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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 767231, 9 pages
Research Article

Study on the Influence of Aerosol Radiation Balance in One-Dimensional Atmospheric Medium Using -Approximation Method

1School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
2School of Civil Engineering and Architecture, Northeast Petroleum University, Daqing 163318, China

Received 17 May 2014; Revised 10 July 2014; Accepted 20 July 2014; Published 27 August 2014

Academic Editor: Hong-Liang Yi

Copyright © 2014 Yuan Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A numerical study, used for the influence of aerosol on the atmospheric radiation transfer, is conducted in this paper. Based on the established atmospheric radiation transfer model, we calculated the effect of species, concentration, and height distribution of aerosols on atmosphere radiation. The calculation results show that the aerosol particles affect the atmospheric radiation balance greatly and thus are the important component of the radiation balance of the earth-troposphere system.

1. Introduction

Atmospheric aerosols are suspensions of small solid or liquid particles with the diameters between 0.001 and 10 μm. Aerosol affects the Earth’s radiation budget by direct and indirect effects and plays an important role in global climate systems. The direct effect is that aerosol particles scatter and absorb the solar radiation and surface infrared radiation [1]; the indirect effect is that aerosol particles act as cloud condensation nuclei and ice nuclei [2, 3]. When the climate system is in balance, its absorbing solar radiation energy is accurately equal to the infrared radiation energy that the earth and atmosphere emit into outer space. Any factor that disturbs this balance and then changes the climate is the radiative forcing factor. Its forcing on the earth-troposphere system is the radiative forcing [4, 5]. Radiative forcing (in W · m−2) is defined as the change in the net radiative flux at the tropopause due to some factors change. Aerosol is a very important correction factor in the atmospheric radiation transfer. So it is significant to study the influence of aerosols particles on the atmospheric transfer.

Many numerical methods have been developed to solve the problem of radiative heat transfer, such as the discrete ordinates method (DOM) [6, 7], finite volume method (FVM) [8], finite element method (FEM), Monte Carlo method (MCM) [9], and spherical harmonics method (SHM). The spherical harmonics method, also known as the -approximation method, was originally proposed by Jeans [10] for interstellar radiative transfer and further developed by Davison [11] and Kourganoff [12] for neutron transport. Khouaja et al. [13] developed the -approximation method for radiative heat transfer. Recently, Larsen et al. [14] used a double- approximation for the transfer equation in neutron transport, but only to low order in the expansion. McClarren et al. [15] analyzed numerical oscillations of the -approximation method and also suggested improvements based on wavelet analysis.

We established atmospheric radiative transfer calculations model based on the spherical harmonics method in one-dimensional media and analyzed the influence of aerosols on the atmospheric radiative transfer. The calculations from the species, number, and distribution height of aerosols were achieved and the results were given in Figure 1.

Figure 1: Radiative heat transfer model diagram of one-dimensional atmospheric layer.

2. Numerical Model

The radiative transfer equation for one-dimensional absorbing, emitting, and scattering gray medium can be expressed as where is the optical thickness given by , is the scattering albedo with , is the medium thickness, and and are the absorbing and scattering coefficients, respectively. The radiative intensity and scattering phase function can be expanded in terms of associated Legendre polynomials . Using recurrence relations between these polynomials, (1) becomes where is an asymmetry parameter. Multiplying (2) by and integrating over , we obtain using the orthogonality properties of the Legendre polynomials differential equations for coefficients that comprise the   -approximation equations. Consider the following: For one-dimensional opaque diffuse emitting and reflecting interfaces, odd terms in the Legendre polynomials form an orthogonal set. The boundary conditions are given by where is the emissivity. The boundary conditions imposed on the spherical harmonics functions are given by The boundary conditions [16] are applicable in the one-dimensional case: where is the circumstance radiation, and are reflectivities of the two sides of the boundary surfaces, respectively, when considering total reflection, and the values of these depend on the medium refractive index .

Separating the large refractive index from small, the diffuse reflectance at the boundary surfaces is given by Siegel and Spuckler [17]: In contrast, separating small refractive index from large, the diffuse reflectance at the boundary surfaces is given by where , . In treating a semitransparent interface, we can use opaque interface treatment as reference.

