## 1. Introduction

In recent years, atmospheric aerosols have been considered important for climate change because their direct forcing and indirect forcing are appreciable to alter the radiative balance. Among all aerosols, sulfate aerosols have been paid most attention, since sulfate aerosols over land are mostly an anthropogenic product and since the optical properties and size distributions of sulfate aerosols could be reasonably well known.

Most of studies on atmospheric aerosol are focused on solar radiation (Chuang et al. 1994; Mitchell et al. 1995; Nemesure et al. 1995; Liousse et al. 1996; Feichter et al. 1997; Haywood et al. 1997; Reader and Boer 1998; Grant et al. 1999; Kiehl et al. 2000; and others). Atmospheric aerosols prevent some solar photons from reaching the earth's surface. The direct forcing at the surface is therefore negative. This negative forcing could partly compensate the positive forcing caused by greenhouse gases. However, in comparison with the number sulfate aerosol solar forcing studies, sulfate aerosol infrared forcing has been paid less attention.

To perform radiative transfer calculations that account for aerosol effects, the related aerosol optical properties are required. From observations, all sulfate aerosol particles grow in size as relative humidity increases (Tang and Munkelwitz 1994; Tang 1996). Therefore, like the parameterization of the aerosol optical properties for solar radiation, the parameterization for aerosol infrared optical properties is also a function of relative humidity.

To understand the mechanics for the aerosol infrared forcing, we will implement the proposed parameterization scheme in a column radiative transfer model to illustrate what the magnitude is for aerosol infrared forcing.

Li et al. (2001, hereafter Li01) proposed a parameterization of sulfate aerosol optical properties for solar radiation, which could be easily implemented within climate models. This work completes Li01 by extending the parameterization to the infrared.

## 2. Parameterization of absorptance coefficient

It is known that, in the troposphere, the dominant sulfate compounds are ammonium sulfate [(NH_{4})_{2}SO_{4}], ammonium bisulfate (NH_{4}HSO_{4}), and sulfuric acid (H_{2}SO_{4}) (Twomey 1971; Nemesure et al. 1995; and others). Recent observations show that sulfate aerosols can mix with organic species. This makes the process for calculation of sulfate aerosol radiative forcing much more complicated. First, the refractive indexes for various organic species are not currently available; second, the proportions of organic species and extent of their mixing is not clear; third, as opposed to nonorganic species, organic species can change easily by chemical reaction. We must admit at the present stage we do not have the ability to properly handle the optical properties of sulfate aerosols mixing with organic species. Therefore, in the works of Kiehl et al. (2000) and others, pure sulfate aerosol without organic species mixture is assumed.

*η*is defined as the ratio of the aerosol particle radius

*r*at a specified

*r*

_{d}:

In our calculations, the aerosol growth is obtained using the traditional Köhler curve with recent development in equilibrium saturation theory considered (see Li01).

*r*is the radius of the aerosol particle,

*N*

_{0}is the total number density,

*r*

_{0}is the geometric mean radius (for the mode), and

*σ*is the geometric standard deviation. The effective radius and effective variance for the lognormal distribution are easily obtained as

*r*

_{e}=

*r*

_{0}exp[2.5(ln

*σ*)

^{2}] and

*υ*

_{e}= exp[(ln

*σ*)

^{2}] − 1.

*n*(

*r*) is related to the dry size distribution

*n*(

*r*

_{d}) in the following way:This prescription constrains the conservation of particle number in the growth process. The number of dry particles in the interval from

*r*

_{d}to

*r*

_{d}+

*dr*

_{d}is required to equal the number of wet particles in the interval from

*r*to

*r*+

*dr*. That is,

*n*(

*r*

_{d})

*dr*

_{d}=

*n*(

*r*)

*dr*. As the particles grow, the wet size distribution

*n*(

*r*) will shift toward larger radius size, and the wet size distribution may be distorted from the lognormal form. However, based on (3), the manner in which the wet size distribution is related to the dry size distribution, an average of a physical quantity

*F*(

*r*) weighted by a wet size distribution can be obtained from the dry distribution:

In most climate models, usually only the size distribution for dry aerosol is specified. Because of growth, wet optical properties will be significantly dependent on the wet aerosol distribution, especially for high relative humidities. But with the relation shown in (4) the wet optical properties are analytically related to the dry size distribution. The exact size distribution weighted physical quantities can be obtained without explicit determination of the wet aerosol size distribution.

