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ΪJ GCۡԪE }rv{2-p%uDeLhتNFvo2dV:$۟;1K'0 KHdDЌǡ~ ;8tx)JZD}fp:1)3$HļI2YX=GV)beW[j)T3jB;YáKxg8Y?2Tef3;%GT>%5ph>YgR43}p[f8mҷF:@w}%Dy5N)0'8䒰ҹk8/JSCyto&CΒ f&80ֽGhkBj,-Y& 1EbBDe2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJs AAjJ" @83ʚ;ʚ;g4YdYd|L  0ppp@ <4dddd w 0T7L  80___PPT10 ppPhysics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity /   h   0 lh(  lr l S s =K    r l S (w V0   l l 6AjJ?" l l 6AjJ?" C l l 6AjJ?" F f l l 6AjJ?" F  p l l 6jJ"`F  H l 0޽h ? )))̙̙380___PPT10.dXP7{   0 p(  pX p C       p S }   0     H p 0޽h ? 3380___PPT10.dXP7{r [ 1q( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ .  @n?" dd@  @@``8U *W3rN+\ ޾};h Cs`ݻg޼y3|ҷg+\+#qR|;"c ĜdNszʷ%``_7W@39;?,QǏWO!ݽ~ VQŸwe!|Sl}g1w r"9@qk#.cm[̧HnOsL}NhqߵO}mbm&\\]+ZO5mޜz)!:%"|vVPo>v>E#ǭE3zkxΧL}qTh5ZHL ucE5rZ@IIE|]~n~_ڪڜ ܸE>#Zۮ}8W>-k kbin\g꾿k+ֵ_iɫUcO"? ķSqRt ؘ(վіXԞm~\ru [r$+W r>UpvXAee=2~3ٝwhKZ '  `,6)5^`5k}9pVڋ| /e\өS>Ti LZr>#W㺩]q_rIkS}> p 8OO$G}/V[H)r_q>7ҋ˳ݰ5.:^?Ռsgjs"㶶9??5ӕ쳧C%פO縿ճ][ԫ!ɎݮVX$#>n+=9{A>%pܢ<&ߧN\po"8s7e'qzu,߹>8Ri.r^'W@>΍T>zVc}۞9`\/ɵKq;Z676?vE["c>I[ϱ'kusccڥ%Ss+qvX{[AZ..;=QXsRz/9-c"=J7{k `){FXR璜i$W^kHrszw|usP\7f/WXŖ)S``9}i?0w_֞OS,[[:[SvS{,:n]J~~Nfw8kcϧH|s|aͧ5˞>>u沶t ^-+ߕǥ׮:Gpg$$}jT=ϗ%\{+PZ#Sᯭ V3g/b×>.9YKs%׾TL; s{ywGnsXK>ۻxzvsD⬥s#>S赘7m7ױ:Oxhcq֚v@I68C;Gn{=NxmI8?W #C9n["}~<;mCdl++W=hsSZ'{ǰ2Hz̧Yf>e#ڑcH;"cuz~10o 5'y5/-靓%} 9='O Ў엖{U{73ҳ-ce#u`-Wb^ޗ{GOբcԵ0?9S+%9LK&vyu{j5(\wnj)|%9zWWay3j91.4I:C;&5֨ì_şrH 蜽Is"Wt;6w{zZ]q1p㮃zz>%qֱҸ鳦%뚳ĵ'.ê{>-<ؚ1X'ghZcWVB^#ldIΚ5:鱼X\*Hy\qG,H~'j9TޖkGr{ 8R8Sqf `8xAiOmЏ |5d*6ƊxX=ykwbc 67Rx;SU]*1l?z[6_n$=J_w"ĵ{po3{jք{8Fr`LYXlۖJ,w{mKt}@V u}~,l K/KX5c^cЇܺ[#H;>Vؗ6u \ F)rQ;uc{[c,q kO\yח.u]]q"z!ۏKēܻ?$%cX*W+rN+%\S |Fr^+p) \+?̴?7ꏇDf翸{o_?7~y%_>!BJBB!| !-BB[!| !-·BB[!o!-·BB[!o!·BB!| !-BB[!| !-·BB[!o!-Тztsss{{;(So!3=>>{֛7o^|i>ڷ]Y±-[:Ș-gB zꕵ.NAD Bhi-ǻ;psscm!\fk-_>ˎ7q/!.v,1L"2O3=z*й{Z֜KMgxB\Ɠ|z3x2G-ЙXNyO).2}/ߑ2Δscŷ:"y_Y.w2}hօo!mĞDKˏJ&.| M,tW~?Mr4-ߊ^X ;-.E~P4ĥ{eHB*gBŤw[!%?fƌ[ygJ_~o!t>X8g qid7n\3'xd^Ro!tbZ U|)FLq؅BǑ1LN?tffJ`S'tqnάM`]"MB\B'8\~D2eɾlCڰ* Ga[Fu٢7c)J ]Nn5Tķڙl-h;@֢ƿb-m\hϿcrRXhBbc.]).|k2-QQG͖KO !'Mi·v`/Tg]u} 2]7yķ*?C ԭTXWiyU]w~E|kbbrg݇2֎5_䈯tѳq[hV Z6Qq"R"ąa3Uw}9Ug w[67)qv.k01wB%:$|ܲ9hl^NiIz:vko)E'Kt|+a>|OrV2]"ݴg/܇lWBk[r2\~ax;Ug\~*ϷeTW0Q!|HݟdsC\-Rߗ2oU옷~V&k۫œv{Ul :8ч+Fؗ/əq\'G/׋ͯ`VZd#|egڄLIݐ"ąoV2I2zڂ̹Yz[%hc[9!ez8lOE`|j!Ηmm̷ g/1}xY ͌|nΗou} S/>6'֑20xʥĺ\2C\~޹_%:o8nj:6Ok[*n#Ě,wf uq^غb-]b| .ydO%_+g.+ݚ94[f?U!a}+c1ŮdWPޒg'GGXSc#m3qŲ PP:2S7y}+'e;lŽ6WE! (־c>%%n/w~xx\T'_ɏ9ȃE濱/w.{dtesc[8qbQћenMkSyc 1lsVVm3~ '&&®Zbz-2j6ibrr TUy"I /=cwh CnƮClu?5cEG*bY^J+FY^!;06~LwʾLll@t (㜴p^F| ^ڧw_ 1 XϑN]Fm)8| uLF~teGyc{(*f")>5x,wo?[>\N eblsؤ'N KݩdJ̌NP^fXva`loir >7C!oip| doC+䄸|bE|jBK73V6(-n蠫dm&Eݩߧlĺuy| -dhA_!ٝݝE*VBm76,E){*vLo,ҙ\5-J24I'(vf(ٝ!wPֶYEkmg:YXBpr ~Yek-j[bBd·BB[!o!-·BB[!o!·BB!| !o!BBB!| !-BB[!o!-·BB[!o]P `!Sθ&ą'<p\( x흿%Gg}6l"< GlfK"/\@`bK./؄̙ 6m$$rR7c޾7{Swyf]]]U]}6|gݳaB~ ~>~LJl#x8 }p6|٠a|%#~z?7=~#/,]Q >ZO|L)w[^c:W wy^w|ׯ\\\ O<yfzܸ3{gZr5 |xgz3l\afc~/zŋظ9h AvUuW-I}յ:/l\r,W x;W= N P;A+B\a -۩9y uV}5z瓓zظj7 jUc~skl(j35sK~?k0K}'o,Φl]#?s~H \RWXs/To*7jĜ){sw|q̣_`nޤvUz5&gxJwPs @k5Zұ:D[k-W>}炚ɧ)fdU_ `~f+~_?K-?dzJy ,B4>oƭpGo3W)r3r=k?Ug)_[IͭyG>ޏ-9ldp*I.ccު^YC1|%ߙ+[?Z^)¯Q !-]F cNC Zkh q{%ظ5s3_?8,)U I\NZq>VSO#_R0^p ist6.sl\am\S ͽU;tUc6׼6E㙚sRʓ2zsbz^f气fʁkv7:6nxNujOZ`9}PIۯayX5KDﶵZSb >*}Ք>ѡ{jsj]Ej.mX3z~)#J}jƕ^}n W~q<~*PoD`{W ۤ~. p \@ÈAbT<.F^|yK&ٳpV+g\#º`;V?pa.L116MG^\\k9bb2 777[ c@0Bh qQ b; _Ӛ!.Ml*ra\CjdX lHɰ.Ðx֧F &5dXsL_7bd0xMN+v$\Xj Vb}rЭՐMݓ>W6%B4Л kv~92d츜w> ҪbS |d  ҙ٭ F9r[$/OA4'?2r†ϙ VVIԁ¾c;萙Vpb|9.A_;8Wpհa:\ϩ^,a.uT#,ƅLP @@{.dɷ,cvGH‰D%7´;isnpGy *!f!?a/e&Cj,X=P&VVNyBvkƼtEa!)3&CH2oLə&\sR34ϸhLZ\VL1f6x]©|Ae3RAKӍ̫).kP2&"2Xx#42νp[!u3Эɐ2m\e\*yW߹:an㚞E_iƈ a":# }Jd# .StG&.M{ \@. p 8CR]WWW^XP"-zlN&:U!KXS7R*.X]}EYS'*LYp3av2F½wIɧn-u:𕉱<ҡ/R5UlFG~#!\jWkVE&:G`B1 :^틫UlhdcSVp%SݘBqaE|k}SXr7IT8z zTϺ}щq(.Ijpzݧr9yDP#S#Bd\827 ;HhIh؜rhԐH `8BIvtiJ`/T|\{76%`xFA涖#r}im>Rymi.ynl%tfV!%/:g;{HN&oGv'=pl Y`p{2ix¡Ŝ'e-+\8&fA:B SՐ~y#Js!v&p :l6Ie2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJ2$-4ƬcMeZ2$Sθ&ą'<4xs AAjJ" @83ʚ;ʚ;g4YdYd|L  0ppp@ <4dddd w 0T7L  80___PPT10 ppPhysics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit/4       0 80 l(  lr l S s =K    r l S (w d>   l l 6AjJ?" l l 6AjJ?" C 8 F   lF  t l 6AjJ?" F f t l 6AjJ?" F p t l 6jJ"`F H l 0޽h ? )))̙̙380___PPT10.dXP7{V   0  tV(  tl  t 6AjJ?" 6 G6 R r t S W =K    x t c $pZ V    r t S  %J  0   F F  t t t t 6AjJ?" F f t t 6AjJ?" F p t t 6jJ"`F l  t 6AjJ?"  