Etude de l’activité électrique des systèmes orageux à l’aide du modèle Méso-NH Christelle Barthe To cite this version: Christelle Barthe. Etude de l’activité électrique des systèmes orageux à l’aide du modèle Méso-NH. Physique [physics]. Université Paul Sabatier - Toulouse III, 2005. Français. �tel-00011490� HAL Id: tel-00011490 https://tel.archives-ouvertes.fr/tel-00011490 Submitted on 30 Jan 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. 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GI}esv~Ôu5n`ØeuA{po>ßäÞáy>u5n`wvÞ3×vn,y PQÞ.å0u=wvsØeu5×C{< : |)or×wvsÞ3~v~Ô}eâ{`wvu!|)Úy Þ3×Ðeu"u{Twvu y Þwu5×vnAor{Úwußhl:Þ3pÐ3uwvuy Þ ØáÞ3~Ôu5s w PQu5Þ3s ÖæyzPQu5Þ3s\×ts:Þ3ÐeusvnpuáÖ_y Þ~y>svoru3ÖÊy>ÞÐey>Þ3ß5u ~vpor|Þáo>puáÖ0y>Þ1×vu5orÐeu3Öu{Ty>u=ÐeAÚ5nAo>y L L 6 AOGI}esv~Ôu5nIØ3u5â{o>ß5Þ3y>unIw:Þá×vny PQÞ.å0u wsØeu×t{ : |o>×)wvu5nI{Þ3s0å)w P Ú5ßhl:Þá×vÐeu=wvu"ßhlvÞ3pÐeu ~:Þ3`yru5n`~vA}0ßu5npnAsvn ×}e×0è7or×vwvsvß{oÙÕ¢nu{To>×wvsvß{porÕ¢n L 6 C»GI}esv~Ôu5nLØeuA{orßäÞ3yru5nLw:Þ3×nbyzP Þ¥åu,wvsØeu×C{wvu,y Þ=wvu×vnpoÙ{ÚTwvu,ßhl:Þ3AÐeuT{p}3{hÞáy>u : |)or× Ö ?&: |)or×u{ C : |o>× L L 6#7 JæØe}eyrs{or}e×Â{u|)~Ô}epuy>yruxwvuy ÞÕ¢AÚ5qtsvu×vß5u)wvu5n!Ú5ßy Þ3orpn"u{wvuy PQÞ3yr{por{swvuwuwÚ5ß5yru5×0è ßhlvu5|u5×C{ L L ;: JæØe}eyrs{or}e×·{u|)~Ô}epuy>yru"wvsßhl:Þ3|~Úy>uß{Ao>qtsvu!|)Þ.å0o>|1sv| L <6 E o ÞáÐeÞ3||u5nÊwu`npÚ5~vÞ3Þá{po>}e× wuTßhlvÞ3pÐeu`nAu5yr}e× @ [email protected] @ D9 6#7 A;C Ö c @3-D8ÔBD6/ 64Ê@3< 6)7&726 u{ c @3-08ÔBD6/ 64_*,6 g 6#7&7 C L L GI}esv~Ôu5n!ØeuA{orßäÞ3yru5nw:Þ3×vny PQÞ.å0uwvsØ3u5×C{ : |)or×§wvu)y>Þwvu5×vnAor{pÚØ3}ey>sv|o>qtsvuwvu ßhl:Þ3AÐeu{}á{hÞ3yru~Ô}esv1HÛZ=Z"ÑÖÊHÛZ=Z"Ñ"o>×vw ÖæW0Z,K R ÖLW0Z,KXR "or×vw Ö_WDZ"Y 7 CÖ u{XW0Z Y 7 C"o>×vw L L = 6 W0}e×vwvÞ3Ðeu=o>×or{o>Þ3y[Þ3w:Þ3~{pÚ!wvu c [email protected]@3/h+ g )[email protected]< ;:&:&: L = S7|ÞáÐeu5nTnÞá{pu5yry>or{pu!w:Þ3×vnXy>u!Ø0ornpo#v y>uwvuxyzPQ}3Þ3Ðeu!wvs 6#: A7 svory>y>u{ 6#7&7&? y>}eAnXwuxy ÞßäÞá|è ~:Þ3Ðe×uW0H,JLMTZTF ;: KTHTGÆu{ :<6#: KXHTG L = B GI}esv~Ôu5nlv}eAo> }e×C{hÞ3yru5n 6#: êD| w P Þ3yÙ{oÙ{svwvuwvu y>ÞØDor{unpnAuØ3u5A{po>ß5Þ3y>uu{wvs Øeu×C{ w:Þ3×vn,y>Þ1ß5}esv~Ôu 6 l u{ l0B : wvu!npor|1svy>Þá{or}e× L 6&6 6#: #6 : = 6#:;? 6#: A 6#:;7 6#:;7 6&6;6 6&6 6&6 = 6&6)? 6&6)? 6&6)7 6 : 6 !6 6 6 6 B 6 A 6 B&A 6 B;C 6B7 * * = IG }esv~Ôulv}epo#>5}e×C{Þ3y>u 6 [email protected]=§êt| w PQÞ3yÙ{or{psvwvuwvuy>Þ{pu5|~[ÚÞá{psvpu~Ô}3{u×C{o>uy>yru3Öæwvu y P Ú5×vupÐeoru"ß5or×vÚ{po>qtsvu={s vsvyru5×C{u!u{Xwu5n,o>nAu5×C{A}e~[un L [email protected]= JæØe}eyrs{or}e×Ä{pu5|~[}eAu5yry>uwvs|)Þ.å0o>|1s| w P Þánpß5u×vw:Þ3×ß5u3Ö`ws|Þ¥åor|1sv| wvu nps+npoÙè wvu5×ß5u3Öwvs|)Þ.å0o>|1s| wvu!ßhl:Þ3|~Ú5yru5ß{pporqDsuu{Twvu!y Þ1y>}3×vÐesvusv`wvu5n,Úß5y Þáo>pn L = ? GI}esv~Ôu5nØ3u5A{po>ß5Þ3y>unw:Þ3×vn`yzP Þ¥åu=ws·Øeu5×C{TwvsÞá~v~[}3A{`wvu=|Ú5y>Þ3×vÐeuXwvsÐeAÚ5npory Ö0wvu"y Þ ×vu5orÐeu3Öwvu!y Þ1Ðey>Þ3ß5u=u{Xwuy>Þ1~vy>so>u L L [email protected] JæØe}eyrs{or}e×·{u|)~Ô}epuy>yru"wvs{hÞ3s0å·wvu~vpÚß5o>~or{hÞ.{o>}3×·o>×vnâ{hÞ3×C{Þ3×vÚ5uÞ3snA}ey L = C GIsv|1svyrn`wvu!~vpÚß5or~vor{Þá{or}e×Þ3snp}ey L L = 7 E u5×npor{pÚ!w PQÚß5y>Þ3o>An`npo>|xsvy>Úu"~:Þ3 Únp}áè7R Ñ(u{T}IvnAu5âØeÚ5u L = 6#: Y_p}::yrnØeuA{orßäÞ3sDåo>×voÙ{o>Þ3s0åwvuG`F u{Tw P F L = 6&6 GI}esv~Ôu5nbØeu5â{orßäÞ3yru5nLw:Þ3×vn^y PQÞ.å0u,wvsØeu×C{wvunIÞá~v~[}3A{nLwvuX|)Úy Þ3×Ðeu`wvsG`F B ?&:&: n,u{ 7&:&:;: n L L = 6 GI}esv~Ôu5n`ØeuA{po>ßäÞáy>u5n`wvÞ3×vn`y PQÞ.å0u=wvsØeu×t{Twvun,Þ3~v~Ô}eâ{n`wvu!|Ú5y Þá×vÐeu"wvuF B ?&:&: n,u{ 7&:&:;: n L L = 6 B»GI}esv~Ôu5nÛØeu5â{orßäÞ3yru5nÊwvÞ3×vnÊyzP Þ¥åubwvsxØ3u5×C{_wvsxÞ3~v~Ô}eâ{Êwvu^|)Úy Þ3×ÐeuLwvuÑ GÑ"F1Ö3Ñ F u{TÑXR"F 7&:&:;: n L L = 6 GI}esv~Ôu5n`ØeuA{po>ßäÞáy>u5n,wvun,Þ3~v~Ô}eâ{n,wvu!|)Úy Þ3×Ðeu"wvunT}äåDmDwvu5nXw PQÞ>5}3{pu ,B ?&:&: n L = 6 O = GI}esv~Ôu5n`ØeuA{po>ßäÞáy>u5n,wvun,Þ3~v~Ô}eâ{n,wvu!|)Úy Þ3×Ðeu"wvunT}äåDmDwvu5nXw PQÞ>5}3{pu 7;:&:&: n L = 6#? GI}esv~Ôu5n)lv}epo#>5}e×C{Þ3y>unwu5npÞ3~v~Ô}eA{pn)wu|Ú5y Þá×vÐeuwvu5nNo R"FXåºu{wvu y ÞØDor{unpnAu ØeuA{orßäÞ3yruL B ?&:&: n,u{ 7&:;:&: n L L = 6 O A GI}esv~Ôu5n)lv}epo#>5}e×C{Þ3y>unwu5npÞ3~v~Ô}eA{pn)wu|Ú5y Þá×vÐeuwvu5nNo R"FXåºu{wvu y ÞØDor{unpnAu ØeuA{orßäÞ3yruL wvo9[8 Ú5Au5×C{unXÞáyr{oÙ{svwu5n`u{Twvo 8 Úpu×t{pu5n,Ú5ßhlÚäÞ3×vßu5nT{u5|~Ô}epuy>y>un( L = 6 O C JæØe}eyrs{or}e×{u5|~Ô}epuy>y>u=wuy>Þyr}e×vÐesu5sv,wvun Ú5ßy Þ3orpn`u{ wvuy Þ |)Þ3npnAuwuxR F ~p}áè wvsvoÙ{n`~:Þ3`yru5n,Ú5ßy Þ3orpn? L L = 6#7 Y_p} 7Auß{or}e×wvu!y Þ1ßu5y>yrsvy>u ∆x Ö ∆y L L L = ;: JæØe}eyrs{or}e×Â{u|)~Ô}epuy>yruxwvuy Þ·wu5×vnAor{Ú wu +vs0å§w:Þ3×nyzPQu×vß5yrsv|u1wvu y PQÞ3or5ÖÔwvs G`F1Ö wvu!y P F Ö:wvunXR F wvu=Õ¢}e×vwu{,wvu!y>Þ {p}3{hÞáy>or{pÚ wvun R F L = <6 JæØe}eyrs{or}e×{pu5|~[}3pu5yry>u=wvuy Þwvu×vnpoÙ{Úxwu +:s0å w:Þá×vnXyzPQu×vß5yrsv|uwu1Ñ"GÑ F1Ö[Ñ F 6 : 6 : 6 6 6 6 6 6 = 6 ? 6 A 6 7 6=6 6= 6= 6= 6 =;= 6 =&= 6 =;A u{TÑXR"F L L 6#?;: 3 3 2 2 2 2 3 x 3 x 3 x 6 , 6 bU Þ3y>usvpnÎwvunÝ~vÞ3Þ3|{ppun α Ö ν Ö.ÞÖ Ö¥ß3Ö¥w ÖáG Öäåu5×Õ¢}e×vß{po>}e×!wvs{7mD~[u^w PQlCmDwvA}e|)Ú{Ú}epu A IG }0uF$ß5o>u×C{n,wvu"[email protected]/AB64 g 6#7 C ? L = : B Ub3Þ y>usvpn`wu5n,ß5}e×nA{hÞá×t{pu5n ] Öv| u{X×s{po>yro>npÚu5n`w:Þá×vn,y>u5n,ÚqDsvÞá{or}e×vn u{ C = J_qDsvÞá{or}e×vnbwvu {pÞ3×vnâÕ¢u5A{^wvu"ßhlvÞ3pÐeuX~[}3sv^y>Þ~:ÞáÞ3|Ú{ppo>npÞá{or}e×wvu c @3-D8ÔBD6/ [email protected]< 6)7&726 L L = ? 6 MTÚnpsv|Ú=wvunT{pu5nA{pnXwunAu5×vnAov ory>oÙ{Ú u 8 uß{psvÚ5nXnpsv,y>un`~:Þ3pÞ3|){Au5n`Õ¢Þ3ß{hÞ3yrn. L 6&6 B MTÚnpsv|Ú=wvunXß5Þ3Þ3ß{Úpo>nâ{orqDsu5n,wvu5n,Úß5y Þáo>pn`~Ô}esv`ßhlvÞ3qtsvunAo>|1sy Þá{po>}e× L 6 B B HÊÞ6v y>u5Þ3spÚ5ß5Þ3~voÙ{svy>Þá{oÙÕÎwvu5n,pÚnpsvyÙ{hÞá{pn,wvu §@38 6<;<Ý[email protected]< ;:;: = L 6 ? = 6 MTÚnpsv|Úwvu5n)u5nA{po>|)Þá{or}e×vn wvunnA}esvpßu5n)u{)wvu5n)o>×vßu5A{por{psvwvu5nwvu ~vp}Dwvsvß{o>}3×Äwvu R"F yru5n,~vyrsvn`o>|~[}3A{hÞá×t{pu5n L L 6 B = GI}0uF$ß5o>u×C{n,wvuÑXu×vAmu 8 uß{oÙÕ¢n K ~Ô}esv`yru5n`u5nA~[ß5un np}3y>s+vyru5n{pÞ3oÙ{Ú5un( L 6 B ? = B HÊÞ6vy>u5Þ3s)ß}e|~:Þ3Þ.{orÕÔwvun^wu5×vnAor{Ún^wu +:sDå}I{u×tsvu5n^~:Þ3Ly>un^}Inpu5âØáÞá{or}e×vnLu{b~:Þ3 y>un`npo>|xsvy Þá{po>}e×n< L L 6 = C x ∗ 6B 6 J_ß5y Þáo>pnu{{}e××vu5ApuLnp}e×C{Ûy>un|Þá×vorÕ¢unA{hÞ.{o>}3×vn ×:Þá{psvpuy>yru5n u{Û{ppnÛnA~[uß{Þ3ß5svy>Þ3o>Au5nwvuLyzPQÚy>u5ß{Ao>ß5oÙ{Ú áÞ {p|)}enA~vlvÚpo>qtsvu`qtsvoÞáß5ß5}3|)~:ÞáÐe×vu5×C{Lyru5næÚØeÚ5×u5|u5×C{n^}eÞ3Ð3u5s0åxØ0or}ey>u×C{n&eNÛÞ"~y>sv~:ÞáA{Lwvs {u|)~n5Öey>Þ Õ¢}esvwpu^Õ¢pÞ3~v~ÔuIwvunÊßovyru5nÊnpÞ3×vnÝß5}e×vnAÚ5qtsvu×vß5u,~Ô}esvÊyru5nÛlv}e||)unÝu{æyru5n o>u5×n&.Z o>×npo Ö¥ßhl:ÞáqDsu`Þ3×v×vÚu u5×xé:Þ3×vßu3Ö¥}e×pu5ßu5×vnAuu5×CØDo>A}e× |)ory>yro>}e×vn[wvubß5}esv~nÊwvuLÕ¢}esvwvAuÞ3yr}epnqts P }e×1unA{po>|ubqtsvu^y>unÝÚß5y Þáo>pn np}3×t{wvÚß5yru5×vßhlvÚnxwv$u = : 6#:&: Õ¢}eorn=~:Þ3!npuß5}e×vwu)w:Þá×vn!y>u |}e×vwv6u ÛW0sv!y>u1{pu5pAor{p}eo>AuxÕ¢Þ3× *äÞ3orn=}e× wvÚ~vy>}eAu,ßhl:Þ3qtsvuXÞ3×v×vÚuXyru`wvÚ5ß5nIwuT~y>svnAo>u5spnLwvo >5Þ3o>×u5n_wvuT~[upnA}e×v×vun^u{^y>Þ=~[uA{u,wvu ;: :&:&: { {un wv9u ÔÚ{Þ3o>yá~:Þ3 Õ¢}esvwp}eoru5|u5×C"{ 5E s!~[}eor×C{wvuæØDsv9u ::×:Þ3×ß5o>uu{Îw PQÞ3~vA5nÎsv×!Þá~v~[}3A{wvu_yzPQSaRXJLMTSâW [nAsv y Þ`Y_A}3{uß{or}e×=wvuLy PQJ_×CØ0orp}e××vu5|u5×C{äÖy>u_ß} {ÛÞ3×v×tsvuyCwvu5nÎwv}3|)|)Þ3Ðeun wv}e|u5nâ{orqDsu5nÎo>×vwu5|×vo>nAÚ5n ~:Þá y>unÝß}e|~:Þ3Ðe×voru5nÎw PQÞ3npnAsvÞá×vß5u^npuIßhl9o 8[pubu5×xßu5×C{hÞáo>×vunÊwvubêto>y>}eJLsp}enÛÞäØeuß ;: :&:&: npo>×o>nA{ppunÝwvÚß5y>Þ3pÚn wv}e×C{ 6 = :&:&: o>×vßu5×vwvoru5n1u{1wvun1wvunA{Asvß{po>}e×n1wvu·|o>yry>o>upn=wvuß5}e|~{usvpnÚ5yru5ß{pporqDsu5n1nAsvor{puÞ3s0å npsA{u×vnpor}e×vn=u{!npsvAo>×C{u×vnpoÙ{Ú5nÖ wvu ßhlvu|)or×vÚ5un5Ö ~CmDy 3×vu5n=u{!ß5y>}Dßhlvup&n JL× ::×§y Þ)Õ¢}eswvpu Au5~vAÚ5npu×C{u sv×vu,ß5Þ3svnpuTnporÐe×vo ::ßäÞá{porØ3u`w P Þáß5ß5orwvu5×C{pn^|)Þ 7Au5spnLu5×|)ory>oru5s1or×vwvsvnâ{poru5y eJ_y>yru{}esvßhluX~vAo>×vßo>~:Þáy>u5|u5×C{ y>unIÞáß{oÙØDor{Ún^wu Þ $)×vÞ3Ðeu wuX~ÔÚ{pp}eyru3Öty>unIpÚnpuAØe}3o>pn^wvu nâ{}Dßhê.Þ3Ðeu u{Iß5Þ3×:Þ3yro>npÞá{or}e×vnLu{^y P o>×vwsvnA{ppo>u ßhlvor|)orqDsuwvu :Þ3nAu DGIu,~Ô}enpnAo vyruy>oru5×1u×t{ppuTÞ3Úp}enA}ey>n_w P }eAo>Ðeor×vuIsv :Þ3or×vu~vA}0wvsor{n_~vÞ3æy Þ"~Ô}eyry>s{po>}e× wvun1|)ÚÐCÞ3ßor{Únxu{1y>u)pu×Õ¢}epßu5|u5×C{ wvuy PQÞ3ß{porØDor{pÚêáÚ5pÞ3sv×vorqDsu i×v}e| vpu)w PQÚß5y>Þ3o>An~:Þáxsv×voÙ{Ú)wvu npsAÕiÞ3ßubu{Ûwvub{pu5|[email protected] ![ÕiÞáor{Îy P I} 7Au{ÎwvuLy Þß5Þ3|)~vÞ3Ðe×vuLw P o>×CØeunA{po>ÐCÞá{po>}e×!ÑXJ_Z_H ;:&: =è ;:&:;? 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W ac`(_ `_ )+ 3 32 3!"# \[ f 3md fog 3 & 3!& [ 3 5 1 ) 1 8&(4*5 ] d "!$#&%')(*"+-,/.01!23*45(601*7! 8:9(*454;(<./!=.#>[email protected]?2A"!="+CB$+CD;E,/(FEG,DHB1*+7F2JI=(*45!=KKLF8*!=.M"!=D;[email protected]+C(J# 0 ( [email protected]+KNBOAP45(RQS%TPVUP(J+45(FWXBOYZ!OACB$+7!=D5A"([2]\_^` A"!=45!Oa=D5(cbed]f"gihjCkckJlmfOnpo5qsrut=t=vwyxzQ{(R01!23*45(F0F,4;+7Dp# 2D;01(*.7D;!=..(*4S|&ru}FqX~O}!O,=}[email protected]+7 a=A"(,.V"?+C3 0 ( 2]\_ EG,NB$+7D5!=.-BO. 45BO"+7D5EG,(*E,/DKZ(*A70 (*+P2( A7 BO45D;7(*Am2/( m7D50F,4B$+7D5!=./S2,,B$+C0 !=7K/9 A"D5EG,(62(<4y\_ 8:9(*4545(60 "!$# M","E,z\ 4y\_ 8:9(*4545(60 D58*A7!$# 78*!=KD;E,/(Ox/>4L8 !O.G+7D5(*.G+S,. &KBO8:BOa=(J[email protected]?7D;EG,(8 !=0 K4;(*++7,A7Y,/45( ./8 (Oqα78:9*0cB2/(-ACBi?=!O..( 0 ( [email protected]+ q KA"!8*( 7",2(",A"B$8 (OqZ01D;8 A"[email protected]?7D5EG,([( .K/9NBO7(F01D+7(OqL78:9*0cB2/(F8 !O.GIO( 8J+CD5!O.VKA7!O!O.2(OqZ8:9Dp# 0 D5(OqBO*A7!="!=45*x;xpxSW\_!=A7!=aOACBOK9/D5(A" (*4545(BOD5."DPEG,(V4;(+>?KZ(V2(V",A"B$8 (VKL( ,/IO( [email protected]+*+CA"(VD5.Dp+CD5BO45D;7 *x }P(*7D;0,4B+CD5!O. 2D;Z A"( [email protected]+C(* 8:9/( 454;( KL( ,I=( [email protected]+J+CA7(A7 BO45D;7 (*BiI=(*8V4B+C( 8:9/.D5EG,(+>{!#{Bu? a=A"D52#.( +CD5./a$bl>kJ¢¡£kJl<fOnpo5q¤~$¥=¥=¥$w¦xW(§8 !2(§(*"+ ( [email protected]+7D53 A"( 0 ( [email protected]+ KNBOA7BO454; 4;D57 (*+RIO( 8J+C!=A"D57$q(*+ 2(* 7D;0,45B$+CD;!=.=}¨2( .G,NBOa=(*[*45(*8*+CA"D5" 67,A<2( a=A7BO.2<2!=01BOD5./( 6BOD;.7DEG,( 2(*["D50,45B$+CD;!=.2( 8iB$A" ( 4;KL( ,I=( [email protected]+P*+CA"([BODp+C(* x©45,ª2([2*+CBOD54;`7,AP*7!$#>%P'«[email protected]+-2/D57KL!=.D;Y45(*`7,Am45(6"D;+C(6( Y |[email protected]+"+CK¬®=¯``6xBO(*A7!x_!=Y&#>0 D5K]x°A:01(*7!=.9N=Jx W(!=A+H8*!=,K45BOa=(¤EG,D(JD5+C({(*[email protected]+CA7(4B`2/?.NBO0 D5EG,($q¯4Bª01D;8 [email protected]?"D5EG,(±( .K/9NBO7(a=4B$8 (¤(*+H4y\_ 4;( 8J# +CA"D58*D;+C-B$+C0 !=7K/9 A"D5EG,(-[email protected]+ª2([*7!$#>%P'²,.!=,/+CD;4zK,/[email protected]+mKZ!=,/A`*+C,/2D5(*A`4y\³BO8*+7D;ID;+7- 4;( 8*+7A7D;E,/( 2(*!=A7BOa=(*`(*+m7(*A"( 4B+CD5!O.`BiI=(*8[4;( `KNBOA7BO0 3*+CA"( ª( [email protected]"!=..(*01(*[email protected]+:BO,be´fOjµl¢¶NkRkJlSfOnpo5qN~O¥=¥@·w¦x ¸S¹"º¤¹y¸ » ]¼ /½ ¾±/¿&À¤¾ÂÁÄÃ{Å`Á À±Æ ½ ½ À W(±"8:9 01Bª01D;8 [email protected]?"D5EG,(¤( +s,.["8:9 01BmmK9BO7(¤0 Dp+C(ªb®Ç¢¡Nl¦È6kJlZÉ/[email protected]ÊJhË¢n¢n5kOq=rutOt=v$w¦qi8O\_( +"#&¯# 2D;A7(KA"( [email protected]+s( .<8 !O01K/+7(¤`4BS!=D5z4y\_(iBO,645D;EG,D52($q 45BSa=4BO8*((*+X4¦\³( BO,67,A!=.2,($x¯>4=7,Dp+X4y\³BOKKA"!8:9/(2( dH¢¡kJlfOnpobprit=v=$wNK/,D5EG,(`+CA"!=D5(*7KL3 8*( {2/(mKBOA"+7D58 ,/45( aO4BO8* ( ¤"[email protected]+A"( KA" "( [email protected]+C*( q(*+±D54/( "+±8*!=,K4; ,.c"8:9 01B62(-ÌRk:Í:ÍJn5kJjRbprit=Î=t$wNKL!=,A4;( ¤KA"!8*( 7",±8:9NBO,2*x/}( , +>?KL( ±2(ªKNBOA"+7D58*,45(*45D5EG,D;2( |4¦\³( BO,V.G,NBOa=(*,7( (J+64BcK4;,D5(i(*++7A7!=D;P8*4BO"7( 62]\[email protected]?2A7!=0 *+7 !=A"( 6a=45BO8 *( |45Bca=45BO8 (K/A7D501BOD;A7(Oq45B Ï"ÐZÑÒ Ó¦ÔyÑÕsֵӦרJÒÖJÙGÚuÑÙÖSÛÑ7ÜݯÑÔÜݯÑÞßӦߨ*Ô¦Ø*Ù³Ø*àJ×®á â¯Ñ ã · .(*D5a=(§(J+14;(§a=A7*7D;4¢F7!O.G+ 8*!=.7D;2 A" (* q±BOD5./7DSEG,(4BI$BOKL( ,A 2]\_(iB$,]x¤W("8:9 01B8 !=0 KZ!OA"+C(§,/.( EG,NB+CD5!O.KA"!=.!=+CD5EG,( KL!=,A<45( ACBOK/KZ!=A+<2(c0 45BO.a=( 2( 8:9NBOEG,((*7KL3 8*( xsWXB8*!=.8*( [email protected]+CA7B$+CD;!=. +C!$+:BO4;(( . .!=0FYA7( 2/( [email protected]?2/A7!=0 *+7 !=A"( KA7*8 D;KD;[email protected]+C|K45,D;(OqH.( D;a=(§(*+ rxa=A" "D54¢6KZ(*,/+ \_ 8*A7D;A7( N 7!O,ª4B!=A70 (-2!O.. (<KNBOA!¤fO¡NfOË "kJl{fOnposbrut=t ã w |y~x;r¯ N = Cλx !# ( +-4;(FKNBOA7BO0 3*+CA"([2/(FKL( [email protected]+C(2/(4B 2D5+CA"D5Y,/+7D5!=.( .+CBOD54;45(2(*-KNBOA+CD;8 ,4;( x (*+ [email protected]+2/( 8 O! .λ"[email protected]+C( F( 0 KD5A"D5EG,( <2*2,D;+7( [2( !OY7(*A"I$B$+CD;!=.c|IO!=D5A6+:BOY/45(iB$,Ä~xpr¯Jxs©s!=,AFC4;( <a=!=x,/+"+C(*45(*+"+C(* .G,NBOa=(*,7(*c(J+c4;( 8 A7D;"+CBO, 2(a=45BO8 (KA"D501BOD5A"(Oq !O. BODp+ 4y\_9G?KL!O+C9/3 7(2(01!O.!2/D57KL( A"7D;!=. KNBOA# ACB$KKZ!OA"+±4BP+CBOD54;45({2/( ±KNBOA"+7D58*,45(* x=WXB60cB$77( KZ(*,/+±*+7A7(ªA7(*45D5*(`BO, 2DB$013J+CA7( KBOA¤4B-A7(*4B$+7D5!=. m D ( .K,D;77BO.8 ( |y~x®~= m(D) = aD b (*+m45BFIDp+C(*77($N.NB$45(2(68:9G,/+C( B$,2D5BO0 3*+CA"( v v(D) = cD d ρ00 ρdref |y~x@ !0.4 ! # ( +ª4BF2(*.7Dp+C2(A"**A7( ./8 (62(4y\³BOD;AS4B[7,ABO8 ((J+ ( "+ª45B2( ./7D;+72(-A"* A"( .8*(62( ρ00 ρdref 4BACB$8*+CD;!=. 2]\B$D5A±7( 8$x/W(*{8 !O."[email protected]+C( q q (J+ [email protected]+S2/!=..*( S2NB$.{4;(m+CBOY45( BO,c~x;rOx&%-.§B$201(J+ EG,(<4B2D;"+CA"D5Y,+CD5!O.( .+:BOD;454;(P7,/D;+ª4BR4;a!=D b c d n(D)dD = N g(D)dD = N α αν αν−1 λ D exp (−(λD)α ) dD Γ(ν) |¦~x ã ( +[,/.( 54 !=D2/( 2/D5"+7A7D;Y,/+CD;!=..!OA701BO45D;7 § ( |45!OD [email protected] 01Bca= ./ ACB$45D5" (iP2 KL( [email protected]+F2(*6KNBOACB¯# 0 3*+CA"( q (*+ 2!=../ 2NBO. 45(§+:B$Y45( BO, ~ x;r=x¤W(K eme 01!O01(*.G+ 2/(45B4;!=D`7(0 (*+ 7!=,/145B !=A"01(R α ν λ Z +∞ |y~x®·= Γ(ν + p/α) 1 G(p) p g(D) M (p) = >4( +BO45!=A"`KZ!O77D;Y45(P2]\³*8 A"D5A7( ρrx = %-.(*.2/ 2,Dp+ª4y\_(JKA"( 7"D5!=.2( D g(D)dD = 0 λ Z +∞ 0 Γ(ν) λp = m(D)n(D)dD = ax Nx Mx (b) λp |¦~xÎ= KL!=,Aª45(*`KNBOA+CD58*,45(*`KA7*8 D;KD;[email protected]+C( <|K45,/D5(Oq/.( D;a=((*+ma=A" 7D;4¢ λ= ρr aN G(b) ! 