unknown variables are cell centered) that accepts meshes with any type of cells. It solves Navier–Stokes equations for flows, steady or unsteady, laminar or turbulent, incompressible, isothermal or not, with scalar transport. Several modules may be activated such as radiative heat transfer for gas or oil combustion and atmospheric flows. The mesh used for fluid dynamics and radiative heat transfer is the same and assumed constant in the cell.

The 1D radiation broadband emissivity code of Code_Saturne used to provide reference solutions was validated on data from Cabauw (Musson-Genon 1987) and the ParisFog field experiment (Zhang et al. 2014). On this dataset, which contains heating rates and radiative fluxes, the 1D code shows its ability to simulate a full life cycle of fog. To verify the implementation of this new radiative property model, we selected atmospheric profiles from the 6-month ParisFog field experiment, which took place in Paris (Haeffelin et al. 2010) at the Site Instrumental de Recherche en Télédétection Atmosphérique (SIRTA) observatory. The dataset contains 13 intensive observation period (IOP) of fog or quasi-fog events in which formation and evolution are extremely sensitive to radiation. We selected profiles of water vapor concentration, temperatures, and liquid water content as inputs to the radiation code and we compared the 1D radiation scheme with the 3D one, based on the discrete ordinates method. The new scheme proposed in this paper is in principle 3D, but, in this part, it is actually being used in 1D mode and compared to a genuinely 1D code. Three-dimensional here refers only to the angular integration.

For the discrete ordinates method, we used the

The simulated domain is rectangular 1 km × 1 km × 11 km with periodic boundary conditions with uniform concentration of gases in *x* and *y* directions. There are 78 cells up to 11 km in *z* direction with a first level at 2 m and the spacing between the levels increases logarithmically.

### b. Isothermal atmosphere

As suggested by Ellingson and Fouquart (1991) and recommended by Varghese et al. (2003), it is very important to check the model’s outputs for an isothermal atmosphere. Upward fluxes are constant and downward fluxes are analytical. Therefore, checking radiative outputs in this trivial case is a code verification. We first took a single column of water vapor, with a constant temperature of 270 K and no temperature gradient with the ground, meaning

#### 1) Heating/cooling rate

*J*defines the cell, and the subscript

*p*defines the discrete direction.

#### 2) Upper boundary layer

An important aspect of the numerical simulation is the boundary conditions. The plane-parallel atmosphere assumption leads to set periodic boundary conditions in the *x* and *y* directions. Studying the NBL often leads to reducing the height of computational domain to 1.0 or 2.0 km, where the radiance is not equal to zero. Considering an isotropic radiance at the top of the domain is an approximation widely used in the IR domain (Garratt and Brost 1981; Siqueira and Katul 2010). Thus,

To check the reliability of their approximation, we tested the boundary condition at the top of the domain height. The results in Fig. 3 show good agreement between the 3D solver and the 1D broadband emissivity scheme. Moreover, taking

*μ*:This formulation of absorption coefficients would take into account the differences between optical pathlength for each discrete direction and reduce the discrepancy near the top and bottom boundary of the domain.

### c. Simulations with IOP 13 profiles

The atmospheric profiles from the IOP 13 on 18–19 February 2007 were selected to test the radiative parameterization. The dataset contains a whole life cycle of radiative fog. Two profiles were chosen from the data.

#### 1) Clear-sky conditions

Because of the higher value of the ground emissivity, the temperature of the surface drops below that of the air immediately above. This leads to a strong inversion of temperature (Fig. 4) in the lowest part of the atmosphere. As the cooling continues, the air reaches saturation conditions to allow water vapor condensation. So the cooling rate during the night in clear-sky conditions has to be computed as accurately as possible to allow the CFD model to produce liquid water.

Figure 5 shows the cooling rate for domain heights equal to 1 and 2 km at 1800 UTC. The absorption coefficients were computed by using the total emissivity that takes into account water vapor, carbon dioxide, and ozone (given in appendix A).

