Research ArticleStudy on the Influence of Aerosol RadiationBalance in One-Dimensional Atmospheric MediumUsing 119875
119899-Approximation Method
Yuan Yuan1 Xing Huang1 Yong Shuai1 and Qian-Jun Mao2
1 School of Energy Science and Engineering Harbin Institute of Technology Harbin 150001 China2 School of Civil Engineering and Architecture Northeast Petroleum University Daqing 163318 China
Correspondence should be addressed to Yuan Yuan yuanyuan83hiteducn and Yong Shuai shuaiyonghiteducn
Received 17 May 2014 Revised 10 July 2014 Accepted 20 July 2014 Published 27 August 2014
Academic Editor Hong-Liang Yi
Copyright copy 2014 Yuan Yuan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich 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 onthe established atmospheric radiation transfer model we calculated the effect of species concentration and height distribution ofaerosols on atmosphere radiation The calculation results show that the aerosol particles affect the atmospheric radiation balancegreatly and thus are the important component of the radiation balance of the earth-troposphere system
1 Introduction
Atmospheric aerosols are suspensions of small solid orliquid particles with the diameters between 0001 and 10 120583mAerosol affects the Earthrsquos radiation budget by direct andindirect effects and plays an important role in global climatesystems The direct effect is that aerosol particles scatterand absorb the solar radiation and surface infrared radiation[1] the indirect effect is that aerosol particles act as cloudcondensation nuclei and ice nuclei [2 3] When the climatesystem is in balance its absorbing solar radiation energy isaccurately equal to the infrared radiation energy that theearth and atmosphere emit into outer space Any factor thatdisturbs this balance and then changes the climate is theradiative forcing factor Its forcing on the earth-tropospheresystem is the radiative forcing [4 5] Radiative forcing (inW sdot mminus2) is defined as the change in the net radiative fluxat the tropopause due to some factors change Aerosol is avery important correction factor in the atmospheric radiationtransfer So it is significant to study the influence of aerosolsparticles on the atmospheric transfer
Many numerical methods have been developed to solvethe problem of radiative heat transfer such as the discreteordinates method (DOM) [6 7] finite volume method(FVM) [8] finite element method (FEM) Monte Carlomethod (MCM) [9] and spherical harmonics method(SHM) The spherical harmonics method also known as the119875119899
-approximation method was originally proposed by Jeans[10] for interstellar radiative transfer and further developedby Davison [11] and Kourganoff [12] for neutron transportKhouaja et al [13] developed the 119875
119899
-approximation methodfor radiative heat transfer Recently Larsen et al [14] useda double-119875
119899
approximation for the transfer equation inneutron transport but only to low order in the expansionMcClarren et al [15] analyzed numerical oscillations of the119875119899
-approximation method and also suggested improvementsbased on wavelet analysis
We established atmospheric radiative transfer calcula-tionsmodel based on the spherical harmonicsmethod in one-dimensional media and analyzed the influence of aerosolson the atmospheric radiative transfer The calculations fromthe species number and distribution height of aerosols wereachieved and the results were given in Figure 1
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 767231 9 pageshttpdxdoiorg1011552014767231
The radiative transfer equation for one-dimensional absorb-ing emitting and scattering gray medium can be expressedas
120583
119889119868 (120591 120583)
119889120591
= minus119868 (120591 120583) + (1 minus 120596) 119868119887
(120591)
+
120596
2
int
1
minus1
119868 (120591 1205831015840
)Φ (1205831015840
120583) 1198891205831015840
(1)
where 120591 is the optical thickness given by 120591 = (120581119886
+ 120581119904
)119871120596 is the scattering albedo with 120596 = 120581
119904
(120581119886
+ 120581119904
) 119871 isthe medium thickness and 120581
119886
and 120581119904
are the absorbingand scattering coefficients respectively The radiative inten-sity 119868(120591 120583) and scattering phase function Φ(120583
1015840
120583) can beexpanded in terms of associated Legendre polynomials119875
119899
(120583)Using recurrence relations between these polynomials (1)becomes
119889
119889120591
(
119873
sum
119899=0
(119899 + 1) 119875119899+1
(120583) + 119899119875119899minus1
(120583)
2119899 + 1
119868119899
(120591))
= minus
119873
sum
119899=0
119868119899
(120591) 119875119899
(120583) + (1 minus 120596) 119868119887
(120591)
+
119873
sum
119899=0
120596119892119899
2119899 + 1
119868119899
(120591) 119875119899
(120583)
(2)
where 119892119899
is an asymmetry parameter Multiplying (2) by119875119899
(120583) and integrating over 120583 isin [minus1 1] we obtain using theorthogonality properties of the Legendre polynomials 119873 +
1 differential equations for coefficients 119868119899
(120591) that comprise
the119873+ 1 119875119899
-approximation equations Consider the follow-ing
1
3
1198891198681
(120591)
119889120591
+ (1 minus 1205961198920
) 1198680
(120591) = (1 minus 120596) 119868119887
(120591) 119899 = 0
2
5
1198891198682
(120591)
119889120591
+
1198891198680
(120591)
119889120591
+ (1 minus
1
3
1205961198921
) 1198681
(120591) = 0 119899 = 1
sdot sdot sdot
119899 + 1
2119899 + 3
119889119868119899+1
(120591)
119889120591
+
119899
2119899 minus 1
119889119868119899minus1
(120591)
119889120591
+ (1 minus
120596119892119899
2119899 + 1
) 119868119899
(120591)
= (1 minus 120596) 119868119887
(120591) 1205750119899
119899 = 2 3 119873
(3)
For one-dimensional opaque diffuse emitting and reflectinginterfaces odd terms in the Legendre polynomials forman orthogonal set The boundary conditions are givenby
int997888n sdot997888s gt0
119868 (120591119882=0
997888s ) 1198752119899+1
(997888s ) 119889Ω
= 120576119882
int997888n sdot997888s gt0
119868119882
(0997888s ) 1198752119899+1
(997888s ) 119889Ω
+ int997888n sdot997888s gt0
(1minus120576119882
)
1
2120587
int997888n sdot997888s lt0
119868 (120591119882
997888s 1015840) 119889Ω1015840119875
2119899+1
(997888s )119889Ω
(4)
where 120576 is the emissivity The boundary conditionsimposed on the spherical harmonics functions are givenby
119873
sum
119899=0
119868119899
(0) [int
1
0
119875119899
(120583) 1205832119899+1
119889120583
minus 120588119889
int
1
0
int
0
minus1
119875119899
(1205831015840
) 1205832119899+1
1198891205831015840
119889120583]
= 1198681198870
(1 minus 120588119889
0
)int
1
0
1205832119899+1
119889120583
119873
sum
119899=0
119868119899
(120591119871
) [int
0
minus1
119875119899
(120583) 1205832119899+1
119889120583
minus 120588119889
int
0
minus1
int
1
0
119875119899
(1205831015840
) 1205832119899+1
1198891205831015840
119889120583]
= 119868119887120591119871
(1 minus 120588119889
0
) int
0
minus1
1205832119899+1
119889120583
119899 = 0 1
119873 minus 1
2
(5)
Mathematical Problems in Engineering 3
The boundary conditions [16] are applicable in the one-dimensional case
int997888n sdot997888s gt0
119868 (120591119882=0
997888s ) 1198752119899+1
(997888s ) 119889Ω
= (1 minus 120588119889
0
) int997888n sdot997888s gt0
1198680
(997888s ) 1198752119899+1
(997888s ) 119889Ω
+ int997888n sdot997888s gt0
120588119889
1
2120587
int997888n sdot997888s lt0
119868 (120591119882
997888s 1015840) 119889Ω1015840119875
2119899+1
(997888s ) 119889Ω
(6)
where 1198680
(997888s ) is the circumstance radiation 120588119889
0
and 120588119889 are
reflectivities of the two sides of the boundary surfacesrespectively when considering total reflection and thevalues of these depend on the medium refractive index 119899
Separating the large refractive index from small thediffuse reflectance at the boundary surfaces is given by Siegeland Spuckler [17]
120588 (119899) equiv 119865 (119899) =
1
2
+
(3119899 + 1) (119899 minus 1)
6(119899 + 1)2
+
1198992
(1198992
minus 1)
2
(1198992
+ 1)3
ln(119899 minus 1
119899 + 1
)minus
21198993
(1198992
+2119899 minus 1)
(1198992
+1) (1198994
minus 1)
+
81198994
(1198994
+ 1)
(1198992
+ 1) (1198994
minus 1)2
ln (119899)
(7)
In contrast separating small refractive index from large thediffuse reflectance at the boundary surfaces is given by
120588 (119899) = 1 minus
1
1198992
[1 minus 119865 (119899)] (8)
where 119899 = (119899ℎ
119899119904
) 119899ℎ
ge 119899119904
In treating a semitransparentinterface we can use opaque interface treatment as reference
Considering one-dimensional unsteady coupled radiativeand conductive heat transfer the general form of the energyequation of the internal radiation and the transient coupledconductive heat transfer is defined as
120588119888
120597119879
120597119905
= div (119896119888
grad119879) minus div q119903 + 120601 (9)
Considering unsteady state no internal heat source andcoupled radiative and conductive heat transfer the energyequation (9) can be discretized as follows
120588119888Δ119909119894
119879119898+1
119894
minus 119879119898
119894
Δ119905
=
119896119890
(119879119898+1
119894+1
minus 119879119898+1
119894
) + 119896119908
(119879119898+1
119894minus1
minus 119879119898+1
119894
)
Δ119909119894
+ Φ119903
119894119898
(10)
In order to make the problem have universal significancethe dimensionless form of the energy equation (10) will beevaluated Introducing the following definitions referencetemperature 119879
119903119891
the dimensionless temperature 119879119898
119894
=
119879119898
119894
119879119903119891
unit heat capacity 119862 = 120588119888 the dimensionless timeΔ = (4120590119899
2
1198793
119903119891
119862119871)Δ119905 = (41205901198992
1198793
119903119891
119862119871)119905 the dimensionlesslength Δ119883
119894
= Δ119909119894
119871 the radiative and conductive parameter119873 = 1205824120590119899
2
1198793
119903119891
119871 the dimensionless convective heat transfercoefficient 119867 = ℎ4120590119899
2
1198793
119903119891
the dimensionless radiative heatflux Φ = Φ(4120590119899
2
1198794
119903119891
) (10) becomes
Δ119883119894
119879119898+1
119894
minus119879119898
119894
Δ
=
119873119890
(119879119898+1
119894+1
minus119879119898+1
119894
) + 119873119908
(119879119898+1
119894minus1
minus119879119898+1
119894
)
Δ119883119894
+ Φ119903
119894119898
(11)