1 x−b |y~x('= }-BO./m4;(8iBOª2(645B aO4BO8*(KA7D;0cBOD;A7($q8*(*+7+7(6(JKA7(*7"D5!=.2(*ID5(*[email protected]+[ λ= ρr aN G(b) ã Î ! −1 b |y~x[email protected] © BOA7BO013J+CA"( α ν Y B 8 2 Q A i ¥x[email protected]~ ~x®· v=¥=¥ r=x¥ A r s r ¥/x³¥@~ r=x³t ·xpr ¥/x~' · r A r g r rut/x³Î ~x³v r¯~ ã ¥/x³Î=Î ·ru¥ 5 #>¥/x®· A A c r ·=~ ã x®~ru¥ ¥x®· 7 rr r ·=~ ã v ã ~ ¥xv ru¥ 7 #Cr X ~xpr!#"%$'&)(*+&,$-.0/213$4+& ( #=)>@?= 13A3?=B("C13D',E)(GF H*DI($-?.0J21 -J ?$4LK 5 U"5 5<N; ,T$#AY.) 7 AZ$ν ,T ,?"%$M+&ONI?"1P1 #!21P1 +&Q=E"RN*#"&@+& ,?"%α$5 S AV 6#572,5 ?"%59 $Q8:WX ,?["$MW\=]#AN* @1 , ?"%c$ 5 r 5 i 7 5 s 5 g !N[$-J+&#AY ^[_ ¸S¹"º¤¹ ` ÀXazcbP Æ4a À¤¼PÃ{Å`4dQbP ©H!=,/A`8:9NBOEG,(+>?KZ(2]\[email protected]?2A7!=0 *+7 !=A"(OqD;4z(µ/D;"+7(,.§!=,§K4;,7D;( ,A"SKA7!8 (*7",ª2]\³D;.D;+7DB$+7D5!=. (*+S2( 8 "A !=D5"CBO./8 (1| $NaO,A7([~xpr¯Jx W( 8 A"D5+:BO, 2(a=4B$8 (KL( ,/IO( [email protected]+1*+CA"(D5.Dp+CD5*FKNB$AR.,/8 45 B$+CD;!=. 9!=0 !=a=3*.(|¦' %%-( . 4y\³BOY"( .8*( 2(./!¯?=BO,a=4Be !=a=3*.( [z4;!=A7"E,/(F4B +C(*01KL A7B$+C,A"(F2(JID5( [email protected]+D;./ A"D5(*,A7(F #&@· o QPqZ45( a=!=,/+"+C(*45(*+"+C(* .G,NBOa=(*,7(*R8 !=./a=3 4;( [email protected][email protected]+:B$. 0 ( [email protected]+ixBOD;F45(101!2(1KA7D;.8 D;KNBO4¤2]\_D5.Dp+CDB+CD5!O.¬(*"+R45B.G,8*45iB+CD5!O. 9J+C A"!=a=3*.(|¦ ' f¤%-Jx Q{( F8 A7D;"+CBO, 2(a=45BO8 ( KL( ,/IO( [email protected]+ BO4;!=A7<a=A7!O77D;A[KBOAF2*KZ!O7D;+7D5!=. 2(1I$BOKL( ,/A |¦g } f±©±c!=, KBOA(*Z(*i + h( A7aO( A7!O. | hQf¤TPµx`Q{(VKA"!8*( "7,( +YNBO"V7,A§4;(BOD;+1EG,(4BK/A7( "7D;!=. KNBOA+CD;( 454;(2(F7B$+C,/ACB$+7D5!=.2(F45B I$BOKL( ,Am( +K4;,mB$D5Y4;(<KBOAA7BOKKL!=A"+P145B aO4BO8*([EG,([KNBOAPA7BOKKL!=A"+ c4¦\³( BO,4;D5EG,D52/(OxN^PD;.7D¦qD54?B *I$BOKL!=ACB+CD5!O.2(*-a=!=,/+"+C(*45(*+"+C(*P2]\_(iBO,(*+-2/ KL!=7Dp+CD5!O.2([4B I$BOKL( ,/A 2]\_(iBO,7,/Am4;( ª8 A"D5+:BO,2(6a=4B$8 (Ox WXB.(*D5a=(¤(*"+ !OA70 (¤KNBOAHBO,/+7!8*[email protected]=( A"7D;!=.F2(* 8*A7D5+:BO,F2/(a=45BO8 (|9 ^ jkmJq(*+H( 4;45(¤8 A"%! l5+sKNB$AH2*KZ!O7Dp# +CD;!=. 2(ªIOB$KZ(*,A¤2]\³( BO,V|m } f¤©±JqGKNBOA¤a=DpIA7BOa=(`2(ªa=!O,/+7+7( 45(J+7+7( ±.,BOa=( ,/7( -|¦Tª[email protected]±BOa=a=A" [email protected]$+7D5!=. 2(6KNBOA+CD;8 ,4;( `2(6a=45BO8 ( |o ^ nn<{!=,KNBOAªBO8*8 A"*+CD;!=.2( ªa=!=,+7+C(*ª2(6K45,D;( |^mQ`Q`µx W(§a=A7*7D;4sM"!=,(§,. +A p=45(D;01KL!=A+:[email protected]+2BO.R4y\_ 4;( 8J+CA7D;CB$+7D5!=.2, .,BOa=(IDBV45(*KA"!8*( 7", D5.2,/8*+CDp (*+[.!=.#>D5./2,8*+7D;&xXQP\_( +KL!=,A"E,/!=D¤45( ,/A6+7ACBODp+C(*01(*.G+62/!=D;+<*+7A7( (*Z( 8J+C,cBuIO( 81,.a=A7BO.2¬7!=D;.]xXWXB a=*. A7B$+CD;!=.2/,a=A" "D54( +-45B 8*!=." EG,( ./8 (2,a=D;IA7BOa=(2(<4B .( D;a=(1|¦TªS!=,2/([45B 8*!=.a=*4B$+7D5!=. 2(*<aO!=,/+7+7( 62(RK4;,D5(R4;!=A7"EG,]\³(*454;( -(*.G+7A7(*.G+[( .8*[email protected]+:BO8J+FBiIO( 8 2( 68 A"D5+:BO,2( a=4B$8 (|yQ Q q TPµxW( a=A" 7D;48 A"%! l5+6KNB$A[B$8 8 A"*+7D5!=.]xsQ{(J+7+7(18*!=454;( 8*+7D5!=.V2/(1KNB$A"+CD;8 ,4;( 645D;EG,D52(*( +[+7A73 <s( r 8 BO8 ($xHW!=A"62( 8 (cKA7!8 (*77,/ q4B+C(*01KL A7B$+C,A"(c2(c7,ABO8 (§2(*a=A7BO,KL( 45[( +R7,KL A"D5(*,A7(45B+7( 0 KZ*ACB$+7,A7(12( 4y\_( [email protected]"!=..(*01(*[email protected]+m|A"( 34 t$8:9( 0 ( [email protected]+2(`8:9BO45(*,A4B+C( [email protected]+7(S45!=A" 2({4y\³BO8 8*A7*+7D5!=.NJx^ 4y\_ EG,D;45D5Y/A7(Oq¯8*(*+¤(J8*3 2(ª+C(*01KL A7B$+C,A"(ª(*+45(ªA7*8:9NBO,/Z( 0 ( [email protected]+8 [email protected]=(*8*+7D;X2(ª4y\³BOD;A¤BO0[email protected]+¤[email protected]+2,c0 0 (`!=A72/A7(m2(aOACBO.# 2(*,A < x uD 4B +C( 0 KL ACB+C,A"(F2,a=A" 7D;4H(*"+6D;./ A"D5( ,/A7(,/.(+C(*01KL A7B$+C,A"(F"( ,D; 4 k qz4;( PaO!=,/+7+7( (J+ t a=!=,+7+C(*45(J+7+C(*S8 !O4545(*8*+7 ( {KZ(*,/I=(*.G+m8*!=.a=(*45( AP!=.§( +ª2NBO.{45(-0 !2/(P2(8*A7!=D;7CB$.8 ( "( 8 |¦}Pw T v<µxNW( 0 !2(<2(8*A7!=D;77BO.8 ([9G,01D;2(§P| xyfOkm[email protected]+C(*A"ID5(*[email protected]+-7Ds4B +C(*01KL A7B$+C,A"(F2(JID5( [email protected]+",KL A7D;( ,A"(z k ,.( $N./(<8*!=,8:9(<2(645D;EG,D52(PKZ(*A77D;"+7([R45B ",A"B$8 (62,a=A7*7D54¦xNQ{(60 !2(2/(<8*A7!=D;77BO.8 (68*!=.2,Dp+mt 4BaO .*ACB$+7D5!=.2/( 45Bca=A" 45($xQ{(*<2/( ,V+CBO,2(R8 A"!=D5"CBO.8*( [email protected]+68 !O01KNB$A7 (J+645(K/45,PB$D5Y4;(F( + ã ' XZ~x;rg9AP N$-.m.)g(*#& ,T$ ? +&'&#"*&.gA $-?+,cH*D &#A2"E+&m2=),cH &@o.mA 1 (*=& 7 7 ; J+&@? o_ KA"D5`( .8 !O01K/+7(Ox ¸S¹"º¤¹ b @Xƾ ÁP¿H¿¼F P b Á¿b f+:[email protected]+<2!=../ GE ,( 45(R"8:9 01B( "+<,.¬78:9/ 01B§K9NB$7( 01D+C(OqzD54H?VBKL!=7"D5YD;45Dp+C<2( 8 !(JD5&# +C(*.8 ([( [email protected]+CA"(F2(*aO!=,/+7+7( 4;(*+7+7( m2]\_(iBO,(J+P2/( mKZ(J+CD;+78*A7D;"+:B$,2(<a=4BO8*(OxN>4BO,/+2!=./8<+7ACBODp+C( AªBiIO( 8 KA" 8iB$,/+CD;!=.4;( m 8:9NBO./a=( -A7BOKD;2( 2(<I$BOKL( ,/AP2]\_(iBO,Ä|8 !O.2( ./CB$+7D5!=.(*+-2/ KL!=7Dp+CD5!O.JxW(FA7BOKKL!=A"+ 2( 01*4BO./a=( 2(1I$BOKL( ,A<CB+C,[email protected]+C( sat ( "+F2 $.DKBOA6,.(1!OA70,4;( YNBOA"?8*( [email protected]+CA"D5EG,(cEG,D±,/+7D545D;7( 45(*-8 !O,A7YL( 62(R7B$+C,A7B$+CD;!=.2(R45B1I$BOrKLvc,i( ,A2]\_(iB$,¬KBOA-ACB$KKZ!OA"+64¦\³( BO,V45D;E,/D52([(*+6KNBOAPACBOKKL!=A+6 4BRa=45BO8 ($qBOD5./7DEG,(<45( ªEG,[email protected]+7D;+C*2/([01BO7"( (*+ xWX(*m+7( A"01(*Pm } f¤© (J+QS%})A7*7,4p+C(*.G+2]\_,. rc ri B¯M",+C(*01(*.G+D50 K4;D58 Dp+C(KNBOA¤ACBOK/KZ!=A+± sat xGW( ±2 $N8 Dp+C¤!=,c4;( ¤(J8*3 {2/(ªI$BOKL( ,/A2]\_(iB$,1"[email protected]+{8*!=0 # KL( ." ª!=,BOY"!=A7YL `KBOA`45(*Sa=!=,/+"+C(*45(*r+"+Cvc,i (*S2]\_(iBO,(*+ªKBOAS4B[a=4BO8*(-KA"D501BOD5A"(KA"!=KL!=A"+7D5!=../( 454;( 0 ( [email protected]+ R45(*,A`EG,[email protected]+CDp+COxLQ{(J+BO45a=!OA7D;+790 (P( +m.!=.#D;+C*ACB$+7D;X(J+2/,"( 8 !O.2!=A"2A7($x ¸S¹"º¤¹ ¿4a:b Ágb ¿Sb ÁazÀ!"& La &À¤¾ ÁPļ-¿ /4a:b& W ([+:B$,2([01BO"7([+7ACBO./" A" ([45!=A"2,KA7!8 (*77,/-2(F8*!=454;( 8J+CD5!O.0 (*+"+:[email protected]+( . M"(*,45(*(*7KL3 8*( # |"!=,A78*(uS(*+Xv |K,D;+7:S( +m2(64BF!=A"01(R 1 RY COLX = ρ BiI=(*8 K(Dx , Dy ) Z +∞ 0 Z +∞ 0 K(Dx , Dy )my (Dy )ny (Dy )dDy nx (Dx )dDx 4;([email protected],2(68 !=4;45(*8*+CD;!=. K(Dx , Dy ) = π (Dx + Dy )2 |vx − vy |Exy 4 ã v |y~x[email protected] |y~xpru¥@ ( +[4¦\³(Rr 8iBO8*D;+CR2/( 8*!=454;( 8J+CD5!O.]x-,NB$.2 4¦\³,/.(R2( <KNBOA+CD;8 ,4;( -D;01K4;D5EG,*( -2NB$.645(RKA"!8*( 7", 2(-8*!=454;( 8J+CD5!O.1(*"+`,./(PKNBOA+CD;8 ,4;(.!O.#>KA" 8*D5KDp+:[email protected]+C(|a=!O,/+7+7( 45(J+7+7( {.G,NB$a=( ,"( C{!=,45(*[email protected]+C( 0 ( [email protected]+{KA7µ# 8 D;KD;[email protected]+C(|a=4BO8*(RKA7D;0cB$D5A7(iJqL45(*6+CBO,2/( 8*!=454;( 8J+CD5!O.Twv<Q %W # [email protected]+<8iBO4;8 ,4; <BO.NB$4;?G+CD;E,/( 0 ( [email protected]+ix W!=A7"EG,(45(*S2( ,§KNBOA+CD;8 ,4;( {"[email protected]+ªKA7*8 D;KD;[email protected]+C( |K4;,D5($q.(*D5a=(!=,§a=A" 7D;4¢µqD;4Z.]\_( +ªK45,/SKZ!O77D;Y45( 2]\[email protected]+C*a=A7(*AmBO.NBO4p?+7D5EG,(*01(*.G+S7,Aª45("KZ(*8*+7A7(<2(+:BOD;4545($q(J+m!=.,/+CD;45D5"(-2(*.!u?=BO,KA"J#>8 BO458*,45* x Exy ¸S¹¸S¹º ! "!# $ ¿ &¿&% & ½ & La 4dQb W( -8:9NB$A7a=(*P 4;( 8J+CA7D;EG,( [email protected]+-KL!=A+C (*PKNB$AP45(*[email protected]?2A"!=01J+C*!=A7(* qz(J+PA" KNB$A"+CD;( P145(*,Am7,ABO8 ($x f454;( ª[email protected]+P2!=.8<7!=,/01D;7( BO,0 0 ( mKA"!8*( 7",P2?.BO01D;EG,( (*+01D;8 [email protected]?"D5EG,(*EG,([45( [email protected][email protected]# 2A"!=01J+C*!=A7(* x{>4PBOKKNB$ACB l;+§BO4;!=A71. 8*( 77BOD5A"(V2]\³BuIO!=D5Ac,.(*E,B$+CD;!=. KA7!O.!="+7D5EG,(A" a=D;[email protected]+§45(* I$BOA7D5B$+CD;!=."KNB$+7DBO4;( (*+±+C(*01KL!=A"( 454;( 2/(45B-2( ."D;+701BO7"D5EG,(ª2(m8:9NBOA"a=(ªKZ!=A+C*(KNB$A8:9NBOEG,(+>?KL( 2]\[email protected]?2/A7!=0 *+7 !=A"(Ox %. 8:9!=D;7Dp+18*!=0 01(§I$BOA7D5BOY4;(§[email protected]?"D5EG,(4B2( ."D;+72(8:9NBOA"a=(0cBO"7D;E,/( |yQPx_Ga −1 2z\BOD;AS7(*8uµqL(J+m45(-?+C3*01(<2]\_ EG,NB$+7D5!=.ªKA"!=.!=+CD;E,/( mKZ!=,/A ( +P2/(<45BF!=A70 (R Qx Qx ∂ → − → − (ρdref Qx ) + ∇ · (ρdref Qx U ) = ρdref Sx ∂t |y~xpr=r¯ ( +4B ID;+7( "7(2(4¦\BOD;Ac(J+ +CD5(*[email protected]+§8 !=0 K/+7(2( §(Jz(J+C§2(V2/D;Z,7D;!=.]q{2/(8:9NBOA"a=( 0 ( [email protected]+iq{2( Sx +CA7BO.( A"+ 2(8:9NBOA"a=((*+ 2(.( ,/+7ACBO4;D57B$+CD;!=.ÄKNBOA 45(* 8 45BOD5A" xW(8:9!=D 2( 8 !=0 0 (IOB$A7DB$Y45( KA"!=.!=+CD5EG,(6(*"+m2D58J+CKNB$A`4BR8 !=./7( AI$B$+CD;!=.2(64B8:9NBOA"a=(Ox ( "+m2 $N.D8*!=Q 01x0 (4BEG,NB$.G+7D;+72( 8:9NBOA"a=(`2(`",Y"+CBO.8*( KNBOA,/.D;+7S2(`01BO7"(`2]\B$D5A"( 8OxGWXB-8:Q 9NB$xA7a=(S+7!O+:BO4;(|( .1Q` (*+¤45B-0cBO"7(S"[email protected]+ x 8 !O.7(*A"I=*( 2,ACB$.G+m4;(P+CA7BO.7KL!=A+mBO45!=A"SEG,(645(-IO!=45,0 (-.(64¦\³(*"+ªKNBO*xN© B$Am8*!=." EG,( [email protected]+ qZ45BF2( ./7D;+7 2([8:9NBOA7aO([I=!O45,0 D5EG,(<9NBOYDp+C,(*454;( 0 ( [email protected]+ª,/+CD;45D;7 (1| ( .QPx_0 −3 m.]\_( +-KNB$P4BRI$BOA7D5BOY4;([B$2 EG,NB$+7( x KL!=,A2*8 A"D5A7(c45( F8:9NBOA7aO( R 4;( 8J+CA7D;EG,( R2NB$..!O+7A7ρ(§ 0 !2/3 45($xQ{(*KZ(*[email protected]+ q4B2/( ."D;+C§2/(8:9NBOA"a=( 01BO7"D5EG,(+C!O+CBO45(.z\³(*"+mKNBOm8 !=./7( AI=*(F45!OA7`2(68 (*A"+CBOD5.ªKA"!8*( "7,D5A7A"*IO( A7"D5Y4;( 6NKA" 8 D;KD;+CB$+CD;!=. 8:9NBOA"a= (* B$+7+7( [email protected]+R45(c7!=4¦q .(*,/+CA7BO45D;CB$+7D5!=. KNBOAR45(*R 8*4BOD;A7RX Q n (*+RN, 2(§8:9NBOA"a=( +CA7BiI=( A" 45(*A"[email protected]+CD53*A7(*P2,V2!O0cBOD;.(OxLWXBc2(*.7Dp+C2/(F8:9NBOA"a=(01BO7"D5EG,([( "+-A7( 4;D5*(c45B12(*.7Dp+C2([8:9NBOA"a=( I=!O45,0 D5EG,(KNB$A`,.(6A7(*4B$+7D5!=.§"D50 K45( x ρx = Qx /ρdref %-.BOD;+4y\[email protected]?KL!O+793 "(-EG,(4B<8:9NBOA7aO(-D5.2/D;ID52,/( 454;(ª2]\³,/.(KNBOA+CD58*,45( |(*.Q`7,Dp+S,./(-45!=D qx (D) ( .K,D;77BO.8 (<2*KZ(*[email protected]+m2,2DBO0 3*+7A7( 2(64BKBOA"+7D58 ,/45(R → − U D qx (Dx ) = ex D fx |y~xpr¯~= W( ªI$BO45(*,A7`2(*8*!(Rr 8 D5(*[email protected]+C [email protected]+mA7(*8 !=0 0cB$.2 (*KBOAP´-k7fOj('kJl*),+:¶@Í1brut=vOÎ$w±|+CBOY4;(iBO,~x~=Jx fx W( 8:9BOA7a=(*+C(*.2( [email protected]+ J+CA"(A7*KNBOA"+7D5(*R4B7,ABO8 (§2/( [email protected]?2A7!O01J+C !OA7( *xW(a=A" 7D;4J+:[email protected]+R,/.( KNBOA+CD;8 ,4;(cEG,NBO"D`7K9*A7D;E,/(Oq!=.Ä8*!=.7D;23 A"( ~xW( Ra=!=,+7+C(*R*[email protected]+ 2*!=A"0cB$Y45(* q 45B8:9NBOA"a=( f EG,]\_( 454;( KL!=A"+7( [email protected]+ 2!=Dp+ *+CA"(D5./*A7D5(*,A7(V45B8:9BOA7a=(4;D50 D;+C( 2(TmBu?4;( D5aO9 x¤©H!=,/A EG,]\_D54? 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B,/.(8:9NBOA"a=(B$77!8 D; (c4B§I$BOKZ(*,AF2z\³( BO,]x >4 ³\ BOa=Dp+(*.BOD;+H2(S45B8:9NBOA"a=(S2(*D5!=./ EG,D/"[email protected]++CA7BOD;+7 H2({B%e*!=.Ra=A7!O77D;3 A"({2NBO./45({0 !23 4;(OxW!=A7H2( 4y\_*I$BOKL!=A7B$+CD;!=.2(*[email protected]?2/A7!=0 *+7 !=A"( 645D;E,/D52(*P!=,V4;!=A7-2/(4B",Y45D;0cB+CD5!O.2(*KNBOA+CD58*,45(*-a=45BO8 *( q 4y\_ 8:9BO.a=(`2(`01BO"7(`7(`BODp+2(*¤KNBOA"+7D58*,45(* I=(*A7¤4BI$BOKL( ,/A [email protected]\_ 8:9NBO./a=(ª2(`8:9NBOA"a=(mBO7"!8*D5S2(JIACBODp+ 7(BOD;A7(H2( XKNBOA+CD;8 ,4;( zIO( A7X4;( D5!O. x © BOA8*!=010 !2D;+7OqJ(*+X( .<B$+"+C(*[email protected]+X2(KA"( .2/A7(¤(JK4;D58 Dp+C(*01(*.G+ ( .c8 !=0 K/+C(45(*{D5!O. qG4B<8:9NBOA"a=(P2(*SD5!O.(*"+ªB$77!8 D; (PF4BI$BOA7D5BOY45( x© B$ASBOY,/{2([email protected]$a=([email protected] 8:9NBOA"a=(62( ªD;!=.`( +ª2!=.8<BOK/KZ(*452NBO./m45B7,Dp+C(P8:9NBOA7aO(2(64BFI$BOKL(Q,vA&:xNW( ªKA"!8*( "7,BO7"!8*D5* 7!O.G+m4¦\³JIOB$KZ!=A7B$+CD;!=.]q4BR7,/Y45D;0cB$+7D5!=.1(*+m4B./( ,/+7ACBO4;D5CB+CD5!O.§KNBOA`45(*` 8*4BOD;A7 x >4{(*"+A"( 8*!=..G, E,/(45(*[D;!=.FKBOA"+7D58 D;KZ(*[email protected]+FKL( ,Ä4y\_ 45(*8*+7A7D57B$+CD;!=.2,Ä.G,NBOa=($xQ{(*KZ(*[email protected]+ q45(* D5!O.8*!=.2,/D57(*[email protected]+c45B !=A"0cB+CD5!O.2/ ( !O.( 2(F8:9NB$A7a=([2]\³(*7KNB$8 (EG,DH8 !=.+CDp+C,(*.G+P,.*8 A7BO. 4;( 8µ# +CA"!="+CB$+CD;EG,(BO, 7!=0 01(J+R2, .G,NBOa=((*+ B$,ÄA7!O.G+7D53*A7( R45B$+C*ACBO4;( R2, .G,NBOaO(Ox¤}P(K45,/ q8*( D5!=./ KL( ,/IO( [email protected]+6BiI=!=D;Aª2([4¦\³D;01KL!=A+:BO.8*(KZ!O,Aª4B KNBOACB$01J+CA7D;CB$+7D5!=.§2/( P*8 45BOD5A" x j./([A"( KA" 7(*[email protected]+:B$+7D5!=.(µG# K4;D58 Dp+C(2(*P*8 4B$D5A7`"( A7BcKA7!8:9NBOD;.( 0 ( [email protected]+mKA"D57(<( .8 !=0 K/+7([2NB$.- 7!#&%P')BiI=( 8[4y\_,/+CD;45D;CB$+7D5!=.12( 2(*,I$BOA"DBOY/45( ª2D;"+7D5.8J+C( ª>D;!=.ªKZ!O7D;+7D;&6(*+m>D5!O../ [email protected]$+7D;&:xLQ{( 8*D KL( A"01(J+7+CA7B12(<+C(*.D5Am8 !=0 K/+7( 2(*KA"!8*( "7,m2]\³B$+7+CBO8:9( 0 ( [email protected]+m(*+m2(6A"( 8 !O0YD5.BOD5"!=.]q/(J+m2,8*!=,[email protected]+2/(<8*!=.2,/8*+CD;!=.]x ·O¥ ¸S¹¸S¹y¸ ½ az¿¤¾P&ÆÁP ½ ¼P¿ ¿/&À¤¾ ÁP a/ÃP¿ ¾À±¾&¾PÁbgaL }P(.!=0Y/A7( ,/7( §(µKZ*A7D5(*.8 (*(*. 4BOYL!=ACB+C!=D;A7(¬bNf Of¶/fÍ:¶:qrit 'Ov ªÉ/fOÈ=fOj"fOl¡zkkJl fOnpo5qPrit=v= Z fOË¡ 'kJj:Í[kJlfOnpo5qzrut=tr u¢npfkJlfOnpo5qzrut=t=·[email protected]+ª01!O.G+7A7-EG,(4;(-0 8iB$.D5"01(2(7*KNBOACB+CD5!O. 2(8:9BOA7a=(.!=.#D5.2,/8*+CDp( +45(K45,1(sr 8iB$8 (VKL!=,A(µ/K/45D5EG,(*Ac4;( cI$BO4;( ,A"c2(*§8:9NBOA7aO( §(*+2, 8:9NBO0 K 4;( 8*+7A7D;E,/(S9NBOYDp+C,(*454;( 0 ( [email protected]+A"( .8*[email protected]+CA7*( ±2NBO.4;( .G,NBOaO( ¤2]\_!=ACBOaO( xO>4 \³BOa=Dp+2(ª8 !O4545D;7D;!=. 45BO"+7D5EG,(*-( [email protected]+CA"(KNBOA+CD;8 ,4;( P2/(a=4B$8 (OxZWXBc8:9NBOA"a=(*8:9NBO.a=*(R45!=A"2/(8 (*8 !=4;45D5"D5!=./BiIO( 8RA7(*YZ!O.2 2*KZ(*.2 2(2Dpz*A7( [email protected]+7KNBOA7BO013J+CA"( 45(+>?KZ(V2]\[email protected]?2A7!=0 *+7 !=A"(Oq4B +7( 0 KZ*ACB$+7,A7(BO0[email protected]+C(Oq±45( 8 !O.G+7( .G,(*.( BO,7,A!=.2,($qN4B+:B$D5454;(P(J+ª4BID;+C(*7"(<2/(<8:9G,/+7(F2(*ªKNBOA"+7D58*,45(*x x x f Of¶/fÍ:¶mbrut 'Ov$wsB 0 ( 7,/A7<4BR8:9NBOA"a=([*8:9NBO.aO (F4;!=A7"EG,([2/( m8 A7D;"+CBO,2(<a=4B$8 (6( [email protected]+CA"( [email protected]+( . 8 !O4545D;7D;!=.BiIO( 8R,.(R8*D5Y4;(Fa=DpIA7 ([EG,DA7(*KA7*7(*.G+7( ,.Va=ACB$,KZ(*4yxL>4 (*. 