Although this parameterization has been developed using analytical solutions in an isothermal atmosphere, it gives good results in this case. The small discrepancy between the 1D broadband emissivity scheme and the DOM with equivalent absorption coefficient shows that the isothermal hypothesis in the expression of equivalent absorption coefficient has a weak influence.

#### 2) Foggy conditions

To simulate a few hours of fog formation in the nocturnal boundary layer, it is important to test the response of our radiative parameterization in the presence of cloud water droplets. Liquid water is produced once saturation is reached. The deepening is due to the strong cooling at the top of the fog layer. The mixing will homogenize the layers of moist air with the dry ones and increase the production of liquid water droplets at the top.

As we said, scattering was left out in our equations but droplets mean spherical particles and scattering. This process can often be neglected in longwave radiation. The first reason is that absorption is dominant and scattering is of secondary importance in longwave radiation (Chou et al. 1999). The single scattering albedo for longwave can be close to 0.5 but, because scattering is dominantly in the forward direction, the net effect is small. The second reason is the computational time needed by the solver (DOM) for scattering. Not considering scattering in some applications—retrieval of equivalent radius of cloud droplets (Dubuisson et al. 2005)—is an approximation that can lead to significant errors (Chou et al. 1999). However, this radiative parameterization is adequate for applications where absorption is the dominant process.

In Fig. 6, the good agreement with the 1D radiative code illustrates the good behavior of our longwave radiative parameterization in the presence of various gases and liquid water in a nonisothermal atmosphere.

^{−1}—is the signature of a radiative fog. At the top of the fog layer, we recover an extreme cooling value that allows fog development owing to the destabilization of temperature profile in the fog layer. The warming of the air layer near the surface reflects the blackbody behavior of the fog and the absorption of much of the radiative energy emitted from the surface within the fog layer. Equations (4.10) and (4.11) define relative error values in Table 1:where the subscripts 3D and 1D refer to the radiative solvers of Code_Saturne described previously.

Mean relative error value of incident flux on the surface

Mostly, mean relative errors in the domain are around 10% for cooling rates and less than 5% for the incident flux on the surface. The more we reduce the domain height, the less is the discrepancy between the 1D and the simulations. A reason may be that the boundary condition at the top of the 3D simulated domain is given by the 1D code

## 5. Potential 3D radiative effects captured by this parameterization

The purpose of is the section is to show the 3D radiative effects that the new parameterization can capture. To introduce 3D effects in radiative transfer computation, we add three idealized buildings at the same surface temperature as the ground temperature. The 3D mesh was designed within the extrusion of the 2D mesh in Fig. 7. The upward and downward gas absorption coefficients are computed on a column of cells where scalars, needed to compute *z*. The liquid absorption coefficient is locally computed in each cell containing liquid water using (4.9). Before the simulation it is possible to compute absorption coefficient for each column of the 3D mesh in the sense of the independent column approximation (Cahalan et al. 1994). But the goal of the simulation is just to show the limits of the 1D broadband emissivity scheme and the ability of the new parameterization to catch 3D effects:

- radiation between the ground and the walls of the buildings and
- the emitted radiation by liquid water, received by the buildings.

In Fig. 9, a clip plane at *z* = 8 m of liquid water is plotted at 2310 UTC. The distribution of liquid water is horizontally heterogeneous, which implies a 3D absorption coefficient [see (4.9)]. For longwave radiation, the medium absorbs and emits radiation, which makes RT 3D. This is the second 3D radiative effect captured by the new parameterization and shown in Fig. 10.