According to [18] in considering one-dimensional non-steady coupled radiative and conductive heat transfers thethermal radiation source termΦ119903(119879) is found to be
Φ119903
(119879) = int
Δ119881
minus div q119903119889119881 = int
Δ119881
minus120581119886
[4120587119868119887
(120591) minus 119866 (120591)] 119889119881
(12)
Here the incident radiative power 119866(120591) = intΩ
119868(120591 Ω)119889Ω canbe transformed to 119866(120591) = 2120587 int
1
minus1
119868(120591 120583)119889120583 According to theproperties of Legendre polynomials and 119875
0
(120583) = 1 the aboveequation is rearranged into the form
119866 (120591) = 2120587int
1
minus1
1198680
(120591) 119889120583 (13)
Meanwhile considering the physical properties ofthe inhomogeneous medium we next develop a 119875
119899
-approximation method for a multilayer medium Weredetermine the boundary conditions of each layer Themodified boundary conditions are
119873
sum
119899=0
119868119899
(0) [int
1
0
119875119899
(120583) 1205832119894+1
119889120583 minus 120588119889
int
1
0
int
0
minus1
119875119899
(1205831015840
) 1205832119894+1
1198891205831015840
119889120583]
=
119873
sum
119899=0
1198681198991198870
(1 minus 120588119889
0
)int
1
0
119875119899
(120583) 1205832119894+1
119889120583
119873
sum
119899=0
119868119899
(120591119871
)[int
0
minus1
119875119899
(120583) 1205832119894+1
119889120583minus120588119889
int
0
minus1
int
1
0
119875119899
(1205831015840
)1205832119894+1
1198891205831015840
119889120583]
=
119873
sum
119899=0
119868119899119887120591120582
(1 minus 120588119889
0
) int
0
minus1
119875119899
(120583) 1205832119894+1
119889120583
119894 = 0 1
119873 minus 1
2
(14)
4 Mathematical Problems in Engineering
Table 1 The main input parameters of the atmospheric radiation calculation
Parameters Values
Initial condition
Initial temperature 1198790
280KReference temperature 119879rf 1000KDimensionless time 119905lowast 1 times 10
6
Relaxation factor 119891re 001
Boundary condition
Terrestrial temperature 119879minusinfin
280KTerrestrial emissivity 120576
minusinfin
07Solar radiation temperature 119879
+infin
5762KAngle factor 119860
+infin 2
55844 times 10minus6
Media physical propertiesLayers of height 7 layers seen in Table 3Bands absorption coefficient 12 bands seen in Table 4Thermal conductivity 119896
119888
0023 Wsdotmminus1sdotKminus1
Aerosol properties Distribution function Multilognormal distributionsHeight distribution 0-1 km
Table 2 Multilayer atmospheric radiative transfer model
Layer number Thickness (km) Upper-lower limit ofheight (km) Density (kgm3) Dimensionless
thermal conductivity Refractive index
1 1 0sim1 115566 751119890 minus 8 102 2 1sim3 096517 589119890 minus 8 103 6 3sim9 060438 374119890 minus 8 104 8 9sim17 030815 143119890 minus 8 105 10 17sim27 002573 323119890 minus 9 106 11 27sim38 0010076 690119890 minus 10 107 12 38sim50 0002195 423119890 minus 10 10
Here 1198681198991198870
and 119868119899119887120591120582
are the emission radiation intensities oftwo adjacent layers and the initial values are obtained fromthe temperature field The total emission radiation inten-sity is determined by an iterative calculation of intensitiesfrom successive layers the last providing the required totalThe convergence condition of the multilayer model calcula-tion is decided by the radiative intensity 119868
119899
After the intensitycalculation has converged the iterative calculation of thetemperature field is then performed
3 Results and Discussion
According to statistics 988 of the total atmospheric massdistributes in the atmospherewithin 30 km above the groundand with the height increasing to 50 km the proportionreaches 9992Most of the atmospheric radiation (involvingsolar radiation absorption and scattering of the groundinfrared radiation and the emission of the atmosphere itself)occurs within certain atmospheric thickness Compared withthe earth radius (6378 km) the atmospheric thickness canbe regarded as one-dimensional radiative media Thereforewe treated the atmosphere within 50 km above the groundas a multilayer one-dimensional media distributing along theradial of the earth The upper surface of the medium layer isnear to space and is considered as a semitransparentmediumThe lower interface is close to ground and is considered asopaque interface The emissivity and radiative temperature
related to the ground are given For the inner of the mediumlayer we mainly consider the gas absorption and emissionof the radiation energy and the aerosol particles scatteringand absorption of the radiation energy and consider theatmospheric coupled radiation and conduction in the verticaldirection
The atmospheric convection is very important for theenergy transfer of the atmosphere But we mainly discussthe influence of aerosols on the atmospheric radiative energyand investigate the atmospheric radiative balance and thevariation in radiation characteristic which is not affected bythe atmospheric convection that acts only as the backgroundTherefore we ignore the influence of the atmospheric con-vection
The atmospheric radiation physical model is shown inFigure 1 and the atmospheric radiation transfer model isshown in Figure 2 The atmospheric internal heat source isset as 120601 = 0
The standard atmospheric condition is given inTables 1ndash4
Table 1 gives themain input parameters for the calculationof atmospheric radiation transfer
Table 2 gives the parameters associated with layersdivided along the vertical height The density of each layer isdetermined by the calculation of the pressure We consider(1205881205880
) = (1199011199010
) (the standard atmosphere pressure 119901 is
Mathematical Problems in Engineering 5
Δxn
S+infinSminus infin S2S1
T+infin
Pnminus1
Tg2
Mt + 1Mtsum nminus1
i=1Mi + In
nnn2n1
sum nminus1
i=1Mi
middot middot middot
M1 + 1 M1 + 2M1I1
Δx2
2
Δxn
2Δx1
Δx1
2
xS1
I1
xP1
I2x
P119899minus1
I119899
xS1
I+11x
P1
I+11x
P119899minus1
I+1119899
xS2
I+1119899
LnL2 + L3 + middot middot middot + Lnminus1L1
Tminus infin
P1
Tg1
h1 h2
10 2
01
Figure 2 Physical model of one plane-parallel slab divided into 119899 sublayers
180 200 220 240 260 280 300 3200
10
20
30
40
50
Alti
tude
(km
)
Temperature (K)
MarineSoot
Water-solubleDust
(a) Atmosphere temperature field (0ndash50 km)
Alti
tude
(km
)
0
2
4
6
8
10
180 200 220 240 260 280 300 320Temperature (K)
MarineSoot
Water-solubleDust
(b) Atmosphere temperature field (0ndash10 km)
Figure 3 Influence of aerosols species on temperature fields
1225 kgm2) with the temperature fluctuating in small rangeTherefore the equation is as follows
120588 =
1205880
1199010
sdot 119901 =
1225
101325
sdot 119901 = 120898 times 10minus5
sdot 119901 (15)
Table 3 gives the wavelength range of each spectral bandThe atmosphere optical thicknesses of the layers at differentspectral bands are obtained from the HITRAN databaseunder the standard atmospheric condition of America in1976 The aerosol optical thicknesses are decided based onthe Mie scattering theory using the typical aerosol particlessize distribution assumption and combining the complex
refractive index of different aerosol species Table 3 liststhe spectral optical thicknesses of the dust aerosol and thescattering albedo parameters in the height of 0-1 km Theother parameters are not given in this paper as there are toomuch data
Table 4 gives the complex refractive index of differentaerosols
31 The Influence of Aerosol Species Variation Figure 3 showsthe influence of four aerosol species (dust soot marine andwater-soluble) on the atmospheric temperature The aerosol
(b) Temperature difference betweendifferent scattering phase functionsand isotropic scattering
Figure 4 Influence of scattering phase function change on atmospheric temperature
Table 3 Spectral bands for the atmospheric radiative transfermodeland radiative characteristic parameters under the standard dustaerosol in the height of 0-1 km
particle size distribution function is the multilognormaldistribution
From the results in the figure we can see that as the dif-ferent scattering effect of the aerosol particles their influenceon the temperature is greatly different Compared with dustaerosol the marine and water-soluble aerosol particles havestronger scattering effect They will scatter upwards moreradiation energy and decrease the atmospheric temperatureThe soot particles have stronger absorption and their exis-tence will cause the atmospheric temperature increasing
32 The Influence of the Scattering Phase Function VariationThe aerosol particles scattering characteristics are considered
as forward and back scatter (the scattering phase functionsare Φ1
(Θ) and Φ2
(Θ) resp see (16) and (17)) Other param-eters are not changed The results are shown in Figure 4
Φ1
(Θ) = 1 + 1198751
(cosΘ) (16)
Φ2
(Θ) = 1 minus 1198751
(cosΘ) (17)
It can be seen from Figure 4(a) that compared with theanisotropic scattering the variation in the scattering phasefunctions causes the atmospheric temperature to change Butthe effect is not obvious Figure 4(b) shows the differencevalue between the atmospheric temperature fields and theisotropic scattering with different scattering phase functionsFrom the figure we can see that the forward and back scattershave opposite effect on the atmospheric temperature fieldsbut the effect degree is similar
33 The Influence of the Aerosol Particles Concentration Theaerosol particles transport with the tropospheric movementwill cause the aerosol particles concentrations to vary InFigure 5 we consider the influence of the variation in totalparticles number concentrations and the particles numberconcentrations are increased or decreased by 50 So theradiation characteristics of the aerosol particles are changedthe atmospheric energy balance is affected and then theatmospheric temperature fields are influenced From theresults it can be seen that the variation in aerosol particlesconcentrations can affect the atmospheric temperature butnot obviously compared with other factors
34 The Influence of the Height Distribution of Aerosol Parti-cles We investigate the effect of the medium layer height onthe aerosol particles size distribution that is the atmosphericradiation temperature fields change with the aerosol particles
Mathematical Problems in Engineering 7
Table 4 Complex refractive index of different wavelength for different aerosol types [19]
Figure 5 Influence of the aerosol concentration on atmospheric temperature