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[email protected]+CD;IO(OqHBO45!OA7<E,/(KZ!O,AF2(*R%W x QB$D5Y4;( 6!=, *45(*IO *q4B8:9NBOA"a=( BO8*E,/D57(1KNBOA[45( a=A7*7D54¤(*"+KL!="D;+CDpI=($xHW(*F(JKL A"D5( ./8 ( R2/(É/fOÈ=fOj"fOl¡zkkJlfOnpo{brut=v=wA"*IO 45(*[email protected]+REG,( 45(Fa=A7*7D54 [email protected]=.(F,.( 8:9NBOA"a=(RKZ!O7D;+7D;IO(KZ!O,A2( -+7( 0 KZ*ACB$+7,A7(*7,KL A"D5(*,A7(*[§,./(+C( 0 KL ACB+C,A"( 2]\[email protected]=(*A7"D5!=.]q{(*+,.(V8:9BOA7a=(V./ [email protected]$+7D;IO(VKZ!=,/A2( §+7( 0 KZ*ACB$+7,A7(*§D5./*A7D5(*,A7(*§Ä8 (J+7+C(+7( 0 KZ*ACB# +C,/A7(62]\[email protected]=(*A77D;!=.]xLQ{(J+7+C(6+C(*01KL A7B$+C,/A7(2]\[email protected]=(*A7"D5!=.2 KL( ./22,%W x Q (*+m( 4;45("(<2/ K45BO8 (I=(*A7m45(* +C(*01KL A7B$+C,/A7( *45(*IO (*cKL!=,AR2(*c%W x Q BOD;Y45(* x{É/fOÈOfOj7fOl¡zkkJl[fOnpobrut=vO$w`[email protected]+1BO,"7Dª01!O.G+7A7§45B 2*KZ(*.2NBO.8*(c2( 4B§8:9NB$A7a=( 8:9BO.a=*(BiI=(*8c4B+CBOD54;45(F(J+F45BIDp+C(*77( 2(*[KNB$A"+CD;8 ,4;( *< x h{D5(*. 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(6KNBOAmÌRkJ¢l¶VkJlZfOË¡ 'kJj:Í1brut=tO¥$wyxLZfOË¡ 'kJjCÍ[kJlSfOnpoXbprit=t/r [email protected]+ [email protected]+7A7!2,D;+ª45B ./!O+CD;!=.2/(F8 !O.G+7( .G,(*.(iBO,45D;E,/D52(R( r18 BO8 (§| f x ªEG,Ds8 !OA7A7(*7KL!=.2V1%W x Q0F,4;+7Dp# K4;D5KNB$A4y\_s( r 8iBO8*D;+7R2( 8 !=4;45D5"D5!=.zxL>4*\B$a=D;+(*.¬B$D;+2( 45BKZ!=A+CD;!=.V2( %W x Q«EG,D( +[",78*( K/+7D5Y4;( 2(a=DpIA"( AR",A14;( Ra=A"!= 8 A"D5"+CBO, 2/(aO4BO8*(OxW!=A" 2/(8*(*+"+C(*+C,/2(Oq`ZfOË¡ 'kJj:ÍkJlfOnpoªbrut=t/r [email protected]+ EG,[email protected]+CD $N-45B<+C(*01KL A7B$+C,/A7(P2z\³D;.GIO( A"7D5!O.§(*+`J+:BOY4;D;+2( { EG,NB$+7D5!=.{KZ!=,/A{45(*{2Dpz*A7(*.G+7( `A" a=D;!=.±2( 8:9NBOA"a=(Oxz>45P!O.G+<B$,7"D BOD;+PBOKKBOACB l;+CA"(F,.(RA" a=D;!=.2NB$.-4B$E,/( 454;(F4;(Fa=A" 7D;4BO8*EG,D5(*A"+,./(8:9NBOA"a=( KL!=7Dp+CD;IO(KZ!=,/A 2/( +C( 0 KL ACB+C,A"( D5./*A7D5(*,A7(* V#&~O¥ o Q¨(*+12( 1%W x Q BOD5Y4;( *xW( (JKZ*A7D;( .8*( 2( 4BOYL!=A7B$+C!=D;A7(RA"iBO4;D57*( <4¦U\ jP' uTk(J+F8*( 4;45( <A7 BO45D;7*( 6KNBO A f Of¶/fÍ:¶[bprit 'Ov$w¤(J + Nf Of¶/fÍ:¶ kJl .¢ÈOf ±f -b³~O¥=¥@~wX0 [email protected]+CA"( [email protected]+m2( 2D;Z A"( .8*( m7D;a=.D $8iB$+7D;I=(*ªE,/DHKL( ,/IO( [email protected]+-J+CA"(FB$+"+CA"D5Y,*( BO, +C(*8:9.D;E,/( S(µ/KL A"D50 ( [email protected]+:B$45( ±,/+7D545D;7*( x/ǪkJjCkJÈj7fckJl¤fOnpozb³~O¥=¥O¥[email protected]+{A"iBO4;D57ª2(*S(JKL A"D5( ./8 ( {( .c,+CDp# 45D;[email protected]+4;( 2(*,+C(*8:9.D;E,/( x uD4;( 8*A7D5+:BO,2( a=4B$8 (.( "[email protected]+[0 45BO.a=*6BiI=( 8 45(*a=!=,/+"+C(*45(*+"+C(* EG,]\³[email protected]+[2]\_( [email protected]+CA"( A<( .8*!=454;D5"D5!=.BuIO( 814B§8 D;Y45(|01D/(*28 45!O,201(J+C9!2Jq4;( 62!=./. (*["[email protected]+FD;2( .# +CD;EG,( ¤8 (*454;( 2 ( Nf Of¶/fÍ:¶¤bprut'Ov$w¦x uD¦[email protected], 8 [email protected]+7ACBOD;A7(Oq$45(*8 A7D;"+CBO,R2/(maO4BO8*(Sa=A7!O77D;7"( [email protected]+±2NBO.¤45( .G,NBOa=(<2([a=!=,/+"+C( 4;(*+"+C( [|7D5./a=45(68 4;!=,201(J+C9!2µqN45( mA7*7,4p+:B$+7"(FA7BOKKA"!8:9(*[email protected]+P2([8 (*,!=Y/+C(*.G, R4y\ j-' uckx Q{( KL( [email protected]+iqs45( 0 8 BO.D;70 ( [email protected]?7D;E,/( FEG,D¤a=!O,/I=(*A7.(*.G+[8 (R+>?KL(12]\_ 8:9BO.a=( 2( 8:9NBOA"a=( .( 7!O.G+§+7!=,M"!=,/A7KNB$8 !=./.,/BuIO( 88 (*A"+7D;+C,/2(Ox{}P(V.!=0Y/A7( ,/7( §[email protected]?KL!O+C93*7(*[email protected]+§J+CVBiI$BO.8* (*¬ KL!O+C(*.G+7D5(*4 2(8*[email protected]+:BO8J+iqZ(*Z(*+C-+79( A"01! 4;( 8J+CA7D;EG,( *qZ2D;74;!8 B$+CD;!=. qa=A7BO2D;( [email protected]+C-2/(F+C(*01KL A7B$+C,A"(Rc45B XÒ¯× JÑ"Ô ¦×³Ó ª!Ø Þ6ÖµÒÜÝÑ Ó¦Ñ# Ô "¢$Ò Óy×_ÓyâuÓ¦Ñ%Ø '&uÜ׳Ñ"ÒÜ&ÑÖJÒ) Ú (Ñ7ÜݯүØJÙØJ*à ·r 7,/A"BO8*(2,Va=A" 7D;4{x*x x;bZfOË¡ 'kJjCÍ@q rit=t ã .Vf0+21hj3 fO¡ kJl5ªË@ÍJlµqrut=t=vwyxZW(F2(*A7.D;( A-0 !2/3 45(<[email protected][email protected]# 7D;EG,(ª2 ( [kJnÍuh¡kJl´f @kJjRb³~O¥=¥=w2(*0cBO./2( A7BOD;+±*+7A7(ª[email protected]+:BOaO(mJ+C,2/D5Ox=}-BO.±4y\³B$+7+7( .2/(`2]\³,/.( KNBOA7BO0 *+CA"D57B$+CD;[email protected]?7D;EG,(6A7( 8*!=..G,($qK45,/7D5(*,A7`A"( KA" "( [email protected]+:B$+7D5!=./`KNBOACB$01J+CA7D;EG,( S2/( `KA"!8*( 7", .!=.#>D5./2,8*+7D;S"[email protected]+P8*!2/ ( ª2NB$. "!$#&%'[x ! "#$&%"'$(*)+ ,-/.10 WXBcKNBOA7BO01J+CA"D5CB+CD5!O.45B1K4;,P"D50 K45(6( +8 (*454;(F2(3[ 2 kJnÍ'h¡ÄkJl54XfOjµn5kJÈVbrut=v'wyqL8iB$AP( 4;45(<KA7(*.2 ,.D;EG,( 0 ( [email protected]+S(*.8 !O01K/+7(45B<2/ KL( .2NB$.8 (-2/(P4B<8:9NBOA"a=(P*8:9NBO.a=*(BiI=(*8P4B6+C(*01KL A7B$+C,A"((J+`BiI=(*8-45( +>?KZ(2z\³[email protected]?2A7!O01J+C !OA7(OxW\³BO0 K45Dp+C,2(m2(45B8:9NBOA7aO( 8:9NBO./a= (62*KZ(*.22(4BF8 B$+C*a=!=A"D5(-2(*ªKNBOA"+7Dp# 8 ,/45( {2(a=4B$8 (PEG,Dz( [email protected]+CA"( [email protected]+m( .§8 !O4545D;7D;!=.]qGBO4;!=A7E,/(45(-"D5a=.(P( "+ª2/*+C(*A70 D5.KNBOAS45B[+C(*01KL A7B$+C,/A7(Ox WXB +7( 0 KZ*ACB$+7,A7(2z\³D;.GIO( A"7D5!O.2(6KL!=4BOA"D;+7 | kmQSTS( "+8 !O."[email protected]+C(<(*+m( 4;45(( + $/ ([R#Cru¥ ◦ QPxW!=A" + graupel − TCR − pristine ice or snow + XZ~x®~076 HJ#.) (*O,T$+AZ= A ,g("L.)J =EAY&+.0o=T?= AY=](" 13A > _98 $4 ,T$4J+&@2=c1 g1 2.S,EJ#$ I13"$4)(GF AZ= 7 7 7 7 1 8 : ; #$+&+AP?= (*,?W[$+A31 J _< ?[$'&9(GF "=] ?Z AY&+AP?= ; 7 $4 6 ?=T( #=T13$-9"= N$-" ,E# 21(*Q\= #AWNo?" (XoN 7 ,T$#AY.)AZ$4 !\N$-"R,# ; C 2"J2$+AZ$ 7 "= H%$PN*g,? &+A313A ; &+AW 1 2.S,EJ#$ I13"$4z. A3[=c1 +,] "& H%["(* 5 7 7 2"E38 21=]JVNI[13A ; &+AwW 1 #.,J2$ [13"%$4). AP=c1 \+&21G,T "&w>@A @"=8 _ 6 F +&21w4F AZ= ; 2$'& ,7 ?"%$ 6 6 81: 8 ,$213A "81!: 5 Q,TU"& ,213A31 _ 7 2]\_,.(8*!=454;D5"D5!=.( [email protected]+7A7(,./(cKNBOA+CD;8 ,4;(12(ca=4BO8*(1K/A7D501BOD;A7(R(*+F2,Äa=A" 7D;4yq45B8:9NBOA"a=(*8:9NBO.a=*(( + a=BO45(6§ ( 7D |y~xpr '= 2 × 10−15 C H 7D T > T CR δqN I = −15 −2 × 10 C T ≤ T CR ( +64B8:9NB$A7a=( A7s( e ,( KNB$A4BK4;,a=A7!O77(RKBOA"+7D58 ,/45(OqZ2NBO.64;(8iBO6KA" "( [email protected]+<45(Ra=A" 7D;4yxz©H!=,A2/( δq 8 !O4545D;7D;!=.{aOA7 "D54#>.(*D5a=($q45B 8:9BOA7a=(6*8:9NBO.a=*([*\³(µKA7D;01(67!O,ª4BF!=A"01(R ( 7D |[email protected] 2 × 10−13 C H 7D T > T CR δqN I = −13 T ≤ T CR −2 × 10 C }P(64BR0 0 (P%B e*!=.]qN45!OA7`2(*m8*!=454;D5"D5!=..( D;a=(J#a=4B$8 (-KA"D501BOD5A"(F ( 7D |[email protected] 2 × 10−14 C H 7D T > T CR δqN I = −14 −2 × 10 C ·=~ T ≤ T CR " )+ ,/.- 0 }-BO.±8 (J+7+7(mKBOACBO0 *+7A7D;CB$+7D5!=.]q45(*A7*7,4p+:B$+7±2/(m45BOYL!=ACB$+7!=D5A"({2( Nf Of¶/fÍ:¶¤bprut'Ov$ws| $Na=,A"(m~x= 7!O.G+{0,4p+CD5K/45D5*KNBOA±,. BO8*+7( ,A±EG,DZ2/ KL( .22/(P4B-+7( 0 KZ*ACB$+7,A7($q2/,8 [email protected]+7( .G,( .c(iBO,145D5EG,D;2(OqO2( 4B-+CBOD54;45(S2(*KBOA"+7D58 ,/45( ¤(J+2/(4;( ,AID;+7( 7"(2/(8:9G,/+7(b f Of¶/fÍ:¶:q/rit=v ã w¦xGW(m"D5a=./(m(J+{4¦\B$01K4;D;+7,2( 2(64B8:9BOA7a=(< 8:9BO.a=*(["[email protected]+2/!=..*mKBOA6 Di,s 2 |vg − vi,s | = fT (T, LW C)M in 10, 5 D0 v0 ",4;+7(¤2]\³,/.(¤[email protected]+C(*A7KL!=4B$+7D5!=.P2(4BS+:B$Y45(2/(SH k BOBO9NB$79Db " T δqN I # |y~x~O¥@ !# A7 h +J ZqOrut=t ã 7.VfO¡/ÍukJn¢n>q fT (T, LW C) ~O¥=¥O¥$wyxZQ{(J+7+C(+CBOY4;([2!O..(45B18:9NBOA"a=(*8:9NBO.a=*(R( .!O.8*+7D5!=.2/(F4B +C(*01KL A7B$+C,A"(F(J+2,8*[email protected]+C( .G, ( . ( BO,R45D;E,/D52($x ( "+¤4;(S2D5BO013J+CA"({2(`45BPK45,/KZ(J+CDp+C(SKNB$A"+CD;8 ,4;(SEG,DD;.G+7( AID;( [email protected]+2NB$.4BP8*!=454;D5"D5!=. Di,s |aO4BO8*(!=,.( D;a=(uµq ( +45BFID;+C(*7"([2(68:9G,/+7([2,a=A" 7D;4](*+ 8*( 454;(-2(64BRK4;,`KL(*+CDp+C(KNB$A"+CD;8 ,4;(Ox W( `I$BO45(*,A7{2( vg(J+ [email protected]+ªA"( 7KL( 8J+CDpI=( 0 ( [email protected]+<ru¥ −4 0 (*+mvvRi,s0x_ −1 xNWXB 4;D50 D;+CB$+CD;!=. ru¥R( "+ª2/,( R4BI$BO45(*,AªBOKKD BOA70(*.G+7(6v20 (6CB$+7,ACB$+7D5!=.§KL!=,Aª4;( `a=A"!=ª8 A7D;"+CBO,§2/(<./( D5aO(1/b .VfOj:Í:¶/fOn¢nskJlSfOnpo5q]rit 'Ov$w¦x XH~xL 9A3 N[$ .m.0(*)&@J ,[$ [13AP?= -"L A2 "%A3(X& #?=8T H% &'HA21 LAYDIw (* H%$PN* #= >2?[= 13AP?=B(*) 1 #.,J#$-[13"%$- 21(" ?=c1 #=E" #= 7 7 sA 7 _ WXB6+CBOY4;(m2/(kHBOBO9NBO"9DN.(m2!=../(45BI$BO4;( ,A¤2(m4B8:9NB$A7a=(m 8:9NB$.a= (EG,(KL!=,A¤2(*+7( 0 KZ*ACB$+7,A7(* BO4;4[email protected]+-2/(1¥ ◦ Q c#>=¥ ◦ QPx©s!=,A64Ba=BO010 (F2/( +7( 0 KZ*ACB$+7,A7(F8 !=0 KA7D;7(R(*.G+7A7(#&O¥ ◦ Q«(*+-# ã ¥ ◦ QPq]45B I$BO45(*,A`2( 8*!=A7A"( [email protected]+C(<( "+m8*( 454;(6D57",(62(64B+:B$Y45(KL!=,Aª,.(6+C(*01KL A7B$+C,/A7(2( fT (T, LW C) #&O¥ ◦ QP:x uD]45B+C( 0 KL ACB+C,A"(( "+mD;./ A"D5( ,/A7(6F# ã ¥ ◦ QPqN4B8:9NB$A7a=(6 8:9BO.a=*([(*"+.,/4545($x ·O " &$& )+ ,-/0 Q{(*+"+C(cKBOACBO0 *+7A7D;CB$+7D5!=.V(*"+YBO7*(c7,/A[4;( 6+CACBiI$BO,¬2/(É/fOÈ=fOj"fOl¡zk§kJlfOnpo{brut=vO$wyxXWXB8:9NBOA"a=( +CA7BO. A7*(-BO,1a=A7*7D;4N45!=A"7EG,]\_D54( [email protected]+CA"(P(*.c8 !O4545D;7D;!=. BiI=( 8,.(KNBOA"+7D58*,45(ª2(ma=4B$8 (ª!=,c2/(P4B6.(*D5a=(ª( + 2!=./. (<KNBOA |y~x®~ru G m n δqN I = kq Di (∆vgi ) (LW C − LW Ccrit )f (∆T ) (*"+,.8*!(Rr 8 D5(*[email protected]+-2(<KA7!OKZ!=A+CD;!=..NBO4;D;+7- [email protected]$4X 'O/qN(J+-4;( ª(J[email protected]+7 (*+ [email protected]+2(F4¦\³!=A"2A7( kq m n 2( ã (*+<A7( "KZ(*8*+7D;I=(*01(*[email protected]+ix G ( "+<(µ/K/A7D50 R( .¬¦QPx (*"+<45(R8*[email protected]+C(*., ( . (iBO,¬45D;E,/D52( δqN I LW Ccrit 8 A"D;+7D5EG,(m(*+S(*"+ $/¥/xprPax_0 −3 x ( +S45(m2D5BO013J+CA"(2(4B<KNBOA"+7D58*,45(ª2(a=4BO8*(.G,NB$a=( ,"( |(*.0 (*+ (*"+<4BcID;+7( "7( 2]\³D;01KBO8*+D A"( i45B$+CDpI=(|(*. 0x_ −1 P(*.G+7A7( 45(RaOA7 "D54(J+<4BKNBOA+CD;8 ,4;(R2( a=4BO8*(Ox WXB∆v !=.gi8*+7D5!=. ( "+m,.KL!=4;?. p=0 (-D;77,2(*m+7ACBiI$BO,§2/( É/fOÈOfOj"fOl¦¡zkRkJl{fOnposbrut=vO$w f (∆T ) |¦~x®~O~= f (∆T ) = a∆T 3 + b∆T 2 + c∆T + d B #:rOx(' [email protected] #&¥x¥=¥O/qO8 #>¥/x¥=·(J+±2 ¥/x;ri<"D ( "+(JKA"D50 `( . ¦QPx ×10−5 T − 273.15 KL!=,A ~ 'O/xpr¯·RxzQ{(*+"+C([!O.8*+7D5!=. \³BO..G,45([KZ!O,A,/δq .([+7( 0 KZ*ACB$+7,A7([2(<#&~∆T r ◦ QPqL=(*+P 4B 8:9NBOA"a=( T 8:9BO.a=*(-( +`2!=.8.G,454;(KL!=,A±8 (J+7+C(+C( 0 KL ACB+C,A"(Ox/Q{( KL( .2NB$.G+ qB $.2(PKL( A"01(J+7+7A7(P45B6I$BOA"DB$+7D5!=. 2(-45B<+7( 0 KZ*ACB$+7,A7(2]\³D;[email protected]=( A"7D;!=.2(-KL!=45BOA7Dp+COq Xyk On5kJjkJl±fOnpo]brut=t/r [email protected]+{A7( 0 K45BO8 KNB$A BuIO( 8 ∆T τ xW\_ EG,NB$+7D5!=.2(+CA7BO."(*A"+m2(68:9NBOA"a=(62(*ID;( [email protected]+m2!=.8 τ = (−21/Tr )(T − 273.15) BiI=(*8 G m n δqN I = kq Di (∆vgi ) (LW C − LW Ccrit )f (τ ) " &$ )+ ,1,"+0 τ= (−21) (T Tr |y~x~O@ − 273.15) f4545([( +6YNBO7*(R7,A2/( (JKL A"D5( ./8 ( 62(F4BOYL!=ACB+C!=D;A7(<E,/D01(J+7+7( [email protected]+( .1M"(*,,.(R4B$A7a=([[email protected] 01( 2(m+C( 0 KL ACB+C,A"( q2(P8*[email protected]+C(*.,/`( .§(iB$,45D;E,/D52([email protected](P+:B$D5454;( 2(PKNBOA"+7D58*,45(m(*+{2(ID;+7( "7( `2/(P8:9G,/+C($x W\³(µKA7(*77D;!=.2/( 45B8:9NBOA"a=( 8:9NBO./a= ( |( .¦Q`B§,.(R!=A70 (R7D;01D;4BOD;A7(<8 ( 4;45(2/( 1ªfOj('O¢¡zkJj ∆q kJl{fOnposb[email protected]·¯w |y~x®~ ã m n W( `I$BO45(*,A7ª2(*ª8 !=.+:[email protected]+7( q m n (J+ ∆q = BDi (∆vgi ) δq B [email protected]+2!=..*( ª2NBO./4;(-+:BOY4;(iBO,~x/x kHBOD;4545(2(64BKNB$A"+CD;8 ,4;(2(6a=4B$8 ( | 0c µ 2 r¯·=· r¯·=· 2 2 ã ≤·=~ ã ·=~ 2 ~=·$ 2≤ «~=·O L~x³ !#"$+&(*+& 7 ?=V[=c1 +& 5 . uD;+7,NB$+CD;!=. # # h ru¥ 13 ã x t=~ ã x³¥ ã × ru¥ 6 ·=~× x³v=¥ ·x~ ã ru¥ 8 ~ã× x³¥=¥ 0 / x 'OÎ r=x³t=¥ ¥/x ã=ã ~x· ã ¥/x·O¥ 21=L"13AYUAY&@J4+&(*=&X#&gJ 2 "[13A3?[=& %_ . ~ x®· ~x®· ~x®· ~xv ~xv 21 %_ _ ^ 8:9NBOEG,(¬A7*a=D5!=. 2/, =a A7BOK9((*"+BO7"!8*D5 (V,/.( * EG,NB$+CD;!=. 2 8*A7DpIOB$.G+45B 8:9NBOA7aO(¬*8:9NBO.aO (¬( . !=.8J+CD;!=.2(64BF+C(*01KL A7B$C+ ,/A7((*+m2,8 !O.G+7( .G,(*.( BO , 45D5EG,D;2( |( .¦Q`Jx δq(T, EW ) ·ã S8 −3 Contenu en eau liquide effectif (g.m ) 1.5 1.0 S4 S1 CEW S7 0.5 S3 S6 S5 S9 S2 S9 0 0 −10 −20 −30 Température (°C) X~x ã ,T$4J+&@#=T1V[13A3?[=N$-+,cH*A2"EX(* H%$PN*X13$-=&Z>@J2$-J4o#=X>2?[= 31 AP?= (*Wm1 2.S,EJ#$ I13"$4 21 (" 7 ( !# &2(I?= 21<!_ 7 ? =T1 #=E" #=z: -[" UA2"AP(*9 13A > I(I#,]1 JX* _< ?"%$ o$ ?"[N*X$- ,]$-J+&T#=c1 9 7 ?=T1 #=E" 2=C-[" UA2"%AP(*m 7 13A > EW 5 7 6 $#A313A2"E _ 7 7 7 *( "+§4;(8*[email protected]+C( .G, ( . ( BO, 45D;EG,D52((*Z( 8J+CDp1|ax_0 −3 Jq±8O\_( "+#D5A"(45BKZ!OA"+CD;!=. 2z\³( BO, 45D;E,/D52( 7,/A"!=.2/,(EG,DmKNBOA+CD58*D5KL(BO, a=DpIACBOa=(2, a=A7*7D54¦x>4ª( "+c [email protected]$4P !# ( +4;( LW C × Ecg LW C 8 !O.G+7( .G,(*.( BO,45D5EG,D;2(-(J+ (*"+4y\_(sr 8iB$8 D;+72(68 !=4;45(*8*+CD;!=.§2(*maO!=,/+7+7( 4;(*+7+7( ª2]\_(iBO,7,/A"!=.2/,( KNBOAm45(6a=A" "D54|*[email protected](<rF2NBO./EPcg45(60 !2/3 45(iJxLQ{([KNBOACB$013J+CA7(6(*"+-,+CD54;D5"614B K45BO8 (<2,8*[email protected]+C(*.,( . (iB$,c45D;EG,D52(`8 BOA¤45(ª0 8iB$.D5"01(ª2(ª+CA7BO."(*A"+2(m8:9NBOA7aO(2*KZ(*.22, +:BO, 2(8 BOK/+7,A7(m2( a=!=,/+"+C(* K4;,/2+ pO+ªEG,([2,8 [email protected]+C(*.G,( .(iBO,45D5EG,D;2(-2,.G,NBOa=($xZW(<8 [email protected]+7( .G,(*.(iB$,45D;EG,D52((Jz(*8*+7D;8*A7Dp+CD5EG,( |( .ax_0 −3 {(*"+m2!=./.6KNBOAª4y\_ EG,NB$+7D5!=.V EW CEW CEW = −0.49 − 6.64 × 10−2 T |y~x~=·= Q{(*+"+C( EG,NB$+7D5!=.[2 8*A7D;[email protected]+ ( + I$BO4B$Y45(¤KL!=,As2( H8 [email protected]+C(*.G,( .F(iBO,4;D5EG,D;2(¤(*Z( 8*+7D;N8*!=0 KA7D; ( @. +7A7(-¥x®~=~6(*+r=xprPax_0 −3 xTuCEW D ¥/x~=~ax³0 −3 q ¥/x[email protected](J+{"D ²r=xprPax_0 −3 q CEW EW CEW EW rOx;r=x W- ( ª EG,NB+CD5!O.ª2 8*A7D;[email protected]+m8:9NBOEG,(<A7*a=D5!O.§2,aOACBOK9/(<"[email protected]+mA7 8 BOKDp+C,4; ( `2BO.m45(-+:B$Y45( BO,~x ã x f. uZrOq±J+:[email protected]+c2!=..EG,]\_D54ª.]\_(JD5+C(KBO12/(A" 7,/4;+:B+c2(4B$YZ!=A7B$+C!OD5A7(cKZ!=,/A12(*1+7( 0 KZ*ACB$+7,A7(* 7,/KZ*A7D5(*,A7(* # 'x³@· o QPq 4;( FI$BO4;( ,A"2(*8:9NBOA"a=( +7ACBO. A" ( "[email protected][email protected]+C(*A7KL!=45*( 4;D5. BOD5A"( 0 ( [email protected]+[ KNBOA+CD;A2(*PI$BO45(*,A72( # 'x³@· o Q (*+-2/(F+C(*4545(<7!OA"+C([EG,( ¥1KL!=,A ¥ o QPx%-.!