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Cl l 6A PPjJ?" % tQ1 l l 6APPjJ?" f G^ YH l 0޽h ? )))̙̙380___PPT10.a1 r6 :1( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` 8Oc    12 CD LMhb$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQows APPAjJ" PP@8wʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0TԢK <4BdBd x 0T80___PPT10 pp?  %]Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/v"AZZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  /  .8<$  0 4$(  4r 4 S =K   r 4 S nV0  H 4 0޽h ? )))̙̙380___PPT10.b!  0 8(  8X 8 C      8 S   0    H 8 0޽h ? 3380___PPT10.b!rN; 2^*r<1f( / 0    _ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\v`abcdefghijklmnopqrstuvwxyz{|}~|ߔ\H|rs_#2I+p*dLzF̝,?WsMXWߋJl97F+1=NYk}Ǝ1{1W,ϻw^Wnc>#V@%?Z;iAgm`B^r}:s(3He,90`dW#k' `k3 S}uJlo+ߜ H8 -p>]u=9/QS7}>5 <&9C>=մ<s]Siކga;{"6`xyM)b3gɵ0>M9c}(Wc60O0e>W恇} +)r)hX0c:#יS5<6{F̘0onݯuq8H?G6 ?|$mHB- h\W~1_8|: %SؘОj+H;B` JqNޏ!vLm } l5y?GDŘYrCc>hi8Wʡ_b$PRu+v^߅{~L>;;L ı{wE.?ן֗׆_m?_W\>ڶ){ޒb.V;xzlےXu! 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G!l8B!l8B6KJƿ"Bnt@ydb%ePNG  IHDRhRsRGB pHYs0PLTE999SSSzzzhhh-mIDAThQh[UOҦ]t*R9Yi }h5O>^IpAR5)A8Ka&`Dt=%UL&iro|&!4G8r9'7/\*HH#q$΂P8$/pD¹U%q$đ8Gtg N\\:iI#|( NyxkkC+ypXC F6A䳄)pd⥏5Db*2Rs8qjahdmu1X,z_=,D:bA'hM4(qtBQ> 7cJ6ڷ7+^&:ʒ:BR 3F\e2۞d滥nɰ0x9]i:"юc|7qnP5UuunFq7q@j>u7Q ]s#C.lDBR^ְN)"M)CAI֨P9k+28kA\D,rgE`>_7\ybƩ ٺ̹*~:K9WA1sT$yM%jzD5o]g|^+AUZfXJVmWMc$]H+|MLk ߹>"*5o!ŗ{ף|8BTlb4֛۳;=GuO nyptTuHhܶje [i\|8[&*r*يS3G pTC*^)ܰ ڳ+-8\iOY*w\Uٲe8PP~g 7_CJx2󴇳Bq}́ߡg&kuelK=t%i=vٵMTa{N!,zJx= Ec|>! xhEH#q]~~@oIENDB``!.Qx茸Hb#UTT #xA x=%G{61r+俰#dZ[$aED6Jd&$ΆdI,&CZΙ(wWUWWߏ+3wǝzO:Tݳ]ww?z|vu|gܻ;77O#eo_׻wdgzAw^>ӇK!B!B!B!𤙮0Z-!B!B!B!nV>B!B!B!BjQ+!B!B!Bx}}=y;??u9ǣGs*={v_+c=榸tϟHs~=.u3tӧO[ 3jÇw6FuRkݦ1vۡP}6רֿuCw1?uoEnYx7 ym'Fc>1=c獵7-k1}5*=W?m5 =%cgv& K~%{ !ꪤ1{S;o >~o||4D[Bs}s{|Onj:qz%5s+eryjG.WVkwJmxsF[gyѩL\}O|X>cLNrEKL4s˜]R]c݉/eV$x@,pxqXJo7K0{uc{H svx,T}?vo1@\3vA_+Y^=zEY?|.!GTjh}!\v\zk j~X,ȵNhMzE}{<"n>w$?=cx栵ƞo }!\~߲7룲=۽NuR6̰r#R3+/O;z>Zᰨkŵ}sm1eע֮ TMZ?T_FW㍭[PU+jˏLgsp%ϿutۑR V?˹|_b=QZ o5}Ń\gW9661]k}Kn]jְrpNkX-m{ܟ)$je斸=s֧4ߦ_Fԋ#7,qދ?GN)Tmujnc6-scx,zND?6i)Zpc;25m8x<94:kaNJK|xksn|1;K;s98R?۞>JﻪVuԎ&N7Զ3㙲޼᎙_9Ζ湶?%֒>g\\O}_:Wp|>ZamU Ƅyƒ8˒b%}wMNaש[κݦӹ౷d|=;8759ֱ粵߉纫{ /K~!\~vnZ^r?1aXU+~I c]g*yݳ^K})m& ӧ;{ncѥs?bB8VBxh?ﷇB!BOV>B!B!B!Bo[kN@AB!B!B!p)|k{gސo{[}ڽ]7tg݃@ޙw %`@[-m@[mh @[-m@[mh @[-V=}z{{KCC[U4=A^ȟQv& "+Z㕟Z.9Ƅ;Uŋ%-&b;nLUviorZypf mF-w;um/*zᩤK޳A%,gIv[ҴeɇHZvء*`ék:-IЖt oa3XdpFںN-rXZ̃]k ߺqX,6ǘGž n>%lSMҖ=}B Ğ܈|BQ+14ʁ}!#k~"aȝju{F$+E!