being in different height medium layers The calculationconditions are as follows the aerosol particles distribute inthe height of 0sim1 km 1sim3 km and 3sim9 km respectively Thecalculation results are in Figure 6
From the calculation results in Figure 6 it can be seenthat with the aerosol particles transporting to the highaltitude the atmospheric temperature is affected greatly ifthe radiation characteristics of the aerosol particles have nochanges With the distribution height of the aerosol particlesincreasing the medium temperature above themedium layerin which the aerosol particles exist increases greatly but thebelow temperature decreases obviously This phenomenonshows the scattering effect of the aerosol particles on the
solar radiation energy that is as the scattering effect of theatmospheric aerosols most of the solar radiation energy tothe ground is scattered back to the high altitude and cannotreach lower atmosphere which causes the lower atmospherictemperature to decrease to about 200K
4 Conclusion
We analyzed the influence of the aerosol particles charac-teristics on the atmospheric radiation transfer model Theresults showed the different effect of aerosol types on theatmospheric radiation The dust marine and water-soluble
Mathematical Problems in Engineering 9
0
10
20
30
40
50A
ltitu
de (k
m)
180 200 220 240 260 280 300Temperature (K)
0-1 km1ndash3 km3ndash9 km
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
The radiative transfer equation for one-dimensional absorb-ing emitting and scattering gray medium can be expressedas
120583
119889119868 (120591 120583)
119889120591
= minus119868 (120591 120583) + (1 minus 120596) 119868119887
(120591)
+
120596
2
int
1
minus1
119868 (120591 1205831015840
)Φ (1205831015840
120583) 1198891205831015840
(1)
where 120591 is the optical thickness given by 120591 = (120581119886
+ 120581119904
)119871120596 is the scattering albedo with 120596 = 120581
119904
(120581119886
+ 120581119904
) 119871 isthe medium thickness and 120581
119886
and 120581119904
are the absorbingand scattering coefficients respectively The radiative inten-sity 119868(120591 120583) and scattering phase function Φ(120583
1015840
120583) can beexpanded in terms of associated Legendre polynomials119875
119899
(120583)Using recurrence relations between these polynomials (1)becomes
119889
119889120591
(
119873
sum
119899=0
(119899 + 1) 119875119899+1
(120583) + 119899119875119899minus1
(120583)
2119899 + 1
119868119899
(120591))
= minus
119873
sum
119899=0
119868119899
(120591) 119875119899
(120583) + (1 minus 120596) 119868119887
(120591)
+
119873
sum
119899=0
120596119892119899
2119899 + 1
119868119899
(120591) 119875119899
(120583)
(2)
where 119892119899
is an asymmetry parameter Multiplying (2) by119875119899
(120583) and integrating over 120583 isin [minus1 1] we obtain using theorthogonality properties of the Legendre polynomials 119873 +
1 differential equations for coefficients 119868119899
(120591) that comprise
the119873+ 1 119875119899
-approximation equations Consider the follow-ing
1
3
1198891198681
(120591)
119889120591
+ (1 minus 1205961198920
) 1198680
(120591) = (1 minus 120596) 119868119887
(120591) 119899 = 0
2
5
1198891198682
(120591)
119889120591
+
1198891198680
(120591)
119889120591
+ (1 minus
1
3
1205961198921
) 1198681
(120591) = 0 119899 = 1
sdot sdot sdot
119899 + 1
2119899 + 3
119889119868119899+1
(120591)
119889120591
+
119899
2119899 minus 1
119889119868119899minus1
(120591)
119889120591
+ (1 minus
120596119892119899
2119899 + 1
) 119868119899
(120591)
= (1 minus 120596) 119868119887
(120591) 1205750119899
119899 = 2 3 119873
(3)
For one-dimensional opaque diffuse emitting and reflectinginterfaces odd terms in the Legendre polynomials forman orthogonal set The boundary conditions are givenby
int997888n sdot997888s gt0
119868 (120591119882=0
997888s ) 1198752119899+1
(997888s ) 119889Ω
= 120576119882
int997888n sdot997888s gt0
119868119882
(0997888s ) 1198752119899+1
(997888s ) 119889Ω
+ int997888n sdot997888s gt0
(1minus120576119882
)
1
2120587
int997888n sdot997888s lt0
119868 (120591119882
997888s 1015840) 119889Ω1015840119875
2119899+1
(997888s )119889Ω
(4)
where 120576 is the emissivity The boundary conditionsimposed on the spherical harmonics functions are givenby
119873
sum
119899=0
119868119899
(0) [int
1
0
119875119899
(120583) 1205832119899+1
119889120583
minus 120588119889
int
1
0
int
0
minus1
119875119899
(1205831015840
) 1205832119899+1
1198891205831015840
119889120583]
= 1198681198870
(1 minus 120588119889
0
)int
1
0
1205832119899+1
119889120583
119873
sum
119899=0
119868119899
(120591119871
) [int
0
minus1
119875119899
(120583) 1205832119899+1
119889120583
minus 120588119889
int
0
minus1
int
1
0
119875119899
(1205831015840
) 1205832119899+1
1198891205831015840
119889120583]
= 119868119887120591119871
(1 minus 120588119889
0
) int
0
minus1
1205832119899+1
119889120583
119899 = 0 1
119873 minus 1
2
(5)
Mathematical Problems in Engineering 3
The boundary conditions [16] are applicable in the one-dimensional case
int997888n sdot997888s gt0
119868 (120591119882=0
997888s ) 1198752119899+1
(997888s ) 119889Ω
= (1 minus 120588119889
0
) int997888n sdot997888s gt0
1198680
(997888s ) 1198752119899+1
(997888s ) 119889Ω
+ int997888n sdot997888s gt0
120588119889
1
2120587
int997888n sdot997888s lt0
119868 (120591119882
997888s 1015840) 119889Ω1015840119875
2119899+1
(997888s ) 119889Ω
(6)
where 1198680
(997888s ) is the circumstance radiation 120588119889
0
and 120588119889 are
reflectivities of the two sides of the boundary surfacesrespectively when considering total reflection and thevalues of these depend on the medium refractive index 119899
Separating the large refractive index from small thediffuse reflectance at the boundary surfaces is given by Siegeland Spuckler [17]
120588 (119899) equiv 119865 (119899) =
1
2
+
(3119899 + 1) (119899 minus 1)
6(119899 + 1)2
+
1198992
(1198992
minus 1)
2
(1198992
+ 1)3
ln(119899 minus 1
119899 + 1
)minus
21198993
(1198992
+2119899 minus 1)
(1198992
+1) (1198994
minus 1)
+
81198994
(1198994
+ 1)
(1198992
+ 1) (1198994
minus 1)2
ln (119899)
(7)
In contrast separating small refractive index from large thediffuse reflectance at the boundary surfaces is given by
120588 (119899) = 1 minus
1
1198992
[1 minus 119865 (119899)] (8)
where 119899 = (119899ℎ
119899119904
) 119899ℎ
ge 119899119904
In treating a semitransparentinterface we can use opaque interface treatment as reference
Considering one-dimensional unsteady coupled radiativeand conductive heat transfer the general form of the energyequation of the internal radiation and the transient coupledconductive heat transfer is defined as
120588119888
120597119879
120597119905
= div (119896119888
grad119879) minus div q119903 + 120601 (9)
Considering unsteady state no internal heat source andcoupled radiative and conductive heat transfer the energyequation (9) can be discretized as follows
120588119888Δ119909119894
119879119898+1
119894
minus 119879119898
119894
Δ119905
=
119896119890
(119879119898+1
119894+1
minus 119879119898+1
119894
) + 119896119908
(119879119898+1
119894minus1
minus 119879119898+1
119894
)
Δ119909119894
+ Φ119903
119894119898
(10)
In order to make the problem have universal significancethe dimensionless form of the energy equation (10) will beevaluated Introducing the following definitions referencetemperature 119879
119903119891
the dimensionless temperature 119879119898
119894
=
119879119898
119894
119879119903119891
unit heat capacity 119862 = 120588119888 the dimensionless timeΔ = (4120590119899
2
1198793
119903119891
119862119871)Δ119905 = (41205901198992
1198793
119903119891
119862119871)119905 the dimensionlesslength Δ119883
119894
= Δ119909119894
119871 the radiative and conductive parameter119873 = 1205824120590119899
2
1198793
119903119891
119871 the dimensionless convective heat transfercoefficient 119867 = ℎ4120590119899
2
1198793
119903119891
the dimensionless radiative heatflux Φ = Φ(4120590119899
2
1198794
119903119891
) (10) becomes
Δ119883119894
119879119898+1
119894
minus119879119898
119894
Δ
=
119873119890
(119879119898+1
119894+1
minus119879119898+1
119894
) + 119873119908
(119879119898+1
119894minus1
minus119879119898+1
119894
)
Δ119883119894
+ Φ119903
119894119898
(11)
According to [18] in considering one-dimensional non-steady coupled radiative and conductive heat transfers thethermal radiation source termΦ119903(119879) is found to be
Φ119903
(119879) = int
Δ119881
minus div q119903119889119881 = int
Δ119881
minus120581119886
[4120587119868119887
(120591) minus 119866 (120591)] 119889119881
(12)
Here the incident radiative power 119866(120591) = intΩ
119868(120591 Ω)119889Ω canbe transformed to 119866(120591) = 2120587 int
1
minus1
119868(120591 120583)119889120583 According to theproperties of Legendre polynomials and 119875
0
(120583) = 1 the aboveequation is rearranged into the form
119866 (120591) = 2120587int
1
minus1
1198680
(120591) 119889120583 (13)
Meanwhile considering the physical properties ofthe inhomogeneous medium we next develop a 119875
119899
-approximation method for a multilayer medium Weredetermine the boundary conditions of each layer Themodified boundary conditions are
119873
sum
119899=0
119868119899
(0) [int
1
0
119875119899
(120583) 1205832119894+1
119889120583 minus 120588119889
int
1
0
int
0
minus1
119875119899
(1205831015840
) 1205832119894+1
1198891205831015840
119889120583]
=
119873
sum
119899=0
1198681198991198870
(1 minus 120588119889
0
)int
1
0
119875119899
(120583) 1205832119894+1
119889120583
119873
sum
119899=0
119868119899
(120591119871
)[int
0
minus1
119875119899
(120583) 1205832119894+1
119889120583minus120588119889
int
0
minus1
int
1
0
119875119899
(1205831015840
)1205832119894+1
1198891205831015840
119889120583]
=
119873
sum
119899=0
119868119899119887120591120582
(1 minus 120588119889
0
) int
0
minus1
119875119899
(120583) 1205832119894+1
119889120583
119894 = 0 1
119873 minus 1
2
(14)
4 Mathematical Problems in Engineering
Table 1 The main input parameters of the atmospheric radiation calculation
Parameters Values
Initial condition
Initial temperature 1198790
280KReference temperature 119879rf 1000KDimensionless time 119905lowast 1 times 10
6
Relaxation factor 119891re 001
Boundary condition
Terrestrial temperature 119879minusinfin
280KTerrestrial emissivity 120576
minusinfin
07Solar radiation temperature 119879
+infin