H t 0޽h ? )))̙̙380___PPT10.dXWN   0 x(  xX x C       x S P)   0     H x 0޽h ? 3380___PPT10.dXWNr h 1( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ]޾!^@.r݅4]ީI a@t'=BI#Fi6%я;B7xzdt@n-CS}w m#kZ=p@tAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAtAt+ҦҾ?Kv莊z{?wtsћO~;yHz{F.Oy4-r얦8%dyMFK"ok?ko?^e\K/=C6d S-`!r.+\rnYf[$I1wE pT1^0皭v_'RT~/r>ݓܪ'^0܇ߖOojk{ߑ$-̓ȓNw7iG/s#z˞wk2N?7}z>6~qzҌQr?HdOjΠhF/lor&:-ͽ pgMqMQnN}9ݖHA?ޥ zbo/c"Gϥ;|Os/ν|ͼWs/^!e4=Ƽ4;To zGQ(FNʏ3aaG7P2|uCO#пg{["[a{Ɲ}w)NR\bZf@[։|U7|g-nqko}\X-`9w]%]4,gn_TYO }vBVѻ&}Ulq48> ?3kq=92?0_6}~x1{')B 3f47rc„oL=Bg. wV*t2r]3p&^Jm,FObh$wfNɝ#={E PtLz9#`h'}(moӗjHq_m0w =̽+Fw_gwh/(:/Pe3n)Jw6S;m)Dld[ wvmTd-w?1o@}{ܼ>IENDB``!h ,9WO_ &lB-<&R6 x흻ZI! 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ .  @n?" dd@  @@`` 0(|           L2$іYZѻX \bH2$"p/ }+_bH2$ .`,k$6R-[2$#FB/@"Xr#xnxb$Փ9e2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJ2$-4ƬcMeZ2$Sθ&ą'<4xb$NͯG2 t1g s AAjJ" @83ʚ;ʚ;g4YdYd|L  0ppp@ <4dddd w 0T7L  80___PPT10 ppPhysics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit/4       0 % t(  tl  t 6AjJ?" 6 G6 R r t S W =K    x t c $pZ V    F F  t t t t 6AjJ?" F f t t 6AjJ?" F p t t 6jJ"`F l  t 6AjJ?"    t rA*magnetic_moment_fluxm#      t 0T s   Drawing of the ring current of a particle in a magnetic field. The ring will conserve the flux which is related to the magnetic moment H t 0޽h ? )))̙̙380___PPT10.dXWNr"  1( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ .  @n?" dd@  @@`` 0(|           L2$іYZѻX \bH2$"p/ }+_bH2$ .`,k$6R-[2$#FB/@"Xr#xnxb$Փ9e2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJ2$-4ƬcMeZ2$Sθ&ą'<4xb$NͯG2 t1g s AAjJ" @83ʚ;ʚ;g4YdYd|L  0ppp@ <4dddd w 0T7L  80___PPT10 ppPhysics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit/4    re#1( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ .  @n?" dd@  @@`` 0(|           L2$іYZѻX \bH2$"p/ }+_bH2$ .`,k$6R-[2$#FB/@"Xr#xnxb$Փ9e2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJ2$-4ƬcMeZ2$Sθ&ą'<4xb$NͯG2 t1g s AAjJ" @83ʚ;ʚ;g4YdYd|L  0ppp@ <4dddd w 0T7L  80___PPT10 ppPhysics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit/4    r#u#>61( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ .  @n?" d>     !"#$%&'()*+,-./0123456789:;<=N?@ABCDEFGHIJKLM}OPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{~d@  @@`` H@ #          !"L2$іYZѻX \bH2$"p/ }+_bH2$ .`,k$6R-[2$#FB/@"Xr#xnxb$Փ9e2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJ2$-4ƬcMeZ2$Sθ&ą'<4xb$NͯG2 t1g s AAjJ" @83ʚ;ʚ;g4YdYd|L  0ppp@ <4dddd w 0T7L  80___PPT10 pp~Physics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. 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One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit  The mirror Theta pinch has end losses But one could use the mirror force to confine particles The mirror has a low B field in the centre and a high field near the coils Particles moving from the centre outward experience a force in the opposite directionZ Mirror configuration From magnetic moment conservation follows the perpendicular energy Energy conservation then dictates that the parallel velocity must decrease Bouncing condition BAssume the particle moving from A to B is reflected in the point B!$The first key problem of the mirror Only part of the particles are confined (Collisional scattering in the loss region will lead to a rapid loss of the particles from the device)) [# Second key problem of the mirrorThe rapid loss of particles makes that the distribution of particles in velocity space is far from the Maxwell of thermodynamic equilibrium The  population inversion can drive all kinds of kinetic instabilities%Tokamak rBend the theta pinch into a donut shape No end losses because the field lines go around and close on themselves 'Toroidal curvature  PThe magnetic field follows form And therefore varies with major radius R as )Toroidal curvature  _The toroidal magnetic field has a gradient Which leads to a drift in the vertical direction T+Toroidal curvature The drift Leads to charge separation Build up