=Y+CD5(*[email protected]+ δq δq T 2!=./8<2BO.ª45(8 BOª2(64y\[email protected]+C(*A7KL!=4B$+7D5!=.12 ( uZr=Gq u/q u~R(* + u/t/qNA"( "KZ(*8*+CDpI=(*01(*.G+[ δqS1 δqS3 δqS2 δqS9 = = = = (−2.75 × EW − 0.007) × T (−57.06 × EW + 12.60) × T (42.78 × EW − 1.08) × T 0 ·=· |y~x~OÎ@ %,0 A"! 2(64y\_ EG,NB$+7D5!=. uZr u~ u/ u ã u· u/Î u ' u/v u/t Z~x ã 2"[13AP?=&(*13$-=&Y> #$21(* fE,B$+CD;!=.2,+7ACBO./"( A+ª2(68:9NBOA"a=( ±BO4;( , A"`[email protected]+C(*A7KL!=45*( `4;D5. BOD5A"( 0 ( [email protected]+ E #>/r ã x ã ¥ f x 'x³[email protected]~ δ × E f x #[email protected]~xÎ ã ã rut/x ã ¥ δ × E ~O¥/x~=~ f x rOx=Î k ru¥/x³¥@· δ × E ~O× ¥ ã r=x 'OÎ f x #mr¯~$v/x('$¥ δ E #~OtO¥=¥/x~=×~ ã Î@~x³t/r δ × E /x³¥@~[#/r=x 'OÎ f x ~$Î/x®·$ fOx δ E ~O¥/x~=~ × f x #{~=~x~OÎ × δ E × ¥/x³¥ δ 7 2 - H$3NS,?"%$9Q,[$ .0J213$#AY&2[13AP?= (* ] 6 "%=T(*#$+&g21_ @_ © !O,A645( -+7( 0 KZ*ACB$+7,A7(*8 !=0 KA"D57(*( [email protected]+CA"( #>=¥ o Q (J+-# ã ¥ o QPq!=.,/+7D545D;7(4;( *E,B$+CD;!=.-I$BO4B$Y45(* H #&O¥ o QPx 4;!=. 2[kJnÍ'h¡ kJlfOnpo±b³~O¥=¥/r*wyqZD54(*"+6.*8 ( "CBOD;A7( 2(RA7*2,D;A7(R2(ru¥ 4y\_ EG,NB$+7D5!=.VKL!=A+:[email protected]+",A 7 Dª4y\_!=. 8 !=./7D52/3 A7(,.((sr 8 BO8 Dp+C2(8*!=454;( 8J+CD5!O.Ä2( 1a=!=,/+"+C( 4;(*+"+C( 2(¬r=x¤>4`7(*0Y4;(EG,(45(* ∆q EG,NB+CD5!O.R2( A" a=D;!=.R8 !OA7A7(*7KL!=.2NB$.G+1V2(§BOD;Y45(*8 [email protected]+7( .G, ( . (iBO, 45D;EG,D52(| u~qu//qSu·(J+ u/Î@R7,/A7( +CD;01(*.G+c4B8:9NBOA"a=(*8:9NBO.a=*(OxQu/BO,.2/( A7B 7,/a=a= A"2(A7*2,D5A"(B$,7"D8*( *E,B$+CD;!=. x WXBA7*a=D5!=.V./ [email protected]$+7D;IO(R( "+<B$D5."D A7*2,Dp+C( ~O¥ 2(*6I$BO4;( ,A"-KA" 2Dp+C( KBOA0u~c(*+u/q](*+645BcA7*a=D5!O. KL!=7Dp+CD;IO(( +A" 2,/D;+C(<§ru¥ ¨2( ªI$BO45(*,A7`K/A7 2/D;+C(*ªKNBOo A u·R(*g + u/Î/x u( " &$ )+ ,1,1- 0 ´ jChuh uÍ kJl`fOnpo brut=t '¯wH!O.G+A7(*KA7D;4;( mA7*7,4p+:B$+7P2(*P(JKL A"D5( ./8 ( P2(F45BOYZ!OACB$+7!=D5A"(-2(1ZfOË¡ 'kJjCÍ ª kJlªfOnpobrut=t/r*wyq( .+CACB$."!=A"0cB$.G+45(<8 !O.G+7( .G,( .( BO,4;D5EG,D52/([(Jz(*8*+CDpm| ª(*.+:BO,2z\BO8*8 A"*+CD;!=. EW KNBOA1a=D;IA7BOa=(| JxWX(+:BO, 2]\³BO8*8 A7J+CD;!=. KNBOA1a=D;IA7BOa=(( "+2 $N./DP8 !O010 (4;(KA"!2/,D;+12(45B ID;+7( 7"(A7(*4B$+7D;I=RAR (2]\_D50 KNBO8*+(*+S2,§8 [email protected]+7( .G,( .§(iB$,§45D5EG,D;2(P(Jz(*8*+7D;¤ x ( + RAR = EW × V RAR (JKA"D50 ( .ax_0 −2 x_ −1 x}-BO.P "!$#&%'[qN4;(-+:BO,§2]\B$8 8 A"*+7D5!=.KNBOA`aOD;IACBOaO(( "+m2 $N.DXKNBOA RAR = ρair × rc × Ecg × Vgmean BiIO( 8 Vgmean = R +∞ V (Dg )n(Dg )dDg 0 R +∞ 0 |y~x~'= n(Dg )dDg 7A (*KA7*7( [email protected]+7({4y\_(sr 8 BO8 Dp+C2(±8 !=4;45(*8*+CD;!=.-2(*HaO!=,/+7+7( 4;(*+7+7( X2]\_(iBO,[.G,NBOa=(*,7(±KNBOAX45(aOA7 "D54=(J+ Vgmean ( +m4BID;+C(*7"(<0 !u?=(*..(<2,aOA7 "D54¦x WXB8:9NBOA7aO(- 8:9NB$.a= (1|¢¦Q`45!=A"{2]\_,.(P8 !=4;45D5"D5!=.1BiI=(*8A7( YL!=.2§(*[email protected]+CA7(2/( ,KNBOA"+7D58*,45(*{2(a=45BO8 ( ( +m4BR01*01(PE,/([KL!=,A`45B KBOACBO0 *+7A7D;CB$+7D5!=.12(RZfOË¡ 'kJjCÍRkJl{fOnposbrut=t/r*w Ecg ∆q = BD m (∆v)n q |y~x[email protected] *( "+ 45(¤2DB$013J+CA7(¤2(4BªK45,/HKL(*+7D;+C(¤KNB$A"+CD;8 ,4;(|0cJq ( + 54 BªID;+7( "7(SA"( 4B+CD;IO(¤2]\³D;01KBO8*+{|0x_ −1 Jx ∆v W ( 8*!(Rr 8 D5(*[email protected]+C q (*+ [email protected]+4;( 0 0 ( HEG,(`KZ!O,AZfOË¡ 'kJjCÍkJlHfOnpobrut=t/r w(*+ [email protected]+A" 8 BOKD;+7,45* D m n B ·OÎ N2 BO./{45(`+CBOY45( BO,c~xx |( .1¦Q`( +S45B<8:9NB$A7a=(ª+CA7BO."*A7*(PKNBOA±8 !O4545D;7D;!=.][email protected]\_( +S,./(!=.8J+CD;!=.12/(45B +C(*01KL A7B$+C,/A7(F(J+2,+CqBO,2]\B$8 8 A"*+7D5!=.VKNB$APa=DpIA7BOa=($xZ>4H(JD5+C(,.+CBO,2]\³BO8 8*A7*+7D5!=.KNBOA-a=DpIACBOa=( 8 A"D;+7D5EG,(1| `BO,2/( 7",-2,/E,/( 4 45BRK45,ma=A"!=7"(FKNBOA+CD;8 ,4;(6I$B 7([8:9NBOA7aO( AmKZ!="D;+7D;I=(*01(*[email protected]+P(J+ RARcrit ( .R2/( 7"!=,2,/E,/( 4yq$( 4;45(I$BP"(S8:9NBOA"a=( A . [email protected]+CD;IO( 0 ( [email protected]+ixOW\³*E,B$+CD;!=.2*8 [email protected]+45(±+:BO,2]\³BO8*8 A7J+CD;!=. KNBOA`aOD;IACBOaO(8 A7Dp+CD;E,/(( "+m45B ",D;[email protected]+7(R RARcrit = a + bT + cT 2 + dT 3 + eT 4 + f T 5 + gT 6 q q q |y~x[email protected] q a = 1.0 b = 7.9262 × 10−2 c = 4.4847 × 10−2 d = 7.4754 × 10−3 e = 5.4686 × 10−4 (*+ x \_(JKA"D50 (( .ax_0 −2 x_ −1 x f = 1.6737 × 10−5 g = 1.7613 × 10−7 RARcrit 4 3.5 RAR (g m^-2 s^-1) 3 + 2.5 2 1.5 − 1 0.5 0 0 -5 -10 -15 Temperature (C) -20 -25 -30 N$-+,cH*A2"E(Q H%[$3N*13$-=&Z>@J#$4J4\M,TU"&ON$-? &+&@ ,$213A "%!g2=o>@?= 31 AP?= 7 (GF $4J213A3?[=o,$<7 NA ; $ N _ S(I+,]1 J(* 6]"%=T(*#$+&@1 _ < ?7 "$ 7 5 7 6 $-?"N$- ,]$-J+& #=c1 g!1V" (:F ; [email protected] A3?= ,$NA ; $- N* $#AZ13A2"E _ ; 7'7 7 X~x·\ ,]$-J+& #=c1V[13AP?= : *( OWo1 #.,J#$-[13"$4X21G(["m1V[" W ( [ EG,NB$+7D5!=.[7,A .(1"[email protected]+FI$BO45BOY45(*[EG,(1KZ!O,A[2/( [+7( 0 KZ*ACB$+7,A7(*F8*!=01K/A7D5"( <( [email protected]+CA"( #>v o Q (J+ q #~$ o QPxZ©H!O,A (J+ qZ45B K4;,PaOA7!="7(FKBOA"+7D58 ,/45(I$BBO8 EG,*A7D;A o o ,.(68:9NB$A7a=(6KL−23 !=7Dp+CD;IOC(R< T < −8 C RAR > RARcrit |y~x³=¥@ q = 6.74RAR − 1.36(−T ) + 10.05 ©H!O,A (*+ q45B[K4;,{aOA7!="7(PKBOA"+7D58 ,/45(ªIOB[7(-8:9NB$A7a=(*A{.*[email protected]$+CDpI=(µ# 0.3 < RAR < 3.3 RAR < RARcrit 0 ( [email protected]+< |y~x³/r¯ 2 q = 3.02 − 10.59RAR + 2.95(RAR ) ' $ ! '/ " !! ©H!=,/AR45( +7A7!=D;8iB$RD52 BO45D;7 [KA7*7(*.G+7 (*12NB$. 4B",D;+7(¬|8:9NBOK/D;+CA"( (J+ ã Jqs+C!=,/+7( R4;( RKNB# ACB$01J+CA7D;CB$+7D5!=./F2(*1KA"!8*( "7, .!=.#D5.2/,8*+7D;[email protected]+ *+7+7( +C (*1KL( ./[email protected]+ 4B+C9/3 7($xQ{( KL( .2NB$.G+ q KNBOA<4B7,/D;+C($qX7( ,/45<45( <A7*7,4p+:B$+7[2/( F"D50,/4B$+7D5!=.,+CD54;[email protected]+64B§KNBOA7BO01J+CA"D5CB+CD5!O.¬2/( Nf Of¶/fÍ:¶ brut 'Ov$wX"( A"[email protected]+-KA" 7(*[email protected]+C *qz7BO,/KL!=,Aª45(*J+C,2/( P2/(["( ."D5YD;45D;+7OxNW(<8:9!=D2([45B KBOACBO0 *+7A7D;CB$+7D5!=. 29 ( kHBOBO9NBO"9D.]\³(*"+( .1BO,8*,.18 BO±,.M",/a=( 0 ( [email protected]+±2(`I$BO45(*,A x=>4N*\B$a=D;++7!=,/+±7D;01K4;( 0 ( [email protected]+2]\_,.1KA7!$# Y4;3 0 (F2(R8*!=010 !2D;+7[2(R8*!2BOa=(Ox f.¬(Jz(J+iq]KL!=,AP4BcKNB$ACBO0 *+7A7D57B$+CD;!=.2( k B$$B$9NBO79/DyqzD;4H(µD5"+7( ·' q ,.( +:BOY/45(b h +J Zq¤rut=t ã w¤E,/D2!O..(45B8:9NB$A7a=(1 8:9NBO./a= ((*.!=.8J+CD;!=.2(14B§+C(*01KL A7B$+C,A"(c(J+ 2, 8 [email protected]+C(*.G,( . (iBO,¬45D;EG,D52($x]>4¤7,]r +[2!=.8 2]\_D5[email protected]+7( A7KL!=4;( A64B§8:9NBOA"a=( 8:9NB$.a= (1KZ!O,A645( I$BO4;( ,A" 2( (J+ A"( .8*[email protected]+CA" R2NBO./45(1.,BOa=(OxH}-BO./45(18iBO2/(c4BKNBOACB$01J+CA7D;CB$+7D5!=.2(ZfOË¡ 'kJjCÍ T LW C kJl`fOnpoXbrut=t/r w¦qN!=.B I,KA7*8 (*2( 0 01(*[email protected]+-EG,]\_D54(JD;"+C(6,./([*E,B$+CD;!=.2/D;Z A7(*[email protected]+C(<KZ!=,/Am4;( mtRA7*a=D5!=./ 2,a=A7BOK9($q(*+ª,./((J+CA7BOKZ!O4B$+7D5!=.1( "+m.*8 (*7CB$D5A7(<45B[!=D5{KZ!=,/A`45( {YNBO7"( `+C(*01KL A7B$+C,/A7( `(J+ª45(* +C(*01KL A7B$+C,/A7( 6",KZ*A7D;( ,A"( # ' ◦ QPxsWXB8*!=01K/45(JDp+CR( +F2/!=.8 K45,/<D;01KL!=A+:[email protected]+C(R(J+<D54 BO,+<B$D5A7( +CA"3 B$+7+7( [email protected]+CD;!=. <45B-IOB$45D52/D;+C2( ± EG,NB+CD5!O.±BO, 45D;01Dp+C(* ( [email protected]+7A7(m2D;Z A"( [email protected]+C(*A" a=D;!=. Ex jm.(ª*+7,2(m2( 7(*.7D;YD54;D;+7",AS45(*S2Dpz*A7( [email protected]+7( SKNB$ACBO0 *+7A7D57B$+CD;!=.±2( {KA7!8 (*77,/m./!=.#>D;.2,8J+CDp(*"+`A"iBO4;D57*(P2NB$. 45(<8:9NBOK/D;+CA"( ã x f454;(6KZ(*A70 (*+2(F0 (*+"+CA7([( .JID52(*.8 ([45( 2D;Z A"( .8*( ( [email protected]+CA"(45(*KNB$ACBO0 *+7A7D57B$+CD;!=. 8 !2*( F2BO.F "!$#&%'[qs(*+[45( ,/A<D;./N,(*.8 ( 7,/AF4y\_*IO!=45,+CD5!O.¬*45(*8*+CA"D5EG,( 2,.G,NB$a=(Oxs© B$AFBOD;454;( ,A" q D54s*+CBOD;+PD50 KZ!OA"+:B$.G+mKL!=,A-4;( 2D;Z A"( [email protected]+C8 BO-2(8*[email protected]=(*8*+CD;!=.D52 BO45D;7*([2(RK/A7 "( [email protected]+C(*A45(*-A7*7,4p+:B$+7 BiI=(*8145B01*01(KBOACBO0 *+7A7D;CB$+7D5!=.2/( 6KA7!8 (*7",[./!=.#>D;.2,8J+CDpKZ!=,/A*+C,/2D5(*A45( 2/D;Z A7(*.8 (*62( 8 !O01KL!=A"+7( 0 ( [email protected]+ª7(*45!=.4;(-+>?KL(<2(68 [email protected]=(*8*+7D5!=.]x / (" " $ % !=,/PBiI=!O.I,2BO.m4BR7(*8*+CD;!=.~x;rOx®~REG,(OqLKA7D;.8 D;KNBO4;( 0 ( [email protected]+iq4BR8 A"!=D5"CBO.8*(F2,a=A" 7D;47(6BOD;+ 7(*45!=.2( ,0 !2( <45(601!2(<2(<8 A7!OD577BO.8*(F73*8:9((J+P4;([0 !2/(<2/([8*A7!=D;7CB$.8 ([9,/01D;2(OxL}-BO./P45( 0 !2(62(<8 A"!=D5"CBO./8 (F9G,0 D52/(OqN,.($N.(<8 !=,/8:9(45D;EG,D52(KL( A"7D;"+C([145BR7,ABO8 ([2,a=A" "D54yx -,NB$.2 45(a=A" 7D;4(*[email protected]+CA7(( . 8*!=454;D57D;!=. BiI=(*8V,.(KBOA"+7D58 ,/45(2(a=45BO8 (|aO4BO8*(KA"D501BOD5A"(!=, .(*D5a=(iJq±D54`? B [email protected]+7( ACB$8*+CD;!=.¬( [email protected]+CA"(,.(17,ABO8 (145D5EG,D;2( (*+F,.(c",A"B$8 (caO( 45*(Ox Q{( KL( [email protected]+iq 4;( <KA7!8 (*7",.!=.# D5./2,8*+7D;X45!OA77EG,]\_D54;[email protected]+ 2 $N./D5sKNBOAX4BªK45,/KNBOA"+s2(*BO,+C( ,/A7b Nf Of¶/fÍ:¶:q=rut 'Ov =ÉfOÈOfOj"fOl¦¡zkªkJlXfOnpo5q rut=vO LZfOË¡ 'kJjCÍ[kJl±fOnpo5qLrut=tr u¢npfckJl±[email protected]· N´ªjChuh uÍ<kJl±fOnpo5qZrutOt 'wZ8 !=.8*( A".( [email protected]+ª45(*S8 !O4545D;7D;!=. ( [email protected]+7A7(<7,ABO8 (*a=45BO8 *( *xXZfOË¡ 'kJjCÍ[email protected]~wsBR0 [email protected]+CA76EG,(<45!=A"7EG,(<45(8*[email protected]+C( .G,(*.( BO,4;D5EG,D;2(-( + 7T, [email protected]+KL!=,AmKA7!uI=!EG,( A4B 8 A"!=D5"CBO.8*(F2,a=A7*7D;4s"( 4;!=.4;([0 !2/(<9G,0 D52/(Oq4B 8:9NBOA"a=([*8:9NBO.aO ( ( + 2]\_,.Ä!=A72/A7(§2(aOACBO.2/( ,AD;./ A"D5(*,A45B8:9NBOA"a=(§ 8:9NB$.a= (§4;!=A7"EG,(45(ca=A7*7D548 A"%! l5+F7( 4;!=.Ä45( 0 !2(c7( 8$x>4`BBODp+F4¦\³[email protected]?KL!O+C93*7(EG,(Oq ( . a= ./ ACB$4yq 4;( 8*A7D;"+:B$, 2/(§a=4BO8*(7(§8*!=454;( [email protected]+RB$,Äa=A7*7D54 45!OA7EG,]\_D54z(*"+ª2NB$.ª45(-0 !2/(-2(8 A"!=D5"CBO./8 (69G,01D;2(Oq(*+ªA"( YL!=.2D;77(*[email protected]+A7BOA7(*01(*[email protected]+ixN©HBOA`8 !=./7 EG,(*[email protected]+iq 45(0 !2(2(68 A"!=D5"CBO.8*(69,/01D;2(62,aOA7 "D54].z\³(*"+KNBOªs( r 8iB$8 (<KZ!=,/A`4BR7*KNBOACB+CD5!O.2/(<8:9BOA7a=(6KNB$A 45(m0 8iB$.D5"01(ª.!=.#>D5./2,8*+7D;&[email protected](J+`BOD;.7D¦[email protected]( .KA"( 0 D53*A7(BOKKA"!uD;0cB$+7D5!=.zq=7(*,4Z4;(aOA7 "D542NBO./{45(m0 !2( 2(68 A"!=D5"CBO./8 (<}-w T v)( +m8 !=."D52*A76KL!=,Aª45(*`KA7!8 (*77,/2(6" KNBOA7B$+CD;!=.2(68:9NBOA"a=([./!=.#>D;.2,8J+CDp&x ¸S¹¸S¹ ` ½ az¿¤¾ÆÁ /¿¤¾ ÁaÃP¿ &¾Ágbga W!=A7"E,z\³,. a=A7BO,KL( 4¦q$8*!=.7D;2 A"S8 !O010 (,/.(`7K9/3 A7({8 !=.2/,8*+7A7D58*(Oq=8:9G,/+7(`2NBO. ,.R8:9NBO0 K 4;( 8µ# +CA"D5EG,(±I=( A+CD;8iBO42D5A"D5a=¤I=(*A745(±YNBO*qO45BPKNBOA+CD5(±7,/KZ*A7D5(*,A7({2(S4BmKNBOA+CD58*,45(7(S8:9BOA7a=({. [email protected]+CD;IO( 0 ( [email protected]+iq (*+-4B1KBOA"+7D5(<D5./*A7D5(*,A7([KZ!O7D;+7D;IO( 0 ( [email protected]+ix:f.V(*Z(*+iqZ45!=A"7EG,]\_,.(KBOA"+7D58 ,/45(<7(KL!=4B$A7D5"([2/,BOD;+2]\_,. 8:9NBO0 K 4;( 8*+7A7D;E,/(-(J+C*A7D;( ,A*q45(*`8:9NBOA7aO( ªR4y\[email protected]+C*A7D;( ,A2(4BFKBOA"+7D58 ,/45(P+7( .2(*[email protected]+m "(A7 BOA7A7BO.a=(*A 7KB$+CD5BO45(*01(*.G+±¯45(±8:9NBO0 KF*45(*8*+CA"D5EG,(±[email protected]+C(*A7.({BOD5./[email protected] A" S*\³!OKKZ!O7(SB$,F8:9NBO0 KF*45(*8*+CA"D5EG,(±(J+C*A7D;( ,A*x W!=A7"EG,(2/( {aO!=,/+7+7( 4;(*+7+7( *[email protected]( 454;( &#>0 0 (`KZ!=45BOA7D;7*( qO( [email protected]+CA"( [email protected]+S(*.c8 !O.G+CBO8*+{BiI=(*8P4BKNB$A"+CD;(ªD5./*A7D5(*,A7( 2,a=A7*7D;4yq/(*454;( SKL( ,/IO( [email protected]+-A"( YL!=.2D;AS(*+m(*01KL!=A"+7( AS,/.(68:9NBOA7aO(<./(*+7+7(6KZ!="D;+7D;I=($q4[email protected]+`45(aOACBO,KL( 4 BiI=(*8[,./(<8:9BOA7a=(6.(J+7+7([.*[email protected]$+7D;I=($x f.«B$D;+iqD54<.(Ä \³BOa=D;ACBODp+KNB$VA7 (*454;( 0 ( [email protected]+2]\_,.«A7(*YZ!O.2]xW( V(µKZ*A7D5(*.8 (* (*.«4BOYL!=ACB+C!=D;A7( ·Ov − − − −−− − − − − − E + + − + +θ + + + + + + + ++ V − + 6 HJ#.) (*,T$#AY= A ,m(["L.0J [=EA &#.0AY=T([" 13A > _1<G H., J#! 13$#A2"E0# ; #$213A @1(AY$#AN*J #$ + 9 & ! @ & _<G7 ([A &213$+A "13A3?=C(*7 +& H$3N+&97 &+"%$9!+&,$213A 7 "!+&m+&217 (*?= ,? &+A37 13A ; 9&+"%$9WX,$213AV7 9AZ= > J#$#AP#"%$4 ; 6 21S=]JPNI[13A ; )&+"%$\g,6 $213AVm&#" ,J2$+AV#"7 %$- _ < NI?"1P1 #@1P1 #=7 c13$4#= ?ZUA 7 &#A3?= $ &2=c1 ; !N$-J#&+AZwDI=c1 7 7 "%=] ; AZ1 #&'&@g(* H"%1 _ $- ,T$-J#&@#=c1 m-F =N!g(* ?Z AY&+AP?=]_ 7 7 θ X]~xÎ 2]\PËJg'kJj43 fOËjkJl±ÉNhu¶¡/Íuh¡ brut '=~w][email protected]+0 [email protected]+CA"EG,([2BO.m,.K/A7( 0 D5(*AS+C(*01K*qN4BRa=!=,+7+C(*45(J+7+C("( 8 !O4545(ªB$,1a=A" "[email protected]/,D5¤( 4;45(`"(P2J+:BO8:9($qG4BOD;7CB$.G+±,./(KNB$A"+CD;(ª2(7B<01BO"7(m(*+2!=.8m2(mCB68:9NBOA7aO(",A 45(ma=A" 7D;4yx %-..(-8*!=."D523*A7(PEG,(P45( KA7!8 (*7",`D5.2/,8*+7D;±D50 K45D;E,[email protected]+4;(aOA7 "D54L(*+{45( a=!=,/+"+C(*45(*+"+C(* 2]\_(iBO,zxNd]fOl¢¶/f 3 kJl .fÍuh¡¬brut=Î@~¯w[email protected]+{0 [email protected]+CA"mEG,(45(ªKA"!8*( 7",SD;.2,8J+CD;Z.]\_( +SKNBO±s( r 8 BO8 (KZ!=,/A 45(*P8*!=454;D57D;!=.`(*.G+7A7(RKNB$A"+CD;8 ,4;( m2(a=45BO8 ($xLWsB1K45,mBOD5Y/45(68 !=./2,8*+7D;ID;+7F2(F4B1aO4BO8*([KNB$A#>A7BOKKL!=A"+ 4¦\³( BO,¬45D5EG,D;2(R(*+F45(*6+7( 0 KF2/(18*[email protected]+:BO8J+F8*!=,A"+7 qs.(cKL( A"01(J+7+7( [email protected]+FKBO[4¦\³*8:9NBO.aO(12]\_,.(18:9NBOA"a=( BOKK/A7 8*DBOY/45(OxQ{(*KZ(*[email protected]+ q/4B68:9NBOA"a=(P" KNB$A7 (45!=A"¤2(P8*(P0 8iB$.D5"01(ª( +SB$77( PBOD;Y45(`K,/D57EG,(45(* 8 !O4545D;7D;!=.6s( r 8iB$8 ( R"[email protected]+ 2( F8 !=4;45D5"D5!=./<"!=,D;.8 D;2( ./8 ([email protected]+7(Vb 1ªfÍ @kJn¢n>qrut=v/r*wyx %-A qsKA73*R2, a=A7BO.28 ( A"8 4;(<*EG,NB$+C!OA7DB$42,aOA7 "D54¦q4BR8:9NBOA"a=(62D5"KZ!O.D5Y4;(( +YL(iB$,8 !=,/KK4;,`BOD5Y4;(-EG,(<7,Am45(* KGp=45(* x Q{(KA7!8 (*7",.(KL( A"01(J+2!O.8cKNB$R4;,D"( ,42]\_(JK45D;EG,( A[4y\_ 45(*8*+7A7D57B$+CD;!=.