Considering one-dimensional unsteady coupled radiative and conductive heat transfer, the general form of the energy equation of the internal radiation and the transient coupled conductive heat transfer is defined as

Considering unsteady state, no internal heat source and coupled radiative and conductive heat transfer, the energy equation (9) can be discretized as follows:

In order to make the problem have universal significance, the dimensionless form of the energy equation (10) will be evaluated. Introducing the following definitions: reference temperature , the dimensionless temperature , unit heat capacity , the dimensionless time , , the dimensionless length , the radiative and conductive parameter , the dimensionless convective heat transfer coefficient , the dimensionless radiative heat flux , (10) becomes

According to [18], in considering one-dimensional nonsteady coupled radiative and conductive heat transfers, the thermal radiation source term is found to be Here, the incident radiative power can be transformed to . According to the properties of Legendre polynomials and , the above equation is rearranged into the form:

Meanwhile, considering the physical properties of the inhomogeneous medium, we next develop a -approximation method for a multilayer medium. We redetermine the boundary conditions of each layer. The modified boundary conditions are Here, and are the emission radiation intensities of two adjacent layers, and the initial values are obtained from the temperature field. The total emission radiation intensity is determined by an iterative calculation of intensities from successive layers, the last providing the required total. The convergence condition of the multilayer model calculation is decided by the radiative intensity . After the intensity calculation has converged, the iterative calculation of the temperature field is then performed.

3. Results and Discussion

According to statistics, 98.8% of the total atmospheric mass distributes in the atmosphere within 30 km above the ground, and with the height increasing to 50 km the proportion reaches 99.92%. Most of the atmospheric radiation (involving solar radiation, absorption, and scattering of the ground infrared radiation and the emission of the atmosphere itself) occurs within certain atmospheric thickness. Compared with the earth radius (6378 km), the atmospheric thickness can be regarded as one-dimensional radiative media. Therefore, we treated the atmosphere within 50 km above the ground as a multilayer one-dimensional media distributing along the radial of the earth. The upper surface of the medium layer is near to space and is considered as a semitransparent medium. The lower interface is close to ground and is considered as opaque interface. The emissivity and radiative temperature related to the ground are given. For the inner of the medium layer we mainly consider the gas absorption and emission of the radiation energy and the aerosol particles scattering and absorption of the radiation energy and consider the atmospheric coupled radiation and conduction in the vertical direction.

The atmospheric convection is very important for the energy transfer of the atmosphere. But we mainly discuss the influence of aerosols on the atmospheric radiative energy and investigate the atmospheric radiative balance and the variation in radiation characteristic, which is not affected by the atmospheric convection that acts only as the background. Therefore, we ignore the influence of the atmospheric convection.

The atmospheric radiation physical model is shown in Figure 1 and the atmospheric radiation transfer model is shown in Figure 2. The atmospheric internal heat source is set as .

Figure 2: Physical model of one plane-parallel slab divided into sublayers.

The standard atmospheric condition is given in Tables 14.

Table 1: The main input parameters of the atmospheric radiation calculation.
Table 2: Multilayer atmospheric radiative transfer model.
Table 3: Spectral bands for the atmospheric radiative transfer model and radiative characteristic parameters under the standard dust aerosol in the height of 0-1 km.
Table 4: Complex refractive index of different wavelength for different aerosol types [19].

Table 1 gives the main input parameters for the calculation of atmospheric radiation transfer.

Table 2 gives the parameters associated with layers divided along the vertical height. The density of each layer is determined by the calculation of the pressure. We consider (the standard atmosphere pressure is 1.225 kg/m2) with the temperature fluctuating in small range. Therefore, the equation is as follows:

Table 3 gives the wavelength range of each spectral band. The atmosphere optical thicknesses of the layers at different spectral bands are obtained from the HITRAN database under the standard atmospheric condition of America in 1976. The aerosol optical thicknesses are decided based on the Mie scattering theory using the typical aerosol particles size distribution assumption and combining the complex refractive index of different aerosol species. Table 3 lists the spectral optical thicknesses of the dust aerosol and the scattering albedo parameters in the height of 0-1 km. The other parameters are not given in this paper as there are too much data.