of an electric field (calculate through the balance with polarization) And then to an ExB velocity - Toroidal curvature has its priceThe ExB velocity Is directed outward and will move the plasma on the wall in a short timescale This effect is no surprise since /Remedy : a plasma current 123The toroidal electric field /       "$&(*,.0456$  0 $(  r  S '=K  ' r  S I'V0 ' H  0޽h ? )))̙̙380___PPT10.X$  0 $(  r  S +=K  + r  S +V0 + H  0޽h ? )))̙̙380___PPT10.X  0 0(  r  S +=K  + r  S p+V  '   pA4(tokamak_transformerm   '  0+ r b2The plasma is the second winding of a transformer.33H  0޽h ? )))̙̙380___PPT10.X   0 (  X  C     '  S H)'  0  '  H  0޽h ? 3380___PPT10.X  0  (  X  C     '  S  '  0  '  H  0޽h ? 3380___PPT10.X  0 @ (   X   C     '   S O'  0  '  H   0޽h ? 3380___PPT10.X r$1`ACFIKMgO61n&( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ .  @n?" dd@     , !"#$%&'()*-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~`bKGDH cmPPJCmp0712HsIDATx^_SY*C4b SF (%%¡PT¤JyՖ 8m Br^{=~vF Y*.g9B0q#;gD?$wR ɟ_@Č;K 17ΧRUc[bK;[c )_[$jkg݇ITs!HԤ]xT&vɧZ*O9钝TdǭTj&/3|#۟&-줢J * AI!4_rK,^OHk8UqLBR\oSZZ*)8J@s$h _;"T8̪z;Bv%rInH@9( =y(qJ_!D$Fˡzn\k&Xvx A^a:?_FH٫ԡ~޲?%$w֟)/$w@Pr37?Ȓ$֞;DZ3(ϲ!Hwޜ<7^g%7 osԊDY3kig⿐@xm1m5 C$_H !s~*%;ӐL@> ~0tB]lP"4S~؟F;`xJvkU߰7(||ub~UЊzĪhq/q`{ ;' ܨJ>-o/.f~Dp}rɏpΣ_/c96"opDf#S\ Yz%8Ӄߞq2:wyw @;)`q2~ >t91Ͳȇz53h.пjt:.+K@¬ " }~{L $޳,3/ᢗsɩXͿ|`Ӝ ]5,mtm5 @$y,S)Kgsn^&bܘ?˪ov/({NixwHl|bsS{_%5T͎yIK>bp-=7DJFx\zPq/&XE~L$"4Krt, LX{G>oߏ$*@\Boob軬llan.0W쒚{&־pgߋ| Hd|io݄d2̠uMk¹_t@D.Nx<B DnB̗ե-Z߿?Tq>}|H="˘D CsL!X+\)R*]}Fl?fz=Jgpŷ_IAɽNdnl{{e!Si<]!P`qu߂% %s6DݭLF3LוA4?gi ǗϊYqydw5cPD+ z|& Mrh\;5Ygd"lc[Gx]\f\f` 21A|(HƐw& /;Vfs*:廠IA+>d4nܠΌ"=c0=&ۘ{9"47ϲ44:$82ʭҳ9"$ı贳K®8 Q'.'=C %/& ?4Di3]Dnt]8ki\ԿD=A wߛ9<a_( =.ܭ||B.I,?lq=XZӛ67F|]1}\1ٮ<^)\zZ.JRjoI8&J!A|WO.`!en.h0x~~>ٰՂѤ1&242vąԱxձձ&=5Z|+3-:ٳk6(pA|󴡽7b¢yaݫ< MW*eI5vl|p\[\_yY)-$_KB0+V/ jAÏ[;4q;麲F]@]7`5  5l>_xy̙oϼB&Z78ƍׯשWz{3 "\eeL,|kgv~hH?*KKn*=AqbqބeVd(e 'wPp+$/`Im[h4faA*4XL~ƈd2lki&R0L8+dk+OBeiͭ7gx>t>OC⻕(я%oxk^w71w+ç+3|@]RY_60s*KLЃ 2X,&DDY[&/_1/JUH7`mҿc\,e&eLx:/g X:;ԁ$uuZ*ZZ#]Vy}H'}M좔nq,?0Ϝ1n(b,MTSk5 ץbϡɤ)Ϧ[;@9Tf}eew4Aq@̥ L |.kywf$]}2"7ޕ/mfe;fx8P'6'R[͞ur X"&b@lt0`"x0=鎮[@XpBet;&6t32FZHYt 6j(HyАf2 ,+ꚊOJdɹ23>^\3S\6>ܡRo_v:hDIJ";+3Cq| 0CC5؞/)//?Wr$p210FC<QGB II3mAjDz#4P-x߂/8gr8Us Ԋ\aHb(7se:^:txb-ߖUN4}.` ~r[Dm Qz$BeHVd3R>$㓘jWu'ܚT{E>6^+|ס*cFx> :aQ[tV" *(!`K<\!;銧|̡I - 2;>UGIIb:rA%b6 qG:m} 6AB +IAocb OtJ_,ݔ&cua0A\P8D褉&xWS$ xpy/L@`\ P"OO YOCPHԈabJ0yʗ%ˮ[J x h?fg 9nxb WE.gQ^I;*Et$ "g5lgEV6ġ3_pàqGr [͕;nFbc\&e=\z];HS{C8-xg-OPiYhnm0ϞBXAޙqb`g&1 [B+ 30TW%c1뺳NOONBW y}薵KB@}@5jqx)oڸإp獣U FXLؑQ~K;2zC/)//*0Mm-$>󠳦sN26'<ߊE#mڂ`!i"HkC**Gn #~~^x-|BHGxlD= /end1VLcg ^̷fi@rWn. v1 R72|JP*6;/}DLd3NB? 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Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit  The mirror Theta pinch has end losses But one could use the mirror force to confine particles The mirror has a low B field in the centre and a high field near the coils Particles moving from the centre outward experience a force in the opposite directionZ Mirror configuration From magnetic moment conservation follows the perpendicular energy Energy conservation then dictates that the parallel velocity must decrease Bouncing condition BAssume the particle moving from A to B is reflected in the point B!$The first key problem of the mirror Only part of the particles are confined (Collisional scattering in the loss region will lead to a rapid loss of the particles from the device)) [# Second key problem of the mirrorThe rapid loss of particles makes that the distribution of particles in velocity space is far from the Maxwell of thermodynamic equilibrium The  population inversion can drive all kinds of kinetic instabilities%Tokamak rBend the theta pinch into a donut shape No end losses because the field lines go around and close on themselves 'Toroidal curvature  PThe magnetic field follows form And therefore varies with major radius R as )Toroidal curvature  _The toroidal magnetic field has a gradient Which leads to a drift in the vertical direction T+Toroidal curvature The drift Leads to charge separation Build up of an electric field (calculate through the balance with polarization) And then to an ExB velocity - Toroidal curvature has its priceThe ExB velocity Is directed outward and will move the plasma on the wall in a short timescale This effect is no surprise since /Remedy : a plasma current 1Plasma current 23The toroidal electric field /       "$&(*,.0456  0 (  r  S L+=K  ' r  S <+V  +   dA5tokamak_fieldm 22  '  0x+ # -4  W'The field lines wind around helically .