¬2( .G,NB$a=( *x BOD; qZ8 (R0 8iB$.D5"01(KL( ,+2(*IO( .D;A67D5aO.D $N8 B$+CDpK4;,P+:B$A72V45!OA77EG,(R4;(.G,NBOa=(F( "+<2>M7 4;( 8*+7A7D;7Ox .VfO¡/ÍukJn¢nskJlfOnpob~O¥O¥@·[email protected]+`01!O.G+7A7-EG,(64;(BODp+S2]\³B¯M"!=,/+7( A`!=,§KBOS45(*SKA"!8*( 7",mD5.2,/8*+CDpS4;( ,A" 7D;0,45B$+CD;!=.±.,/01*A7D5EG,(*ªKZ(*A70 (*+`2]\_!=Y/+7( .D;A`2( `8*!=0 KZ!=A+C(*01(*.G+7` 4;( 8*+7A7D;E,/( `2Dpz*A7( [email protected]+7 x>4;`[email protected]+ A7(*0cB$A7EG,m45(`A7(*./!=A"8 ( 0 ( [email protected]+2(m4BA7*a=D5!=. KL!=7Dp+CDpI=(ªD5./*A7D;( ,A"(`2(4¦\³!OACBOa=(`2 64BKA7D;7(ª( .18 !=0 K/+7( 2(*SKA7!8 (*7",SD5./2,8*+7D;*q8*(PEG,DZD50 K45D;E,/(`,.§BOYNB$D57"( 0 ( [email protected]+S2/(P4y\³BO4p+CD;+7,2(ª2(2 8*45( ./8:9( 0 ( [email protected]+`2(* 8*4BOD;A7`(J+ª,.([BO,a=0 ( [email protected]+:B+CD5!O.§2,.!=0YA"(62([X Q n)KA7!2,Dp+C*x WXBm!=A"0,4B+CD5!O.2,F+:B$,<2(+7ACBO./"( A+ 2/(8:9NB$A7a=(KNBOAs01*8iBO.D;70 (¤D5.2,/8*+CDpEG,(.!=,H,/+CD;45D5"!=. ·Ot ( +mD57",(62( Xyk On5kJjFkJlSfOnpoXbrut=t/r w¦x |y~x@~O ∂Q(D) π = Ecg Er D 2 V (D)Nc αδq ∂t 4 BiI=(*81 |y~x³=@ d2 π3 2 π2 d Ez cosθ − Q(D) 2 2 6 D ( +4Bm8:9NBOA"a=(SD5./2,D;+7(",A ,/.a=ACB$,KZ(*42/({+:BOD;454;( q ( +45BmID;+7( 7"(S2(8:9,+C(`2,aOA7 "D54¦q Q(D) D V (D) ( +4;(S2D5BO013J+CA"(S2(`4B-KL(*+7D;+7(SKNBOA+CD58*,45($x (*"+¤4B-8*!=0 [email protected]+C(SIO( A"+7D58 BO45(S2/, 8:9BO01K *45( 8J+CA"D5EG,(Oq d (*"+R4¦\³(Rr18 BO8 Dp+C2/(8 !=4;45D;7D5!O. (*.G+7A7(§4;(cEa=zA" 7D;4{(J+R45(*a=!=,/+"+C(*45(*+"+C(* q (*"+R45BKA7!OYNBOYD;45D;+7 2( Ecg Er A7(*YZ!O.2]x 8 !=A"A7( "KZ!O.2 4y\³BO.aO45(2(8 !O4545D;7D;!=. KNBOA#ACBOK/KZ!=A+§45B¬IO( A"+7D58 BO45(2( "8 ( ./[email protected]+C(¬(*+ A7(*KA7*7(*.G+7θ( 4y\_(sr 8iBO8*D;+7F2(A"( YL!=.2]x Xyk On5kJj kJlmfOnpobrut=tr w !O.G+-B$+7+7A7D;Y,F45B I$BO4;( ,AP¥/xv ã (*α+ cosθ ¥/x³¥@~=~R x^KA73*D;[email protected]+C aOACB$+7D5!=.§",Aª.X|¦}µqN!=.!=Y/+7D5( [email protected]+6 α δq = 1 π3 ρ00 0.4 1 ∂Q = Ecg Er Nc Ng αcg d2 ( ) [πEz cosθM (2 + dg ) − eg M (fg + dg )] ∂t ρdref 8 ρdref 3 ©H!O,A ¸S¹¸S¹ d !=.KA"( .28 !=0 0 (-IOB$45( ,/A-r¯· ¿¤¾ µ 0q(*+mKL!=,A Nc q ã ¥=¥R8 0 −3 |y~x³ ã x ¿¤¾ ¼[À Á a ÿ j.(!OD5cEG,(45( c8:9NBOA7aO( §[email protected]+§J+C" KNBOA" (*KBOAc4;( KA"!8*( "7,.!=.#>D5./2,8*+7D;*q±(*454;( c"[email protected]+ +CA7BO. A7*( [2]\³,/.(cKNB$A"+CD;8 ,4;( 4¦\B$,/+CA"( 4;!=A762(*[K/A7!8 ( "7,F01D;8 [email protected]?"D5EG,(* xs©H(*[email protected]+8*( [KA7!$# 8 (*7",| $Na=,A"(<~xpr¯Jq2/(4B01BO7"(( "+m*8:9NBO.a=*([(*[email protected]+CA7(4;( `KNB$A"+CD;8 ,4;( `EG,D]D;[email protected]+C( A7BOa=D;77(*[email protected]+iq(J+ª2!=.8$q 4B8:9BOA7a=(§*45(*8*+CA"D5EG,(§J+:[email protected]+ A7*KNBOA+CD5(45B7,/A"BO8*(2/(§4BKNBOA+CD;8 ,4;(Oqs,.(8 (*A"+:B$D5.(§EG,[email protected]+7D;+Cc2( 8:9NBOA"a=(m( "+SBO,"7DN+CA7BO."*A7*(Ox/Q{(+:B$, 2(ª+CACB$."(*A"+2(8:9BOA7a=(m( +S8 BO458*,45`(*.[email protected]+C*a=ACB$.G+±2(ªB%e*!=. (JK4;D58*D;+C(P45(-+:B$,2(+CA7BO.( A"+m2(601BO7"(60,4;+7D5K4;D5mKNBOA`45B A"( 45B$+CD;!=.8:9NBOA"a=(J#2DB$013J+CA7($x 1 Z +∞ QY COLX = ρ 0 Z %-.BR4;( ` EG,NB+CD5!O.ª7,DpIOB$.G+7( 6 0 K(Dx , Dy )qy (Dy )ny (Dy )dDy nx (Dx )dDx |y~x³@·= G(b) λb 0 Z +∞ G(f ) ρQ = q(D)n(D)dD = eN f λ 0 |y~x³=Î@ 1 dλ ρdr = aN G(b)d b = aN G(b)(−b) b+1 λ λ 1 dλ ρdQ = eN G(f )d f = eN G(f )(−f ) f +1 λ λ |y~x³[email protected] ρr = uD!=.2Dpz*A7(*.8 +∞ Z +∞ m(D)n(D)dD = aN |y~x³ '= D;(8 ( m2(*, EG,NB$+7D5!=./ qN!=.B§ Î=¥ |y~x³[email protected] \ %-.2/D;ID5"(4BKA"( 0 D53 A"( EG,NB$+7D5!=.KNB$A r (*+m4B2(*,D53*01(6KNBOA ∆r ∆λ = aN G(b)(−b) r λ ∆λ ∆Q = eN G(f )(−f ) ρ Q λ ρ uD!=.8*!=0YD;.(45(*m2/( Q |y~x ã ¥@ |y~x ã r¯ ,*EG,NB$+CD;!=.6 ∆Q = f ∆r Q b r |y~x ã ~= ^ D;.7D¦qHD;4¤( "+ KL!="7D5Y/45(c2z\³(µ/K/A7D50 ( A[7D;01K4;( 0 ( [email protected]+[45(1+CBO,2(8:9NB$A7a=(c+7ACBO./" A" (§(*. !=.8*+7D5!=.2, P +:B$, 2(01BO7"(+7ACBO./" A" ($x`Q{( KL( ./[email protected]+iq45!=A"7EG,(45( 12( , KBOA"+7D58 ,/45( EG,[email protected]+C*ACBOa=D;7"( [email protected]+c"[email protected]+ KA" 8 D;KD;[email protected]+C(* q 4;( +CBO,2(§8 !=4;45(*8*+CD;!=../(§KZ(*,/I=(*.G+ K4;,R*+7A7(D;[email protected]+C aOA7 B$.NBO4;?G+7D5EG,( 0 ( [email protected]+R",AR45( 7KL( 8J+CA"([2(+CBOD54;45(Ox}-BO./m8*(68iBO&#>4q!=.,/+CD;45D;7(-2(*./!¯?=BO,K/A7J#8iBO4;8 ,4; *x W( <9G?2A"!=0 *+C*!=A7(*[+7ACBO."KZ!OA"+C(*[email protected]+F8*( <8:9NBOA7aO( <2NBO.<2]\³BO,/+CA"( <KNBOA+CD5(*62,.G,NBOaO(1"!=,<4y\_D5.# N,(*.8 (P2(P45B[7*2D50 ( [email protected]+CB$+CD;!=.][email protected](4B<+7,A7Y/,45(*.8 ((*+S2/(P4y\³BO2/IO( 8J+CD5!O.]xW¤\³BO2/IO( 8J+CD5!O.( +SA7 BO45D;7*(KNB$A 45(78:9*0cB ©¤}`O ^ ks^ |¦,4p+CD52/D50 ( ."D5!=.NB$4©H!="D;+CDpI=(}( $N.Dp+C(-^`2I=( 8J+CD;!=. kHA7BO.7KL!=A+`^P4;a=!=A7Dp+C90c EG,DzKL( A"01(J+{45B[8*!=.7(*A"I$B$+7D5!=.2/( S2(*.7Dp+C*S2(P8:9BOA7a=(PKZ!="D;+7D;I=(*2z\³,./(-KNBOA+S(*+{. a=B$+CDpI=( {2]\B$,/+CA"( KNBOA+ixHW( FKBOA"+7D58 ,/45( <45(*FK4;,FaOA7!="7( F8:9,+C( [email protected]+ B$45!=A"[EG,(4;( [KNBOA"+7D58*,45(*[4;( FK/45,<45*a=3 A"( F"[email protected]+ +CA7BO."KZ!=A+C*( `2NBO./`4BFKNB$A"+CD;(P",KZ*A7D;( ,A"(2,§.G,NBOa=((J+`2NBO.`4¦\³(*.8 4;,0 (Ox}P( `A" a=D;!=.S2(PKZ!=45BOA7Dp+C !=KKL!=" (R7!O.G+<B$45!=A"P!=A70 (* qZ(*+6,.8:9NBO0 KV 4;( 8J+CA7D;EG,([email protected]+C(*.7(R(*"+<a=*. A" 4y\[email protected]+C(*A"BO8*(2(R8 (* A7*a=D5!O. x ! ! " W(V8:9NBO0 K 45(*8*+7A7D5EG,( → − (*"+7!O45,/+7D5!=. 2]\³,/.( EG,NB$+7D5!=. 2( n<BO,/7§[email protected]+45B7!=,/A78 (V(*"+45B 2(*.7Dp+CI=!=4;,0 D5EG, (2(68:9NBOA"Ea=(+C!O+CBO45( ρtotal ρtotal → − ∇· E = 0 |y~x ã @ → − E → − 0 ∇V = − ρ̃ |y~x ã ·= ρtotal → − GDIV (ρ̃ ∇V 0 ) = 0 |y~x ã Î@ ( "+ 45B8*!=.+:[email protected]+C(2D; 45(*8*+7A7D5EG,(2(4¦\BOD;A x WX(KL!O+C(*[email protected]+CD5(*4` 4;( 8*+7A7D;E,/( ( + 2 $N./DKNBOA+CD;AR2, V 8:9NBO0 K 45(*8*+7A7D5EG,(KBOA6 |y~x ã=ã → − → − E = − ∇V f.§BOD;+ qD;4Z(*"+ª8 !O010 !2/(P2NBO./`4BF",D;+7(-2]\[email protected]+CA"!2,/D5A7(P,.§K7(*,2!$#KZ!$+C( [email protected]+7D5( 4z 45(*8*+7A7D5EG,( 2!=./. V0 KNBOA 0 iB I=(*8 x (*"+4;(M7B$8 !=YD;( .[2({45Bª+7ACBO.!=A701B$+7D5!=.c|IO!=45,0 (¤2({45B01BOD54;45(iJq (J+ A"( K A" "( [email protected]+C( ρ̃ = ρdref J J 2!=./8<45B0cB$77(62]\³BOD;A`7( 868*[email protected]+C( .G,2NBO.m,.(601BOD54;45(OxG^D5."Dyq4¦\³*E,B$+CD;!=.§2(n<B$,7ª2(Jρ̃ID;( [email protected]< + Î/r n6}P ( +X4y\_!=KZ*ACB$+7( ,Az2D;IO( A7aO( .8*(±2BO.X45(?"+C3*01(¤2(8*!!OA72!=./. (*8 ,AID;45D;a=.( z.!=.#!=A"+79!=a=!=.BO4yx ©A7*7( [email protected]+7 (<BOD5./7Dyq8 (J+7+7( EG,NB$+7D5!=.(*"+ª8*!=01KBOACBOY/45(8 ( 4;45(,/+CD;45D5" (mKZ!O,AS45(8iBO4;8 ,4L2(45B[KA"( 7"D5!=.zx © B$AR8 !=." EG,(*.G+ q¤45("!=4;IO( ,AR2(§K/A7( "7D;!=. ( 454;D5K/+7D5EG,(1"+:B$.2NBOA"2 2/( "!$#&%' KZ(*,/+ *+7A7(§,/+7D545D;7 KL!=,AªA7(*.GIO!u?=( AP45(K"( ,2!#>KL!O+C(*.G+7D5(*4 0 x %-.A"(*+7A7!=,/IO(F4;(8:9NBO0 K*45( 8J+CA"D5EG,(<( .,/+CD;[email protected]+ª4y\_ EG,NB# V +CD;!=.~x ã · ux W( F8*!=.2Dp+CD;!=.[BO, 45D;01Dp+C( [email protected]+RB$2NBOK/+7 ( BO, +7ACBODp+C( 0 ( [email protected]+<2, 8:9BO01K 4;( 8*+7A7D;E,/(Ox %.¬BOD;+ 4y\[email protected]?KZ!O+793 "(VEG,(4B4;D50 D;+C(",KZ*A7D;( ,A"(V2, 2!=01BOD5.(8 !=A"A7(*7KL!=.2£ 4y\_D5!=./!=7K9/3 A7((*+45B45D;01Dp+C( D5. A7D;( ,A"(< 45B ",A"BO8*(<+7( A"A7( +CA"(OxZWX(*4;D50 D;+7( S",KZ*A7D;( ,A"([(J+PD;./ A"D5( ,/A7(62,2/!=0cB$D5.("[email protected]+-2!=./8 8 !O.7D;2 A" ( ±8 !O010 (S*[email protected]+±2( , 8*!=.2,/8*+C(*,A7±KNBOABOD;+7 [email protected]©HBOA¤8 !=./7 EG,(*[email protected]+iq45(`KL!O+C(*.G+7D5(*4 4;( 8*+7A7D;E,/( ( +`8 !O."[email protected]+`BO,1A7!O.G+7D53*A7( 7,KL A"D5(*,A7(*S(*+{D5./*A7D5(*,A7(* q(*+{45(m8:9NBO0 K 4;( 8*+7A7D;E,/(P( +`!=A+C9!=aO!=.NBO4 R8 (*",A"BO8*( *x %-.BOKK/45D5EG,(2/!=.862( m8 !O.2D;+7D5!=./S2(<%(*,0cB$. |y~x ã '= → − → − → − − n · ∇V 0 = → n · E fw ( "+1,. I=(*8*+C(*,A ,.D;+CBOD5A"(.!=A"0cB$4`V4BV7,ABO8 ($q(*+ −−→ (*"+ 45(§8:9BO01K 4;( 8J+CA7D;EG,(2(YZ( BO, +C(*01K/ xG©H!O,A4;( ¤45D;01Dp+C(* 4B+C A7BO45(* q=!O. BODp+¤4y\[email protected]?KL!O+793 "(EGE,f(mw 45(`8:9NB$01K1 45(*8*+7A7D5EG,(ª(*"+±A7(*4B`I=(*A7 CBRI$BO45(*,A`2(6YL(iBO,§+C(*01K/ xW(68:9NBO0 K2/([YL(iBO,§+7( 0 KmB KL!=,A`(µKA7(*77D;!=. → − n → − → − E f w = E 0 exp −KE H z − zs H − zs |y~x ã [email protected] ( "+ [email protected]/[email protected]~xpru¥ − → (*"+IO( A+CD58 BO4/(J+¤2D5A"D5a=¤I=(*A7¤45(YNBO O7!=.R0 !2/,45(±( +ªru¥=¥ Fx³0 −1 x E0 (*"+`45B[9NBO,+C( ,/A{01BD501BO45(ª2,2/!=0cB$D5.($q(J+ 4B[9BO,/+C(*,AS2,cA7( 4;K D5(JE&x^,c45D50 D;+7( ¤4B+C H zs KL!O+C(*.G+7D5(*4] 4;( 8*+7A7D;E,/(<2/!=D;+`2/!=.86!=YL D5Aª§ ∂V 0 =0 ∂n 0 −1 x A7BO45(* Gq 45( −6 |y~x ã t= 4 W \B¯M"!O,/+¤2( *8 45BOD5A"2NBO.45BPKNBOA7BO0 *+CA"D57B$+CD;!=.F2(*!=ACB$a=( (*"+¤D5./2D5"KZ(*.CBOY/45(Ox%f. (Jz(J+iq=D;[email protected]+ 2(*, (*Z(*+701B¯M"( ,/A7m@D545 45D50 D;+7( [email protected]+4¦\BO0 K4;D;+C,/2({2,18:9NBO0 K1*45(*8*+CA"D5EG,(ªKA"!2,/D;+¤2NBO./4;(`.,BOa=(OqG(J+ D54;{A"( 2D;"+CA"D5Y,/( [email protected]+ª45(*`8:9NBOA7aO( mR4y\[email protected]+C*A7D5(*,A{2/,.G,NB$a=(OxGuDz!=..]\³D;[email protected]+CA7!2,Dp+SKNB$`45( `*8 45BOD5A"S2NBO.`8*( +>?KZ(`2/(`KNBOACB$01J+CA7D;CB$+7D5!=.zq4;({8:9NB$01K 4;( 8J+CA7D;EG,(`B$+"+C( D;[email protected]+2( I$BO4;( ,A"7,/A7A7 BO45D;"+7( q+CA"3 4B$A7a=(*01(*.G+ 7,/KZ*A7D5(*,A7(*PB$,"( ,D;42(6A7,K+C,A"(<2/(<4¦\B$D5AS8*4BOD;A x&%.2D;"+CD;.a=,(2/( ,8 B$+C*a=!=A"D5( ª2]\_ 8*4BOD;A7- 4;( ª 8*4BOD;A7SD;.G+7ACB#.G,NBOa=(*F|.!O+7 ªQ`ªEG,D.z\B$+"+C(*D5a=.(*[email protected]+KBOª45(67!=4¦x>45`8*!=.+CD;+7,( [email protected]+( [email protected]"!=. v=¥ [email protected]· 2(645BF+C!O+CBO45Dp+C-2(*ª 8 45BOD5A"`7,[email protected]+ª4BR4B$+7D;+7,2(Ox 4;( 6 8*4BOD;A7.G,NBOa=(µ#>"!=4m|.!$+C [X Q n<JxXW( < 8*4BOD;A7<X Q n "[email protected]+FKL!="D;+CDp-!=,¬. a=B$+CDp7,DpIOB$.G+<45B 8:9NB$A7a=(<.( ,+CACB$45D5" (*m2BO.m45(.G,NBOa=($xLWX(*-X Q n )"[email protected]+PYL(iBO,/8 !=,K0 !=D5.`./!=0YA"( ,EG,(<45(* Q nP#mD54z7(6KA7!2,Dp+ª( [email protected]"!=.,/.X X Q n KL!=,Aru¥ X Q nP#:x GÑ Jß"Ԧ׳ÓyÖ ¯ ÙÑ OصÓyÑÒ Ó¦×ÑÙ§ß"Ù³Ñ7Ü>Ó¦Ôy×®á*âÑ → − → − GDIV (ρ̃2 ∇V 0 ) = GDIV (ρ̃ E ) V0 Ø ¯ Ó¦×ÑÒ Ó ÖµÙØJÔ ¬Ñ"Ò)ÔyߦØJÙ:ÖµÒ ÓÙ ß"á âÖ:Óy׳Ø*Ò Î@~ → − GDIV (ρ̃ ∇V ) = ¸S¹ m¹º £½ a ¾gaà Xƾ Á VbP¾ ½ a &¿ W(R8:9NBO0 KV2(A7,/K/+C,A"(R2D5*45(*8*+CA"D5EG,([( "+62/(R4y\_!=A72A"(F2(=¥=¥=¥c x_0 −1 BO,V./D;I=( BO,V2(45Bc0 ( A x Q ( KL( ./[email protected]+iq=4;( K/A7! $N4;2(S8:9BO01KR*45(*8*+CA"D5EG,(S0 ( ",A7*2NBO. 45( .,BOa=( 2z\³!=A7BOa=($q$0 0 ({*\³D;45 "[email protected]+ { KL!=.8*+7,( 4; [email protected]+m,.([BO0 K45Dp+C,2/(-0cB¯/D;0cB$45(P2/(<4¦\³!OA72A"(2([45B8 (*.G+CBOD5./([2(6 x_0 −1 xN>4"( 0Y/45( A7BOD;+ 2!=./8`E,/(m4;(`8:9NBO0 K 2(`A7,K+C,A"(m2/D5 4;( 8J+CA7D;EG,(S0 (*[email protected]+( .-M"( , 2(*¤ 45(*8*+7A7!=.2/(ªB$D5Y4;({ ./( A7aOD5(<| =¥m( 6s./({KL( A70 (*+"+C(±KNBOH2]\³(µK45D;E,/( A 4;(¤2 8*45( ./8:9( 0 ( [email protected]+2(* 8*4BOD;A7*x .f0+ !¤fOjµl¢¶ÈFkJlZÇSfOj uÍ[email protected]∼ ·w [email protected]+*+C,/2D5[4y\_ 0 D57"D5!=.2(<ACBi?O!=. # M","+7([email protected]+P4;([2/ 8 4;( .8:9/( 0 ( [email protected]+2(*P 8*4BOD;A7*xN>[email protected]+P0 [email protected]+CA" EG,(<2( 45(*8*+7A7!=.ªA"( 4B+CD;ID;"+C(*[| r( 6ªBO8 8* 4; A7*KBOAª45(68:9NBO0 K*45(*8*+CA"D5EG,([ 4¦\³D;[email protected]+C A"D5(*,A`2, .G,NBOa=(KL!=,/I$BOD;( [email protected]+`J+CA7(64¦\³!OA7D5aOD5.∼ (m2(8 (J+7+7( 0 D57"D5!=.c2(ACBi?=!O. #x jm.§ 8 45BOD5A7(*ACBODp+`2!=.8D;.D;+7D5 45!OA77EG,( 2(*6 45(*8*+7A7!=.6*.( A"a=*+7D5EG,( [BO8*8 4; A" 6KNBOA4;(R8:9NBO0 K *45( 8J+CA"D5EG,(R2( 4y\_!=A7BOa=(RB$+"+C(*D5a=.(*[email protected]+ 2(*P ./( A7aOD5( m7T, [email protected]+C(*-KZ!O,AªKA7!2,D;A7(62(F./!=,/I=( BO, 4;( 8J+CA7!O.(*.8 BO78 BO2(FKBOA8*!=454;D5"D5!=.§BiIO( 8 2(*0 !=4; 8 ,/45( c2]\B$D5A x{W\³D;!=.D57B$+CD;!=. 2(*B$+C!=0 ( §2z\B *!O+C(V(J+2]\_!u?a=3*.( 4;!=A7§2/(V8 ( 8 !O4545D;7D;!=. KA"!2,/D;+12( A7Bi?=!=. # b 1SËjC%k u¢l +:¶£kJlRfOnpo5qmrut=t=~ .0f + !¤fOjµl¢¶È kJl<ÇSfOj uÍ@[email protected]~w¦x±©H!=,/AEG,]\³,/.( +C(*454;(§BiIOB$4BO.8:9/(*45(*8*+CA"!=.D;E,/(B$D;+R4;D5( ,zqD54{B$,/+ EG,(4;(§8:9NBO0 K *45(*8*+CA"D5EG,(§"!=D;+ ",KZ*A7D;( ,A ,. 8:9NBO0 K 2/(A7,/K/+C,A"( |&Y/A7(iB$O(*IO( . $N(*452&Jx>4m8 !=A"A7(*7KL!=.2 B$, 8:9NBO0 K *45( 8J+CA"D5EG,(. 8*( 77BOD5A"( KL!=,Am8 !=0 KZ(*.7(*AP45B1KLE( beA"+7([2z\³*.( A"a=D5([45!=A"2(*P8*!=454;D57D;!=.ªBiI=(*8D5!O.D57B$+CD;!=.BuIO( 84;( m01!O45 8*,45(*ª2( 4y\³BOD;A x$WX({8:9NBO0 K *45(*8*+CA"D5EG,(`2(`2/ 8 4;( .8:9/( 0 ( [email protected]+±( "+¤D;7", 2/( +CA7BiIOB$,R2* ( .VfOjCÍ:¶/fOn¢nZkJl fOnpobrut=t=·$Y/w] Ebe = ±167ρA (z) BiI=(*8 ρA (z) = 1.208exp −z 8.4 |y~x·O¥@ *\³(µ/K/A7D50 (V( . x_0 −1 (*+4¦\B$4;+CDp+C,2/( V(*.£G0x -,(*4545( 45(8:9NBO0 K£*45(*8*+CA"D5EG,(2/(VA7,K/+7,A7( EG,(67!=Dp+`4y\³BO4p+CD;+7,2($q ( +mD5./*A7D5(*,EA`beBO,8:9NB$01K 4;( 8J+CA7D;EG,(9NBOYDp+C,(*454;( 0 ( [email protected]+S.*8 (*7CB$D5A7(6KL!=,A`45B A7,/K/+C,A"(62D5*45( 8J+CA"D5EG,(OEx be.VfOjCÍ:¶/fOn¢nHkJl{fOnpoHb[email protected]·[email protected]+m,/+CD;45D5"6~OR7!O.2NBOa=(*ª2([8:9BO01K*45( 8J+CA"D5EG,(< 4y\[email protected]+7 A7D;( ,A{2]\³!OACBOa=(*`KZ!O,A`01!O.G+7A7(*A`E,/( 45D;01Dp+C(mYD5(*.45( SI$BO4;( ,A"`2,8:9NBO0 K*45(*8*+CA"D5EG,(2NB$. Ebe 45(.G,NB$a=(Ox ^PKA"3 §BiIO!=D5A 2 $.D4;(2!=01BOD5.(KL!=,A 45(*E,/( 4m45(2 8*45( ./8:9( 0 ( [email protected]+§2]\_,. *8 45BOD5A ( +KL!=7"D5Y4;(Oq !=. 8 BO458*,45(45(8:9NBO0 K *45( 8J+CA"D5EG,((*. +C!O,[email protected]+C*x %. 8 !=0 K/+:B$YD54;D57(4;( KZ!=D;[email protected]+Cc! # 45(8:9NBO0 K 4;( 8J+CA7D;EG,(F(*"+-",KL A7D;( ,APBO,8:9NB$01K*45(*8*+CA"D5EG,(<2(A",K/+C,/A7(F0F,4;+7D5K4;D5-KNB$AP,.B$8*+C(*,A-¥xt KZ!=,/A +C(*.D5A[8 !=0 K/+7(c2(14BI$BOA"DBOYD;45Dp+C"!