大气物理第四章课后习题答案.pdf

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1、Chapter44.44Showthatforoverheadparallelbeamradiationincidentonanisothermalatmosphere,inwhichthemixingratiooftheabsorbinggas,r,andthevolumeabsorptioncoefficient,kλ,arebothindependentofheight,theabsorptionperunitvolume(i.e.,dIλ/dz)isstrongestatthelevelofu

2、nitopticaldepth.Solution:FromEq.(4.17)ifrandkλarebothindependentofheight,dIλ=Iλ∞Tλkλρr(1)dzwhereIλ∞istheintensityoftheradiationincidentonthetopoftheatmosphere,Tλ=Iλ/Iλ∞isthetransmissivityoftheoverlyinglayer,andρisthedensityoftheambientair.From(4.28)fo

3、ranoverheadbeam−τTλ=eλ(2)Foranisothermalatmosphere,p∝ρ,thereforefrom(3.26)ρ=ρe−z/H(3)0where,Histhescaleheight.Substituting(2)and(3)into(1)weobtain,dIλ−z/H−τ=Iλ∞kλrρ0eeλ(4)dzFrom(4.32)Z∞τ=(kρr)e−z/Hdzλλ0z=H(krρ)e−z/H(5)λ0Using(5)toexpresse−z/Hin(4)inte

4、rmsofopticaldepth,weobtaindIλIλ∞−τ=τλeλdzHAtthelevelwheretheabsorptionisstrongest,µ¶ddIλIλ∞d−τλ=(τλe)=0(6)dzdzHdzPerformingthedifferentiationontheright-handsideof(6),weobtaindτ−τλeλ(1−τλ)=0dz14fromwhichitfollowsthatτλ=1.Althoughthisresultisstrictlyap-p

5、licableonlytoanisothermalatmosphereinwhichkλandrareinde-pendentofheight,itisqualitativelyrepresentativeofconditionsinplan-etaryatmospheresinwhichthemixingratiosoftheprincipalabsorbingconstituentsdonotchangerapidlywithheight.Itwasoriginallydevel-opedby

6、SidneyChapmanforunderstandingradiativeandphotochemicalprocessesrelatedtothestratosphericozonelayer(seeSection5.7).4.48Provethattheexpressionfortheheatingrate,asapproximatedbycoolingtospace,namely,eqn.(4.54)intext:µ¶Z1dT2π−τν/µ=−kνrBν(z)edµ(1)dtνcp0can

7、beintegratedoversolidangletoobtaineqn.(4.56)inthetext,thatis:µ¶dTπe−τν/µ=−kνrBν(z)(2)dtνcpµwhereisasdefinedinEq.(4.47)inthetext.Solution:Equation(1)canberewrittenintheformµ¶Z1dTπ−τν/µ=−kνrBν(z)×2edµ(3)dtνcp0Theintegralin(3)canberewrittenZ1Z1de−τν/µ2e−τ

8、ν/µdµ=2(−µ)dµ00dτνZ¡¢1de−τν/µµ=−2dµ0dτνExchangingtheorderofthedifferentiationandtheintergrationyieldsZ1∙Z1¸2e−τν/µdµ=d2e−τν/µµdµ0dτν0=e−τν/µ/µSubstitutingfortheintegralin(3)yields(2).4.51ShowthatSchwarszchild’sequation[(4.41)inthetext,whichisdI