*μ*= cos

*θ*,

*θ*is the zenith angle,

*τ*is the optical depth,

*ω̃*

*B*(

*T*) is the blackbody emission at temperature

*T*.

*κ*=

*τ*(1 −

*ω*) Eq. (5) is simplified asLike optical depth, we also can separate out the wet aerosol content (WAC) from the absorptance depth:

*κ*= WAC

*ξ*, where we call

*ξ*the specific absorptance, and

*ξ*

*ψ*

*ω*

*ψ*is the specific extinction.

*κ*

*ξ*

*ρ*

_{d}is the mass density of the dry aerosols, andwhere

*ρ*is the mass density of wet particles, a function of

*λ*iswhere

*Q*

_{ext}is the extinction efficiency and

*Q*

_{sca}is the scattering efficiency.

For our calculations, we use the refractive indexes of water and sulfate the same as Li01.

Measurements show that the aerosol distributions usually have an effective radius in the range from 0.1 to 1 *μ*m (Lacis and Mishchenko 1995) in a lognormal distribution. As in the parameterization for solar radiation, three effective radii are considered: 0.166, 0.5, and 1 *μ*m. Note that these values of effective radius correspond to dry aerosol sizes. To obtain the optical properties for other values of effective radius, an appropriate interpolation technique can be employed. An effective variance of 0.693 is used in all calculations for the dry aerosol distribution (Chýlek and Wong 1995; Li01).

_{4})

_{2}SO

_{4}, the parameterization is valid for 0.35 <

_{4}HSO

_{4}and H

_{2}SO

_{4}is valid for 0.05 <

_{2}gas that will react to produce sulfate aerosols and the dry loading of the sulfate aerosol, we can use

*i*is defined aswhere

*B*

_{λ}(

*T*) is the Planck function,

*T*is the temperature, and Δ

*λ*

_{i}is the spectral interval for band

*i*. Equation (13) is physically similar to the Chandraskhar mean (Liou 1992).

^{−1}. In the range of 0–340 cm

^{−1}, the refractive index for sulfate aerosol is not currently available. Also, the fractional weight of the Planck function in this range is very small. The specific absorptance for each band (

*i*) is parameterized in the following way:The parameterization coefficients

*a*

^{i}

_{1}

*a*

^{i}

_{2}

*a*

^{i}

_{3}

In Fig. 1, the specific absorption for the eight-band scheme of sulfuric acid is shown with the effective radii of 0.166, 0.5, and 1.0 *μ*m, respectively. These results show that the parameterizations of (14) are able to provide accurate values for the sulfate aerosol specific absorptance.

In Fig. 1, opposite to the solar case shown in Li01, in which the specific extinction shows completely different behavior in response to changes in

For different bands, the behaviors in response to changes in

The absorptance depth is a product of WAC and the specific absorptance. By comparing the behavior of WAC (Fig. 2 in Li01) and the behavior of specific absorptance with the change of

*λ*represents the wavenumber. Any band scheme can be obtained through the sum of the results for individual wavelengths with respect to the Planck function, as in (15).

The results of the specific absorptance for 32 individual wavenumbers are listed in Table 3 covering the range of 400–2500 cm^{−1}. The distribution of the chosen wavenumber is not even. We consider more cases near the window region, since higher radiative energy is weighted for the window region at the atmospheric temperature conditions. Since the specific absorptance is not sensitive to the effective radius, only the results of *r*_{e} = 0.5 *μ*m are presented.

## 3. Sulfate aerosol surface infrared radiative forcing

For the upward flux in the infrared, the aerosols act like greenhouse gases to prevent radiation loss directly to space. However, this effect is very small (see later discussion). Aerosols can reduce the downward flux and also can enhance the emission to the surface. The aerosol direct forcing for the downward flux is determined by these two opposing factors.

We investigated the aerosol infrared forcing using a one-dimensional radiative transfer model instead of using general circulation models (GCMs), since it is much easier for us to understand the crucial factors determining aerosol infrared direct forcing.