=,#0cB$D5454;(R2,Ä8:9NBO0 K *45( 8J+CA"D5EG,(O x f. (*Z(*+ q !=. ./(KZ(*,/+RKNB$ 2J+C( A"01D;.( ABiI=(*8-KA7*8 D;7D5!O.1EG,(*[email protected]+B["!=.0 !2,/45(ª45(ªK4;,{KA"!8:9/(P2,10cBD;0,0KNB$A70 DL8*( , [email protected]@+<,.0 !2,/45(F2/,¬8:9BO01K*45( 8J+CA"D5EG,(8*!=01KBOACBOY/45(OxzW(RKZ!=D;[email protected]+2(R2*8 45(*.8:9(*01(*.G+<2(R4¦\³*8 45BOD5A ( +m8:9!=D5"DXBO,9NBO7BOA72KNBOA"01D]4;( `[email protected]+7mEG,DA" KL!=.2(*[email protected]+P 4B8*!=.2Dp+CD;!=.§KA7*8 *2( [email protected]+C($x % &Á a &À±¾¾ j.(<KA"( 0 D53 A"(6KNBOA"+7D5(62(64B KNBOA7BO0*+CA"D7B$+CD;!=.2( 8*4BOD;A *\BOK/K,D5(7,/A45BR+79 !OA7D5(62,45( BO2(*A YD;2D5A"( 8*+7D5!=./.( 4<b®ÌFfÍukR 3 ¢jqmrut=Î=¥qmrit=v=$w¦x{W(8 BO.NBO4m2(4¦\*8 45BOD5A*[email protected]+§8 !O.2,8J+C( ,/A qS,./(8:9NBOA"a=( I$B *+7A7(VD5./2,D;+7( 45B7 ,ABO8 (V2/, 8iBO.NB$4PK45BO8 2BO. § 45(8:9NBO0 K 45(*8*+7A7D5EG,(2, .,BOa=(OqEG,DPI$B a*. A"( AF,/.KL!O+C(*.G+7D5(*4" ( 8*!=.2NBOD;A7($x %-.B2!=./8c,. KZ!O+7( [email protected]+CD;( 4¤8 !O. " [email protected]+4B7 ,ABO8 (12(14¦\* 8 45BOD5A EG,D ( " +645B1A 7 ,4p+:[email protected]+7(2,KZ!$+C( [email protected]+7D5( 4HBO0Y/D[email protected]+45DF BO,8:9NB$01K* 45( 8J+CA"D5EG,(2,V.G,NB$a=((*+-2,V8:9NBO0 K 4;( 8J+CA7D;EG,(7 (*8 !=./2NBOD5A"(OxWXB0 !=YD;45D;+ 2( KNBOA+CD;8 ,4;(F .G,NB$a=( , " (* [email protected]++CA"!=K BOD5Y/45(Oqs8 (" [email protected]+ 45( 4;( 8J+CA7!O. ` 45D;YA7( q/K/A7!2,D;+ S KNBOA{D5!=.DC B$+7D5!=.145!=AS 2(8*!=454;D7 D;!=. ± 2NBO. ` 4;(-8iB$.NBO4yqEG,D7 !O.G+mR4¦\³!=A"D5a=D;.( 2( < 0 !=,/IO( 0 ( [email protected]+6 2( 8:9NB$A7a=( BO,7 (*D5.2,¬8iBO.NB$4yx u!=, 6 4y\³BO8*+7D5!=.V2/,¬8:9BO01K 4;( 8J+CA7D;EG,( BO0[email protected]+iq ¸S¹ m¹y¸ ¿¤Á Î= 8 ( *45(*8*+CA"!=. 7( 2 K/4BO8*( [email protected]+F2z\³,./(1(µ+7A *01Dp+ 2( 4y\ 8 45BOD5A-4¦\BO,+CA7($q8 ( EG,D¤8 A *(1,/.7,/A7K4;, 62( 8:9NBOA"a=(KL!7Dp+CD;IO(-2]\_,.8 pO+ (*+`. [email protected]$+7D;IO(2(4y\³BO,/+7A7(OxQ{( KL( [email protected]+iqN4B[8:9NBOA7aO(a=45!=YBO45(2,8iBO.NB$4z(+ .G,454;([email protected]+ E,/(§4y\ 8 45BOD5A<.(+7!=,8:9(§KNB R45("!=4yxQP\³( "+ 8 (§0 !=,/IO( 0 ( [email protected]+ 2( 45(*8*+7A7!=. R4;D5YA"([E,/D (+c4¦\³!OA7D5aOD5.(2,KZ!O+7( [email protected]+CD;( 4 7(*8 !=./2NBOD5A"(OxZ}-BO. P45(<8iB 2]\³,/. 8*4BOD;AmEG,D B$+7+7( D;.G+45(7!O4yq4B 8:9NBOA"a=( .(J+7+C(( +S2Dp *A7( [email protected]+7(P2( A"!qK,D "EG,(Oq/BOKA73 {8*[email protected]+:BO8J+`BiI=(*8-45( 7!=4¦qG45(ªKZ!$+C( [email protected]+7D5( 4L2,8 BO.NBO4 "( ACBFBO, KL!O+C(*.G+7D5(*4]2, 7!=4¦x }-BO. [.!O+7A7( 01!23 4;(Oq4;( 8:9BO01K 4;( 8*+7A7D;E,/(1,/+7D54;D RKL!=,A62 *8 A"D5A7( 4BKA"[email protected]+CD5!O.¬2/,45( BO2(*A YD;2D5A"( 8*+7D5!=./.( 4H( "+<4;(R8:9NBO0 K *45(*8*+CA"D5EG,( BO0FYD[email protected]+(J+6.!=.¬45(8:9BO01 K 4;( 8J+CA7D;EG,(+C!O+CBO4yxz%!=, .( +C(*.!=. <2!=./8 KB [8*!=0 K/+C( 2,¬8:9NBO0 K *45(*8*+CA"D5EG,( 45!8iBO4S|8*A KNBOA64;( 68:9NBOA"a=( <D5.2/,D;+7( 62NBO. <45( 8iB$.NBO4¢µqZ(J+62,VA7(*./!=A"8 ( 0 ( [email protected]+62,a=A7BO2D;( [email protected]+-2(RKL!O+7( [email protected]+CD;( 4 (J+2!=./8R2,8:9NBO0 K 45(*8*+7A7D5EG,(BO,V(µG# +CA 0 D;+ {2/,§45(iB$2( A*x uDZ!=.I=(*,/+`+C(*.D5AS8*!=0 K/+C(2,8:9NBO0 K 45(*8*+7A7D5EG,(P45!8iBO4KL!=,[email protected]$+7D5!=. 2,45(iB$2( A*qLD;4]BO,/+m8iB$458 ,/45( Aª8*(F8:9NBO0 K 8:9NBOEG,(F(J+CA 0 D;+ 62( 7(*a=01(*[email protected]+ixZQ{( 45B 7D;a=.D $N(62!=.8<A µ# 7!O,2A7(P4y\ EG,NB$+7D5!=.2/(©H!=D "!=.]q/(J+`KNBOA8 !=. EG,(*[email protected]+,+CD54;D "( A±45( "!=4;IO( ,A{2(-KA"( "D5!=.§2/( &%' 8:9NBOEG, ( "( a=0 ( [email protected]+RA7B¯M =,/+ OxW(8 /+ ( . +7( 0 K R2(8 BO458*,4{( "+ BO4 =A <IACBOD;01(*.G+(J =A7Y/D;+:B$.G+ q (J+ . =, BiI =. mKA J A <BOaOD5A ",Aª45( 7(*,D542/(<K/A [email protected]$+CD =.]x W =A "E,/(( .§,.K=D;.G+{2,§.G,NBOa=($q4;(P8:9NB$01K*45( 8J+CA"D5EG,(P2/(*ID5(*.G+ 7,K A"D5( ,/A`F,.(IOB$45( ,/A 7(*,D54 q{2(*, "( a=0 ( [email protected]+§2/(VK=45BOA7Dp+=KK *(7(VKA=KNBOaO( [email protected]+2NBO. §4;(2/D5A7(*8*+7D=. 1KNBOA7BO454;3 45( c(J+ Etrig [email protected]+CD;KNBOA7BO454;3 45( `BO,8:9NBO0 K 4;( 8J+CA7D;EG,([01BD50F,0x/W([4;(iBO2(*AK"D;+CDpHI$B7([2 K45BO8 (*AP 4y\[email protected]+*A7D;( ,A 2(45B =.(2(F8:9NBOA7aO(. a=B$+CDpI=(Oq+:B$.2D EG,(45([45( BO2( A. a=B$+CDpI$B 7(2 K45BO8 (*A-2NBO. 4B =.([2( 8:9NBOA"a=(K 7Dp+CDpI=(OxHW(8:9NBO0 K 4;( 8J+CA7D;EG,( J+:[email protected]+R2D;A7D;a 12/(§4BA a=D =. 2(c8:9NBOA7aO(K 7D;+7D;IO(1I=(*A R45B A *a=D O.§2(68:9NBOA7aO(. [email protected]+CD;IO(Oq4;(45( BO2( A`K "D;+CDpX( +ªKNBOACB$45453*45(PBO,8:9NBO0 K *45(*8*+CA"D5EG,($q/+:BO./2D `EG,(645( 45( BO2(*A`. [email protected]+CD;H( [email protected]+CD5KBOACBO4;453*45(BO,8:9BO01 K *45( 8J+CA"D5EG,(1| $NaO,A7([~x '=Jx − + − − − − − − − − − − − E + + + + + + + + + $4JPNAP?= $-?"[N* $4 ,T$4J+& #=c1 "=] ?=] H%$PN*J4\,? &#AZ13A ; #.0#=T1 219W $4JPNAP?= 2" &+A313"J' " (*+&'&@?"& "%=] ?[=] H%$PN*J4 =]JPNII13A ; 2.)2=c1 _ <: ; 7 1 #"%$ H%., J#! 13$#A25"L+X(I?[= (AZ6 $+AN*J ; #$+&0! & _< $ ?5A $ ?["N*9$4 ,T7 $4J+&@#=T1 ,?AZ=c1w(*(*J !#= H#.7 0#=c1 7(*9-F J AZ$ 7 21<W013$ 1V?[AY$47 g(*X4F J AY$# 6.)[1 J#$#A3[ 7 AY&@J4,; $M#&UAWN= +&=T?[AY$4+&@_<G#& $-= H7 +&w,7 ? &+A313A +&S& w,T$-?+7, N*2=5 c1(I?[= (I=E7 &SW ?=]9(* 7 H%[$3N* ; 7 7 =]JPN*[13A ; 21 F +-F AZ= ; #$+&@O,?"%$9!+& $-6 = H+&7 o=]JPN*[13A ; +&@_ 5 7 6 7 X¤~x 'i <G W( 4;(iBO2/( AYD;2D5A"( 8*+7D=./.( 4 7( KA OKNBOa=(RKNB$A 7( aO01(*.G+ <[email protected]+6EG,( 45(R8:9NBO0 K 4;( 8*+7A7D;E,/( 4¦\³(µ +CA 0 D;+ [2,2( A".D5(*A 7(*a=0 ( [email protected]+-( "+ ",K *A7D;( ,APBO, 7(*,D54s2([KA [email protected]$+7D=. xz©=,A-A7B¯M=,/+7( A,. Eprop 7(*a=0 ( [email protected]+SB$,c8iB$.NBO42(m4y\ 8 45BOD5A*q=.18iBO4;8 ,4;(m4¦\B$.a=45(`EG,(mBOD;+¤4;(`I=( 8J+C(*,AS8:9NBO0 K *45(*8*+CA"D5EG,(BiI=( 845( I=(*8*+7( ,Am2/,"? "+73 0 (2(P8 OA72 =./. ( x^ KNBOA"+7D5A±2(-8=. 7D;2 A7B$+CD=. 7,A{45({BO.a=4;(*q =.cKZ(*,/+`2 Î ã $N.D;A 7D/45( "( a=0 ( [email protected]+I$B 7(ªKA=KNBOaO( A "( 4O.R4BIO( A"+7D58 BO45($q 7(*4=.R4¦\³9 =A7D [email protected]+:B$45( =,7( 4=.R4B2DBOa =.NBO4;(Oq (*+m2 O.8=.KZ(*,/+m2 *+7( A70 D5./( A`45( ª8 =A72 O.. ( ª2(64B $N.2, 7( aO01(*.G+ x W( +7ACBODp+C( 0 ( [email protected]+<2( *8 4B$D5A .G,NB$a=( =4±B*+ 7D;01K4;D $ $x f+:B$.G+[2 =./. 1EG,( 45( [D=. 6(*+<4¦\³(Jz(J+ 8 OA =.NB[.( [email protected]+`KNB ±+CACB$D;+ 2BO. S4;(0 2/3 45($qG45(m8:9NBO0 K 4;( 8*+7A7D;E,/( O, S45(m.G,NBOa=((+S+CA=KcBOD;Y45( K =,AKL( A"01(J+7+CA"(S4BKA [email protected]$+7D =.[2, 8iBO.NB$4/2(S4¦\*8 45BOD5AM", 7EG,]\³BO, =4¦xOW(S4;(iBO2/( A( "+¤2 =./8`8 =../( 8*+ BOA+CD $8 D5(*454;( 0 ( [email protected]+`BO , =44 =A 7EG,]\_D54zB$+7+7( [email protected]+ª4¦\BO4p+CDp+C,2(2([~0 KNBOA`ACB$KKOA"+mBO, O4yq(J+ 7BRK=45BOA7Dp+ 2 J+C( A"01D;.(m8 ( 4;45(2,X Q nRx/©HBOA{8 =. *EG,( [email protected]+iq 7Dz,/.45( BO2( A. a=B$+CDpB$+"+C( D;[email protected]+S45( O4yqG4y\ 8*4BOD;A±A",4;[email protected]+ ( +m,.XQ n :q/(*+ 7D,.45( BO2(*A`K "D;+7D;HB$+7+7( D;.G+ª4;( =4¦q =.BR,.XQ n x WXB K/A [email protected]$+CD =.2,45( BO2(*ASYD52/D5A7(*8*+7D =../( 4z( "+ "+ =KK (64 OA 7EG,(4;(8:9NBO0 K 45(*8*+7A7D5EG,(64¦\³(µ x +CA" 0 D;+ -2,2( A".D5(*A "( a=0 ( [email protected]+m2(JID;( [email protected]+mD5./ *A7D;( ,AªBO , "( ,D;42(6KA [email protected]$+7D =. Eprop ¿a/¿! Á ½ a &¿ % ¿¤¾gaà Xƾ Q{( KL( [email protected]+iqz4=A "EG,]\ O . = Y "( A"IO(,. *8 45BOD5A*q =. 7([A7(*.28 O01K/+7(FEG,]\_D54X.]\_(+KNB ,.D5EG,(*01(*[email protected]+ 8 O. "+7D;+C, [ 2]\_,.8 BO.NBO4IO( A"+7D58 BO4yx %-.A"( 01BOA7EG,([K45,/+#pO+m,.( "+CA",8*+7,A7(<I=(*A"+CD;8iBO4;([Y/ACBO.8:9 ((J+P2( (J+7( . " D= . ¤ 9 O A7D = [email protected]+:BO4;(H D50 K= A+:[email protected]+7( x ]Íuh¡N5ÍPkJlnÍJ¡zkJjbrut=v'w O.G+±B$.NBO4;? S2(K/9 O+ Oa=ACBOK/9D5( 2]\ 8*4BOD;A |y~O}-(J+ = [email protected]+0 = [email protected]+CA ¬ EG,(¬45( 8*4BOD;A = [email protected]+,.(¬a = 0* +7A7D;(ACBO8J+:BO4;( B 8 D ( ,/.( 2D;01(*. 7 D= .ACBO8J+:BO4;(2( r=x³ ã x W\ = Y 7 ( AI$B$+CD= . 2( 0 D7 DO . 'qKBOA¤45( * 8 45BOD5Aª b¶/fhkJlXÌ<jCk:¶/Ê ykJn>qrut=t=Î Ì<jCk:¶/Ê ykJnzkJlfOnpo5q ~O¥=¥O¥ ª 5 5Íuh¡ kJlfOnpo5q¤rut=tOt ]¶Nh73 fͧkJlfOnpo5qH~O¥=¥/r*w{BA J I 4c 45B.NB+C,A"(cYD;2D5A"( 8*+7D= ./.( 4;45(R2( F 2 8:9NBOA"a=(s [email protected]+7ACB .,BOa=(¤BOD;. 7 D=EG,(±4B +CA7,/8*+C,/A7(m2(*,<.DpI=(iB$,62(±8 (+>?KZ(±2]\ 8*4BOD;A x¯Q{(J+7+C(" +7A7,8J+C,A"( BOKKBOACBl;+8 45BOD5A"( 0 ( [email protected]+ " ,A¤4B $a=,A7(6r=xpr¯·xOW(m4;(iBO2/( AYD;2D5A"( 8J+CDO ..( 4.]\_(" +±2 O .8`KNB 7 ,Tr 7 [email protected]+±K= ,/A A7(*KA 2,D;A7(14B +CA",8*+7,A7(§a=4= YNBO4;( (*+R4¦\³(µ+7( . " D= .9 = A7D = [email protected]+CBO45( 2( * 8 45BOD5A xX>4{( " +RD;01K= A"[email protected]+<2( A7(*KA 2,D;A7([BO,0 D5(*,4;( 8 45BOD5A K/,D" E,/(F45(68:9(*01D;.EG,]\_D54 " ,D;IO( [email protected]+PI$B12 * +C(*A70 D5.(*A45B EG,NB$.G+7D;+ 2(S8:9BOA7a=(.( ,+CACB$45D ($q$45BmA * EG,( .8*(Oq=45B4 8 BO45DC B$+7D= .(J+4y\_(J+7( . " D= .2/( * 8 4B$D5A * xO},KO [email protected]+ 2/({I,( 2(45B18:9D;01D;(FB+C0 7 K9 A7D;EG,(OqZ4B1KBOACBO0* +7A7DC B$+7D= .2( 8 45BOD5A Bc,/.VD50 KNBO8*+ " ,A-4B1KA 2,8J+CDO . 2( % % x<u( 4O . RfO¡ kJlfOnpobprit=t=vOY/w¦qz4;(F+:B$,V2( % %¨K/A 2,D;+PKZ(*,/+6*+7A7(R( " +CD;0 F KNBOA-,./D;+F 2( x 4O .a=,(*,A2(ª4y\ 8*4BOD;A x$© BOA8= . EG,( [email protected]+ q = A"8 (`( " +±2(ª8 = . " +:B+C( AEG,(ª45B-4= .aO,( ,A+ O +:B$45(S2(`4¦\* 8 45BOD5A ( +m,. * 4* 01(*.G+ª8*46 K= ,A`( +CD;01(*AªBO,01D;( ,4BRKA 2,8J+CD= .§2(<% % KNB$Am4;( * 8 4B$D5A * x ¸S¹ m¹ ` ½ À±Æ ½ x -yk3ckJÈ@kJj§kJlfOnpobrut=v ã w [email protected]+F0 [email protected]+CA1EG,]\_,. 0 23 4;( "+ 8:9NB +CD;E,/(c2( A7,K+C,A"(c2D 45(*8*+7A7D5EG,( 8 O.2,Dp+.NB$+7,A7(*4545(*01(*[email protected]+-2( "+7A7,8J+C,A"(ACBO8J+:BO4;(*xGÌFf±fÍ f kJl.fOl¢ÍJËËj"fcb³~O¥=¥=¥$w [email protected]+¤BOKK/45D5EG, 8 (ª0 2/3 45(2(ªA7,K+C,A"(m2/D 4;( 8J+CA7D;EG,(`4y\³B$+C0 "K93 A"(S(*+ [email protected]+¤2 O.8`[email protected]+CA2/,D;+45B-. O+7D=.2/(ªA7BO8*+CBO45( 2NBO. -45(*,AP0 2/3 45(<2(K/A =a=A"( 7D O.2,V45( BO2(*APK=,A "D50,/45( Am4B + =A+C, 7D;+ [(J+45( PYA7BO.8:9(*01(*[email protected]+ 2,45( BO2( A*xQ{( m2(*,0 23*45( \³BOKK,D;( [email protected]+ 7,Am4B0 0 (KA 8 2,A"( 45B $Na=,A"(<8*A %l;+ *E,/( [email protected]+CD;( 454;( 0 ( [email protected]+[| $a=,A7([~x³[email protected] ,./(<K/A =YNB$YD54;D;+ 2 KL( ./[email protected]+P2/, K O+C(*[email protected]+CD5(*4]4 8iB$4z( +-B 8 D (6 8:9NBOEG,(<45D5BOD =.K "D5Y4;( £ 8:9NBOEG, ( *+CBOKZ(6(J+ J+:[email protected]+m2 =.. (<4BR2D +CA"D5Y,/+7D =.§2/(<K/A =YNB$YD54;D;+ $q/,./(<4;DBOD =.§( "+m8:9 =D 7D;( BO,9NB CBOA"2(J+A7B¯M =,/+ (<BO,0 O+CDp&q¯M =D;[email protected]+`,.K [email protected]+ª2(64B $a=,A7(<R,.. =,/IO(iBO,K [email protected]+ix W\BO2BOK/+:B+CD O. 2(8*(*+7+7(VKA 8 *2,A"(IOB . =, KL( A"01(J+7+CA"(2(VA"( KA 7(*[email protected]+C( A45B + =A+C, 7D;+ 2( 8*4BOD;A -(J+62]\³B$+7+7( D5./2A7(R2( 6A aOD =. 2/,¬.G,NB$a=( 2NBO. 4;( 7EG,(*454;( 4B2/( . "D;+ 2(R8:9BOA7a=( ( "+ 4;(*I *( 01BOD # 4;(8:9NBO0 K 4;( 8*+7A7D;E,/(§( "+ BOD;Y45($x %-.ÄI$B2 =.8§8 O0YD5./( A,./(B$KKA 8:9(§ACB$8*+:B$45(2( 8*4BOD;A {4B[+7( 8:9.D;EG,(P x D;452 qHD5A7(µ2( .0f +21h4j 3 fO¡kJl{fOnpo]b³~O¥=¥r wy]x j.( =D SEG,(4;(P4;(iBO2(*ASYD;2D Î@· XN~x³v\ $#AZY! .)?=c13$-=c1 "%=]w,?$213AP?=L(*o(*J H%[$3N*R_1<G+& #$ !+& = &Q$- ,T$-J#&@#=c1 2=c1<!+&<,?AY=T1Z& (*oN$+AZZg=]?= ?=c=] 1 J+& (*J H%$PN* _"<:+& #$ +&Q7 ,]#AZ=&)21O!+&7 UAW7 N[=]+&6 ,]#A7 Z=]+&mAZ=T(A2"2=c1 H#.mAZ= 7 7 7 AZ$-J ( (* C(*J H%$PN* _9<G#&0&@VN.0#=c1Z& #=7 z,?AZ=c13AZY!J+&07 $4 ,T7 $4J+&@2=c1 #=c1o+&=T?" ; #Z+& UA3AY&2?[=&Q,? &+&+A #&@_ 8: 6 z=E&@#Z:7 21 _ @_ 5 A7(*8*+7D=../( 4B $N.D2( "({KA=KNBOaO( A q4y\ 8*4BOD;A]I$BPKO,/I=D;A "({YA7BO.8:9(*A(*+ \ *+7( .2A"(S9 =A"D [email protected]+:BO4;( 0 ( [email protected]+ix W( <K [email protected]+EG,DI [email protected]+FJ+CA7( A7B$+7+CBO8:9 [BO,¬8iBO.BO42 =DpI=( [email protected]+=Y D;A62/(<8 A7Dp+C3*A7( ",A[45B2( . 7D;+ 2( 8:9NBOA"a=(R(*+ ",A45(FKO+7( [email protected]+CD;( 4 45(*8*+7A7D5EG,($x]W\ 8 45BOD5A8 ( 7( 2( "(RKA =KBOa=( AP4=A7EG,]\_D54B+7+C(*[email protected]+2( A a=D =. 2(BOD5Y4;({2(*. 7Dp+ `2(`8:9BOA7a=( O,R2(A a=D=. 2NB$. 45( 7EG,( 4;45( 45(S8:9BO01K*45( 8J+CA"D5EG,( *\ =K/K"( CB KA [email protected]$+7D =.zx %O+7A7( 78:9 01B 2/(<Y/ACBO.8:9/( 0 ( [email protected]+P8=0 K=A+C(-K/45, "D5( ,/A *+CBOKL([| $NaO,A7(<~x³[email protected]µx f.KA"( 0 D5(*AS45D;( ,]q =.8 =. +CA",D;+m,.([0cB+CA7D;8 (2 *8 A"D;[email protected]+P45B +7ACB¯M"(*8*+ =D5A"(<2/(F4y\ 8*4BOD;AS(*+,.([0cB+CA7D;8 ((*.GID;A =../( 0 ( [email protected]+ EG,DA7(*8 ( . 7(-4;( ±K =D;[email protected]+ ¤A7(*01K4;[email protected]+±2(±8 O.2D;+7D=. ",A±4B62( . "D;+ 2(m8:9NBOA"a=(P(J+ ",A±45(ªK$+C( [email protected]+7D5( 4 4;( 8J+CA7D;EG,(OxLQ{( m8 =.2Dp+CD =. O.G+m4;( 7,/D;[email protected]+C( [ ( |Q(i, j, k)| |V (i, j, k)| ≥ ≥ Qexcess |V0 | |y~x·r¯ 8 =A7A"(7K=.2BO,7(*,D54z2(68:9NBOA7aO(K=,/A`45(`Y/ACBO.8:9/( 0 ( [email protected]+<q(*+ V (+4;(K$+C( [email protected]+7D5( 4Z0 u?O( . 2,8iBO.NB$4s4;(iBO2/( A x %-Am45(6. =0Y/A7(<2([Y/ACBO.8:9/(-2(<4y\ 8*4BOD;A`2 =pD +OY *D5Am 0,/.(F4=D2(6+>?KL([A7BO8*+CBO45( b -yk3ckJÈ@kJjkJlSfOnpo5q]rit=v ã w |y~x·=~= N (d) ∼ dχ−1 # N ( "+¤45(`. =0YA"(S0cB¯/D;0,0 2(`YA7BO.8:9( ,./(`2D +:BO.8*( d 2, K [email protected]+2z\³D;.D;+7DB$+7D =.<2(`4y\ 8*4BOD;A q (*+ ( +64B2D;01(*. 7D =.A7BO8*+CBO45([2(R4y\ 8*4BOD; A =,2D;01(*. 7D =.2( 'BO, "2 =A"HqZ+7( 4;45(EG,(§r kJ~ l 2NBO. χ,.(a = 0 *+7A7D5(1~2D;01(*. 7D =. <(*+ ~ 2NBO. F45(c8 B R2D50 ( . "D =. b®Çm kJlχjCh ǪkJl¦jCh fNqzrutOt=$w¦x>4BO,/+{2 =.8P2D;ID "( A45(m2 =01 BOD5.χ(m2 ( "D50,/4B$+7D =. ( . "K93*A7( <| =,8*( A78*45( S( .~O}- 8 (*[email protected]+CA *( 6BO,K =D;[email protected]+P2]\_D5./D;+CD5B$+CD =.§2(4¦\ *8 45BOD5A[| $a=,A7(~x³[email protected]JxLW(ACBi? =.2(8*( "K93*A7( ( "+ *[email protected]c45B K4;, `KZ(J+CDp+C(A =4;,/+CD = . 7KNB+CDB$45(-0,/4;+CD;K45D (mKNBOAª,.( [email protected]+CD;( A*x }-BO. ,. KA7(*01D;( A+7( 0 K q =. I$BV(*Z( 8*+7,( A 45( YACBO./8:9( 0 ( [email protected]+ 12(ACBO./a rOq8O\_( "+ & 2D5A"(§45( YA7BO.8:9(*01(*.G+ ª( [email protected]+CA"(45(-4;(iBO2(*ASYD;2D5A"( 8*+7D =./.( 4(*+ª4;( SK =D;.G+ `BO2M7B$8 ( [email protected]+ ª8 ( 4;,D 8 D¦x&%-.§A"( 8:9(*A78:9( BO4 =A S+ =,/+C( ª45( `. =,/I=(*454;( ªYACB$.8:9( mK "D5Y4;( 6 =.(*Z( 8J+C,(<,.2 8 BO4BOaO(2(64B01B$+CA"D58 (8 [email protected]+C( [email protected]+ 4B+7ACB¯M"(*8*+ =D5A"(§2(4y\ 8 45BOD5A[2NBO. R+ =,/+C( 45( R2/D5A7(*8*+7D =. |&ruvV( . =}µxW(A ",4;+CB$+ ( +1,/.( Qexcess Î=Î (n+1.5) mesh_size Positive branches (n+0.5) mesh_size 4 3 3 4 1 3 2 2 1 1 3 E 2 1 2 3 1 2 2 1 3 2 Negative branches Xª~x³t # <GL!-I(*2$ AP(AZ$- 13A3?[=E=]#X+g$4 ,T$4J+&@#=T1 Ji#= =T?AZ$ _+O&@),]$ ?+, N* ,[$#13AZ$L(" ,?AZ=c1g( 6 7 (*J #= H#.0#=c1K $-?A =T?[AY$4 ^[_ =C$-?"[N* ?= (AY