## 6. Conclusions and perspectives

A new parameterization based on broadband emissivity scheme has been developed here for the representation of absorption in the radiative transfer equation. This radiative property model is based on the mathematical calculations derived from considering an isothermal atmosphere and proposes an alternative to narrow band models. By distinguishing the upward and the downward directions, a broadband emissivity scheme suited for the discrete ordinates method was built. The distribution of gases in the atmosphere is assumed horizontally uniform, so the absorption coefficients only depend on height (Evans 1998). The spectral representation of the upward and downward fluxes leads us to consider two different equivalent absorption coefficients as suggested by Edwards (1996). Several tests with various meteorological conditions have been done to test the reliability of the parameterization. Under clear skies, in both stable and unstable conditions, and under cloudy skies, coupling the discrete ordinates method with the equivalent absorption parameterization gave good results for heating rates and fluxes in comparison with the broadband emissivity scheme. The sensitivity study to the boundary condition at the top of the computed domain showed that it is possible to reduce the domain height to 1 or 2 km. This will save computational time for 3D simulations.

Using Ponnulakshmi et al. (2012)’s formulation of directional fluxes in equivalent absorption coefficient would reduce the discrepancy between 1D and 3D results in plane-parallel atmosphere configuration. This allows us to test the sensitivity to the quadrature scheme (i.e., the number of discrete directions).

This fast parameterization may be useful for a dynamical study to simulate 3D radiative effects of complex terrain or vegetation in the NBL. Computing real three-dimensional radiative transfer implies making “no assumption about the translational or rotational symmetry of the optical media nor about the sources of radiation” (Davis and Knyazikhin 2004). In our paper, we captured only some of the three-dimensional radiative exchanges by assuming horizontally homogeneous layers of gases.

With the correction of the reflected flux on a nonblack surface, it would be possible to compare data simulation from 1D and 3D radiation codes with measurements from the ParisFog field experiment. This parameterization is independent from the emissivity function and may be tested with those of Zdunkowski and Johnson (1965) or Shaffer and Long (1975).

The research reported in this paper has been supported by the CEREA, École des Ponts Paristech, and EDF R&D. The authors are thankful to Pr. Dubuisson for many valuable discussions and Dr. Ponnulakshmi and Pr. Subramanian for useful scientific literature. The authors are thankful to the people who have worked during the ParisFog field experiment. The authors are thankful to the reviewers for their helpful remarks, comments, and suggestions, which considerably helped us to improve the content.

# APPENDIX A

## Broadband Emissivity Scheme Formulas

### a. Optical path

*c*chosen as

### b. Emissivity functions

# APPENDIX B

## Simulated Temperature Profiles

Figure B1 features temperature profiles simulated during the fog formation.

## REFERENCES

Archambeau, F., , N. Méchitoua, , and M. Sakiz, 2004:

*Code_Saturne*: A finite volume code for the computation of turbulent incompressible flows—Industrial applications.,*Int. J. Finite Vol.***1**, 1–62.Cahalan, R. F., , W. Ridgway, , W. J. Wiscombe, , and S. Gollmer, 1994: Independent pixel and Monte Carlo estimates of stratocumulus albedo.

,*J. Atmos. Sci.***51**, 3776–3790, doi:10.1175/1520-0469(1994)051<3776:IPAMCE>2.0.CO;2.Carlson, B. G., , and K. D. Lathrop, 1965: Transport theory: The method of discrete ordinates. Los Alamos Scientific Laboratory of the University of California, 92 pp.

Chandrasekhar, S., 1950:

*Radiative Transfer*. Oxford University Press, 393 pp.Chou, M.-D., , and A. Arking, 1980: Computation of infrared cooling rates in the water vapor bands.

,*J. Atmos. Sci.***37**, 855–867, doi:10.1175/1520-0469(1980)037<0855:COICRI>2.0.CO;2.Chou, M.-D., , K.-T. Lee, , S.-C. Tsay, , and Q. Fu, 1999: Parameterization for cloud longwave scattering for use in atmospheric models.

,*J. Climate***12**, 159–169, doi:10.1175/1520-0442-12.1.159.Davis, A. B., , and Y. Knyazikhin, 2004: A primer in 3D radiative transfer.

