MASSACHUSETTS INSTITUTE OF TECHNOLOGY HAYSTACK OBSERVATORY WESTFORD, MASSACHUSETTS
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1 VSRT MEMO #05 MASSACHUSETTS INSTITUTE OF TECHNOLOGY HAYSTACK OBSERVATORY WESTFORD, MASSACHUSETTS Fbrury 3, 009 Tlphon: Fx: To: VSRT Group From: Aln E.E. Rogrs Subjct: Simplifid rdition blnc ndd to xplin Globl wrming nd uppr tmosphric cooling. 1] Plnt without n tmosphr Considr plnt without n tmosphr. In this cs th nrgy from th Sun is blncd by th rdition from th plnt s surfc out into spc. ( 1 ) π σ π S A r = T r (1) Whr S = solr flux 1 = 1370 wtts/m r = rdius of th Erth A = Erth s lbdo frction of solr nrgy rflctd ~ σ = Stfn-Boltzmnn constnt = wtts/m /K πr is th cross-sction r of th pln wv rys from th Sun intrcptd by th Erth. π r is th totl surfc r of th Erth from which th Erth s ht is lost into spc going out rdilly in ll dirctions. σ T is known s th Stfn-Boltzmnn lw nd is th intgrl of th blck body rdition givn by Plnck s lw 3 hν 1 B = () h / KT c ( ν 1) ovr ll frquncis (or quivlntly ll wvlngths) nd ll dirctions. B = wtts/m /solid ngl/hz h = Plnck constnt ν = Frquncy Hz c = vlocity of light K = Boltzmnn s constnt If w solv qution (1) w gt ( 1 ) ( ) T S A σ T = 9 K = - C A tmprtur wll blow th Erth s vrg tmprtur. = (3) 1 Th vrg ovr th sphr norml to th surfc is 1370/=3 W/m 1
2 ] Adding blnkt wrms th Erth Now if w dd n tmosphr which bsorbs som of th rdition lving th Erth, whil hrdly bsorbing ny of th Sun s rdition th Erth will wrm up to mintin th nrgy blnc. Constitunts of th tmosphr, lik crbon dioxid, r known s grnhous gss bcus thy hv spctrl lins which bsorb in th infrrd prt of th spctrum (s figur 1) whr th rdition from th Erth domints whil thy r quit trnsprnt in th visibl prt of th spctrum whr th Sun s nrgy is concntrtd. Ths grnhous gss wrm th Erth in much th sm wy s blnkt kps you wrm in bd t night. 3] In wrming th Erth th uppr tmosphr cools To look in mor dtil t wht hppns whn w dd grnhous gs w nd to considr th Rditiv trnsfr in lyr of th tmosphr contining th rgion whr th infrrd is bsorbd. Whn gs contins spctrl lins ths lins not only bsorb rdition but lso mit rdition in th sm frquncy bnd. If wv trvls through n bsorbing gs th powr in this wv dcrss xponntilly. Tht is th powr is multiplid by th fctor τ, whr τ is th opcity of th gs. Equivlntly th frction 1 τ. Howvr, this lost powr is compnstd by mission of powr lost is ( ) from th gs which is proportionl to ( 1 τ ). If th tmprtur of th gs is th sm s th tmprtur of th Erth s surfc thn th nrgy lost by th rdition lving th Erth is xctly compnstd by n qul mount of nrgy from th gs going out into spc. Th mission from th lyr in th tmosphr gos out in both dirctions, up nd down, s illustrtd in Figur. Now rwriting th nrgy blnc (s illustrtd in Figur ) qutions for 3 css rmoving th common fctor π r : A] Abov th tmosphr S( 1 A) = ( 1 f ) σt + fσt () whr T = Erth surfc tmprtur T = tmosphr tmprtur f = frction bsorbd in tmosphr = ( 1 τ ) B] In th tmosphr f σt + q= fσt (5) whr q is th powr bsorbd by lyr in th tmosphr in th ultrviolt frquncy rng. q is vry smll in th troposphr but is significnt in th strtosphr nd msosphr whr O nd O 3 hv spctrl lins in th uv.
3 C] Blow th tmosphr S ( 1 A) + f σt q= σt (6) Not: Eqution 6 cn lso b drivd from qutions nd 5 s ths 3 qutions r not indpndnt. Solving th qutions for T nd T w gt som rsults r s follows T T ( 1 ) σ ( f ) S A q = 1 1 ( ) 1 σ T + q f = (7) (8) f q (w) T (K) T (K) Th rsults for q = 0 r rlvnt to th troposphr nd show tht incrsd opcity rsults in wrming of both th Erth nd th tmosphr. Th rsults for q = 50 w r pproprit for lyr much highr in th tmosphr. In this cs T should b considrd s th lowr tmosphr tmprtur. In this cs th uppr tmosphr tmprtur T, dcrss with incrsd opcity. In rlity th nrgy budgt involvs th dditionl procsss s illustrtd in Figur 3. This simplifid nlysis mks th following simplifictions: 1] Only singl uniform lyr is considrd. A mor ccurt modl should considr mny lyrs of diffrnt tmprturs. ] Only rditiv trnsfr is considrd. This is only good pproximtion for th uppr lyrs of th tmosphr for which circultion is lss importnt. 3] Th powr is ssumd to b proportionl to T. An ccurt modl nds to tk into ccount th ctul bsorption lin shps nd frquncis. Howvr it is good ssumption tht th powr mittd will b monotoniclly incrsing function of tmprtur s in qution () which is ll tht is rquird to show tht th Erth will wrm whil th uppr tmosphr cools in th cs of q > 0. 3
4 In ddition th chngs in tmprtur which r xpctd to rsult from chngs in opcity of th grnhous gss nd to tk into ccount fdbck procsss. For xmpl mor CO my lso rsult in th mlting of snow nd ic which in turn will rduc th lbdo of th Erth rsulting in furthr incrs in th vrg tmprtur. Th dtils of ll ths fdbck procsss long with mor ccurt modl of globl wrming is givn in Th Physics of Atmosphr, 3 rd Edition by John Houghton, publishd by th Cmbridg Univrsity Prss, 00. Figur 1. Rdition spctrum trnsmittd by th Erth s tmosphr
5 Figur. Enrgy blnc for lyr in th tmosphr Figur 3. Illustrtiv nrgy budgt for th Erth from Th Physics of Atmosphrs, by John Houghton. 5
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