We used the radiation model for CCC GCM in the Canadian Center for Climate Modeling and Analysis model, which is a model using a correlated-*k* distribution for gaseous transmission. The continuum scheme is based on numerical calculations from LBLRTM (Mlawer et al. 1997). For cloud and aerosol optical properties, there are nine bands for the infrared, covering from 0–2500 cm^{−1}.

Usually it is believed that aerosols are well mixed within boundary layer (usually 1 ∼ 2 km) due to turbulence. Above the boundary layer, the aerosol concentration decays exponentially. In Fig. 2 the surface forcing via aerosol loading is shown. For simplicity, aerosol is assumed to be homogeneously loaded within 2 km above the surface. Two atmospheric profiles of middle latitude summer (MLS; upper panels) and subarctic winter (SAW; lower panels) are considered. The global averaged loading of ^{− −}_{4}^{−2} (Feichter et al. 1995; Kiehl et al. 2000). If the sulfate aerosol is specified to be H_{2}SO_{4}, the loading will exhibit almost no change, since the molecular weights for ^{− −}_{4}_{2}SO_{4} are close. If the sulfate aerosol is specified to be (NH_{4})_{2}SO_{4}, the loading will increase about 30%. And if the sulfate aerosol is specified to be NH_{4}HSO_{4}, the loading will increase about 15%.

For a sulfate aerosol loading of 2.5 mg m^{−2}, the surface forcing is about 0.06 ∼ 0.1 W m^{−2} for (NH_{4})_{2}SO_{4}. The forcing for H_{2}SO_{4} is higher. We present the corresponding surface solar forcing in Fig. 3 with the solar zenith angle set to the diurnal mean of 50.6°. We find that unlike aerosol surface infrared forcing, the aerosol surface solar forcing is very sensitive to the aerosol size. Aerosols with a small effective radius can produce a much larger forcing than aerosols with a larger effective radius.

Comparing Fig. 2 and Fig. 3, we see the surface infrared forcing could be about 6% ∼ 10% of the surface solar forcing for (NH_{4})_{2}SO_{4}, and the surface infrared forcing could be about 12% of the surface solar forcing for H_{2}SO_{4}. These are relatively small. However, the aerosol solar forcing only exists in the daytime, but the aerosol infrared forcing exists all day. As a result, the aerosol infrared forcing has to be doubled when comparing it with the aerosol solar forcing. Therefore, the aerosol surface infrared forcing can actually cancel about 12%–24% surface solar forcing.

The aerosol infrared surface forcing for tropical profile is not shown in Fig. 2. By using a tropical sounding the results are similar to MLS except about 10% lower in magnitude.

As mentioned before, the shade of cloud over aerosols will reduce the aerosol surface solar forcing, since the downward solar flux could be dramatically reduced, thus the difference in flux caused by existing aerosols will also be reduced. However, the influence of cloud to aerosol surface infrared forcing could be smaller than that of solar, since the downward infrared flux contributed from the emission below the cloud will not change. Figure 4 shows the aerosol surface forcing change with a cloud located between 4 and 5 km. The aerosol conditions are the same as those in Figs. 2 and 3. In comparison with Figs. 2 and 3, the surface infrared forcing is about 30% of the results without clouds, while the surface solar forcing is only about 2% of the results without clouds. Therefore, the existence of cloud could enhance the ratio of aerosol surface infrared forcing to the aerosol surface solar forcing.

*N*is the surface, and layer

*i*is between level

*i*and level

*i*+ 1. For simplicity, we use isothermal source, that is, constant Planck function

*B*

_{i}determined by the temperature at the middle of the layer

*i*. The solution for

*i*th layer iswhere

*F*

^{+}

_{i}

*F*

^{−}

_{i}

*i*, 1/

*μ*

_{1}=

*e*

^{1/2}= 1.648 721 3 (Li 2000) is the diffusivity factor, and

*B̃*

_{i}=

*π*

*B*

_{i}. Note that in the column radiative transfer model we use a nonisothermal source function. The isothermal source of a constant Planck function presented here is only for the purposes of illustrating the related physics for aerosol forcing.