|h|wA[6h Z<0udnޮp^B l\ ⟽guR5a 4[+&DOv%pmܳ, e{ex&;;cJDF-J05a0I[YX9FnUJv B#%옜j~9z;4<)E[iSǴ!NX8h8A|VW@C.bKO; f+sI{ZO{ u ~BF$LvrooneÄujkΐ+ GSAףkkkMJGl6i+N෵ dP6Z-׎xY Cb0o MlJvbұ9YK5E͎kZZw!kɤd޼roG;Nh0 }ܻmV'+EO6Ydxote|) Yr;஺-%G4f`ڌрLA,#"^q#g8(4*;`V(`uBn<G9l-W[ZtUbPlV4̊v#DRwlm҂,a ^o!a0Hfh+ j2a0P?V=*OynC\’BCSDgYv+n ~ ^eJجJ)6nroWCU_i //wl+uj0B[9ola^2*'zT޴hL񹂼Ci+kX&PtS&1l7Qq:Sk4sjf1RXcaafe:L"Ӵ[ZOY^yHd#}dXXY47u~x*fĞU[+dnufX+h0rDbgGL2roS#a6iVԬZSiGq4Krf l1bx,S@IܖJş+AR!tY3E4ĿM[B-݅zs0~F4$ɞipX-E/: 'K"A)=E[`= /A/ڒVŐڷg&`a}qh$ebV{2B[`c.zìm@[-mh -h @[-m.mݻ[% }7`!*-sdUyGphZ j!7R,x{ו' $Hj@ $x$r ܍=.C۔^clf6bzܭ靘i{C:V}yyN}yDArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` @Qn    12 CD L MNO/b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J$b$t@ydb%e2$.Qx茸Hb#UTs#s APPAjJ" PP@8wʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0TԢK <4BdBd x 0T80___PPT10 pp?  % Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/v"AZZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember /  .8<>  0 4(  4r 4 S =K   r 4 S nV0  l 4 6APPjJ?" "0 l 4 6APPjJ?" S t A  4 6PPjJ" F t+4 #At constant electric field the heating power goes up with temperature, but also the current is increased. It is the current that is limited by the kink stability limit, and at constant current the heating power decreases with temperature  2l 4 6APPjJ?" S pZ l 4 6APPjJ?"`t KH 4 0޽h ? )))̙̙380___PPT10.b!$  0 <$(  <r < S =K   r < S lV0  H < 0޽h ? )))̙̙380___PPT10.b  0 @(  @X @ C      @ S   0    H @ 0޽h ? 3380___PPT10.b(r; = >1( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` @Qo    12 CD L MNO/b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J$b$t@ydb%e2$.Qx茸Hb#UTs#s APPAjJ" PP@8wʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0TԢK <4BdBd x 0T80___PPT10 pp?  %k Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/v"AZZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF/  .8<>  0 <(  <r < S =K   r < S lV0  l < 6APPjJ?"` e H < 0޽h ? )))̙̙380___PPT10.brC=>1( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` @Qo    12 CD L MNO/b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J$b$t@ydb%e2$.Qx茸Hb#UTs#s APPAjJ" PP@8wʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0TԢK <4BdBd x 0T80___PPT10 pp?  %k Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/v"AZZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF/  .8<>r>1v( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` LTv    12 CD L MNOPQ/b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J$b$t@ydb%e2$.Qx茸Hb#UTs#s APPAjJ" PP@8wʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0TԢK <4BdBd x 0T80___PPT10 pp?  % Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/v"AZZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today /  .8<>@  0 D(  Dr D S =K   r D S RV   r D S , 0  H D 0޽h ? )))̙̙380___PPT10.b  0 H(  HX H C      H S 3  0    H H 0޽h ? 3380___PPT10.br? m!'#@1T( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` LTv    12 CD L MNOPQ/b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J$b$t@ydb%e2$.Qx茸Hb#UTs#s APPAjJ" PP@8wʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0TԢK <4BdBd x 0T80___PPT10 pp?  % Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/v"AZZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today /  .8<>##<@1T( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` LTv    12 CD L MNOPQ/b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J$b$t@ydb%e2$.Qx茸Hb#UTs#s APPAjJ" PP@8wʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0TԢK <4BdBd x 0T80___PPT10 pp?  % Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/v"AZZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. 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 %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today ASaw-tooth instability  /4  .8<>@B  0    P(  Pr P S =K    P  APPjJ ?"     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Pw/krϝ*y_JN~Y[]$K-( Woh/fD'ɡ*)?,+Sˌ7%Cn=wP/+tk-2=)/Aۑ^$\YJlI.stLIX-}բjiGu,Gi;:x,^ cagVX4c3y9L]n@\?6#3~ZSxL fCH*T7[Mni50GPOgZYB7 S%coW3v*uGmA^kNkJκ>=T GK)kͼ(K$IF&ѫJٮjD+/#JVSǥ>tW53zſh(s/^k#ڗReY*cԑz-rX;VyeZY^{gJR'aX5Sf:J m]?Ǐ m06K[ 5B|K?$($Ɉe᨝4MjwyP*I|'{#ob$ɟF^^%WzSy8)LdyMU%S=r$|%1{+fx3ߘĦk妏bðrD MlnVJ[_I|$O@U"0M֤2=ՠ-^D%PڌOn=LæSKr qKp*HbiXgr$iONiT|4$y $ d|rHi'I>ٙcsȫEǁCc|LnbYI3-}=;;*d7.//I|rNNNI|I|I''|I|I'|O)]+}:F~DžSIAJFIFC     C c !1"2AQa#7BUVWq346RrstvӁ  / 0DArialngsRomanTTKܖ 0ܖDTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` 0(h[    12 CD L MNOPQR SVWOpb$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8Xs APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %dPhysics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today ASaw-tooth instability  CA real discharge again /P  .8<>@BD  0  ``(  `r ` S y=K    `  APPjJ ?"  H ` 0޽h ? )))̙̙380___PPT10.ccPR  0 pd(  dX d C      d S ŷ  0    H d 0޽h ? 