AZ=N"Eg!+& $ = H+&,? &#AZ13A ; +& 2"A& S,T$-?+, N*2=c1S(I=E& "%=]7 ?7 =](* H%[$37 N*= ; JPNI[13A ; 21 2= !#" 5 !+& $-= H#&S=]JPNII136 A ; #&M7 (I?=c1GW,T$-?+,RNI[13AP?= &@G>2[AZ1G(*=&O"= $4JPNAP?= H$3NJ47 ,? &#AZ13A ; #.0#=c1 _5 <G#& H*6 A O$4+&5 #=c6 1V?"%$-J#7 &9(:F "%= #$ !oAY=T([A2"E#=c1 !Q=cA ; 4" (* $-= H#.0#=c1 _ 7 7 7 7 6 7 01B$+CA"D58 ([email protected]+C(*[email protected]+F45B+CACBuM"( 8*+ =D5A"( 2(14¦\*8 45BOD5A(*+[45([email protected]+ [B$2M7BO8 (*[email protected]+[BO,EG,( 4 [email protected]+B$+"+CA"D5Y, ,.. =0YA"(P8 OA7A7( 7K=.2NB$.G+`BO,. =0FYA7(2(-45D5BOD=. ¤K "D5Y4;( ",A{8*(`[email protected]+{FKNBOA+CD;A2(4y\ 8*4BOD;A x uDG4y\ =.<A7(J+CACB$.8:9(Sm8 (J+7+C(0cB+CA7D;8 (45Bª+7ACB¯M"(*8*+ OD5A7(¤2(4y\ 8*4BOD;A](*+HEG,]\ O.4Bª8 [email protected] =45,(BiI=(*8S4B`01B$+CA"D58 ( ( [email protected] =..(*01(*[email protected]+iq =. =Y+CD5(*[email protected]+{,./(P01B$+CA"D58 (ª[email protected]+C( [email protected]+S45( K=D;.G+ EG,DLKL( ,/IO( [email protected]+`*+7A7(YACBO./8:9 *x %-. 8iB$458 ,/45(RBO4 =A 4BcK/A =YNB$YD54;D;+ <2(RYA7BO.8:9(*01(*[email protected]+[2/(R8 ( [email protected]+ qzEG,D ("+62 $N.D;(R8 =0 0 (F4y\[email protected]( A "( 2,. =0Y/A7(2(4;DBOD =. {K 7D;Y45( 7,Aª8*( mK =D;[email protected]+ S2(6a=A"D54;45(Ox&%-.§IOB BOD;. 7DzYNBO4Bi?O( A`45(2 =0cB$D5.(2( 7D;0,45B$+CD =.KNB$A"+CD;A62,K [email protected]+<2]\_D5.Dp+CD5B$+CD =.( .,/+CD;45D [email protected]+645( "K93 A"( 2 $N.D;( <K45, [9NB$,/+ixs© O,A 8:9NBOEG,( "K93*A7(Oq =.8 BO458*,45(45(m. =0YA"(P2(PYACBO./8:9( `8 =01K/A7D "( ( [email protected]+CA"(-4 B 7K93*A7(8 =. 7D;2 A (-(J+S45B xN>4]?§B B$4 =A S2( ,K 7D;YD54;D;+ - 7K/93 A"([D;01 0 2D5B$+C(*01(*[email protected]+SK4;, `KZ(J+CDp+C( Nexistantes =D;+R4;(§. =0Y/A7(§2(YA7BO.8:9( c8*(*+7+7(2/D "+CBO.8*(§( "+ 2 >M7 7,K A"D5( ,/A O, *[email protected]`B$, . =0YA"( 01BD50,/0 2(6YA7BO.8:9( 2 8 A"D;+mKNBOA`45B4 =DZACBO8J+:BO4;( =D;+ª4;(. =0YA"(2(6YACB$.8:9( 4B2D "+:B$.8 ( ( "+mD;./ A"D5( ,/Aª x d }-BO. F45( KA"( 0 D5(*A<8 B qsBO,/8 ,. YACBO./8:9( 0 ( [email protected]+.z\³( "+ A7B¯M =,/+ OqX(*+ =.¬KNNB (d) 7(4 B 7K/93 A" ( 7,[email protected]+C(Ox }-BO. 4;( 7( 8 =.28iB *qL4;( ªK [email protected]+ `2/([a=A"D54;45(PEG,D "( A O.G+mA"( 45D mR4y\ 8*4BOD;A [email protected]+m8 (*,EG,D [email protected]+m4BRK45, a=A7BO.2(KA =YNBOY/D545Dp+ m2(-YA7BO.8:9(*01(*[email protected]+ix %-.Y =A".(-4;( SKA =YNBOY/D545Dp+ S4y\³BOD;2(P2]\_, . 7(*,D54 7,K A"D5( ,/A (J+`2]\_, . 7(*,D54]D;./ A"D5(*,A (*+ =.§8 BO458*,45(45(-. =0YA"(2(K [email protected]+ 2 [email protected]+ª4BFKA =YNBOY/D545Dp+ pnew 2(6YA7BO.8:9(*01(*.G+( + 7,K A"D5( p,/old A7(< xN©4;, 7D;( ,A m8 B ª2( $Na=,A"(6KZ(*,/NI=(*seuil .G+ 7(<KA 7(*[email protected]+C( A< pnew r=x ( "+ a=BO4]BO,§. =0YA"(2(-K [email protected]+ {. 8*( 7BOD5A"( ªK =,A`BOA"A7DpI=( ASBO,§. =0Y/A7(-01BD50,/0 2( Nseuil YA7BO.8:9( ~x ( + ",K A7D;( ,Am Nseuil N (d) − Nexistantes Î ' / x (+D;./ A"D5(*,A` x Nseuil N (d) − Nexistantes W(-8iB r("+S45(-K4;, "D50 K45(< =.§Y/ACBO.8:9/(45({K=D;[email protected]+S2 O.G+S4BFKA=YNBOY/D545Dp+m(+ 7,K A"D5(*,A7(6 (*+ =.1KNB "([45B"K93*A7( ",D;[email protected]+7(Ox}-BO. 45(ª8iB S~qG8 =0 0 ( O.B[K/45, ±2(PK=D;.G+ ¤K7D;Y45( EG,p(new 2( YA7BO.8:9(*01(*.G+ BO,/+ =A7D q=.BO,/a=01(*[email protected]+C(<45(7( ,/D54s2(FKA=YNBOY/D545Dp+ 2( O~qN(*+ =. pnew |pnew − pold | A *I$BO4;,([4;(6. =0YA"([2/([K [email protected]+ [email protected]$.G+P,.(<KA OYNBOYD;45D;+ 7,/K *A7D5(*,A7(F18*([. O,/I=( BO, 7(*,D54¦x© =,/AP45( +CA =D "D53*01(-8 B q8$\³( "+4y\[email protected]=(*A 7( =.BOYNBOD "(<4;( 7(*,D542(<KA =YNBOYD;45Dp+ 2/( O~x %-. pnew |pnew − pold | KA 8 2(PBOD5. "DNKNBOA±2D58:9 O+ =0 D5(`K =,A =K+CD50 D "( A4;(m. =0YA"(2(mYA7BO.8:9( `<,/.(P2D "+CBO.8 (m2 =./. ($x uDK/45, "D5( ,/A 6 K [email protected]+ -2/( aOA7D54;45( [email protected]+64B0 0 (RKA =YBOYD54;D;+ [2( YA7BO.8:9(*01(*[email protected]+F(J+6E,/( 45B2D; A"( .8*( ( [email protected]+7A7(4;( P2/( ,2( A".D5(*A "( ,D;4 2(KA =YNBOYD;45Dp+ ( +D5. A7D;( ,A"(c,/.(FKA =YNBOY/D545Dp+ 60 D5.D;0,0q45( K [email protected]+ ¤8 =..(*8*+ [email protected]+{8:9 OD 7D ¤2(mB e =.1BO4 B$+ OD5A7($x j./(m =D 45( YA7BO.8:9(*01(*[email protected]+ {(Jz(*8*+7, {<8*(*+7+7( 2D "+:B$.8 ($q =. KNB "(45 B "K93*A7( ",D;[email protected]+C($x %-. KB "(VBO, YA7BO.8:9(*01(*.G+ 2(ACBO. a 7,K A"D5( ,/A 4 OA 7EG,(64;( `45D;01Dp+C( {2,.G,NBOa=( O.G+B$+"+C(*[email protected]+C( O,EG,(<K4;, ªBO,8*,.Y/ACBO.8:9/( 0 ( [email protected]+P.z\³( "+K 7D;Y45($x Q{( 78:9 01B2(PYA7BO.8:9(*01(*[email protected]+`KZ(*A70 (*+{2 =.82]\ OY/+C(*.D5A,.(-(µ+7( . "D =.c9 =A7D [email protected]+CBO45(m2(P4¦\ *8 45BOD5A EG,D`.(KL( ,/+ *+7A7( =Y/+7( .G,(BiI=( 845(§Y/D >4;(iBO2/( A "( ,4¦x¤}P(K45, *q4y\[email protected]+CA 2/,8*+7D =. 2/(45B2D50 ( . 7D O. ACB$8*+:B$45(-KL( A"01(J+2/(<A"( KA 2/,D5A"(4BF+ OA"+C, 7Dp+ (*+m4;( `YA7BO.8:9( 2( *8 45BOD5A x 6 ! ! # W 7a \ "! Xyk On5kJj[kJl&.f0+21hj43 fO¡Äbrut=t ã w#$"! . f0+21hj3 fO¡kJlfOnposb³~O¥=¥/r*wyxW%&' "a " "() #)*+! , ! ' "a V ,<- , q xN} 1 excess |y~x·O@ |q(i, j, k)| > qexcess ,.*/'(7a $01 2 3Y a! a 341 !2¦\ *x q(i, j, k) } -) -! * 5 67 Oa Oq8{9 :07;< ! = >! ? §a =Y 4! y\ * @ A $x %? ! B' "a C ( C! ¦\ *( q1$ #C]\"' &1 Oq ?! ,5Y /' 7a 2,! δq . & " & ! y\ * x ( q(i,j,k) δq(i, j, k) = |q(i,j,k)| × (|q(i, j, k)| − qexcess) δq(i, j, k) = 0 neut δqneut = X i,j,k δq(i, j, k) × ,. 3Y ,! *a 34 "! y\ * xW5' 7a !,;Y !D71 q')' */ ! = )! :( a OY' ! y\ * 6|QE 01 34 µ- N ! |y~x· ã 1 N dq(i, j, k) = δq(i, j, k) − δqneut dt " 34 |y~x·=·= (±.¦\F , %! B ' 1! Gy\ * HI D -|9! Gy\ !J! =¥=¥=¥=¥ <Jq .! +y\F *xLW $ C .3 =Y "01 B. K:! 3 , qL M! 8 !'( [email protected] '(! &¦\*(%D J!]\FOY !5 8M M: Zx%5,! ,,5Y ! G O $ ! J*P' "a % #7N(ZqNQ*P ,M,, % K7! 3 ,*x W.34 3 Î=v W5' 7a +(3Y *(1 <5*/ 7a ! ¦\*( q734 P#.;Y 01 7N(3- */ 7a 2! . "! 3 3- * ,01 +/' "a ! y\ *x % ! a ! ! =' P "*/ ( ! . ,8*4' 7a 5! : ; ' "a ! ,5Y ! / * , $q# &7N() AD ,L`5*5 ( ]xW5' 7a +,! ,,5Y C3- * Ama ! J-D( GOY " C' 7a ! .*D( A J *'(7a ,! ,,5Y xQ W 5'(7a , D7 5 :(,&&' "a .! . K7! 3 $ x ,Q. .: $C. " ! = C! +3 3- * ,$q "7N(? +' , A ' X& *P ]q $*@ 8! P' 7a &,! ,,5Y ,,H& /' 7a 2!.1K7! 3 x W/N C! P ' @P K7! 3 $ &! A !) , 3- &! C* ,$ )sr-N ! '(, *x W ,$ Rr-N + ,.! . 01 2 K7A!]F\ K7! 3 $ [ W5' 7a σ(i, j, k) = 2 ,.! Z +∞ 0 πD(i, j, k)2 N (i, j, k)dD & ,LI2X |¦~x·OÎ@ $ dq(i, j, k) σ(l, i, j, k) = [δq(l, i, j, k) − δqneut ] (1 − α) PN species dt σ(l, i, j, k) l=1 ?sr-N 1 |y~x·'= 3 07&! L3-J! ,3/8P$K:&! &[email protected] 3- $ ! G 7a $01 .!' K ,I34 $a | $Na ,~x;r=ruJxW%' 7a !]\FOY ! ' " 7 C ! ;&q:3 &y\ ( !;1'Oa +Am$2/ ,'(. , J ! p'01 !5! Py\3- , J! I M 801]\ 5'(34 ,01 P P ,a !' +7 Oa Ox j q%' 7a 1 ' /a 2 B1K7! 3 C! 1D @ 3- K 01 *q7$ , G' J2 D $ [email protected] ND7$xGW ,07C ' 3- $,01 -y\ "! 1'$a B! $D7 , P4 BD( , yq " ! $ x/ A,@ 3- . !]\ ! Y ! ,$ ! 1 .,I.! Oa ( -SY' B 34 yxX^ $' ! 4 , 3- *) , = '(-! * q! Y 1<a < 34 , : , 01 -07 = Y <? O $ 5! ! ! . 3 7 ! [email protected]!2, ! 07OxW% & ' 7a " = P C a5! ' ! ¦\ *( x} & -Y :"! 3B @CK72!D:! . 7a q' D (*$Y , 01 7 < 01 2 K7A+!z\F K7! 3 $ Ox W\$ A;01 , -) 7 !'Oa ?! 3 7 ! 07 , -* D( !'( ! 3 7 ! I)- ' I,3- ! 5' 01 ' 3 , ( ;34 K OxW%/ ! 5*45a +!Ra ~$} ! Q %©Okªv/r , ) 7$Y ! ?! Y * !; )'(,) D D , p 3-/! 9 =! a 3-/L34 12! K , [email protected]<Y A,34$ +!-3-,,5 ND( + : , B 3- ! 1!-D7 07Oq%! 5D $'2*)*( ,I $ 4 ,)* !K7' 3-01 OqH* 3- 1K , 01 $5y\ O q34 + P!]\ ! / D( OY / ! / 7 , * a ! ¦\ Oa $xXWQ 3/ * ! , K I3-5 D p ! N =}FqA 3-3-+ /3/ = A, Oq A3-C!]\ ! y\ D W% Î=t 07Oq J, a ,!' & *Oa ND 5* 3- K 01 /y\ OxL}C C! N C ! 3 $01 /3=1 ! ! @! 5Y=1y\ $( 5! 1'Oa 34B!]\ xQ 3-3- D M!'( J&' ,Oq 3 3- , P 7 G ! ; , G ' 3-3- ! & 3-34 2 : ,.21'Oa /! C /! +! /01 01 3-1*qL! [a ( ! C , ! ,Y , m/ @! *2 K ,01 ! ) : " P! , / 3 ( ]x}C , ! D ! F$B*L \F K C! ; , ' 3 $ , " ! , ;Y @! 7 x©8(& * q7! ! & . 3 , "01 7 , " ! p A 2 #p 4!'( + 4! 3- 5!]\ * [ QXn xHW\F34 ! 3 N 3- ! p ! /( , ! O xZWQ-! I3-B'(,+! 3 40783 = ! /! 5Y= . 34-!]\ * * q #]q' 343- ?A.¦\F3 Oa q1# , . C!]\ D( &¦\F34 ! Y ( 3- q B! D $'B* =Y )! , 34 D7 & D:+ ! * 3 ( , ! O((G! HY 3- * x= 1M3- , [email protected]/ = #M ! ;Y 3-C* =Y ! 34 3 8 CD $ 3- & ! 3 / 34 K x % @! 3/ + / N 5 !; 5 ! ) D $ ! N ( 4 ! 5) 3- 341!:K: 3401 OqM3- 1K , 01 ? $01 )! 3 7 ! IOqH : @ 3-' N , 01 [email protected]!P! p 1 , K I3-*x j. ,3-I,. =D 2! +D( !' )! 3 : ! I2ND B! = Y , ,D(( , [a 2t F-* 3/ * )! y\ Oa .! ru¥,ªrutOt=Î = Y , D ! \uTk9f¤Tª^ %[email protected] P &y\ 7 N 4! A , . N( +! 3 07&07. &* 3B *( +! "* :! 5!]\ i 7K:! !]\F ( !' " Oa *x 34 ,@$ '(* @! &¦\ Oa !]\ 4 18! &D7 'O¥ Branchements de rang n Branches positives (puis branches négatives) Points possibles tels que |q| > qexcess et | ϕ| > ϕ0 n=n+1 Calcul de la probabilité de branchement p pour ces points Pas de branchement possible n=n+1 oui Σp = 0 oui non non Sphère de rayon A*r A*r = Rmax Nexistants = nombre de points entre les sphères de rayon (A−1)*r et A*r oui Nexistants > Nmax Pas de branchement possible A=A+1 non Définition de 2 seuils de probabilitép:new et pold Nseuil = nombre de points tels que p > pnew Nseuil = Nmax −Nexistants Nseuil < Nmax − Nexistants Nseuil > Nmax − Nexistants Augmentation du seuil de probabilité pstep = pnew − pold 2 Diminution du seuil de probabilité pstep = − pnew − pold 2 Nseuil tel que p > pnew + pstep oui 0.75 ( Nmax − Nexistants ) < Nseuil < ( Nmax − Nexistants ) non pmax = max(p) et pmin = min(p) p max = pmin non oui pold = pnew pnew = pnew + pstep Tirage aléatoire de ( Nmax − Nexistants ) points Branchements de ces points A=A+1 non A*r = Rmax oui 6 HJ#.)C(*9,T$#AZ= A , (*+& $ [= H#.0#=T1Z&@_ +M"%= #=c13AV#$X,? &+A313A > +M0,TU"*&X,E213AZ1 7 7 6 7 A 5 r $4J+&2?U"13A3?=L&V,[13A3[ #<!9$ DI?=z.) AZ.m". (+&Q&V,cH/2$-+& 21 +&21<!X=]?. $-o.) AZ.m". ( N&2max $-= H+&g"1V?$#A & J4+&5 R0max W (AYV= _<G; +&["%13$4+& $-J ; A3[135 AP?=&o ?=c1(*J $#A31 +&g(I6 =E&Q!1 ; 1 _ 6 7 7 A×r 6 6 7 ; X~xpru¥ 'r Séparation et échange de charge Processus microphysiques et dynamiques Transport des charges E > Ebe Production de NOx T+dT Calcul du champ électrique non oui Leader bidirectionnel E > Eprop Extension verticale du canal de l’éclair oui non Schéma pseudo−fractal Extension horizontale du canal de l’éclair Neutralisation partielle des charges XZ~xpr=rg76 7 HJ#.) (*O,T$+AZ= A ,g(" .)? 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D & 23 34'(, /W : , J 01 ! 1 !3-1 H* 5 P!' @1' &01 ' l ! 3 3 ( $&*2 I 07 ! = B 34( ◦ ! { Q> ; I* [email protected] 3Z?Na j* A 5* n* X/W @ _ W ( a s[[email protected]+021*[email protected] ^ * @ * gA n * m W w[ 5 \ @ / , Y , 5 * @ ^ * @ m W n * A +> [[email protected] [4* 3Y*[email protected] ^v* @ 6 /@ B a n * A :[ 5 k @ 4 [ , *[email protected] ^ * @ A C-02[4[4[/[BA A /* /@ 7 lC [4[/[ a haBA /A p> * Q> /@ _W i* +* * Ba n* * BA * BA h* /@ rA /@ 7 p> Q> 4NW6 * NW * ha L* A p> +> R ~NGIH KNMjE/OQPa 02141/. U @ ~ GIH dKNMYE/OQP RQ021/14. U BA BA p> -* [email protected] * > * h* 4 [ 4 [ [ 7 p> 34[4[/[ ; I* [email protected] [email protected] A BA j[ a > [email protected] iA h* Y* [email protected] * >A f* > :C l> ]/[4[/[ 9* 0m]/[4[4[ @ VA w]/[/[4[ 90m,/[4[/[ @ * h*4* i* sl]C</> [4[4[ l0x[Z]/[4[ +* @ 6 <] h 4 [ 4 [ [ m 0 / , 4 [ 4 [ [ /a +* D* * / * [email protected] I * 9. _ k @ / 1 r 6 a f6 * /* [4[4[ ]4]/[4[ a (* AkCd34]/[4[ a C [email protected] * * /* gA p> |* /@ }* / * * B a * /* * [email protected]{ C5a 7 9* /* A * + * /a * * B @ NW f6 * h* C sC l> (* /* /@ * > 7 a 56 B A [ ` C s > j* [email protected] 350 ; 4? sublimation de la neige (DEPS) sublimation du grésil (DEPG) évaporation de la pluie (REVA) ajustement de la vapeur d’eau (DEPI) autoconversion de l’eau nuageuse (AUTO) accrétion d’eau nuageuse (ACCR) ;J6 ? givrage de la neige et du grésil (RIM) croissance humide du grésil (WETG) croissance sèche du grésil (DRYG) effet Bergeron−FIndeisen (BERFI) ajustement de la vapeur d’eau (DEPI) autoconversion de l’eau nuageuse (AUTO) ; x? accrétion d’eau nuageuse (ACCR) 9 évaporation de la pluie (REVA) accrétion de gouttes (ACC) fonte du grésil (GMLT) [email protected] X ³ÎX AY=.gA $-?+,cH*D # & A2"E(I=&W ,$213AP ?