*3D Radiative Transfer in Cloudy Atmospheres*, A. Marshak and A. Davis, Eds., Springer, 153–242.Davis, J. M., 1994: Methods of modeling radiant energy exchange in radiation fog and clouds. Army Research Laboratory Tech. Rep. ARL-CR-103, 97 pp.

Dubuisson, P., , V. Giraud, , O. Chomette, , H. Chepfer, , and J. Pelon, 2005: Fast radiative transfer modeling for infrared imaging radiometry.

,*J. Quant. Spectrosc. Radiat. Transfer***95**, 201–220, doi:10.1016/j.jqsrt.2004.09.034.Edwards, J. M., 1996: Efficient calculations of infrared fluxes and cooling rates using the two-stream equations.

,*J. Atmos. Sci.***53**, 1921–1932, doi:10.1175/1520-0469(1996)053<1921:ECOIFA>2.0.CO;2.Edwards, J. M., 2009: Radiative processes in the stable boundary layer: Part I. Radiative aspects.

,*Bound.-Layer Meteor.***131**, 105–126, doi:10.1007/s10546-009-9364-8.Ellingson, R. G., , and Y. Fouquart, 1991: The intercomparison of radiation codes in climate models: An overview.

,*J. Geophys. Res.***96**, 8925–8927, doi:10.1029/90JD01618.Elsasser, W. M., 1942: Heat transfer by infrared radiation in the atmosphere. Harvard Meteorological Studies 6, 107 pp.

Evans, K. F., 1998: The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer.

,*J. Atmos. Sci.***55**, 429–446, doi:10.1175/1520-0469(1998)055<0429:TSHDOM>2.0.CO;2.Ferziger, J. H., , and M. Perić, 2002:

*Computational Methods for Fluid Dynamics*. 3rd ed. Springer-Verlag, 426 pp.Fleck, J. A., 1961: The calculation of nonlinear radiation transport by a Monte Carlo method. University of California Lawrence Radiation Laboratory Tech. Rep., 43 pp.

Fu, Q., , and K. N. Liou, 1992: On the correlated

*k*-distribution method for radiative transfer in nonhomogeneous atmospheres.,*J. Atmos. Sci.***49**, 2139–2156, doi:10.1175/1520-0469(1992)049<2139:OTCDMF>2.0.CO;2.Garratt, J. R., , and R. A. Brost, 1981: Radiative cooling effects within and above the nocturnal boundary layer.

,*J. Atmos. Sci.***38**, 2730–2746, doi:10.1175/1520-0469(1981)038<2730:RCEWAA>2.0.CO;2.Goody, R. M., 1952: A statistical band model for water vapor.

,*Quart. J. Roy. Meteor. Soc.***78**, 165–169, doi:10.1002/qj.49707833604.Green, A. E. S., 1964: Attenuation by ozone and the earth’s albedo in the middle ultraviolet.

,*Appl. Opt.***3**, 203–208, doi:10.1364/AO.3.000203.Haeffelin, M., and Coauthors, 2010: PARISFOG: Shedding new light on fog physical processes.

,*Bull. Amer. Meteor. Soc.***91**, 767–783, doi:10.1175/2009BAMS2671.1.Hoch, S. W., , C. D. Whiteman, , and B. Mayer, 2011: A systematic study of longwave radiative heating and cooling within valleys and basins using a three-dimensional radiative transfer model.

,*J. Appl. Meteor. Climatol.***50**, 2473–2489, doi:10.1175/JAMC-D-11-083.1.Lacis, A. A., , and V. Oinas, 1991: A description of the correlated

*k*distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres.,*J. Geophys. Res.***96**, 9027–9063, doi:10.1029/90JD01945.Lenoble, J., 1993:

*Atmospheric Radiative Transfer.*Hampton Series, A. Deepak Publishing, 532 pp.Liou, K.-N., 2002:

*An Introduction to Atmospheric Radiation*. International Geophysics Series, Vol. 84, Academic Press, 583 pp.Malkmus, W., 1967: Random Lorentz band model with exponential-tailed line-intensity distribution function.