5762KAngle factor 119860
+infin 2
55844 times 10minus6
Media physical propertiesLayers of height 7 layers seen in Table 3Bands absorption coefficient 12 bands seen in Table 4Thermal conductivity 119896
119888
0023 Wsdotmminus1sdotKminus1
Aerosol properties Distribution function Multilognormal distributionsHeight distribution 0-1 km
Table 2 Multilayer atmospheric radiative transfer model
Layer number Thickness (km) Upper-lower limit ofheight (km) Density (kgm3) Dimensionless
thermal conductivity Refractive index
1 1 0sim1 115566 751119890 minus 8 102 2 1sim3 096517 589119890 minus 8 103 6 3sim9 060438 374119890 minus 8 104 8 9sim17 030815 143119890 minus 8 105 10 17sim27 002573 323119890 minus 9 106 11 27sim38 0010076 690119890 minus 10 107 12 38sim50 0002195 423119890 minus 10 10
Here 1198681198991198870
and 119868119899119887120591120582
are the emission radiation intensities oftwo adjacent layers and the initial values are obtained fromthe temperature field The total emission radiation inten-sity is determined by an iterative calculation of intensitiesfrom successive layers the last providing the required totalThe convergence condition of the multilayer model calcula-tion is decided by the radiative intensity 119868
119899
After the intensitycalculation has converged the iterative calculation of thetemperature field is then performed
3 Results and Discussion
According to statistics 988 of the total atmospheric massdistributes in the atmospherewithin 30 km above the groundand with the height increasing to 50 km the proportionreaches 9992Most of the atmospheric radiation (involvingsolar radiation absorption and scattering of the groundinfrared radiation and the emission of the atmosphere itself)occurs within certain atmospheric thickness Compared withthe earth radius (6378 km) the atmospheric thickness canbe regarded as one-dimensional radiative media Thereforewe treated the atmosphere within 50 km above the groundas a multilayer one-dimensional media distributing along theradial of the earth The upper surface of the medium layer isnear to space and is considered as a semitransparentmediumThe lower interface is close to ground and is considered asopaque interface The emissivity and radiative temperature
related to the ground are given For the inner of the mediumlayer we mainly consider the gas absorption and emissionof the radiation energy and the aerosol particles scatteringand absorption of the radiation energy and consider theatmospheric coupled radiation and conduction in the verticaldirection
The atmospheric convection is very important for theenergy transfer of the atmosphere But we mainly discussthe influence of aerosols on the atmospheric radiative energyand investigate the atmospheric radiative balance and thevariation in radiation characteristic which is not affected bythe atmospheric convection that acts only as the backgroundTherefore we ignore the influence of the atmospheric con-vection
The atmospheric radiation physical model is shown inFigure 1 and the atmospheric radiation transfer model isshown in Figure 2 The atmospheric internal heat source isset as 120601 = 0
The standard atmospheric condition is given inTables 1ndash4
Table 1 gives themain input parameters for the calculationof atmospheric radiation transfer
Table 2 gives the parameters associated with layersdivided along the vertical height The density of each layer isdetermined by the calculation of the pressure We consider(1205881205880
) = (1199011199010
) (the standard atmosphere pressure 119901 is
Mathematical Problems in Engineering 5
Δxn
S+infinSminus infin S2S1
T+infin
Pnminus1
Tg2
Mt + 1Mtsum nminus1
i=1Mi + In
nnn2n1
sum nminus1
i=1Mi
middot middot middot
M1 + 1 M1 + 2M1I1
Δx2
2
Δxn
2Δx1
Δx1
2
xS1
I1
xP1
I2x
P119899minus1
I119899
xS1
I+11x
P1
I+11x
P119899minus1
I+1119899
xS2
I+1119899
LnL2 + L3 + middot middot middot + Lnminus1L1
Tminus infin
P1
Tg1
h1 h2
10 2
01
Figure 2 Physical model of one plane-parallel slab divided into 119899 sublayers
180 200 220 240 260 280 300 3200
10
20
30
40
50
Alti
tude
(km
)
Temperature (K)
MarineSoot
Water-solubleDust
(a) Atmosphere temperature field (0ndash50 km)
Alti
tude
(km
)
0
2
4
6
8
10
180 200 220 240 260 280 300 320Temperature (K)
MarineSoot
Water-solubleDust
(b) Atmosphere temperature field (0ndash10 km)
Figure 3 Influence of aerosols species on temperature fields
1225 kgm2) with the temperature fluctuating in small rangeTherefore the equation is as follows
120588 =
1205880
1199010
sdot 119901 =
1225
101325
sdot 119901 = 120898 times 10minus5
sdot 119901 (15)
Table 3 gives the wavelength range of each spectral bandThe atmosphere optical thicknesses of the layers at differentspectral bands are obtained from the HITRAN databaseunder the standard atmospheric condition of America in1976 The aerosol optical thicknesses are decided based onthe Mie scattering theory using the typical aerosol particlessize distribution assumption and combining the complex
refractive index of different aerosol species Table 3 liststhe spectral optical thicknesses of the dust aerosol and thescattering albedo parameters in the height of 0-1 km Theother parameters are not given in this paper as there are toomuch data
Table 4 gives the complex refractive index of differentaerosols
31 The Influence of Aerosol Species Variation Figure 3 showsthe influence of four aerosol species (dust soot marine andwater-soluble) on the atmospheric temperature The aerosol
(b) Temperature difference betweendifferent scattering phase functionsand isotropic scattering
Figure 4 Influence of scattering phase function change on atmospheric temperature
Table 3 Spectral bands for the atmospheric radiative transfermodeland radiative characteristic parameters under the standard dustaerosol in the height of 0-1 km
particle size distribution function is the multilognormaldistribution
From the results in the figure we can see that as the dif-ferent scattering effect of the aerosol particles their influenceon the temperature is greatly different Compared with dustaerosol the marine and water-soluble aerosol particles havestronger scattering effect They will scatter upwards moreradiation energy and decrease the atmospheric temperatureThe soot particles have stronger absorption and their exis-tence will cause the atmospheric temperature increasing
32 The Influence of the Scattering Phase Function VariationThe aerosol particles scattering characteristics are considered
as forward and back scatter (the scattering phase functionsare Φ1
(Θ) and Φ2
(Θ) resp see (16) and (17)) Other param-eters are not changed The results are shown in Figure 4
Φ1
(Θ) = 1 + 1198751
(cosΘ) (16)
Φ2
(Θ) = 1 minus 1198751
(cosΘ) (17)
It can be seen from Figure 4(a) that compared with theanisotropic scattering the variation in the scattering phasefunctions causes the atmospheric temperature to change Butthe effect is not obvious Figure 4(b) shows the differencevalue between the atmospheric temperature fields and theisotropic scattering with different scattering phase functionsFrom the figure we can see that the forward and back scattershave opposite effect on the atmospheric temperature fieldsbut the effect degree is similar
33 The Influence of the Aerosol Particles Concentration Theaerosol particles transport with the tropospheric movementwill cause the aerosol particles concentrations to vary InFigure 5 we consider the influence of the variation in totalparticles number concentrations and the particles numberconcentrations are increased or decreased by 50 So theradiation characteristics of the aerosol particles are changedthe atmospheric energy balance is affected and then theatmospheric temperature fields are influenced From theresults it can be seen that the variation in aerosol particlesconcentrations can affect the atmospheric temperature butnot obviously compared with other factors
34 The Influence of the Height Distribution of Aerosol Parti-cles We investigate the effect of the medium layer height onthe aerosol particles size distribution that is the atmosphericradiation temperature fields change with the aerosol particles
Mathematical Problems in Engineering 7
Table 4 Complex refractive index of different wavelength for different aerosol types [19]
Figure 5 Influence of the aerosol concentration on atmospheric temperature
being in different height medium layers The calculationconditions are as follows the aerosol particles distribute inthe height of 0sim1 km 1sim3 km and 3sim9 km respectively Thecalculation results are in Figure 6
From the calculation results in Figure 6 it can be seenthat with the aerosol particles transporting to the highaltitude the atmospheric temperature is affected greatly ifthe radiation characteristics of the aerosol particles have nochanges With the distribution height of the aerosol particlesincreasing the medium temperature above themedium layerin which the aerosol particles exist increases greatly but thebelow temperature decreases obviously This phenomenonshows the scattering effect of the aerosol particles on the
solar radiation energy that is as the scattering effect of theatmospheric aerosols most of the solar radiation energy tothe ground is scattered back to the high altitude and cannotreach lower atmosphere which causes the lower atmospherictemperature to decrease to about 200K
4 Conclusion
We analyzed the influence of the aerosol particles charac-teristics on the atmospheric radiation transfer model Theresults showed the different effect of aerosol types on theatmospheric radiation The dust marine and water-soluble
Mathematical Problems in Engineering 9
0
10
20
30
40
50A
ltitu
de (k
m)
180 200 220 240 260 280 300Temperature (K)
0-1 km1ndash3 km3ndash9 km
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
The boundary conditions [16] are applicable in the one-dimensional case
int997888n sdot997888s gt0
119868 (120591119882=0
997888s ) 1198752119899+1
(997888s ) 119889Ω
= (1 minus 120588119889
0
) int997888n sdot997888s gt0
1198680
(997888s ) 1198752119899+1
(997888s ) 119889Ω
+ int997888n sdot997888s gt0
120588119889
1
2120587
int997888n sdot997888s lt0
119868 (120591119882
997888s 1015840) 119889Ω1015840119875
2119899+1
(997888s ) 119889Ω
(6)
where 1198680
(997888s ) is the circumstance radiation 120588119889
0
and 120588119889 are
reflectivities of the two sides of the boundary surfacesrespectively when considering total reflection and thevalues of these depend on the medium refractive index 119899
