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  • View in gallery

    Water vapor content profile from IOP 13 at 1200 UTC.

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    Simulated domain with three different domain heights—1, 2, and 11 km—and periodic boundary conditions in x and y directions, .

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    Heating rate (K day−1) for three different mesh heights: , water vapor only, and the semianalytic radiative source term.

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    Temperature profile at 1800 UTC.

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    Heating rate (K day−1), semianalytic radiative source term, atmospheric profiles from IOP 13 data at 1800 UTC; , , , and .

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    Heating rate (K day−1), semianalytic radiative source term, atmospheric profiles from IOP 13 data at 0300 UTC; liquid water and , , , and .

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    The 2D mesh extruded to obtain the 3D mesh with idealized buildings.

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    Clear-sky conditions at 2100 UTC: incident flux on the walls and the ground and clip planes in the x and y directions, showing liquid water contents equal to zero.

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    Foggy conditions at 2310 UTC; clip plane at z = 8 m of liquid water content.

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    Foggy conditions at 2310 UTC: incident flux on the walls and the ground and clip planes in the x and y directions, showing liquid water contents.

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    Temperature profiles simulated by Code_Saturne used to perform radiative transfer computation.

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A New Method for Fast Computation of Three-Dimensional Atmospheric Infrared Radiative Transfer in a Nonscattering Medium, with an Application to Dynamical Simulation of Radiation Fog in a Built Environment

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  • 1 CEREA, Marne la Vallée, France
  • 2 CEREA, Marne la Vallée, and EDF Research and Development, Chatou, France
  • 3 EDF Research and Development, Chatou, France
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Abstract

The atmospheric radiation field has seen the development of more accurate and faster methods to take into account absorption. Modeling fog formation, where infrared radiation is involved, requires accurate methods to compute cooling rates. Radiative fog appears under clear-sky conditions owing to a significant cooling during the night where absorption and emission are the dominant processes. Thanks to high-performance computing, high-resolution multispectral approaches to solving the radiative transfer equation are often used. Nevertheless, the coupling of three-dimensional radiative transfer with fluid dynamics is very computationally expensive. Radiation increases the computation time by around 50% over the pure computational fluid dynamics simulation. To reduce the time spent in radiation calculations, a new method using analytical absorption functions fitted by Sasamori on Yamamoto’s radiation chart has been developed to compute an equivalent absorption coefficient (spectrally integrated). Only one solution of the radiative transfer equation is needed against Nband × Ngauss for an Nband model with Ngauss quadrature points on each band. A comparison with simulation data has been done and the new parameterization of radiative properties proposed in this article shows the ability to handle variations of gas concentrations and liquid water.

Denotes Open Access content.

Current affiliation: ARIA Technologies, Boulougne-Billancourt, France.

Corresponding author address: Laurent Makké, ARIA Technologies, 8 rue de la ferme, Boulougne-Billancourt 92100, France. E-mail: lmakke@aria.fr

Abstract

The atmospheric radiation field has seen the development of more accurate and faster methods to take into account absorption. Modeling fog formation, where infrared radiation is involved, requires accurate methods to compute cooling rates. Radiative fog appears under clear-sky conditions owing to a significant cooling during the night where absorption and emission are the dominant processes. Thanks to high-performance computing, high-resolution multispectral approaches to solving the radiative transfer equation are often used. Nevertheless, the coupling of three-dimensional radiative transfer with fluid dynamics is very computationally expensive. Radiation increases the computation time by around 50% over the pure computational fluid dynamics simulation. To reduce the time spent in radiation calculations, a new method using analytical absorption functions fitted by Sasamori on Yamamoto’s radiation chart has been developed to compute an equivalent absorption coefficient (spectrally integrated). Only one solution of the radiative transfer equation is needed against Nband × Ngauss for an Nband model with Ngauss quadrature points on each band. A comparison with simulation data has been done and the new parameterization of radiative properties proposed in this article shows the ability to handle variations of gas concentrations and liquid water.

Denotes Open Access content.

Current affiliation: ARIA Technologies, Boulougne-Billancourt, France.

Corresponding author address: Laurent Makké, ARIA Technologies, 8 rue de la ferme, Boulougne-Billancourt 92100, France. E-mail: lmakke@aria.fr

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 4π. 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:

  1. 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.
  2. 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.
  3. 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).
  4. 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