((H  0޽h ? )))̙̙380___PPT10.XrHP1v$PUz61A,( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ . 8  ʿΔιƔܯkkڵ{sرֽ层߹ASESX^R\ﶩغȲνƜΜɲ¶̲ǽƽި筜ٳ䵙ֽֽ޽ƜƥΔΜΥƠƜТټػҸ+1)1@Zccghs|w,0-R;@v긖/S 4$m&-c;lg i! 6ep{λΚH`Νxߒ$-[dI{~φ߬כoW/+7eU^_Y|UrJArv/z+Pb6LFߪ,Fl8ZZGY&zL<_.Wcr8&iVWjf9h1#ö \]mY]mQ K?̇=EZMf}WW;Tsd80 X5ivZiÑJEx|/8 HmF>b ƊJ `y}'Hʛ@d;W).E$eګ,A5himגH&Gl䞃{uҩpӲPG?Kgi.0beU$W$z>ga'D7 P׊zb7WWnyUw>;?1NB#!#ɲܱ-޶ᎽHД}ٻc϶5!1<>t(ӟe 4?d?u.{eeEEeœ|'Ǭ?jDlM+P(˷JMJ3rY 6`y}د*Xpe\64g_ȡlc{!ۘj?f\&mʡ٤UE7|-ap~CDewuwT2"wYP$,df.\続LUVXǻ-Vqb# #EqYvo9њHxmY͐4Bd!# ሽ1 =CwMo3h%4b2{ PYV x\٠WkX ˰ aJ;0̶>ɯ3LHNoxQi]Stm׭NNWxOVoSs4~ M(0D!>1k$@$GX$2aI[$ΐ"(qiB.4IIbB٬vV/˦ej?/r!-{ fWE^i0dn>yv;vݾmbvKsLc ;<RҤɘ'CPqw*4nϴ 4KJNRw Cd){lV59BZ3t&*E?L %[_}$lvlH%daA)qar[bef[DBN`ȝKW#p? ?^@c x2L7x͑бꂧEV9`Xɴ-L[$-?'m쎀8.w,琼UFBrzi{СB*#xi%\!ki9@27(UTdڳ k$isK"SrShJJ^﬉W/i;" 86]Kɍ}Cp0n v9>O#6{2l}I> q=J MBb *d ;ۘ0z݉ IXM!Kܨ'c({]aRd*Dū!)!MxC-n@P@oĖnW~ʆ)tG&D<ƃJv{5EAƵ̀ңPhHr|ӿ_V`xx7ZddU _p!9Dxr̃~Ͻ[h/h)FkF$%)l{f7 -9ђҶgdߣ"F'B(W#?Of܅R꒵x&sC>4MDdp7Y5dOf.C nt޺b 6 `+;qp*B,l.wٛ~Q\ "C;oKssSKK"r`GM7,"I RNqNZP:ϷmtصvL3۔-l`[pU- NrF,|RH!+M!ג=[$%Iv|NBS76=>7)O8FB2٨-N&E?XT 뒃Иz 6-Phpp0`3LfbOن9qv'ӸSM=9 lypL46d6" se̓zX~WcPzIcJ3Hd6l0*]+ːu?Ss|y0T2}POPʯ?᳅ VP[[,4~6q:ﹸmđ!ujY٠@JM1Śwõ~:/y39BUrwQߨ)u{nbp)WE%7{(H Hat; in8"M 󌏏ݏPV*X-ٌ=Eo$&TUbd@P&C~L~($Q6Uh*''3muպ jdq2|aEzRy20BK-uAǀ CAbЬa=bFc^tUjp@#9o7i"ޚ)IT/ *,=YFOw^{n{Kf6<^f3٪X{?ݭfŁFe]%CiyPĒƉ @@U`.09rdi;`A{kU3Y9Ҝqc3!Ǜ;w?wok[~ ={_u֟KmѬ8(|8՚Ǡi(XlI`rhnCQ=,bH# Zԗ(C n$ m'G xjU `!I~*j 3H9m0e-ub~W s)Z[̡T2`hh0b2pZX ( iԅMq`'7MX*U'܃U{@5BU8t0v[[ⵌmI~+[c =ОbKVl7̲\^s\ei,Y7$/{Hue-p%ل=04'rnRw핑=ɐD998rٹ{!?Ͳ&,/ 3JEM3DҡxWBc!^ħp2d'ÏnΣ&btEWO=jf rN}ӘL>)>izF2FD!HBvA 2Xm _HD~ URxdžFw9`0TaM^5 C, y*;!YIHUv*R?myUlXa;Wx-!fZE3#'5 Z GMF1@!Y8:Xm)_74܅PkQd$Bء(VW*GU#)26*E|5iU2i BTpq06P!2|~ %^kWrhErٙʼntwLQ KidÛ3TӞHPF @-|s$^5 FWne?o.7bWQ70:e/ k#9~AX\lͥS1ѳR6ػ4U{ZI( >妑ODd1ŋ'͑ tv҆)P L1myﺫXoo@ki/awk[z=H=/m @|Vn! 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Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit  The mirror Theta pinch has end losses But one could use the mirror force to confine particles The mirror has a low B field in the centre and a high field near the coils Particles moving from the centre outward experience a force in the opposite directionZ Mirror configuration From magnetic moment conservation follows the perpendicular energy Energy conservation then dictates that the parallel velocity must decrease Bouncing condition BAssume the particle moving from A to B is reflected in the point B!$The first key problem of the mirror Only part of the particles are confined (Collisional scattering in the loss region will lead to a rapid loss of the particles from the device)) [# Second key problem of the mirrorThe rapid loss of particles makes that the distribution of particles in velocity space is far from the Maxwell of thermodynamic equilibrium The  population inversion can drive all kinds of kinetic instabilities%Tokamak rBend the theta pinch into a donut shape No end losses because the field lines go around and close on themselves 'Toroidal curvature  PThe magnetic field follows form And therefore varies with major radius R as )Toroidal curvature  _The toroidal magnetic field has a gradient Which leads to a drift in the vertical direction T+Toroidal curvature The drift Leads to charge separation Build up of an electric field (calculate through the balance with polarization) And then to an ExB velocity - Toroidal curvature has its priceThe ExB velocity Is directed outward and will move the plasma on the wall in a short timescale This effect is no surprise since /Remedy : a plasma current %A toroidal current in the plasma will generate a poloidal field Top and bottom are connected by the magnetic field line A vertical electric field would have a component along the field and leads to acceleration of the ions / electrons Drift will be balanced by a return flow along the field&Z&,'1Plasma current Because of the plasma current the field lines wind around hellically Resistivity is small but finite Note that the toroidal electric field that drives the current can not be electro-static >:  %D7Electric field induced An iron core on the axis of symmetry is added The magnetic flux through the iron core is increase This generates a toroidal electric field y3The toroidal electric field 29/       "$&(*,.04568:  0 *"(  r  S '=K  ' r  S d                           ! 