Separating the large refractive index from small thediffuse reflectance at the boundary surfaces is given by Siegeland Spuckler [17]
120588 (119899) equiv 119865 (119899) =
1
2
+
(3119899 + 1) (119899 minus 1)
6(119899 + 1)2
+
1198992
(1198992
minus 1)
2
(1198992
+ 1)3
ln(119899 minus 1
119899 + 1
)minus
21198993
(1198992
+2119899 minus 1)
(1198992
+1) (1198994
minus 1)
+
81198994
(1198994
+ 1)
(1198992
+ 1) (1198994
minus 1)2
ln (119899)
(7)
In contrast separating small refractive index from large thediffuse reflectance at the boundary surfaces is given by
120588 (119899) = 1 minus
1
1198992
[1 minus 119865 (119899)] (8)
where 119899 = (119899ℎ
119899119904
) 119899ℎ
ge 119899119904
In treating a semitransparentinterface we can use opaque interface treatment as reference
Considering one-dimensional unsteady coupled radiativeand conductive heat transfer the general form of the energyequation of the internal radiation and the transient coupledconductive heat transfer is defined as
120588119888
120597119879
120597119905
= div (119896119888
grad119879) minus div q119903 + 120601 (9)
Considering unsteady state no internal heat source andcoupled radiative and conductive heat transfer the energyequation (9) can be discretized as follows
120588119888Δ119909119894
119879119898+1
119894
minus 119879119898
119894
Δ119905
=
119896119890
(119879119898+1
119894+1
minus 119879119898+1
119894
) + 119896119908
(119879119898+1
119894minus1
minus 119879119898+1
119894
)
Δ119909119894
+ Φ119903
119894119898
(10)
In order to make the problem have universal significancethe dimensionless form of the energy equation (10) will beevaluated Introducing the following definitions referencetemperature 119879
119903119891
the dimensionless temperature 119879119898
119894
=
119879119898
119894
119879119903119891
unit heat capacity 119862 = 120588119888 the dimensionless timeΔ = (4120590119899
2
1198793
119903119891
119862119871)Δ119905 = (41205901198992
1198793
119903119891
119862119871)119905 the dimensionlesslength Δ119883
119894
= Δ119909119894
119871 the radiative and conductive parameter119873 = 1205824120590119899
2
1198793
119903119891
119871 the dimensionless convective heat transfercoefficient 119867 = ℎ4120590119899
2
1198793
119903119891
the dimensionless radiative heatflux Φ = Φ(4120590119899
2
1198794
119903119891
) (10) becomes
Δ119883119894
119879119898+1
119894
minus119879119898
119894
Δ
=
119873119890
(119879119898+1
119894+1
minus119879119898+1
119894
) + 119873119908
(119879119898+1
119894minus1
minus119879119898+1
119894
)
Δ119883119894
+ Φ119903
119894119898
(11)
According to [18] in considering one-dimensional non-steady coupled radiative and conductive heat transfers thethermal radiation source termΦ119903(119879) is found to be
Φ119903
(119879) = int
Δ119881
minus div q119903119889119881 = int
Δ119881
minus120581119886
[4120587119868119887
(120591) minus 119866 (120591)] 119889119881
(12)
Here the incident radiative power 119866(120591) = intΩ
119868(120591 Ω)119889Ω canbe transformed to 119866(120591) = 2120587 int
1
minus1
119868(120591 120583)119889120583 According to theproperties of Legendre polynomials and 119875
0
(120583) = 1 the aboveequation is rearranged into the form
119866 (120591) = 2120587int
1
minus1
1198680
(120591) 119889120583 (13)
Meanwhile considering the physical properties ofthe inhomogeneous medium we next develop a 119875
119899
-approximation method for a multilayer medium Weredetermine the boundary conditions of each layer Themodified boundary conditions are
119873
sum
119899=0
119868119899
(0) [int
1
0
119875119899
(120583) 1205832119894+1
119889120583 minus 120588119889
int
1
0
int
0
minus1
119875119899
(1205831015840
) 1205832119894+1
1198891205831015840
119889120583]
=
119873
sum
119899=0
1198681198991198870
(1 minus 120588119889
0
)int
1
0
119875119899
(120583) 1205832119894+1
119889120583
119873
sum
119899=0
119868119899
(120591119871
)[int
0
minus1
119875119899
(120583) 1205832119894+1
119889120583minus120588119889
int
0
minus1
int
1
0
119875119899
(1205831015840
)1205832119894+1
1198891205831015840
119889120583]
=
119873
sum
119899=0
119868119899119887120591120582
(1 minus 120588119889
0
) int
0
minus1
119875119899
(120583) 1205832119894+1
119889120583
119894 = 0 1
119873 minus 1
2
(14)
4 Mathematical Problems in Engineering
Table 1 The main input parameters of the atmospheric radiation calculation
Parameters Values
Initial condition
Initial temperature 1198790
280KReference temperature 119879rf 1000KDimensionless time 119905lowast 1 times 10
6
Relaxation factor 119891re 001
Boundary condition
Terrestrial temperature 119879minusinfin
280KTerrestrial emissivity 120576
minusinfin
07Solar radiation temperature 119879
+infin
5762KAngle factor 119860
+infin 2
55844 times 10minus6
Media physical propertiesLayers of height 7 layers seen in Table 3Bands absorption coefficient 12 bands seen in Table 4Thermal conductivity 119896
119888
0023 Wsdotmminus1sdotKminus1
Aerosol properties Distribution function Multilognormal distributionsHeight distribution 0-1 km
Table 2 Multilayer atmospheric radiative transfer model
Layer number Thickness (km) Upper-lower limit ofheight (km) Density (kgm3) Dimensionless
thermal conductivity Refractive index
1 1 0sim1 115566 751119890 minus 8 102 2 1sim3 096517 589119890 minus 8 103 6 3sim9 060438 374119890 minus 8 104 8 9sim17 030815 143119890 minus 8 105 10 17sim27 002573 323119890 minus 9 106 11 27sim38 0010076 690119890 minus 10 107 12 38sim50 0002195 423119890 minus 10 10
Here 1198681198991198870
and 119868119899119887120591120582
are the emission radiation intensities oftwo adjacent layers and the initial values are obtained fromthe temperature field The total emission radiation inten-sity is determined by an iterative calculation of intensitiesfrom successive layers the last providing the required totalThe convergence condition of the multilayer model calcula-tion is decided by the radiative intensity 119868
119899
After the intensitycalculation has converged the iterative calculation of thetemperature field is then performed
3 Results and Discussion
According to statistics 988 of the total atmospheric massdistributes in the atmospherewithin 30 km above the groundand with the height increasing to 50 km the proportionreaches 9992Most of the atmospheric radiation (involvingsolar radiation absorption and scattering of the groundinfrared radiation and the emission of the atmosphere itself)occurs within certain atmospheric thickness Compared withthe earth radius (6378 km) the atmospheric thickness canbe regarded as one-dimensional radiative media Thereforewe treated the atmosphere within 50 km above the groundas a multilayer one-dimensional media distributing along theradial of the earth The upper surface of the medium layer isnear to space and is considered as a semitransparentmediumThe lower interface is close to ground and is considered asopaque interface The emissivity and radiative temperature
related to the ground are given For the inner of the mediumlayer we mainly consider the gas absorption and emissionof the radiation energy and the aerosol particles scatteringand absorption of the radiation energy and consider theatmospheric coupled radiation and conduction in the verticaldirection
The atmospheric convection is very important for theenergy transfer of the atmosphere But we mainly discussthe influence of aerosols on the atmospheric radiative energyand investigate the atmospheric radiative balance and thevariation in radiation characteristic which is not affected bythe atmospheric convection that acts only as the backgroundTherefore we ignore the influence of the atmospheric con-vection
The atmospheric radiation physical model is shown inFigure 1 and the atmospheric radiation transfer model isshown in Figure 2 The atmospheric internal heat source isset as 120601 = 0
The standard atmospheric condition is given inTables 1ndash4
Table 1 gives themain input parameters for the calculationof atmospheric radiation transfer
Table 2 gives the parameters associated with layersdivided along the vertical height The density of each layer isdetermined by the calculation of the pressure We consider(1205881205880
) = (1199011199010
) (the standard atmosphere pressure 119901 is
Mathematical Problems in Engineering 5
Δxn
S+infinSminus infin S2S1
T+infin
Pnminus1
Tg2
Mt + 1Mtsum nminus1
i=1Mi + In
nnn2n1
sum nminus1
i=1Mi
middot middot middot
M1 + 1 M1 + 2M1I1
Δx2
2
Δxn
2Δx1
Δx1
2
xS1
I1
xP1
I2x
P119899minus1
I119899
xS1
I+11x
P1
I+11x
P119899minus1
I+1119899
xS2
I+1119899
LnL2 + L3 + middot middot middot + Lnminus1L1
Tminus infin
P1
Tg1
h1 h2
10 2
01
Figure 2 Physical model of one plane-parallel slab divided into 119899 sublayers
180 200 220 240 260 280 300 3200
10
20
30
40
50
Alti
tude
(km
)
Temperature (K)
MarineSoot
Water-solubleDust
(a) Atmosphere temperature field (0ndash50 km)
Alti
tude
(km
)
0
2
4
6
8
10
180 200 220 240 260 280 300 320Temperature (K)
MarineSoot
Water-solubleDust
(b) Atmosphere temperature field (0ndash10 km)
Figure 3 Influence of aerosols species on temperature fields
1225 kgm2) with the temperature fluctuating in small rangeTherefore the equation is as follows
120588 =
1205880
1199010
sdot 119901 =
1225
101325
sdot 119901 = 120898 times 10minus5
sdot 119901 (15)
Table 3 gives the wavelength range of each spectral bandThe atmosphere optical thicknesses of the layers at differentspectral bands are obtained from the HITRAN databaseunder the standard atmospheric condition of America in1976 The aerosol optical thicknesses are decided based onthe Mie scattering theory using the typical aerosol particlessize distribution assumption and combining the complex
refractive index of different aerosol species Table 3 liststhe spectral optical thicknesses of the dust aerosol and thescattering albedo parameters in the height of 0-1 km Theother parameters are not given in this paper as there are toomuch data
Table 4 gives the complex refractive index of differentaerosols
31 The Influence of Aerosol Species Variation Figure 3 showsthe influence of four aerosol species (dust soot marine andwater-soluble) on the atmospheric temperature The aerosol
(b) Temperature difference betweendifferent scattering phase functionsand isotropic scattering
Figure 4 Influence of scattering phase function change on atmospheric temperature