3380___PPT10.ccPRrC ^͓rD1( / 0DArialngsRomanTTKܖ 0ܖDTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` 0(h[    12 CD L MNOPQR SVWOpb$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8Xs APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %dPhysics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today ASaw-tooth instability  CA real discharge again /P  .8<>@BDruD1( / 0DArialngsRomanTTKܖ 0ܖDTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` 0(h[    12 CD L MNOPQR SVWOpb$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8Xs APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %dPhysics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. 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The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile Temperature profile is peaked Therefore conductivity profile is peaked Since the electric field is roughly constant The current profile will be peaked This means that the safety factor profile will not be constant  ASaw-tooth instability  /l  .8<>@BDF    0 t(  tr t S t·=K   r t S շV   r t S  0  H t 0޽h ? )))̙̙380___PPT10.tcSd   0 x(  xX x C      x S X  0    H x 0޽h ? 3380___PPT10.tcSdrE a$' )F1( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` ||`    12 CD L MNOPQR SVW[\_b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,s APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile Temperature profile is peaked Therefore conductivity profile is peaked Since the electric field is roughly constant The current profile will be peaked This means that the safety factor profile will not be constant A&Saw-tooth instability / Internal kink  /l  .8<>@BDF   0    P(  Pr P S =K    P  APPjJ ?"     P  APPjJ ?" / 6  H P 0޽h ? )))̙̙380___PPT10.bc]˿r[)AE7)IF1( wÈ̙Ԫݻ""33DDUUwwffCX_IDATx [H mֲՂX_^33ϮL&s=GԜں݃N-%G*Nͩ95ԜSkRmǩ ` >ڒd Ƈmܥd5p :}Uk %Ѐ)WIc GC޻ ]!*7VrOzðsYe=Gmi=ٸO Tn2Fh#O^LCw D"|g=E7whQjM.#95ڼj@h(*(H)+!YT&nC_/L) 䔗`8^ce+}?R\pjNͩxΔ=ߙ0)}d]`anX\ 6W5DztwDXhV}$f^ W(U{#`ygPSs*btR{ؠӖNm}SsjڜSsjNͩ95ԜSsjNbݖna*yq)Iq WMm'u8l QӉ_庳~j~j~nZZ]h'$!Itnѝ8m4yڷu_'|E7P9/*M l-ZZr$$ {y?jO)ʱ\?#y ?}E"=<^db؀~ ^m'8OeK٩ol V&7PbeܜūyH6 H`%TiHM\S"oj'D^oxf#qLtѨPISmi%]_/q;eB!'(S\CYpW/~Dۅ6"<$FCLie_ ⿁|?=Dur6] +đt^Ay&PÔd,yO>j5ާ;N#jh4z]]5Qe5v >Pi Xµ%ԞhN99;Y\ͳ>W^JSC {SjU^ljJsmn!-Id/Zd]@932^i[P/hʴܰ䵬h6埶۰љ5Ջe'(堊Ȱ386'"9<3泩ef^rLدQ;lnVr@W3aSa!oȹ5T 9Tj!6lʙytm2l*e Tpwu(y#hMU jjhI;bf%>Ģx|~}5Wb&ٜZ5Բ9j-c˥hN5ǖ]ͱRse4cˮr2SͱeWsl\ͩز̖SͱeWsl\ͩr|wcˣhNUbvWRlsvW[2z ; cn˧*l]0^I}kj#Z]WYg6@'NmqjFf [A/dfTn5؂@uX9M6oTl5@ܖ;LMSSb(Sfk VH-+@5KMEC<-peyjfijC0 φRC֢fGJ 3s7Z6H6\z_ljyBh($}Q&W VEZ5SؒЌU32532["jF9R* ٜh%k7M٩0[hN,ak-A-\j͠%ЌrͱU)ЖiZlMy,kB-0\ÙC]gk-`SV7[hyơInN-[IsqnuVŹՙE+4qY!7ǭ|/Hy5ݭ[hKj[mjѽ/ߐ]OYoKS:2[Bv>\Q:ْjj fH~;h/۔