Table 4 gives the complex refractive index of different aerosols.

3.1. The Influence of Aerosol Species Variation

Figure 3 shows the influence of four aerosol species (dust, soot, marine, and water-soluble) on the atmospheric temperature. The aerosol particle size distribution function is the multilognormal distribution.

Figure 3: Influence of aerosols species on temperature fields.

From the results in the figure we can see that as the different scattering effect of the aerosol particles, their influence on the temperature is greatly different. Compared with dust aerosol, the marine and water-soluble aerosol particles have stronger scattering effect. They will scatter upwards more radiation energy and decrease the atmospheric temperature. The soot particles have stronger absorption and their existence will cause the atmospheric temperature increasing.

3.2. The Influence of the Scattering Phase Function Variation

The aerosol particles scattering characteristics are considered as forward and back scatter (the scattering phase functions are and , resp.; see (16) and (17)). Other parameters are not changed. The results are shown in Figure 4:

Figure 4: Influence of scattering phase function change on atmospheric temperature.

It can be seen from Figure 4(a) that, compared with the anisotropic scattering, the variation in the scattering phase functions causes the atmospheric temperature to change. But the effect is not obvious. Figure 4(b) shows the difference value between the atmospheric temperature fields and the isotropic scattering with different scattering phase functions. From the figure we can see that the forward and back scatters have opposite effect on the atmospheric temperature fields but the effect degree is similar.

3.3. The Influence of the Aerosol Particles Concentration

The aerosol particles transport with the tropospheric movement will cause the aerosol particles concentrations to vary. In Figure 5 we consider the influence of the variation in total particles number concentrations, and the particles number concentrations are increased or decreased by 50%. So the radiation characteristics of the aerosol particles are changed, the atmospheric energy balance is affected, and then the atmospheric temperature fields are influenced. From the results it can be seen that the variation in aerosol particles concentrations can affect the atmospheric temperature but not obviously compared with other factors.

Figure 5: Influence of the aerosol concentration on atmospheric temperature.
3.4. The Influence of the Height Distribution of Aerosol Particles

We investigate the effect of the medium layer height on the aerosol particles size distribution; that is, the atmospheric radiation temperature fields change with the aerosol particles being in different height medium layers. The calculation conditions are as follows: the aerosol particles distribute in the height of 0~1 km, 1~3 km, and 3~9 km, respectively. The calculation results are in Figure 6.

Figure 6: Influence of temperature fields on different aerosol distribution altitude.

From the calculation results in Figure 6 it can be seen that, with the aerosol particles transporting to the high altitude, the atmospheric temperature is affected greatly if the radiation characteristics of the aerosol particles have no changes. With the distribution height of the aerosol particles increasing, the medium temperature above the medium layer in which the aerosol particles exist increases greatly but the below temperature decreases obviously. This phenomenon shows the scattering effect of the aerosol particles on the solar radiation energy, that is, as the scattering effect of the atmospheric aerosols, most of the solar radiation energy to the ground is scattered back to the high altitude and cannot reach lower atmosphere, which causes the lower atmospheric temperature to decrease to about 200 K.

4. Conclusion

We analyzed the influence of the aerosol particles characteristics on the atmospheric radiation transfer model. The results showed the different effect of aerosol types on the atmospheric radiation. The dust, marine, and water-soluble aerosols had negative effects on the atmospheric absorption, but the soot aerosol could increase the atmospheric absorption. The scattering phase functions affected the ground temperature. The variation of the aerosol particles distribution height could determine the atmospheric temperature; that is, the ground temperature decreased greatly with the aerosol particles existing in the high altitude stable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


This work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant no. 51121004) and the National Natural Science Foundation of China (Grant nos. 51176040 and 51276193). A very special acknowledgement is made to the editors and referees who made important comments to improve this paper.