For a nonscattering medium, the RTE may be written as follows:
e2.1
where S is the propagation direction of the radiance , is the absorption coefficient (m−1) for the λ wavelength, and the source term is given by the Planck function , where h is the Planck constant, is the Boltzmann constant, c is the velocity of light in vacuum, and T is temperature.
Under the plane-parallel hypothesis, (2.1) becomes
e2.2
where 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:
eq1
eq2
to obtain from (2.2) the system
e2.3
e2.4
By integrating these equations over a solid angle and with respect to z, expressions for the fluxes are obtained:
e2.5
e2.6
where is the monochromatic diffuse transmittance,
e2.7
and the monochromatic transmittance
e2.8

b. The broadband emissivity

Radiative quantities should now be integrated over the wavelength to compute the net flux and the heating rate. Most of radiation applications in the atmospheric community are performed using the correlated-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 is defined as follows:
e2.9
For a nonhomogeneous path, it is common to introduce the corrected pathlength of water vapor, carbon dioxide, and ozone:
eq3
where is a scaling function eliminating the dependence of the absorption coefficient on pressure and temperature. These corrections, known as the one-parameter scaling approximation, are attributed to Chou and Arking (1980). The emissivity for each gas is now a function of the scaled path . The total emissivity for the three dominant atmospheric gases in the infrared domain is
e2.10
where and are the transmissivity of water vapor in the 15-μm band and rotation band, respectively, and is the emissivity of water vapor dimers (H2O)2 (the self-broadened continuum).
The problem of gas overlapping has been treated by using integrated emissivity and transmissivity functions. The expression for , , , and are available in appendix A. Fluxes, integrated over the spectrum, may be written as follows (Stephens 1984):
e2.11
e2.12
For a nonblack wall, is replaced by —where is the ground emissivity—which leads to
e2.13
e2.14
Actually, the reflected term at the ground, , is subject to controversy. Edwards (2009) and Ponnulakshmi et al. (2009) pointed out a spurious near-surface cooling that results from an erroneous reflected flux. Physical considerations about the spectrum of downward flux prove that the reflected flux should not be attenuated by —because what has not been absorbing along z to 0 remains transparent along 0 to z after reflection—and should be rewritten as in Ponnulakshmi et al. (2012):
e2.15
The term can be seen as a downward radiation along and . As we work on idealized cases, we will consider the ground as a blackbody; that is, for all simulations.
By assuming an isothermal atmosphere with a temperature discontinuity between the ground and the air, the integral terms vanish (Lenoble 1993):
e2.16
e2.17
This analytical solution is exact for an isothermal plane-parallel atmosphere.

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

By replacing (2.16) and (2.17) for an isothermal atmosphere in the “flux form” of the RTE, the emissivities are linked to equivalent absorption coefficients that appear in the gray RTE. To formulate the upward and downward equivalent extinction, let us consider the gray RTE in a nonscattering medium:
e3.1
where x is the position in space and with
eq4
are the radiative quantities spectrally integrated.
Under the hypothesis of a plane-parallel atmosphere, we can write the equations governing the upward and downward radiances [(2.3) and (2.4), respectively] as
e3.2
e3.3
To obtain the equations for fluxes, an integration over the zenith angle must be completed. A widely used approximation is to eliminate the zenith angle dependence in favor of a constant inclination so that , well known as the “diffusivity factor” (Elsasser 1942). This is equivalent to taking a zenith angle equal to zero but an optical path expanded by a factor of . Relying on hemispheric isotropy, (3.2) and (3.3), written in z coordinates, become
e3.4
e3.5
respectively. By using the analytical expressions of fluxes for an isothermal atmosphere, (2.16) and (2.17), the above lead to
e3.6
e3.7
respectively. 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 and . This formulation brings four remarks:
  1. 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:
    eq5
  2. 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 between and z. This gives upward and downward coefficients their nonlocal nature.
  3. 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:
    e3.8
    Numerical problems may appear and we may have negative extinction. Separating upward and downward direction provides the positivity of the equivalent absorption coefficients.
  4. 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 quadrature (80 directions) provided by Carlson and Lathrop (1965).

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 . The vertical profile of water vapor taken from the ParisFog dataset (IOP at 1200 UTC 18 February 2007) is given on Fig. 1.

Fig. 1.
Fig. 1.

Water vapor content profile from IOP 13 at 1200 UTC.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

1) Heating/cooling rate

As a source term of the energy balance equation, the heating rate
e4.1
must be computed. A natural approach consists in first computing the radiative flux density and then its divergence. Another formulation of the heating rate can be found by integrating the RTE (3.1) over the solid angle:
e4.2
The permutation between the integral and the divergence operator gives the semianalytic form of the radiative source term:
e4.3
The spatial and angular discretization of (4.3) leads to the following expression:
e4.4
where the subscript 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, , where (see Fig. 2) is computed once at the beginning of the simulation. Up to , the 1D radiation code assumes an isothermal layer so that , where is the temperature of isothermal layer between and . If is less than 11 km, is computed thanks to (2.12). The upper boundary condition on the radiance for the 3D code is provided by the 1D broadband emissivity scheme.

Fig. 2.
Fig. 2.

Simulated domain with three different domain heights—1, 2, and 11 km—and periodic boundary conditions in x and y directions, .

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

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 in (2.16) leads to . The values obtained for upward fluxes in Code_Saturne with the 1D and 3D code are consistent with its analytical expression in an isothermal atmosphere with no temperature discontinuity between the ground and the air. Although we imposed the analytical downward flux at the upper boundary, we have small differences at the top of the domain (1- and 2-km heights).