wi5VZA-*ȯ{>wߪlɝ_Ɂ$X5ݪe8ᷡ(mg$2ʔn-<}BV$Zo /Q~au2%xEd#RV1ۂ2ga,Զg98j肔FjEWZo'Q?S`umavJaOپ`VsmD]4~0K n4@ (!{T<.y jm6P|2'zZԠӃڂ*|X.[峭θV6[wMjd`UIhɖ-@ oB56Q [Z P[ZqTZR5M k NXV[J4|X/ˣ`BW '(k>\/ t>35rŷ樕Fz[Vplλ~9Yjr!7[-=J> ?1{َz"B~LL04#_9p ĬXV-@*^WB/h @M]dVG5x[keG*V Ԟ jڪr&lCP^@F RrPfՠPf.Z=j؂W?0dԲVZV6jSEO-YlƠ5f+?+RUP<m$oZ lUiejirPu z z(e lX-NcK-m(SG3Bm9[mYK ;""؊itRMYOX5 [j"+|Cyleբ  /p5 {Ԓ؊PY`QZȆ5jIl;XB=ydf+> W"<ȆEj1OBYjD͏p CUW Q\q5r\y5?!fЖЌdS4RlZ;`[Sjj 5ְ5t8lakL q4KؚS֔Zp (ZWֈKmek@MjW&}KjV /tXVZP9ffV!dK-yFz.w[Vڲ$U֍VKQhji?c>dYVZcbvUmed]-BjAO#ZVZ6%>jQPZ=e,dЦ 2 FmԂR(-Z%ǒ2UB1R Y,եȆl')̖ͨA4ϖJDg%IPlWi3[j?2DaPFK*PޙS&>E]v6_?7-+,.OB-m"r7xOcn[?ixF'ءĖ/{3o/z~#?7_1֓$bZe{nf@{?mGAϛx;-R[b٢eޡ!GZXms[jξ}W2áS 2_'Ij_u;EMPPtO!wlJkpQv z \ƁdGLxfP.NwB~ !0Z?5yIQWFٿwq%2YWU?Wl~5T FQ=Lq-[)9 6?`02~ajA(}}=WOC)+e^òЩiAx#[تW Fsl0RpnԗΘ4_ FЦIڀqbb Rݏ3!c>J Yk[- 1'iYc8#carc0;6sPG Eꉄ[&ؚ[O0t-6"lN-%TfN-һ5f^ 6fjj5lUSJS@fXM9l~#j5r3jͩYffsj9jE=ʉfJ TB˩95\pjNͩ95S+C a;_j*X/}f E>/'/bf%~vm@7 @~5qȯgVg󪉋$!qƟM U?}?HU wQ-|~g  '?Ok(`uXJ*5K,i8A]1CD:"j<&I v9{zq MC((+ LƶAgR!F.h {dxecTc3ځ<]E} _#RKdPlOj+OvqcVyrm.ھ5tG*MJkR[ )q v2 n ئ85teY5HdܝPJ_}MXwǣB5bT׈WD--o7!/洄BVy".+j]0y( JPq g5Eh%iYnޡ]YkMmU@gF5kt5J@FN=19"uG HC@7Tp&a5+yY\C) qvNͩ95SsjNͩ95SsjNͩ@}@IENDB`n a&ozAPNG  IHDRsD6IgsRGB pHYsh 0PLTE999SSSzzzhhhf kIDATx[mh[>o'JÚ,]~4,mnmg,K;YJ` Ƙ4;P*nVf-ݟ@]0^Hۄ1$0/uR,Y/ 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` |`    12 CD L MNOPQR SVW[\ob$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,b$MP%#;bVVs APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile 7Temperature / conductivity / current profile is peaked A&Saw-tooth instability / Internal kink  /l  .8<>@BDF    0 B:t(  tr t S t·=K   r t S շV    t fAcurrentprofile +  H t 0޽h ? )))̙̙380___PPT10.tcSdrIEe IhF1( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` |`    12 CD L MNOPQR SVW[\ob$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,b$MP%#;bVVs APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile 7Temperature / conductivity / current profile is peaked A&Saw-tooth instability / Internal kink  /l  .8<>@BDF rh hF1( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New R|y9WGv(t{X/@jaR@$|p sÛw7PZȪ1‰Vb1Rϒ/@JQcq  %&2m4/8XG@ae 1af1຿B: @+׋bHMqj>gdzЄ<2iB޺/X8þ!WIdp ؔ]OY5VVQzE4.{bZ30Fi(D*%[OҘc!H~'  HoQ>8f(Yl”n}Cc?gMyfˠ =WݦTv[M0U4ZȉX~*oZ| { 7>9-Fdu;tɲA@o!:cÀ{ qz*~vF'@+a"6N7.@J0rrLm&(,Nس5&Xh#@gH YU2ef] [ЛJޓ A벵N L1s,$P[ԧ!:X/^Tq2E~E_涺}M%[\ hX~δB-ʹb5rz1>#@6Hhx 6~vX#RWVPGX a\ٟ?btr1N_w~ Pk?