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An iron core is added through which the magnetic flux is increased\\El   6A<jJ?" p h l   6A>jJ?" c  H  0޽h ? )))̙̙380___PPT10.X7`  0 F>(  r  S x =K  ' r  S  V0      jA6"tok_and_stell_gb   'H  0޽h ? )))̙̙380___PPT10.X  0 D<$(  $r $ S PO+=K  + r $ S  V     $ hA7 stellarator_w7x"   +H $ 0޽h ? )))̙̙380___PPT10.Xuq  0 (  X  C     '  S `#'  0  '  H  0޽h ? 3380___PPT10.X7`  0 ((  (X ( C     ' ( S v   0  '  H ( 0޽h ? 3380___PPT10.Xqr0z/ڦ1 7@̶mzн:1-( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ .  @n?" dd@  @@``  x4N69          !"$ %' () *, -/ 02 35 67 8: ;=>? @ABCDG HJK 2$іYZѻX \bH2$"p/ }+_bH2$ .`,k$6R-[2$#FB/@"Xr#xnxb$Փ9e2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJ2$-4ƬcMeZ2$Sθ&ą'<4xb$NͯG2 t1g 2$,9WO_ &lB-p b$#UoK1GUO&2$M"|H543rAW{4ub$bJ,/5[b$hb7B\( 2$0$=9!R2$s$p{Yyw`!Sn2$,b$$%أ7jC b$XV'0(b^# b$:k-j;BI%Y b$RkO=l,xZP~ 2$go#D 3z{ 0 2$W}"P+ b$Q#1&qEgW"i 2$FJ܋t $b$h!*2:rS?4 s AAjJ" @83ʚ;ʚ;g4YdYd|L  0ppp@ <4dddd w 0T7L  80___PPT10 pp?  %JPhysics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit  The mirror Theta pinch has end losses But one could use the mirror force to confine particles The mirror has a low B field in the centre and a high field near the coils Particles moving from the centre outward experience a force in the opposite directionZ Mirror configuration From magnetic moment conservation follows the perpendicular energy Energy conservation then dictates that the parallel velocity must decrease Bouncing condition BAssume the particle moving from A to B is reflected in the point B!$The first key problem of the mirror Only part of the particles are confined (Collisional scattering in the loss region will lead to a rapid loss of the particles from the device)) [# Second key problem of the mirrorThe rapid loss of particles makes that the distribution of particles in velocity space is far from the Maxwell of thermodynamic equilibrium The  population inversion can drive all kinds of kinetic instabilities%Tokamak rBend the theta pinch into a donut shape No end losses because the field lines go around and close on themselves 'Toroidal curvature  PThe magnetic field follows form And therefore varies with major radius R as )Toroidal curvature  _The toroidal magnetic field has a gradient Which leads to a drift in the vertical direction T+Toroidal curvature The drift Leads to charge separation Build up of an electric field (calculate through the balance with polarization) And then to an ExB velocity - Toroidal curvature has its priceThe ExB velocity Is directed outward and will move the plasma on the wall in a short timescale This effect is no surprise since /Remedy : a plasma current %A toroidal current in the plasma will generate a poloidal field Top and bottom are connected by the magnetic field line A vertical electric field would have a component along the field and leads to acceleration of the ions / electrons Drift will be balanced by a return flow along the field&Z&,'1Plasma current Because of the plasma current the field lines wind around hellically Resistivity is small but finite Note that the toroidal electric field that drives the current can not be electro-static >:  %D7Electric field induced An iron core on the axis of symmetry is added The magnetic flux through the iron core is increase This generates a toroidal electric field y3The toroidal electric field HPlasma is the second winding of a transformer Flux in the iron core cannot be increased forever. The tokamak is necessarily a pulsed machine That is not good for energy production Also thermal stresses are associated with the pulsed character One can either: live with it / drive current another way / use a different concept IZIe29/       "$&(*,.04568:  0 0(  r  S +=K  + r  S p+V  '   pA4(tokamak_transformerm   '  0+ r b2The plasma is the second winding of a transformer.33H  0޽h ? )))̙̙380___PPT10.X r,3:1-( / 0DArialngsRomanTT7L ܖ 0ܖ"DTimes New RomanTT7L ܖ 0ܖ DWingdingsRomanTT7L ܖ 0ܖ@ .  @n?" dd@  @@``  x4N69          !"$ %' () *, -/ 02 35 67 8: ;=>? @ABCDG HJK 2$іYZѻX \bH2$"p/ }+_bH2$ .`,k$6R-[2$#FB/@"Xr#xnxb$Փ9e2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJ2$-4ƬcMeZ2$Sθ&ą'<4xb$NͯG2 t1g 2$,9WO_ &lB-p b$#UoK1GUO&2$M"|H543rAW{4ub$bJ,/5[b$hb7B\( 2$0$=9!R2$s$p{Yyw`!Sn2$,b$$%أ7jC b$XV'0(b^# b$:k-j;BI%Y b$RkO=l,xZP~ 2$go#D 3z{ 0 2$W}"P+ b$Q#1&qEgW"i 2$FJ܋t $b$h!*2:rS?4 s AAjJ" @83ʚ;ʚ;g4YdYd|L  0ppp@ <4dddd w 0T7L  80___PPT10 pp?  %JPhysics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit  The mirror Theta pinch has end losses But one could use the mirror force to confine particles The mirror has a low B field in the centre and a high field near the coils Particles moving from the centre outward experience a force in the opposite directionZ Mirror configuration From magnetic moment conservation follows the perpendicular energy Energy conservation then dictates that the parallel velocity must decrease Bouncing condition BAssume the particle moving from A to B is reflected in the point B!$The first key problem of the mirror Only part of the particles are confined (Collisional scattering in the loss region will lead to a rapid loss of the particles from the device)) [# Second key problem of the mirrorThe rapid loss of particles makes that the distribution of particles in velocity space is far from the Maxwell of thermodynamic equilibrium The  population inversion can drive all kinds of kinetic instabilities%Tokamak rBend the theta pinch into a donut shape No end losses because the field lines go around and close on themselves 'Toroidal curvature  PThe magnetic field follows form And therefore varies with major radius R as )Toroidal curvature  _The toroidal magnetic field has a gradient Which leads to a drift in the vertical direction T+Toroidal curvature The drift Leads to charge separation Build up of an electric field (calculate through the balance with polarization) And then to an ExB velocity - Toroidal curvature has its priceThe ExB velocity Is directed outward and will move the plasma on the wall in a short timescale This effect is no surprise since /Remedy : a plasma current %A toroidal current in the plasma will generate a poloidal field Top and bottom are connected by the magnetic field line A vertical electric field would have a component along the field and leads to acceleration of the ions / electrons Drift will be balanced by a return flow along the field&Z&,'1Plasma current Because of the plasma current the field lines wind around hellically Resistivity is small but finite Note that the toroidal electric field that drives the current can not be electro-static >:  %D7Electric field induced An iron core on the axis of symmetry is added The magnetic flux through the iron core is increase This generates a toroidal electric field y3The toroidal electric field HPlasma is the second winding of a transformer Flux in the iron core cannot be increased forever. The tokamak is necessarily a pulsed machine That is not good for energy production Also thermal stresses are associated with the pulsed character One can either: live with it / drive current another way / use a different concept IZIe29/       "$&(*,.04568:r:10|,,SKq('  U '/=J !֗n_` +&\"'ZA;x7 }#no6M(n WBh]Bo&eD.: NaqaLap 7OnakdX :FOٕWBlarA !0ar2T m\V#i![ 0wj 9B0ܗ-BaxÀrBX !L0C !0agvDhnaYBaxOK 07BC[B0<5c+#42B30BuB!< !7 !̃0LB} 7 !̅0LB! G8naaa>f2Bh#p0%Hœd40+I"!$!a ! #n 3Sd! ´  #1 #L/!0!+A!aja!Z3A!aZa!;A!Fw:R”B!![c C! OK8,!C!Ba$qO &a\5aa!1BS"B!d!&FOB!3B"B!e!&F00pqe2Ba0˜B!t0B#T L8 !,aD !300BJ a!t Z0u_!A# '8H(ڲ;\խ1^3lIaP ue9B&l_\wd۹B'8IX\w[Py^Wzg'v?њYVӡ: <i/v5{m׻j]ې’ڞ]#5]չ~n%kwʰ}) .L OPN(vvZSmծzweܿ'aZ[o/]v,%P̽:6~+DzoP)%mUn/{6$n%U}hpeoP?i&|:+nW5ɣϚn̓zl.}hLy|/5roNGm9J88oNfo Զ#:ټ==&W‘Y Gs c{/pt5!4:#ڪ[u݉kgbO$ DžL)T5+ um^ӝ#،=aw!RZ;5Ku*5uݽ[SVe=BFo+PioYj;Bd֑mgIq4:"z6xPDh1sy0BG@r8(rōpDB[C!