Fig. 3.
Fig. 3.

Heating rate (K day−1) for three different mesh heights: , water vapor only, and the semianalytic radiative source term.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

Differences close to the ground come from the diffuse approximation used to formulate gray absorption coefficient. Thus, it means that the optical paths for each discrete direction are equal ( is a constant), which is generally not true. Ponnulakshmi et al. (2012) proposed a formulation of directional emissivity. Here is the formulation of directional equivalent absorption coefficients for the discrete ordinates method:
e4.5
e4.6
where is defined with an optical path divided by μ:
e4.7
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.

Fig. 4.
Fig. 4.

Temperature profile at 1800 UTC.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

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).

Fig. 5.
Fig. 5.

Heating rate (K day−1), semianalytic radiative source term, atmospheric profiles from IOP 13 data at 1800 UTC; , , , and .

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

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.

The absorption in the cloud may be parameterized using the transmission through a layer containing gases and liquid water and is nothing else than the product of the gases’ transmission and the liquid water transmission. Fu and Liou (1992) have shown that this treatment of carbon dioxide and water vapor overlapping does not lead to significant errors:
e4.8
The equivalent absorption coefficients keep the formulation (3.6) and (3.7). If the absorption coefficients are assumed horizontally constant, the distribution of liquid water is not. The problem is that the formulation (4.8) implies calculations for each atmospheric column containing liquid water. It is more convenient to separate the absorption caused by cloud droplets and gases. This leads us to consider an equivalent absorption coefficient equal to the sum of gaseous equivalent absorption coefficient and liquid water absorption coefficient:
e4.9
where is the absorption coefficient for liquid water and is the volume fraction of liquid water in the cell. The absorption cross section of cloud droplets is commonly assumed constant in infrared domain and computed according to Stephens (1984).

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.

Fig. 6.
Fig. 6.

Heating rate (K day−1), semianalytic radiative source term, atmospheric profiles from IOP 13 data at 0300 UTC; liquid water and , , , and .

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

The shape of the cooling rate and its highest value—around 200 K day−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:
e4.10
e4.11
where the subscripts 3D and 1D refer to the radiative solvers of Code_Saturne described previously.
Table 1.

Mean relative error value of incident flux on the surface and mean relative error of cooling rate between 1D and 3D radiation codes.

Table 1.

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 . Another reason may be the mesh used in the radiation fog code. It has to be very fine near the ground and coarser toward the top of the domain. Reducing the domain height leads to keep the finest cells of mesh and may limit the false scattering induced by the upwind scheme, used for the spatial integration (Ferziger and Perić 2002, chapter 4). A different scheme to compute radiances on faces should be used like the one used in SHDOM (Evans 1998). The radiance on an internal face may be interpolated with the radiance in the entering cell and the radiance in the exiting cell.

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 , such as gas concentrations, temperature, and pressure, are horizontally averaged, so that the absorption coefficient only depend on 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:

  1. radiation between the ground and the walls of the buildings and
  2. the emitted radiation by liquid water, received by the buildings.
In Fig. 8, we can observe that the incident flux on the sidewalls is greater than the rooftops. It is the first 3D radiative effect we enhance in this case. The difference results from the additional flux incoming from the ground that the rooftops do not receive. Under clear-sky conditions, the 1D model will yield horizontally homogeneous fluxes, even if computations are performed for multiple columns. Thanks to nonvertical discrete directions used by the DOM, the new parameterization takes into account the radiative exchanges between the ground and the sidewalls. Note that the nonhomogeneity of the incident flux on the roof is due to an artifact of the contouring algorithm.
Fig. 7.
Fig. 7.

The 2D mesh extruded to obtain the 3D mesh with idealized buildings.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

Fig. 8.
Fig. 8.

Clear-sky conditions at 2100 UTC: incident flux on the walls and the ground and clip planes in the x and y directions, showing liquid water contents equal to zero.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

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.

Fig. 9.
Fig. 9.

Foggy conditions at 2310 UTC; clip plane at z = 8 m of liquid water content.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

Fig. 10.
Fig. 10.

Foggy conditions at 2310 UTC: incident flux on the walls and the ground and clip planes in the x and y directions, showing liquid water contents.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

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).

Acknowledgments

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

The formulas for water vapor, carbon dioxide, ozone, and water vapor dimer are as follows:
eq6
where and with STP, , and c chosen as (Green 1964).

b. Emissivity functions

Sasamori’s (1968) formulas for water vapor, carbon dioxide, and ozone are as follows:
eq7
eq8
eq9
eq10
Veyre et al.’s (1980) formula for water vapor dimer is
eq11
where , , , , , , and , and
eq12
where , , , , , , , , , , and .

APPENDIX B

Simulated Temperature Profiles

Figure B1 features temperature profiles simulated during the fog formation.

Fig. B1.
Fig. B1.

Temperature profiles simulated by Code_Saturne used to perform radiative transfer computation.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-15-0012.1

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