~ 1&D /B`e-G6 HӎR+NgzNlQY[h~vD:7/q6-}-w= f1z'ǎZm)Ǒ2I#|&ꁼ([ʀ=L:OE,3io*b@ş4"1bjY2/1iH>NṬ>)q\ ⳀTQxM457(p/ r|O溵 =<$osT `9 NLVz2S>Ry n6o-2Z/,;7Ni:fb/4= xpmrkF&s=]4`2'7="$Ds}{t?[lF9Κqa/gqQ[>ǜڲƶ,&13ᝧٲ´YNZ<Б[cѣ-Rs׆$&szO$>Ed1Dbqqay̸V|n /Edptq͊UJ ,,S)Ec) e}ֱA|TU&KzE7{ [{09,>@؋ŰpZ; ӬCՕs@ fl0yWBss $#$(393FӁk\[b}gQw\4R#L>cGbaM#>\]c@&%-f Y|ģ0*V*яsyư=EeI|ژ BiR o`Wi@ ⏱ J%GY|nJ0/@|#Al|N8ҍ ~H}³9eBru50T/c' >T> QXz{R  Erq=F?0r:CTB3t0e"t4Ƶhp0"_fEuH+!KͥVRXݔ2{3Cfr! VkR빕;ZWP |;D rMSԞuP Uji Z'\>6s Աnc-r] @v@.bFIENDB``!T ^!gIza_I68"x흿oEqB QƢ@':!EHAI:BdP WYV҄%{w77@BDF 2   0 t2(  tr t S t·=K   r t S շV    t fAcurrentprofile +  X t 0A?}  H t 0޽h ? )))̙̙380___PPT10.tcSd  0 [S  P(  Pr P S =K    P  APPjJ ?"     P  APPjJ ?" / 6  I  P 6=PPjJ"  m  Temperature and current increase in the centre -> q drops below 1 An internal kink becomes unstable and throws out particles density and current -> q again above 1  2H P 0޽h ? )))̙̙380___PPT10.bc]˿r;AME@F1( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n<?" dd@  @@`` |`    12 CD L MNOPQR SVW[ \b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,b$MP%#;bVVb$a&ozA s APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile Temperature / conductivity / current profile is peaked Safety factor is smaller at smaller radius A critical value of 3 at the edge does lead to the stability of the plasma as a whole but not locally A&Saw-tooth instability / Internal kink  /l  .8<>@BDF r`\F1( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` b    12 CD L MNOPQR SVW[ \^_b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,b$MP%#;bVVb$a&ozA s APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. 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 %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile Temperature / conductivity / current profile is peaked Safety factor is smaller at smaller radius A critical value of 3 at the edge does lead to the stability of the plasma as a whole but not locally A&Saw-tooth instability / Internal kink  G!I"*Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0J#*Transport in a homogeneous magnetic field ++ LTypical values for a reactor The particles satisfy a diffusion equation /  .8<>@BDF H!K"L# " 0 \T (  X  0A?PAr  S ī=K   r  S V   r  S D 0  X  0A?  X   0A?6 Z / f   0APP?"` <XB   0DPPjJ   XB @ 0DPPjJ  H  0޽h ? )))̙̙380___PPT10.c6 # 0 @ 6(  r  S =K   r  S 8V0  f  0APP?"`"Zl  6APPjJ?" }j" hl  6APPjJ?" ^ dl  6APPjJ?" Fq l  6A PPjJ?"  