$g86b]rB!)ag88SoX>~`ӄ~~Va˄>`ɓ0 o?UBi9p 9%^)ᘡO&-Re:\ 9OI(ĥ!,G ܫˋP;y(/C1'pP '@8Jӄs\-$lBٍ jf'Z B#Ce!4!أϑ]ui p9pGlovv1ӻ8/k\ 5{cBgG!rppB/G!] B)B!!0 E9IENDB`F4!(1'JFIFHH XI/( / 0DArialngsRomanTTPKܖ 0ܖDTimes New RomanTTPKܖ 0ܖ DWingdingsRomanTTPKܖ 0ܖ@ .  @n?" dd@  @@`` @Q>;          !"$ %' () *, -/ 02 35 67 8: ;=>? @ABCDG HJKLM 2$іYZѻX \bH2$"p/ }+_bH2$ .`,k$6R-[2$#FB/@"Xr#xnxb$Փ9e2$kHu.#7Gb$/hcҎdQb$>7wP!U!b$cc Q<'+&2$ف̬,+2$iP(^v{z{5(b$_/%QyO\Cb$yo*JNӥe\ 72$'eN02ֻ)H2$b|ሲ B4|0r2$>nȡ zA-2$Y;+\T h2.2$[d.\lE,*-2$վtLp>Et =2$խ Q=^yJ2$-4ƬcMeZ2$Sθ&ą'<4xb$NͯG2 t1g 2$,9WO_ &lB-p b$#UoK1GUO&2$M"|H543rAW{4ub$bJ,/5[b$hb7B\( 2$0$=9!R2$s$p{Yyw`!Sn2$,b$$%أ7jC b$XV'0(b^# b$:k-j;BI%Y b$RkO=l,xZP~ 2$go#D 3z{ 0 2$W}"P+ b$Q#1&qEgW"i 2$FJ܋t b$oQ)f8m' b$h!*2:rS?4 s AAjJ" @83ʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0T,K 80___PPT10 pp?  %Physics of fusion power:Lecture 6: Conserved quantities / Mirror device / tokamak 2 Reminder 5Perpendicular forces lead to drifts of the particles  #Dynamics perpendicular to the fieldrForce in the x-direction Balanced by polarization drift Generating an electric field, and consequent ExB drifti NOTE & & $Does not apply to a uniform plasma.  Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit  The mirror Theta pinch has end losses But one could use the mirror force to confine particles The mirror has a low B field in the centre and a high field near the coils Particles moving from the centre outward experience a force in the opposite directionZ Mirror configuration From magnetic moment conservation follows the perpendicular energy Energy conservation then dictates that the parallel velocity must decrease Bouncing condition BAssume the particle moving from A to B is reflected in the point B!$The first key problem of the mirror Only part of the particles are confined (Collisional scattering in the loss region will lead to a rapid loss of the particles from the device)) [# Second key problem of the mirrorThe rapid loss of particles makes that the distribution of particles in velocity space is far from the Maxwell of thermodynamic equilibrium The  population inversion can drive all kinds of kinetic instabilities%Tokamak rBend the theta pinch into a donut shape No end losses because the field lines go around and close on themselves 'Toroidal curvature  PThe magnetic field follows form And therefore varies with major radius R as )Toroidal curvature  _The toroidal magnetic field has a gradient Which leads to a drift in the vertical direction T+Toroidal curvature The drift Leads to charge separation Build up of an electric field (calculate through the balance with polarization) And then to an ExB velocity - Toroidal curvature has its priceThe ExB velocity Is directed outward and will move the plasma on the wall in a short timescale This effect is no surprise since /Remedy : a plasma current %A toroidal current in the plasma will generate a poloidal field Top and bottom are connected by the magnetic field line A vertical electric field would have a component along the field and leads to acceleration of the ions / electrons Drift will be balanced by a return flow along the field&Z&,'1Plasma current Because of the plasma current the field lines wind around hellically Resistivity is small but finite Note that the toroidal electric field that drives the current can not be electro-static >:  %D7Electric field induced An iron core on the axis of symmetry is added The magnetic flux through the iron core is increase This generates a toroidal electric field y3The toroidal electric field HPlasma is the second winding of a transformer Flux in the iron core cannot be increased forever. 