*i*th layer is Δ

*κ*

_{i}, the changes in upward flux at level

*i*and downward flux at level

*i*+ 1 areFigure 2 shows that the aerosol surface infrared forcing is always linearly proportional to the aerosol loading, which supports the perturbation theory. The downward flux is very small at the high altitude and it gradually increases with a decrease in height. Therefore, generally,

*F*

^{−}

_{i}

*B̃*

_{i}< 0 and Δ

*F*

^{−}

_{i+1}

*F*

^{+}

_{i+1}

*B̃*

_{i}> 0 and the forcing is negative. However, if temperature inversion occurs in the lower atmosphere, the incoming flux

*F*

^{+}

_{i+1}

*B̃*

_{i}and forcing could be positive.

In Fig. 5 the changes of upward and downward flux due to the existence of aerosols are shown. The cases are the same as Fig. 2 with aerosol loading 2.5 mg m^{−2}. We find that the model results are consistent with the above analysis. Since the temperature profile is slightly inverted below 2 km in SAW, the forcing for the upward flux is slightly positive.

*i*layer produce forcing Δ

*F*

^{+}

_{i}

*i*, the flux at level

*i*− 1 isTherefore, the forcing of upward flux at level

*i*− 1 becomes Δ

*F*

^{+}

_{i}

*e*

^{−κi−1/μ1}

*F*

^{+}

_{i}

*e*

^{−(κi−1+κi−2+···+κ1)/μ1}

The aerosol surface infrared forcing is also determined by the location of the aerosol. In Fig. 6, the aerosol loading is set to a constant 2.5 mg m^{−2}, but the vertical stretch of the distribution varies from 0.25 to 5 km. The effective radius is 0.5 *μ*m for all types of sulfate aerosol. It was found that for the same amount of aerosol loading, the surface forcing is larger when the aerosol is more concentrated near the surface. However, we must ask whether this is generally true and what are the relevant physics for this.

*N*− 1:

*F*

^{−}

_{N}

*F*

^{−}

_{N−1}

*e*

^{−κN−1/μ1}

*B̃*

_{N−1}

*e*

^{−κN−1/μ1}

*κ*

_{i}is the perturbation of absorptance depth by aerosol for layer

*i*.

*N*− 1 isand the surface forcing isSince the total loading for the two cases is the same, Δ

*κ*

_{N−1}(=Δ

*κ*

_{N−2}) in (22) is equal to 0.5Δ

*κ*

_{N−1}in (20), we denote it as 0.5Δ

*κ*. Thus,

Since Δ*κ* is very small, the first term dominates. As we mentioned, generally, *F*^{−}_{i}*B̃*_{i} < 0. Therefore, Δ*F*^{(1)} − Δ*F*^{(2)} is positive. The results of (23) hold true for the multilayer case. Therefore, the aerosol surface forcing is larger for the case with more aerosols located near the surface. Of course, in the particular case with very strong temperature inversion, which leads *F*^{−}_{i}*B̃*_{i} > 0, the results could be opposite.

In summary, a simple parameterization of the sulfate aerosol infrared optical properties as a function of relative humidity is presented in this note. It is found that sulfate aerosol absorptance is sensitive to relative humidity but not sensitive to particle size distribution. It is found that the aerosol surface infrared forcing can cancel about 12%–24% aerosol surface solar forcing in clear sky conditions. Also in this note the mechanics for aerosol infrared forcing is discussed.

We are grateful to Drs. J. Fyfe and L. Harrison, and two anonymous reviewers for their helpful comments. Part of this research was supported by the Office of Science, Biological and Environmental Research Program (BER), U.S. Department of Energy, through the Northeast Regional Center of the National Institute for Global Environmental Change (NIGEC) under Cooperative Agreement No. DE-FC03-90ER61010.

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Coefficients in (12) for the ratio of wet to dry aerosol content

Coefficients in Eq. (14) for H_{2}SO_{4}, (NH_{4})_{2})SO_{4}4, and NH_{4}HSO_{4} for eight-band scheme; α_{1}, α_{2}, and α_{3} units: m^{2} g^{−1}

Coefficients in Eq. (15) for H_{2}SO_{4}, (NH_{4})_{2}SO_{4}, and NH_{4}HSO_{4} with *r*_{e} = 0.5 μm; α_{1}, α_{2}, and α_{3} units: m^{2}g^{−1}.