L XB   0DPPjJ  l   6A!PPjJ?"  q% H  0޽h ? )))̙̙380___PPT10.cT " 0  (  X  C       S L/  0    H  0޽h ? 3380___PPT10.c # 0 P(  X  C       S   0    H  0޽h ? 3380___PPT10.cTrI@ @T#hL1!( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@1 ͯ^:}G|4Ix@og^^3=4_޴15Oʿ<>IıyĶ5?W;ϟ?V+S}8yv_M>΍w 0KDM׈W.%_B|z-=2Upyu"4WrM2wLѹ9YlG4i1 n_lH b6XnئkŰ=G?S]b eCkk4P14G)5sĨ1W@%O%ҙZdwO;S'rO@_>.2߬?5ɾ~z:/ֹ78F>1K}Į氇<ީ9Jmߐ#jX SCnFLRĔSnsk5rĦi6(}ULmsKjZ9b4b~a]՚61x\Sq2X;y}~eSw2]wKյZq1&CQs;W@i.oZ}ߙK\FcCOۧϵNʱ>>G25g t}j50Į_q}x_:cE)vXC7rmpnoҏc-ێF~F ֘v; IjTi߾vwN#~ 4vvOr;LՖ9"!r+`1F7&O-s0ƶŭԽJsD' hԈ9b5Ǧk)8 |>{s4&p>JIwXhΉp>1)WOs=M͘'F>[۞Jɚcb祲FG">}cwyϘp z9toyϩMi;ѿ hǯܘuߵsmd^99n$`1vk w0W~0S`w'i11O ,0C`ͷ~zN\'0gp3^p{No[o=4ݧ|qȬ#z)9@a~kFn) >&!߽\.0YܐvQ=%. G"Gٙ}DRU޸|?VQEPeQPt-Ԗip鞏PW +jl{j5❻=#0֯$Po{l o,7r;mlo5+*w WJEA?[e^}k/_JCWHIo֯͞)TR#4둽,ty2ZFy%zIQS]̈́"j;TzhN!ʸ?>੯%)87FggoB[AgKyګ"Ī.1ī $ewPuicu%W=R %j "mZ0_7z:a91m\.QepHz]!jdSeJ,W1) Qk)a?(6P򂵮D%k2Tn__0 &Mj.oۥ/Iѕ"MzUc}%t}+ |oT%*yRS#*#o tg†R6앚%;[ڮum)K-1A*A\nZBmX!m!Jx*%jX=/C4rI1SW8zÒ.rڦNצ.Q x:!Z(Ql.QJx:&+_'Q6[ߙ$QFt.5{IJD\aͺٗ$QziԤ~r_ȫґm@J6S+L K%<;9 >I 6G> ȼ<YR7}5KA!;.k{k~*|?Rg CO`H# >v1$ ^\h%p>8F)VJ!-FW!G!Zddw?mA6# ͞ EƹСVeVe@ E.98ր\e}M^[56ȃ\SdyuG"ݓߚѴ;2i'[, [ VD9܈dcKn5InBt.:@M`Wjڭtr]w1ܜ2`ջfdŤgݐqZ!o֐6 :M9V,# @_U>izbE#L:3w% +I^G{>c1!`OѴAO9$au7's XJ[oo)ʃKo< Sd)П'~Y,ciRl)|c0R :GzGzGzGOWdIENDB``!ΥSMp({,Hx?o$Ň:IYgK`8<800}RJ8aNfh̙)H su;{fv{f~xGrwg}]U]]}_Ѳ|X~(@BDF H!K"L# # 0 @(  r  S =K   r  S 8V0  f      !"#$%&'()*+,-./0123456789:;I=>?@ABCDEFGH\JKLMNOPQRSTUVWXYZ[]^_`abcdefghijklmnopqrstuvwxyz|}~  0APP?"`"Zl  6APPjJ?" }j" hl  6APPjJ?" ^ dl  6APPjJ?"  A l  6A PPjJ?"  L XB   0DPPjJ  l   6A!PPjJ?"  q%    6 PPjJ"  E  e1Rough estimate for r = a gives confinement time T2 22PPl   6APPjJ?"  `(l   6A#PPjJ?"  B XB  0DPPjJf]>f  6 PPjJ" t$ v[ @ For T = 3 s  2 PPH  0޽h ? )))̙̙380___PPT10.cTrJ:#BL1!( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` $  @g#    12 CD L MNOPQR SVW[ \^_abde?b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,b$MP%#;bVVb$a&ozA 2$ ^!gIza_\b$~IWĂ-Syz2$vʒ\xb\A| b$cH%]A_˭po+2$./@}U%.2${=8rIbԅ2x62$[sX7Fa%4 Bub$ mɝYUXtE2$aX`̍,d(L2$0] H{ ]e$b$8a|%ȥAaUbvs APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile Temperature / conductivity / current profile is peaked Safety factor is smaller at smaller radius A critical value of 3 at the edge does lead to the stability of the plasma as a whole but not locally A&Saw-tooth instability / Internal kink  