Table 3 Spectral bands for the atmospheric radiative transfermodeland radiative characteristic parameters under the standard dustaerosol in the height of 0-1 km
particle size distribution function is the multilognormaldistribution
From the results in the figure we can see that as the dif-ferent scattering effect of the aerosol particles their influenceon the temperature is greatly different Compared with dustaerosol the marine and water-soluble aerosol particles havestronger scattering effect They will scatter upwards moreradiation energy and decrease the atmospheric temperatureThe soot particles have stronger absorption and their exis-tence will cause the atmospheric temperature increasing
32 The Influence of the Scattering Phase Function VariationThe aerosol particles scattering characteristics are considered
as forward and back scatter (the scattering phase functionsare Φ1
(Θ) and Φ2
(Θ) resp see (16) and (17)) Other param-eters are not changed The results are shown in Figure 4
Φ1
(Θ) = 1 + 1198751
(cosΘ) (16)
Φ2
(Θ) = 1 minus 1198751
(cosΘ) (17)
It can be seen from Figure 4(a) that compared with theanisotropic scattering the variation in the scattering phasefunctions causes the atmospheric temperature to change Butthe effect is not obvious Figure 4(b) shows the differencevalue between the atmospheric temperature fields and theisotropic scattering with different scattering phase functionsFrom the figure we can see that the forward and back scattershave opposite effect on the atmospheric temperature fieldsbut the effect degree is similar
33 The Influence of the Aerosol Particles Concentration Theaerosol particles transport with the tropospheric movementwill cause the aerosol particles concentrations to vary InFigure 5 we consider the influence of the variation in totalparticles number concentrations and the particles numberconcentrations are increased or decreased by 50 So theradiation characteristics of the aerosol particles are changedthe atmospheric energy balance is affected and then theatmospheric temperature fields are influenced From theresults it can be seen that the variation in aerosol particlesconcentrations can affect the atmospheric temperature butnot obviously compared with other factors
34 The Influence of the Height Distribution of Aerosol Parti-cles We investigate the effect of the medium layer height onthe aerosol particles size distribution that is the atmosphericradiation temperature fields change with the aerosol particles
Mathematical Problems in Engineering 7
Table 4 Complex refractive index of different wavelength for different aerosol types [19]
Figure 5 Influence of the aerosol concentration on atmospheric temperature
being in different height medium layers The calculationconditions are as follows the aerosol particles distribute inthe height of 0sim1 km 1sim3 km and 3sim9 km respectively Thecalculation results are in Figure 6
From the calculation results in Figure 6 it can be seenthat with the aerosol particles transporting to the highaltitude the atmospheric temperature is affected greatly ifthe radiation characteristics of the aerosol particles have nochanges With the distribution height of the aerosol particlesincreasing the medium temperature above themedium layerin which the aerosol particles exist increases greatly but thebelow temperature decreases obviously This phenomenonshows the scattering effect of the aerosol particles on the
solar radiation energy that is as the scattering effect of theatmospheric aerosols most of the solar radiation energy tothe ground is scattered back to the high altitude and cannotreach lower atmosphere which causes the lower atmospherictemperature to decrease to about 200K
4 Conclusion
We analyzed the influence of the aerosol particles charac-teristics on the atmospheric radiation transfer model Theresults showed the different effect of aerosol types on theatmospheric radiation The dust marine and water-soluble
Mathematical Problems in Engineering 9
0
10
20
30
40
50A
ltitu
de (k
m)
180 200 220 240 260 280 300Temperature (K)
0-1 km1ndash3 km3ndash9 km
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
Table 1 The main input parameters of the atmospheric radiation calculation
Parameters Values
Initial condition
Initial temperature 1198790
280KReference temperature 119879rf 1000KDimensionless time 119905lowast 1 times 10
6
Relaxation factor 119891re 001
Boundary condition
Terrestrial temperature 119879minusinfin
280KTerrestrial emissivity 120576
minusinfin
07Solar radiation temperature 119879
+infin
5762KAngle factor 119860
+infin 2
55844 times 10minus6
Media physical propertiesLayers of height 7 layers seen in Table 3Bands absorption coefficient 12 bands seen in Table 4Thermal conductivity 119896
119888
0023 Wsdotmminus1sdotKminus1
Aerosol properties Distribution function Multilognormal distributionsHeight distribution 0-1 km
Table 2 Multilayer atmospheric radiative transfer model
Layer number Thickness (km) Upper-lower limit ofheight (km) Density (kgm3) Dimensionless
thermal conductivity Refractive index
1 1 0sim1 115566 751119890 minus 8 102 2 1sim3 096517 589119890 minus 8 103 6 3sim9 060438 374119890 minus 8 104 8 9sim17 030815 143119890 minus 8 105 10 17sim27 002573 323119890 minus 9 106 11 27sim38 0010076 690119890 minus 10 107 12 38sim50 0002195 423119890 minus 10 10
Here 1198681198991198870
and 119868119899119887120591120582
are the emission radiation intensities oftwo adjacent layers and the initial values are obtained fromthe temperature field The total emission radiation inten-sity is determined by an iterative calculation of intensitiesfrom successive layers the last providing the required totalThe convergence condition of the multilayer model calcula-tion is decided by the radiative intensity 119868
119899
After the intensitycalculation has converged the iterative calculation of thetemperature field is then performed
3 Results and Discussion
According to statistics 988 of the total atmospheric massdistributes in the atmospherewithin 30 km above the groundand with the height increasing to 50 km the proportionreaches 9992Most of the atmospheric radiation (involvingsolar radiation absorption and scattering of the groundinfrared radiation and the emission of the atmosphere itself)occurs within certain atmospheric thickness Compared withthe earth radius (6378 km) the atmospheric thickness canbe regarded as one-dimensional radiative media Thereforewe treated the atmosphere within 50 km above the groundas a multilayer one-dimensional media distributing along theradial of the earth The upper surface of the medium layer isnear to space and is considered as a semitransparentmediumThe lower interface is close to ground and is considered asopaque interface The emissivity and radiative temperature
related to the ground are given For the inner of the mediumlayer we mainly consider the gas absorption and emissionof the radiation energy and the aerosol particles scatteringand absorption of the radiation energy and consider theatmospheric coupled radiation and conduction in the verticaldirection
The atmospheric convection is very important for theenergy transfer of the atmosphere But we mainly discussthe influence of aerosols on the atmospheric radiative energyand investigate the atmospheric radiative balance and thevariation in radiation characteristic which is not affected bythe atmospheric convection that acts only as the backgroundTherefore we ignore the influence of the atmospheric con-vection
The atmospheric radiation physical model is shown inFigure 1 and the atmospheric radiation transfer model isshown in Figure 2 The atmospheric internal heat source isset as 120601 = 0
The standard atmospheric condition is given inTables 1ndash4
Table 1 gives themain input parameters for the calculationof atmospheric radiation transfer
Table 2 gives the parameters associated with layersdivided along the vertical height The density of each layer isdetermined by the calculation of the pressure We consider(1205881205880
) = (1199011199010
) (the standard atmosphere pressure 119901 is
Mathematical Problems in Engineering 5
Δxn
S+infinSminus infin S2S1
T+infin
Pnminus1
Tg2
Mt + 1Mtsum nminus1
i=1Mi + In
nnn2n1
sum nminus1
i=1Mi
middot middot middot
M1 + 1 M1 + 2M1I1
Δx2
2
Δxn
2Δx1
Δx1
2
xS1
I1
xP1
I2x
P119899minus1
I119899
xS1
I+11x
P1
I+11x
P119899minus1
I+1119899
xS2
I+1119899
LnL2 + L3 + middot middot middot + Lnminus1L1
Tminus infin
P1
Tg1
h1 h2
10 2
01
Figure 2 Physical model of one plane-parallel slab divided into 119899 sublayers
180 200 220 240 260 280 300 3200
10
20
30
40
50
Alti
tude
(km
)
Temperature (K)
MarineSoot
Water-solubleDust
(a) Atmosphere temperature field (0ndash50 km)
Alti
tude
(km
)
0
2
4
6
8
10
180 200 220 240 260 280 300 320Temperature (K)
MarineSoot
Water-solubleDust
(b) Atmosphere temperature field (0ndash10 km)
Figure 3 Influence of aerosols species on temperature fields
1225 kgm2) with the temperature fluctuating in small rangeTherefore the equation is as follows
120588 =
1205880
1199010
sdot 119901 =
1225
101325
sdot 119901 = 120898 times 10minus5
sdot 119901 (15)
Table 3 gives the wavelength range of each spectral bandThe atmosphere optical thicknesses of the layers at differentspectral bands are obtained from the HITRAN databaseunder the standard atmospheric condition of America in1976 The aerosol optical thicknesses are decided based onthe Mie scattering theory using the typical aerosol particlessize distribution assumption and combining the complex
refractive index of different aerosol species Table 3 liststhe spectral optical thicknesses of the dust aerosol and thescattering albedo parameters in the height of 0-1 km Theother parameters are not given in this paper as there are toomuch data
Table 4 gives the complex refractive index of differentaerosols
31 The Influence of Aerosol Species Variation Figure 3 showsthe influence of four aerosol species (dust soot marine andwater-soluble) on the atmospheric temperature The aerosol
(b) Temperature difference betweendifferent scattering phase functionsand isotropic scattering
Figure 4 Influence of scattering phase function change on atmospheric temperature
Table 3 Spectral bands for the atmospheric radiative transfermodeland radiative characteristic parameters under the standard dustaerosol in the height of 0-1 km
particle size distribution function is the multilognormaldistribution