  1. M. Schulz, C. Textor, S. Kinne et al., “Radiative forcing by aerosols as derived from the AeroCom present-day and pre-industrial simulations,” Atmospheric Chemistry and Physics, vol. 6, pp. 5225–5246, 2006. View at Publisher · View at Google Scholar
  2. U. Lohmann and J. Feichter, “Global indirect aerosol effects: a review,” Atmospheric Chemistry and Physics, vol. 5, pp. 715–737, 2005. View at Google Scholar
  3. I. Koren, Y. J. Kaufman, L. A. Remer, and J. V. Martins, “Measurement of the effect of Amazon smoke on inhibition of cloud formation,” Science, vol. 303, no. 5662, pp. 1342–1345, 2004. View at Google Scholar
  4. W. W. Qing, R. S. Chen, and W. M. Sun, “Estimation of global radiation in China and comparison with satellite product,” Environmental Earth Sciences, vol. 70, no. 4, pp. 1681–1687, 2013. View at Publisher · View at Google Scholar · View at Scopus
  5. X. S. Wang, “Metal geochemical and mineral magnetic characterization of the <2.5 lm fraction of urban soils in Xuzhou (China),” Environmental Earth Sciences, vol. 71, no. 8, pp. 3491–3501, 2014. View at Publisher · View at Google Scholar
  6. L. David, B. Nacer, B. Pascal, and J. Gérard, “Transient radiative and conductive heat transfer in non-gray semitransparent two-dimensional media with mixed boundary conditions,” Heat and Mass Transfer, vol. 42, no. 4, pp. 322–337, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. S. Dembele, K. L. M. Lima, and J. X. Wen, “Assessment of an unstructured exponential scheme discrete ordinates radiation model for non-gray media,” Heat and Mass Transfer, vol. 47, no. 11, pp. 1349–1362, 2011. View at Publisher · View at Google Scholar · View at Scopus
  8. P. Talukdar, F. V. Issendorff, D. Trimis, and C. J. Simonson, “Conduction-radiation interaction in 3D irregular enclosures using the finite volume method,” Heat and Mass Transfer, vol. 44, no. 6, pp. 695–704, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. C.-Y. Wu, “Monte Carlo simulation of radiative transfer in a refractive layered medium,” Heat and Mass Transfer, vol. 46, no. 6, pp. 607–613, 2010. View at Publisher · View at Google Scholar · View at Scopus
  10. J. H. Jeans, “The equations of radiative transfer of energy,” Monthly Notices Royal Astronomical Society, vol. 78, pp. 28–36, 1917. View at Google Scholar
  11. B. Davison, Neutron Transport Theory, Oxford University Press, London, UK, 1958.
  12. V. Kourganoff, Basic Methods in Transfer Problems, Dover, New York, NY, USA, 1963.
  13. H. Khouaja, D. R. Edwards, and N. Tsoulfanidis, “Spherical harmonics—finite element treatment of neutron transport in cylindrical geometry,” Annals of Nuclear Energy, vol. 24, no. 7, pp. 515–531, 1997. View at Publisher · View at Google Scholar · View at Scopus
  14. E. W. Larsen, G. Thömmes, A. Klar, M. Seaïd, and T. Götz, “Simplified Pn approximations to the equations of radiative heat transfer and application,” Journal of Computational Physics, vol. 183, no. 2, pp. 652–675, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. R. G. McClarren and C. D. Hauck, “Robust and accurate filtered spherical harmonics expansions for radiative transfer,” Journal of Computational Physics, vol. 229, no. 16, pp. 5597–5614, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. M. F. Modest, Radiative Heat Transfer, Academic Press, Amsterdam, The Netherlands, 2nd edition, 2003.
  17. R. Siegel and C. M. Spuckler, “Refractive index effects on radiation in an absorbing, emitting, and scattering laminated layer,” Journal of Heat Transfer, vol. 115, pp. 194–200, 1993. View at Google Scholar
  18. Q. Z. Yu, Principles of Radiative Heat Transfer, Harbin Institute of Technology Press, Harbin, China, 2000.
  19. G. Y. Shi, Atmospheric Radiology, Science Press, Beijing, China, 2007.