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Conserved quantities {In the absence of an electric field Perpendicular energy is conserved And consequently the total energy is conserved  More tricky & .. uConsider a changing magnetic field. An electric field is generated Integrating over the area of the Larmor orbit f  AccelerationDerive a second equation for the integral of the electric field from Solve through the inner product with the velocity Integrate towards time  Acceleration Integrate in time Note the integration has the opposite orientation compared with the one from Maxwell equation. One is minus the other  Magnetic moment is conservedAThe equation The magnetic moment is a conserved quantity  Flux conservation NThe magnetic moment is conserved Calculate the flux through the gyro-orbit  The mirror Theta pinch has end losses But one could use the mirror force to confine particles The mirror has a low B field in the centre and a high field near the coils Particles moving from the centre outward experience a force in the opposite directionZ Mirror configuration From magnetic moment conservation follows the perpendicular energy Energy conservation then dictates that the parallel velocity must decrease Bouncing condition BAssume the particle moving from A to B is reflected in the point B!$The first key problem of the mirror Only part of the particles are confined (Collisional scattering in the loss region will lead to a rapid loss of the particles from the device)) [# Second key problem of the mirrorThe rapid loss of particles makes that the distribution of particles in velocity space is far from the Maxwell of thermodynamic equilibrium The  population inversion can drive all kinds of kinetic instabilities%Tokamak rBend the theta pinch into a donut shape No end losses because the field lines go around and close on themselves 'Toroidal curvature  PThe magnetic field follows form And therefore varies with major radius R as )Toroidal curvature  _The toroidal magnetic field has a gradient Which leads to a drift in the vertical direction T+Toroidal curvature The drift Leads to charge separation Build up of an electric field (calculate through the balance with polarization) And then to an ExB velocity - Toroidal curvature has its priceThe ExB velocity Is directed outward and will move the plasma on the wall in a short timescale This effect is no surprise since /Remedy : a plasma current %A toroidal current in the plasma will generate a poloidal field Top and bottom are connected by the magnetic field line A vertical electric field would have a component along the field and leads to acceleration of the ions / electrons Drift will be balanced by a return flow along the field&Z&,'1Plasma current Because of the plasma current the field lines wind around hellically Resistivity is small but finite Note that the toroidal electric field that drives the current can not be electro-static >:  %D7Electric field induced An iron core on the axis of symmetry is added The magnetic flux through the iron core is increase This generates a toroidal electric field y3The toroidal electric field HPlasma is the second winding of a transformer Flux in the iron core cannot be increased forever. The tokamak is necessarily a pulsed machine That is not good for energy production Also thermal stresses are associated with the pulsed character One can either: live with it / drive current another way / use a different concept IZIe;&Do we really need the plasma current? It might at first appear obvious that the answer is yes since without current inside the plasma But a positive as well as negative poloidal field does not necessarily mean that the field line on average does not go around poloidally Z,S =Toroidal symmetry At some point the poloidal field must be zero In the case of toroidal symmetry this field line closes upon itself Regions of positive and negative field are not connected A field line can not wind around poloidally Then top and bottom can not be connected >$ ,2The stellarator  0If the field is not toroidally symmetric the motion in the toroidal direction will move the field line from regions of positive poloidal field into regions of negative field Then a net poloidal turn of the field line can be achieved Steady state operation is possible at the cost of greater complexity 1Z1P =2o> Stellarator  KInside the device it looks something like this Picture from LHD in JAPAN A!Hellical coils can be simplified tThe picture shows how the combination of helical coils and toroidal field coils can be changed to use modular coils ;29Applied in W7XModular coils of W7x There is a large disadvantage in the use of the modular coils. 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