,*J. Opt. Soc. Amer.***57**, 323–329, doi:10.1364/JOSA.57.000323.Modest, M. F., 2003:

*Radiative Heat Transfer.*Academic Press, 822 pp.Musson-Genon, L., 1987: Numerical simulations of a fog event with a one-dimensional boundary layer model.

,*Mon. Wea. Rev.***115**, 592–607, doi:10.1175/1520-0493(1987)115<0592:NSOAFE>2.0.CO;2.Ponnulakshmi, V. K., , V. Mukund, , K. R. Sreenivas, , and G. Subramanian, 2009: The ramdas layer remains a micro-meteorological problem. Jawaharlal Nehru Centre for Advanced Scientific Research Tech. Rep. JNCASR/EMU/2009-1, 41 pp.

Ponnulakshmi, V. K., , V. Mukund, , D. K. Singh, , K. R. Sreenivas, , and G. Subramanian, 2012: Hypercooling in the nocturnal boundary layer: Broadband emissivity schemes.

,*J. Atmos. Sci.***69**, 2892–2905, doi:10.1175/JAS-D-11-0269.1.Rothman, L. S., and Coauthors, 1992: The HITRAN molecular database: Editions of 1991 and 1992.

,*J. Quant. Spectrosc. Radiat. Transfer***48**, 469–507, doi:10.1016/0022-4073(92)90115-K.Sasamori, T., 1968: The radiative cooling calculation for application to general circulation experiments.

,*J. Appl. Meteor.***7**, 721–729, doi:10.1175/1520-0450(1968)007<0721:TRCCFA>2.0.CO;2.Savijärvi, H., 2006: Radiative and turbulent heating rates in the clear-air boundary layer.

,*Quart. J. Roy. Meteor. Soc.***132**, 147–161, doi:10.1256/qj.05.61.Shaffer, W. A., , and P. E. Long Jr., 1975: A predictive boundary layer model. NOAA Tech. Memo. TDL-57, 44 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, doi:10.1007/s10546-009-9440-0.Stephens, G. L., 1984: The parametrization of radiation for numerical weather prediction and climate models.

,*Mon. Wea. Rev.***112**, 826–866, doi:10.1175/1520-0493(1984)112<0826:TPORFN>2.0.CO;2.Varghese, S., , A. S. Vasudevamurthy, , and R. Narasimha, 2003: A fast, accurate method of computing near-surface longwave fluxes and cooling rates in the atmosphere.

,*J. Atmos. Sci.***60**, 2869–2886, doi:10.1175/1520-0469(2003)060<2869:AFAMOC>2.0.CO;2.Veyre, P., , G. Sommeria, , and Y. Fouquart, 1980: Modélisation de l’effet des hétérogénéités du champ radiatif infra-rouge sur la dynamique des nuages.

,*J. Rech. Atmos.***14**, 89–108.Viskanta, R., , R. W. Bergstrom, , and R. O. Johnson, 1977: Radiative transfer in a polluted urban planetary boundary layer.

,*J. Atmos. Sci.***34**, 1091–1103, doi:10.1175/1520-0469(1977)034<1091:RTIAPU>2.0.CO;2.Yamamoto, G., 1952: On the radiation chart.

,*Sci. Rep. Tohoku Univ. Ser. 5***4**, 9–23.Zdunkowski, W. G., , and F. G. Johnson, 1965: Infrared flux divergence calculations with newly constructed radiation tables.

,*J. Appl. Meteor.***4**, 371–377, doi:10.1175/1520-0450(1965)004<0371:IFDCWN>2.0.CO;2.Zhang, X., , L. Musson-Genon, , B. Carissimo, , M. Milliez, , and E. Dupont, 2014: On the influence of a simple microphysics parametrization on radiation fog modelling: A case study during ParisFog.

,*Bound.-Layer Meteor.***151**, 293–315, doi:10.1007/s10546-013-9894-y.