From the results in the figure we can see that as the dif-ferent scattering effect of the aerosol particles their influenceon the temperature is greatly different Compared with dustaerosol the marine and water-soluble aerosol particles havestronger scattering effect They will scatter upwards moreradiation energy and decrease the atmospheric temperatureThe soot particles have stronger absorption and their exis-tence will cause the atmospheric temperature increasing
32 The Influence of the Scattering Phase Function VariationThe aerosol particles scattering characteristics are considered
as forward and back scatter (the scattering phase functionsare Φ1
(Θ) and Φ2
(Θ) resp see (16) and (17)) Other param-eters are not changed The results are shown in Figure 4
Φ1
(Θ) = 1 + 1198751
(cosΘ) (16)
Φ2
(Θ) = 1 minus 1198751
(cosΘ) (17)
It can be seen from Figure 4(a) that compared with theanisotropic scattering the variation in the scattering phasefunctions causes the atmospheric temperature to change Butthe effect is not obvious Figure 4(b) shows the differencevalue between the atmospheric temperature fields and theisotropic scattering with different scattering phase functionsFrom the figure we can see that the forward and back scattershave opposite effect on the atmospheric temperature fieldsbut the effect degree is similar
33 The Influence of the Aerosol Particles Concentration Theaerosol particles transport with the tropospheric movementwill cause the aerosol particles concentrations to vary InFigure 5 we consider the influence of the variation in totalparticles number concentrations and the particles numberconcentrations are increased or decreased by 50 So theradiation characteristics of the aerosol particles are changedthe atmospheric energy balance is affected and then theatmospheric temperature fields are influenced From theresults it can be seen that the variation in aerosol particlesconcentrations can affect the atmospheric temperature butnot obviously compared with other factors
34 The Influence of the Height Distribution of Aerosol Parti-cles We investigate the effect of the medium layer height onthe aerosol particles size distribution that is the atmosphericradiation temperature fields change with the aerosol particles
Mathematical Problems in Engineering 7
Table 4 Complex refractive index of different wavelength for different aerosol types [19]
Figure 5 Influence of the aerosol concentration on atmospheric temperature
being in different height medium layers The calculationconditions are as follows the aerosol particles distribute inthe height of 0sim1 km 1sim3 km and 3sim9 km respectively Thecalculation results are in Figure 6
From the calculation results in Figure 6 it can be seenthat with the aerosol particles transporting to the highaltitude the atmospheric temperature is affected greatly ifthe radiation characteristics of the aerosol particles have nochanges With the distribution height of the aerosol particlesincreasing the medium temperature above themedium layerin which the aerosol particles exist increases greatly but thebelow temperature decreases obviously This phenomenonshows the scattering effect of the aerosol particles on the
solar radiation energy that is as the scattering effect of theatmospheric aerosols most of the solar radiation energy tothe ground is scattered back to the high altitude and cannotreach lower atmosphere which causes the lower atmospherictemperature to decrease to about 200K
4 Conclusion
We analyzed the influence of the aerosol particles charac-teristics on the atmospheric radiation transfer model Theresults showed the different effect of aerosol types on theatmospheric radiation The dust marine and water-soluble
Mathematical Problems in Engineering 9
0
10
20
30
40
50A
ltitu
de (k
m)
180 200 220 240 260 280 300Temperature (K)
0-1 km1ndash3 km3ndash9 km
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
Figure 2 Physical model of one plane-parallel slab divided into 119899 sublayers
180 200 220 240 260 280 300 3200
10
20
30
40
50
Alti
tude
(km
)
Temperature (K)
MarineSoot
Water-solubleDust
(a) Atmosphere temperature field (0ndash50 km)
Alti
tude
(km
)
0
2
4
6
8
10
180 200 220 240 260 280 300 320Temperature (K)
MarineSoot
Water-solubleDust
(b) Atmosphere temperature field (0ndash10 km)
Figure 3 Influence of aerosols species on temperature fields
1225 kgm2) with the temperature fluctuating in small rangeTherefore the equation is as follows
120588 =
1205880
1199010
sdot 119901 =
1225
101325
sdot 119901 = 120898 times 10minus5
sdot 119901 (15)
Table 3 gives the wavelength range of each spectral bandThe atmosphere optical thicknesses of the layers at differentspectral bands are obtained from the HITRAN databaseunder the standard atmospheric condition of America in1976 The aerosol optical thicknesses are decided based onthe Mie scattering theory using the typical aerosol particlessize distribution assumption and combining the complex
refractive index of different aerosol species Table 3 liststhe spectral optical thicknesses of the dust aerosol and thescattering albedo parameters in the height of 0-1 km Theother parameters are not given in this paper as there are toomuch data
Table 4 gives the complex refractive index of differentaerosols
31 The Influence of Aerosol Species Variation Figure 3 showsthe influence of four aerosol species (dust soot marine andwater-soluble) on the atmospheric temperature The aerosol
(b) Temperature difference betweendifferent scattering phase functionsand isotropic scattering
Figure 4 Influence of scattering phase function change on atmospheric temperature
Table 3 Spectral bands for the atmospheric radiative transfermodeland radiative characteristic parameters under the standard dustaerosol in the height of 0-1 km
particle size distribution function is the multilognormaldistribution
From the results in the figure we can see that as the dif-ferent scattering effect of the aerosol particles their influenceon the temperature is greatly different Compared with dustaerosol the marine and water-soluble aerosol particles havestronger scattering effect They will scatter upwards moreradiation energy and decrease the atmospheric temperatureThe soot particles have stronger absorption and their exis-tence will cause the atmospheric temperature increasing
32 The Influence of the Scattering Phase Function VariationThe aerosol particles scattering characteristics are considered
as forward and back scatter (the scattering phase functionsare Φ1
(Θ) and Φ2
(Θ) resp see (16) and (17)) Other param-eters are not changed The results are shown in Figure 4
Φ1
(Θ) = 1 + 1198751
(cosΘ) (16)
Φ2
(Θ) = 1 minus 1198751
(cosΘ) (17)
It can be seen from Figure 4(a) that compared with theanisotropic scattering the variation in the scattering phasefunctions causes the atmospheric temperature to change Butthe effect is not obvious Figure 4(b) shows the differencevalue between the atmospheric temperature fields and theisotropic scattering with different scattering phase functionsFrom the figure we can see that the forward and back scattershave opposite effect on the atmospheric temperature fieldsbut the effect degree is similar
33 The Influence of the Aerosol Particles Concentration Theaerosol particles transport with the tropospheric movementwill cause the aerosol particles concentrations to vary InFigure 5 we consider the influence of the variation in totalparticles number concentrations and the particles numberconcentrations are increased or decreased by 50 So theradiation characteristics of the aerosol particles are changedthe atmospheric energy balance is affected and then theatmospheric temperature fields are influenced From theresults it can be seen that the variation in aerosol particlesconcentrations can affect the atmospheric temperature butnot obviously compared with other factors
34 The Influence of the Height Distribution of Aerosol Parti-cles We investigate the effect of the medium layer height onthe aerosol particles size distribution that is the atmosphericradiation temperature fields change with the aerosol particles
Mathematical Problems in Engineering 7
Table 4 Complex refractive index of different wavelength for different aerosol types [19]
Figure 5 Influence of the aerosol concentration on atmospheric temperature
being in different height medium layers The calculationconditions are as follows the aerosol particles distribute inthe height of 0sim1 km 1sim3 km and 3sim9 km respectively Thecalculation results are in Figure 6
From the calculation results in Figure 6 it can be seenthat with the aerosol particles transporting to the highaltitude the atmospheric temperature is affected greatly ifthe radiation characteristics of the aerosol particles have nochanges With the distribution height of the aerosol particlesincreasing the medium temperature above themedium layerin which the aerosol particles exist increases greatly but thebelow temperature decreases obviously This phenomenonshows the scattering effect of the aerosol particles on the
solar radiation energy that is as the scattering effect of theatmospheric aerosols most of the solar radiation energy tothe ground is scattered back to the high altitude and cannotreach lower atmosphere which causes the lower atmospherictemperature to decrease to about 200K
4 Conclusion
We analyzed the influence of the aerosol particles charac-teristics on the atmospheric radiation transfer model Theresults showed the different effect of aerosol types on theatmospheric radiation The dust marine and water-soluble
Mathematical Problems in Engineering 9
0
10
20
30
40
50A
ltitu
de (k
m)
180 200 220 240 260 280 300Temperature (K)
0-1 km1ndash3 km3ndash9 km
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
(b) Temperature difference betweendifferent scattering phase functionsand isotropic scattering
Figure 4 Influence of scattering phase function change on atmospheric temperature
Table 3 Spectral bands for the atmospheric radiative transfermodeland radiative characteristic parameters under the standard dustaerosol in the height of 0-1 km
particle size distribution function is the multilognormaldistribution
From the results in the figure we can see that as the dif-ferent scattering effect of the aerosol particles their influenceon the temperature is greatly different Compared with dustaerosol the marine and water-soluble aerosol particles havestronger scattering effect They will scatter upwards moreradiation energy and decrease the atmospheric temperatureThe soot particles have stronger absorption and their exis-tence will cause the atmospheric temperature increasing