= ; 13A ; Q(*XK #^ ; # ,#"%$S(GF 4" K 219K ^ Wm,TU"AV _ < JP7 N#=T(* *( +&,T$-? +&+&+"& +&21 &+A37 13"J'7 #=>2 0( H 2E " N"$4 _ <5 F 7 7 7 7 N$-I("EJ\#= #N*_ #N −1 _& −1 _ 3Z, ^4F -"\=E" N*#"&@ 6 gH%?$#A ?=c1V[<# ; ; 4? aggrégation de la glace primaire (AGGS) autoconversion de la glace primaire (AUTS) effet Bergeron−Findeisen (BERFI) ajustement de la vapeur d’eau (DEPI) sublimation de la neige (DEPS) aggrégation de la glace primaire (AGGS) ;J6 ? autoconversion de la glace primaire (AUTS) givrage de la neige (RIM) accrétion de pluie et de neige (ACC) conversion−melting (CMLT) croissance humide du grésil (WETG) croissance sèche du grésil (DRYG) sublimation du grésil (DEPG) givrage lourd de la neige (RIM) ; x? accrétion de pluie et de neige (ACC) 9 conversion−melting (CMEL) congélation de la pluie (CFRZ) croissance humide du grésil (WETG) croissance sèche du grésil (DRYG) fonte du grésil (GMLT) [email protected] X ('z AY= .mA $-?+,cHDs&#[email protected]" I( [=&\W),$213AV ?= ; 13A ; (*zK #^ W N o,]$+AZ.)AY$4 K ^ = #AWN 219K ^!9N[$-J+&#AY!_ < !JPN*7 2=T(*(*+&,T$ ? +&'&#"*&+&21 &+A313"E7J40#=7 >2 )(* H% @ " :N7 "%$-R_ <F gH%5 ?[$+6 A ?T = 1V # 7 7 7 7 ; N$-I("EJ\#= #N*_ #N −1 _& −1 _ 34. [email protected] 3 o AY= (*+&M+& ,/ +&M.mA $-?+,cHDs&#[email protected]"+&(*=& WO,$213AP&213$ [13A >@?$+.0 _ <G ?["$ M$-?"[N*$- ,T$-J#&@#=c1 7 ,TU"AV 7 !#"E >@?= JXoN O,T$#AZ.\[AY$4 W !#" 7 AZ$Q6 W)=]#AN* 21 \$-? & 4F -[" =E"RN*#"&@ ; #$#1 9M 5 6 #= #N*_ #N 7 6 _ 7 5 !N[$-J+&#AY!_1<F QH%?[$+A ?=T1V:+ N$-5 I("EJ\ 21-F 7 #$213A 5 2=L.0 ; −1 ; ; 7 X ` ` "&¿±¾ Tb H¿ &¿ % ½ a 4dQb ' / / W/' ' P+D( A P! F )' ! * , P! ã B3 D / 6 ! (=D 5( ! , $ ZWB $ 8D ,N H! * ;! N( 1' ) ' *l+' , / $ ;N( ! ,4! '( 4,BI 9 34'( /! ' , f %eN(1 * '( 3- D 01 N( 1' C,P, D .! +3 WQ+ ",P,D ã +3 @ D C! ã ?3 $ ã ?3 sW * 4 ,34 ,/ ,4 ' 3 K (=D - , +3 23 ! @ 01 +' C! L , & (D , ã +3 23 W , $ , $,01 B!+- 2 . 3- : 4' BC,=D 2 , ã -3 $ Î .3 Q G!D71H (=D , 01 ' "3 $8,! $D: H, D G +! ¯W%Q3 3 3 K Q ! Q ! , H !G G!' Q ,G 8! J ,! ! G F3 −3 uW"D( A J J ,LI @ ! 3- 8! 4 1M! -D7 "! =& ! "'( $01 ' , P3 K (= ! 7 @ 01 C & :2p @* * C ,34 ,.$&*2 C01 L ! 3- 3-34$! 1' zW ,01 + , +,C ! D F , 6 ! , - '( 4 ==D 3- 3 -! - (=D $ XW%4Gp = Q! ;! C B : "! ã 73 ! == ! .2Gp 8 "! ' 73 ! F ! J =( G- ,!- D . ,D(( !! JÈ J 4 D $=D @! ! , C ,&ND G ! C'( , D 7 N( + ( +! ã 13 13 $/*? (D 41 ) 2! ' 13 sW , $ ,$01 501 : B , 5! , , *)D ,N( 3 3- % . 2 D " ' " , :! ==D f* rA n * _ W / ] 4 [ [ i* ; I* [email protected][ [email protected] d6 BA /* rA I* <[email protected]+02[ f* l> +6 C ( @ Y6 ( a B A /* .4[4[4[sC [4[/[ (* 4[ [4[sCl34[/[4[ @ , / / ] 4 [ [ / ] 4 [ [ @ * p* i * ]4]/[/[ 02[4[4[4[ a p* <* i* [4[/[ 34[4[4[ @ * W f* ]h[4[ Z]/[/[ a if** A C /(]/* [4[ a gA 5 C @ m W _[ @+0 Cj0 7 @ @ `* f* :C 9 6 /a s* +* /* [email protected] /a hA yC i+> * f* A i * i * ; n * 5 . Q @ 2 0 / [ 6 ? @ @k]^ rA ha (* +> Y^ iA * /@ 7 /a 46 FE/(~q* K YKNM E/OQP R ,/[4[4f*] U @/{ @k] ^ [email protected]\]^ a i* i* ^ @ L* 46 46 <6 /a [email protected] 3ã ; 4? aggrégation de la glace primaire (AGGS) autoconversion de la glace primaire (AUTS) effet Bergeron−Findeisen (BERFI) ajustement de la vapeur d’eau (DEPI) sublimation de la neige (DEPS) aggrégation de la glace primaire (AGGS) ;J6 ? autoconversion de la glace primaire (AUTS) givrage de la neige (RIM) accrétion de pluie et de neige (ACC) conversion−melting (CMLT) croissance humide du grésil (WETG) croissance sèche du grésil (DRYG) sublimation du grésil (DEPG) givrage lourd de la neige (RIM) ; x? accrétion de pluie et de neige (ACC) 9 conversion−melting (CMEL) congélation de la pluie (CFRZ) croissance humide du grésil (WETG) croissance sèche du grésil (DRYG) fonte du grésil (GMLT) 5 _ . k @ m AY= .mA $ ?#,cH*D &+A2"E9(I=E&SWM,$213AP&213$ I13A >@?$#.)9(*mK 2^oWoNW ,T$#AZ.\[AY$4 K ^om=]2AWN*21 1 K ^)MN$4J+&+AZ!_< !JPN*#=]7 (*\(*+& ,]$ ? +&'&#"*&)+w&+A313"J'\#=m>2 m(* H% @" N"$47 _/<oF 9H%?5 $#A R6 ?=c1VG9,?["$ 7 7 7 ; "%7 =EAZ1 Jg(*#& 2N*_ 2N _& _ 3Z] X −1 −1 ; Z? ;J6 ? 9 [email protected] 02[ _ 6c"O$\AZW[ =i(*N+&"0$4LK+&V,#/ ^ 7 +& W J#! ?7 "%13$#$ [email protected]$"+&?"[KN*#0$^ 421,T$4(J+m&@#=T 1 (*#=WL&#AZ(*1 #=J)(*&#AZ1 J7 H%(*[$3N*01H%V?[$PN*1V! (*)K 6 W^ (*+=,&o2"$(:,F -$2[13"AV X ; ; 6 M,TU"%AP !#"E W[AY$QoN O,T$#7 AY.)AZ$4 W)$ ? & ; X\= #AWN 215 7 ; #$27 1 9-F -" =E" N*2"*& !5 #"E<>@7 ?= JX 2["] = moN$-J#&+AZ_ <oF 5 gH%?6 $#A R?=c1V +&7 #1wN[$ I(["J 5 #= 6 !_ . −37 21 4F ; 7 #$#13A 7 2= .05 _ 6c"%$ N"$4 K 5 6 ^ 5 ; ; 8 4 F XH? $+A ?[=c1V:+&21 N$-I("EJ\#= _. −3 _ ; 8 ?= P!A h6 ,! ' ' PP 7 " ! $+> [email protected] &ND PD7? ! 3-3- G07.J3N( ,34 34$ J ,-! J ,01 J1"! G , ** * "2 j* 3 [email protected] W%` B* D77 [email protected] W% . 364 !I "+>MA34I01''( 1f*P* +* 1+.$! C +* * * +*(,34, 34, ,/a ) 7 4 ! 3-'(1 |* * ) 3 =; $I* [email protected]+02,h6 [email protected] . f* ?|* * ) 3 i * (=D 341P ,w]4]/[4[/3 '/]/[4[B3 a$& D 3- & '4]/[/[B3 $}14[4[4[[email protected] == ! '4]/[/[43 ,, !|C A ,Ij C gAF ,34 ,/[ ◦ 7 @8H 01 xa% ]r!A 1 ;N ;07! ! s[[email protected] .*[email protected] −3 "01 +*5 +* +' f* + i * (D 34 $C* * 3 =D 3- [email protected]±W% '(n* )! ' p* ! : , ) 3- K 07 L>9 +6 34'( ?! 3 N 3- 6 ! +>[email protected] %ia'!' . * $I* [email protected]+02, 5a ' lP07B*Y* ,34 ,&,@' p* ( * (D 3- P01 .01 BAF = ! [email protected]@W ! 3- 4! *' p* , 2' C.,, _W ! s* *(2,C ( * * N/6 /a'( .01 B C 3 N( , 34 . !+ , 1 ! .BA !&! }* ( ! ! 7 6 ! +> ( @ 7 A !'(1(a:&( ,GA ' 5 ,, /6 )! )*; = Q> 3- ! )* 34 /@ 7 A4gA , , 3- 5! D( /6 07.07Q 3-L 22' p* 3- 6 ! Q>8, Q>8 == ! !+* ,34 ,/@ 7 $ , 341CL3-C!A$ D7!rA ND C!C f* $,01 3- ,(1 _W ! = # ,"!23 n* C,:>9 +6 [email protected] u: +* $n* [email protected],h6 , , 6 * /*? +* /a :D 3 N( , 34 ! +>L +* 6 ,, _W!}* * .01 i A , W 3- `C.3 34& 7 !' G :6 * ! * * ,34 ,/@W ) * 01 ) ,<* * $?=* , D7 01 ; W/' f* ,@ 3=Î p* +> i* (+>G! 5* * YC6' ,- ! /@ D N ra% 13 ,34 ! 01 3- ,(1 (@ZWVA : 7 D , <! 5L* *(/ ,34 ,+< +* / #3 N 3-C3- 1K ,01 01 #1D: &!'( & f* 34 "! *B* /a34 @A 02[> 7 &3 P 34 , P01 2 & 7 C! ' p* 3- . ! +>[email protected] 3-34/ C*L* + 3 (2 * ha +3 N 3-/ ! $+>M,/ : , ! 3- !' 6 $01 5! * [email protected]]W% ,01 1 ,, _W ! * P$ o* : "C +* P$ * , :L3- G Y* , ! &' p* ,=D 341 ã [4[4[53 34[4[4[53 ; $I* ,[email protected],([email protected]W%3 mW3 /!C 7 6 ! +> & 3 K 0? , * ? D D -! )gA ,! )! ± ,[email protected]+0x[ −12 7 @3 −3 @ −1 / ,LI * (@W%j* % 2' p* + " ( * D 3- . ! /a:N "S>9 ! "01A#K = : W:, C * , ga *j* * " 3 4* +* /a1$P! BAN > ! ! , ! BA 3- ,h[ ◦ 7 @Gj; : , C 34 P! . 96 * )! * #Cl> 2! * [email protected] D ( AF/ * &=2*/! AF [email protected] A4! 4 f* - =D +* , J+! 4!A * ! -' p* %) :D j6 & 6*( ! & f* & ! B C * 4! ' p* D C! C/ > B! d* H =Q>}; $I* [email protected]+0/0N2? @ ( AF/* C! M : $,01 /34 , . ' 3- /@ BA /* /a /A ; $n* }.5 @Q 040x [email protected] (A /Q*> > +> gA [/ [4[ ªÎ4[/[4[ f* XW rA @ % ][email protected]+02[ 7 @ @ C]/[4[4[ @NWVA /* ã f* (* ia 5> p* * !CY6 YR/.dE/G/Í2KNO ObKNM E/OQPaI,/[4[Z] U @ W BA f* rA a f* * BA A /* @]f D. 4[/[4[ gA [4 [4[ a BA / * ã A ra > p* BA h* /@ 7 > p* w> +6 [email protected] { f* p> A * /* A BA ã ^ C gA D[ 7 @ /6 D6 / 6 (a p* rA /@sW L6 p* $I* [email protected]+02. >[email protected] { ; ha * w* ?Na @ 8 l* Ba * f* Q y > a h A yC g A 5 [ k @ , 2 0 [ ± × 7 @ @ @W jC l> f6 gA _ W > @!{ * ha 7 @ @ ? a 5> _W `* A :;n* ∼ ± [[email protected]Î ×f* 02/@[ XW * B A /a #p f6 [email protected]W * /* A 3' { */ Q ! J N 1' 3 7 F =J , 8 −3 −1 01 % N ?1' ,*5' B3 D ! +1+*/D( L .. N ;1' , B ! -' ,BI :$ P1' 13 01 , ,34 07 B !' *-' ,2 = 9 C ' ( ' B ! 3- 4 ("01 "! &! : 1' K-+ 01 ' =K AB *( '( .!+ ! ( 4! D( A 1' , / ! -*)D( L 2! F )! % 7 A *( C 5 ! !' * ! , +! 5 , 3 , .! F = ! 34 3 34' * !B! 6! , ! K 07 , $! 01 !P! ,"! &1' ! 341( ( ! C01 /23-2! P*B ! 3- 3- C K7! , + 2' ,! 34 5* J! 53 '(,/3- ! , ! −11 " !:D $ 2N ! -* ! 2 , 1 ,34 +! '(( < 2! -! -' −3 −1 %3 ( ,34 , ,C ' 3- & 2' . F !D $ ;D ,N( AND B ; B , P! *B O D + +N ! AI ! $ Q F , ! 34 ( ! 3 3- −14 A ◦ ," ,' . −3 −1 3-! , B ,01 4! 3 34 , 01 -* ,M ! 3 =D ,+ 3 3- M!P( :3/ 3 ( ?! 2 N $ ! 3- ; ,! ' B 7 )3 (D 3- " ! ( )! *2D (L P! N D D - , , '+3-+! ( 01 ! + = ! 3 D( $&01 =D B! B*4D( A D N /! 5 (01 ! C- ,((1! ' , 5 P! *+ F D( ! "D( AG! F 2 &! + $D( C-*?' " ! M ! A B3 ! ' + O! /' 9 ! ( 07! 2D( A P! F !*2' ! ( 4! D( A ! N( ! N #! '( P! F , 3- &!*2' H! J 9 ! , ,34 !.' , " ! " 1 /3 7 % −12 , ."! *BD( A&! F ; Z? sublimation du grésil 9 ajustement vapeur d’eau neutralisation givrage du grésil ;J6 ? croissance sèche du grésil processus inductif ajustement vapeur d’eau neutralisation ; x? 9 fonte du grésil neutralisation [email protected]+040 X \ OAZ=i(*+&mJ H%=N+&\( H%$PN*+& K #= _!. _& −1 ^ (*=&9o,$213AP ?= ; 13A ; M,?"%$ K #^0 7 K ^4F -" =E8 "RN*−3 7 !JPN*#=]7 (*+&Q&2?=T1G&#AZ13"EJ4+& ; ,E#"$o(:F -[" 21 !+&w,$213A 7 "%!+&MUA2"AP(*+& + #"&@g21OK ^gWS,] "%APR_ <:+& 7 6 7 #=g>2 m(* H% @" :N["$4 _ 6 #" &X#& ,T$ ? +&'&#"*&g=T?=z=E"% &Q&2?[=c1<$4 ,T$4J+&@2=c1 J+&o&#"$ +&MN$-+,cH#&@_ 7 7 7 7 343 processus non−inductif glace−neige ; Z? autoconversion de la glace primaire 9 processus non−inductif glace−grésil ajustement vapeur d’eau neutralisation ;J6 ? processus non−inductif glace−neige autoconversion de la glace primaire processus non−inductif neige−grésil neutralisation processus non−inductif glace−grésil ; x? et neige−grésil 9 sublimation du grésil neutralisation processus inductif [email protected]+02, X B AZW= (*+&LJ H=N*+& (* H%$PN*+& K #= _!. −3 _& −1 ^i(I=& ,[$#13AV ?= ; 13A ; ,?"%$ #& , $213A "!+&MN[W J4+& K #^\g7 N[W S,T$#AY.)AZ7 $- K ^)=]#A8 N*\21MK ^)!MN$-J#&+AZ_1<:+&9!JPN*#=T7 (+&o&@7?=c1 &#AZ31 "EJ4+&\#= 5 =T6 ?=z=E"% &Q&2?[=c1<$4 7 ,T$4J+&@2=c1 J+&o&#"$ +&N$-+,cH#&@_ >@ g(7* H% @"7 N"$4 _/6 #" &9!7 +& ,T$-? +&+&+"&o 7 7 7 7 341 (A L$ [email protected]_!W 8 l6 $+?D( A !AN ra'&3-&!2 , , 1.5> , " ,34 "!A' n* .! ' f* .' P >O&=3-D 4 K2BAF07 !/@D u7$ ) D ,N! (I,/a D,7*B,D! ,N*( ,i * B(=D 3 - @B* !4 ! 34 , 1 ( $ 6 ?; !;ND + =C!2/' p* 01 [email protected] 4' + !D D ,N + + ' 01 ! %; ( ; 3-' " $I* [email protected]+0 ã $ [email protected]+02[01 *5 ( )! f* "' " * (A wC/*+' f* [email protected] ,07rAkC Î4[4[/[/3 !AF ! /a7*+ ( ,! ,,6 ! ) ' f* ( * (D )- >! C5aH! ?* '(f* ?=D /@ u: > ! I, ' p* ! 1'h* /aA*- ( 4 C $! +! 3-1 "*' f* /NW: BC! +2' 3- $01 /@ 8 B*; '(,/!5 K7! 3 ,5 343--*;D( A B!AF aH* ha** ,34 , Y* Ba#) ( ! C?! 3-1 *)' p* NW: /@ 2 * 01 B, =D , , 07rAkCÎZ]/[4[)3 $B( ! , B ! '4]/[/[?3 2 (* (=D - ,42! _W = ! ( aQ* ( !C n* 3- / ' p* )NW: /@W 01' = ! ' p* B' 01 ) K7L !rAF K7! 3 $ )! L !>! * sr- )! 5 ' ( a C! 2! +. /@/W4 ( ?4 Rr-N C ,. 343-+-* +$w* [email protected] {( 2N C ! +* /a *' p* B + , D>9(6 N D ! ; - 134 B gAF ,34 [ ◦ 7 abC ! BL (1+' + * w ;HD $I* [email protected] ã [email protected] '7 1 . *5'(, ,( 3- :*/ !' B " &! .D( ,* ,01 $ , @ , .01' 34 "*/3 3-072! "*/'(, D D LW5D( A. ,.=D ' , = ! 44! 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Y*yni,i. ;)6\*<76%5=U$%R:;R `7MH:CU 2 2 5Ad3I</;[email protected]/%[email protected]BMH6v*+:;?$HA%[email protected]*+^;MHA%A%CUR?968*Z&)[email protected]:;J b*/&5)MH*+R43Ad )M%:V:5A%:5R568*+:/CQ&<j&EAH*+M%:5 $%S;A I<4<ZG%=U?=U[\M%iAB&E:/CUSA%:;R<[email protected]*Z&W*+=U)6%[email protected]=U[\M%[email protected]$%zW({_R[\MB&E*;=U76h$%<76%5:5KE&E*;=U76hA 7M%:iM%6 ;<&)CQ&E=U:;i;[email protected]/%CUS{'R<:[email protected]*^;7MH^CU&%¦o7:;? /&)6%wCUX<j&)$%:5g$%gCQ&S;[email protected]?!M%CQ&W*+=U)6D$%gC¥{_7:;&&}7O$% d(ρCg ) ∂(ρCg ) ∂(ρCg ) = |T + |M T dt ∂t ∂t d(ρCl ) ∂(ρCg ) ∂(ρCl ) ∂(ρCl ) ∂(ρVl Cl ) = |T + |M T + |M P + |SED dt ∂t ∂t ∂t ∂z ¡Z ' ' !"$#*%&$'( "*)" Y * ;)6\*9CU:+&)AHAd7:b*+c$%?RCQ&E6/}7N$%Mr;<&)CQ&)[email protected]:;N57CUM/HCU$B&E6%9CU&pA%GB&)5 }7&)(*$B&)6%cCU& Cl A%GB&E;[email protected]=U[\M%[email protected]$%S:;;A <*;[email protected])?68*zWt&c<j&W*+R})7:;[email protected]$%iAB&E:5*[email protected]<M%[email protected][email protected][\M%=U$%g5*V<)6%5*;[email protected]*+MHRD$H })7MH*5F *+CUY*;*+S6\MB&&}7M%;4Y*J$%})7MH*;*;S$%AHCUM%[email protected]&]W!:;&)A%A 7:5*/$%?9RCQ&)6g}7D$%M }8&E($%=U5;7MH*/$B&)6%SCU& A%GB&E;9&)[\M%M%595*4<)?9A 7;Rc$%M :+&EA%Ad):5*J$%c?RCQ&E6/}7!$%cC|{';A I<<ZG%[email protected][email protected][#MH$B&)6%JCUY}77MH*5*+YF CUY*;*;c6\MB&&}7M%; Y*$%M :;&)A%A 7:5*c$% ?RCQ&E6/}7N$% C¥{_5AdI< <ZG%=U?=U[\M%N$B&E6%[email protected],}77MT*;*+c$% Cc A%[email protected]%=U * 5768*/[email protected]*+:5?9i$%4*+:;&)6%;A 7:b*i$%/;<&)CQ&E=U:;g<ZG%=U?=U[\M% Cr ∂(ρCg )/∂t|T ∂(ρCl )/∂t|T $B&)6HSCQ&>A%G%&); }8&E(MH;c*S$%&)6%SCQ&9A%GB&)5c&)[\M%M%5c:;5Ad<*;[email protected]?968* :5A%:5R;68*+ CUP*+:+&E6%2¦o:5*+^$5{_M%6%J5AdI<<ZG%=U?=U[\M%S$%4CQ&A%G%&);Y}8&)(M%5K):;^CQ&A%∂(ρC G%&);l&))/∂t| [\M%MHM;&TZWP*+:;&)6%F ¦o:b*+4$5{_;A I<<ZG%=U?=U[\M%468*+:;9CUQ})7MH*;*;[email protected]*;*;J6\MB&&}7M%;DY*[email protected]}77MH*5*+$%cA%[email protected]%[email protected]:;S$% ?R<j&E6%=U5?94?=U<:;7AHG\"#5=U[\M%4<):;:;;A 76%$%68*c&)M *+:5?9 &jK7< F 5*CQ&PK#[email protected]*+;5X$%f<ZG#MT*+O$%AB&E:5*[email protected]<M%[email protected]∂(ρC [email protected][#MH=U$%l )/∂t| ¯WM f5T<76%$!*+∂(ρC :;?fr )/∂t| $HfC|{'RM}8&)TCU=l*+0 R ∂(ρCc )/∂t|M T Vl <):;:;;A 76%$q&EMe~ *+:+&E6%2¦o:5*+$%>?9&)5;>$HD;A I<;)CUM/%[email protected]tY*[email protected]*;:;[email protected];[email protected]?c*;:5?9yvCU& 5R$%= F ?68*Z&E*;=U76 WS*+:+&E6%2¦o:5*^$%J?>&E;;J6\*;:;JCUgA%GB&)5 }8&)(S*^[email protected]=U[\M%=U$H/{_~TA%:5=U?S;7M%^CQ&%¦o):;? 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