32 The Influence of the Scattering Phase Function VariationThe aerosol particles scattering characteristics are considered
as forward and back scatter (the scattering phase functionsare Φ1
(Θ) and Φ2
(Θ) resp see (16) and (17)) Other param-eters are not changed The results are shown in Figure 4
Φ1
(Θ) = 1 + 1198751
(cosΘ) (16)
Φ2
(Θ) = 1 minus 1198751
(cosΘ) (17)
It can be seen from Figure 4(a) that compared with theanisotropic scattering the variation in the scattering phasefunctions causes the atmospheric temperature to change Butthe effect is not obvious Figure 4(b) shows the differencevalue between the atmospheric temperature fields and theisotropic scattering with different scattering phase functionsFrom the figure we can see that the forward and back scattershave opposite effect on the atmospheric temperature fieldsbut the effect degree is similar
33 The Influence of the Aerosol Particles Concentration Theaerosol particles transport with the tropospheric movementwill cause the aerosol particles concentrations to vary InFigure 5 we consider the influence of the variation in totalparticles number concentrations and the particles numberconcentrations are increased or decreased by 50 So theradiation characteristics of the aerosol particles are changedthe atmospheric energy balance is affected and then theatmospheric temperature fields are influenced From theresults it can be seen that the variation in aerosol particlesconcentrations can affect the atmospheric temperature butnot obviously compared with other factors
34 The Influence of the Height Distribution of Aerosol Parti-cles We investigate the effect of the medium layer height onthe aerosol particles size distribution that is the atmosphericradiation temperature fields change with the aerosol particles
Mathematical Problems in Engineering 7
Table 4 Complex refractive index of different wavelength for different aerosol types [19]
Figure 5 Influence of the aerosol concentration on atmospheric temperature
being in different height medium layers The calculationconditions are as follows the aerosol particles distribute inthe height of 0sim1 km 1sim3 km and 3sim9 km respectively Thecalculation results are in Figure 6
From the calculation results in Figure 6 it can be seenthat with the aerosol particles transporting to the highaltitude the atmospheric temperature is affected greatly ifthe radiation characteristics of the aerosol particles have nochanges With the distribution height of the aerosol particlesincreasing the medium temperature above themedium layerin which the aerosol particles exist increases greatly but thebelow temperature decreases obviously This phenomenonshows the scattering effect of the aerosol particles on the
solar radiation energy that is as the scattering effect of theatmospheric aerosols most of the solar radiation energy tothe ground is scattered back to the high altitude and cannotreach lower atmosphere which causes the lower atmospherictemperature to decrease to about 200K
4 Conclusion
We analyzed the influence of the aerosol particles charac-teristics on the atmospheric radiation transfer model Theresults showed the different effect of aerosol types on theatmospheric radiation The dust marine and water-soluble
Mathematical Problems in Engineering 9
0
10
20
30
40
50A
ltitu
de (k
m)
180 200 220 240 260 280 300Temperature (K)
0-1 km1ndash3 km3ndash9 km
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
Figure 5 Influence of the aerosol concentration on atmospheric temperature
being in different height medium layers The calculationconditions are as follows the aerosol particles distribute inthe height of 0sim1 km 1sim3 km and 3sim9 km respectively Thecalculation results are in Figure 6
From the calculation results in Figure 6 it can be seenthat with the aerosol particles transporting to the highaltitude the atmospheric temperature is affected greatly ifthe radiation characteristics of the aerosol particles have nochanges With the distribution height of the aerosol particlesincreasing the medium temperature above themedium layerin which the aerosol particles exist increases greatly but thebelow temperature decreases obviously This phenomenonshows the scattering effect of the aerosol particles on the
solar radiation energy that is as the scattering effect of theatmospheric aerosols most of the solar radiation energy tothe ground is scattered back to the high altitude and cannotreach lower atmosphere which causes the lower atmospherictemperature to decrease to about 200K
4 Conclusion
We analyzed the influence of the aerosol particles charac-teristics on the atmospheric radiation transfer model Theresults showed the different effect of aerosol types on theatmospheric radiation The dust marine and water-soluble
Mathematical Problems in Engineering 9
0
10
20
30
40
50A
ltitu
de (k
m)
180 200 220 240 260 280 300Temperature (K)
0-1 km1ndash3 km3ndash9 km
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
Figure 5 Influence of the aerosol concentration on atmospheric temperature
being in different height medium layers The calculationconditions are as follows the aerosol particles distribute inthe height of 0sim1 km 1sim3 km and 3sim9 km respectively Thecalculation results are in Figure 6
From the calculation results in Figure 6 it can be seenthat with the aerosol particles transporting to the highaltitude the atmospheric temperature is affected greatly ifthe radiation characteristics of the aerosol particles have nochanges With the distribution height of the aerosol particlesincreasing the medium temperature above themedium layerin which the aerosol particles exist increases greatly but thebelow temperature decreases obviously This phenomenonshows the scattering effect of the aerosol particles on the
solar radiation energy that is as the scattering effect of theatmospheric aerosols most of the solar radiation energy tothe ground is scattered back to the high altitude and cannotreach lower atmosphere which causes the lower atmospherictemperature to decrease to about 200K
4 Conclusion
We analyzed the influence of the aerosol particles charac-teristics on the atmospheric radiation transfer model Theresults showed the different effect of aerosol types on theatmospheric radiation The dust marine and water-soluble
Mathematical Problems in Engineering 9
0
10
20
30
40
50A
ltitu
de (k
m)
180 200 220 240 260 280 300Temperature (K)
0-1 km1ndash3 km3ndash9 km
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
Figure 6 Influence of temperature fields on different aerosoldistribution altitude
aerosols had negative effects on the atmospheric absorptionbut the soot aerosol could increase the atmospheric absorp-tionThe scattering phase functions affected the ground tem-perature The variation of the aerosol particles distributionheight could determine the atmospheric temperature that isthe ground temperature decreased greatly with the aerosolparticles existing in the high altitude stable
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 51121004) and the National NaturalScience Foundation of China (Grant nos 51176040 and51276193) A very special acknowledgement is made to theeditors and referees who made important comments toimprove this paper
References
[1] M Schulz C Textor S Kinne et al ldquoRadiative forcing byaerosols as derived from the AeroCom present-day and pre-industrial simulationsrdquoAtmospheric Chemistry and Physics vol6 pp 5225ndash5246 2006
[2] U Lohmann and J Feichter ldquoGlobal indirect aerosol effects areviewrdquo Atmospheric Chemistry and Physics vol 5 pp 715ndash7372005
[3] I Koren Y J Kaufman L A Remer and J V MartinsldquoMeasurement of the effect of Amazon smoke on inhibitionof cloud formationrdquo Science vol 303 no 5662 pp 1342ndash13452004
[4] W W Qing R S Chen and W M Sun ldquoEstimation of globalradiation in China and comparison with satellite productrdquoEnvironmental Earth Sciences vol 70 no 4 pp 1681ndash1687 2013
[5] X S Wang ldquoMetal geochemical and mineral magnetic char-acterization of the lt25 lm fraction of urban soils in Xuzhou(China)rdquo Environmental Earth Sciences vol 71 no 8 pp 3491ndash3501 2014
[6] L David B Nacer B Pascal and J Gerard ldquoTransient radiativeand conductive heat transfer in non-gray semitransparent two-dimensional media withmixed boundary conditionsrdquoHeat andMass Transfer vol 42 no 4 pp 322ndash337 2006
[7] S Dembele K L M Lima and J X Wen ldquoAssessment of anunstructured exponential scheme discrete ordinates radiationmodel for non-gray mediardquoHeat and Mass Transfer vol 47 no11 pp 1349ndash1362 2011
[8] P Talukdar F V Issendorff D Trimis and C J SimonsonldquoConduction-radiation interaction in 3D irregular enclosuresusing the finite volume methodrdquo Heat and Mass Transfer vol44 no 6 pp 695ndash704 2008
[9] C-Y Wu ldquoMonte Carlo simulation of radiative transfer in arefractive layeredmediumrdquoHeat andMass Transfer vol 46 no6 pp 607ndash613 2010
[10] J H Jeans ldquoThe equations of radiative transfer of energyrdquoMonthly Notices Royal Astronomical Society vol 78 pp 28ndash361917
[11] B DavisonNeutron TransportTheory Oxford University PressLondon UK 1958
[12] V Kourganoff Basic Methods in Transfer Problems Dover NewYork NY USA 1963
[13] H Khouaja D R Edwards and N Tsoulfanidis ldquoSphericalharmonicsmdashfinite element treatment of neutron transport incylindrical geometryrdquo Annals of Nuclear Energy vol 24 no 7pp 515ndash531 1997
[14] E W Larsen G Thommes A Klar M Seaıd and T GotzldquoSimplified 119875
119899
approximations to the equations of radiative heattransfer and applicationrdquo Journal of Computational Physics vol183 no 2 pp 652ndash675 2002
[15] R GMcClarren and C D Hauck ldquoRobust and accurate filteredspherical harmonics expansions for radiative transferrdquo Journalof Computational Physics vol 229 no 16 pp 5597ndash5614 2010
[16] M F Modest Radiative Heat Transfer Academic Press Ams-terdam The Netherlands 2nd edition 2003
[17] R Siegel and C M Spuckler ldquoRefractive index effects onradiation in an absorbing emitting and scattering laminatedlayerrdquo Journal of Heat Transfer vol 115 pp 194ndash200 1993
[18] Q Z Yu Principles of Radiative Heat Transfer Harbin Instituteof Technology Press Harbin China 2000
[19] G Y Shi Atmospheric Radiology Science Press Beijing China2007