G!I"*Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0J#*Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NN/  .8<>@BDF H!K"L#rB#BdL1q"( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` 4 , Pi%    12 CD L MNOPQR SVW[ \^_abdefg?b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,b$MP%#;bVVb$a&ozA 2$ ^!gIza_\b$~IWĂ-Syz2$vʒ\xb\A| b$cH%]A_˭po+2$./@}U%.2${=8rIbԅ2x62$[sX7Fa%4 Bub$ mɝYUXtE2$aX`̍,d(L2$0] H{ ]e$b$8a|%ȥAaUbvs APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile Temperature / conductivity / current profile is peaked Safety factor is smaller at smaller radius A critical value of 3 at the edge does lead to the stability of the plasma as a whole but not locally A&Saw-tooth instability / Internal kink  G!I"*Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0J#*Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NNM$Particle motion in a tokamak/  .8<>@BDF H!K"L#N$$ $ 0 `$(  r  S xI=K   r  S LV0  H  0޽h ? )))̙̙380___PPT10.cC, $ 0 p(  X  C       S ^  0    H  0޽h ? 3380___PPT10.c,rdM b$dN1m#( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@  @@`` < 4 Xj%    12 CD L MNOPQR SVW[ \^_abdefg?b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,b$MP%#;bVVb$a&ozA 2$ ^!gIza_\b$~IWĂ-Syz2$vʒ\xb\A| b$cH%]A_˭po+2$./@}U%.2${=8rIbԅ2x62$[sX7Fa%4 Bub$ mɝYUXtE2$aX`̍,d(L2$0] H{ ]e$b$8a|%ȥAaUbvs APPAjJ" PP@8wʚ;ʚ;g4YdYdtrK 0ppp@ <4dddd w 0TK <4BdBd x 0T80___PPT10 pp?  %~Physics of fusion power "Lecture 11: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile Temperature / conductivity / current profile is peaked Safety factor is smaller at smaller radius A critical value of 3 at the edge does lead to the stability of the plasma as a whole but not locally A&Saw-tooth instability / Internal kink  G!I"*Transport in a homogeneous magnetic field ++ Particles undergo scattering. The diffusion coefficient Typical step size is the Larmor radius, typical time the collision frequency S0J#*Transport in a homogeneous magnetic field ++ MTypical values for a reactor The particles satisfy a diffusion equation NNM$Particle motion in a tokamakDrift moves the particles of the surface Distance can be estimated from the canonical toroidal angular momentum This quantity is conserved due to toroidal symmetry >W5 /  .8<>@BDF H!K"L#N$ $ 0  `(  r  S xI=K   x  c $LV   r  S N 0  H  0޽h ? )))̙̙380___PPT10.cC,rMW$N1#( / 0DArialngsRomanTTKܖ 0ܖ"DTimes New RomanTTKܖ 0ܖ DWingdingsRomanTTKܖ 0ܖ@ .  @n?" dd@ ?wϏK~gr~OknO5t'+ǯ/~{Z{,޼y!B!B!B!Z+_|%tvIn^~箮x j'VNO!\كhsP777'VNc${a䀳 Vwww9PlŋlLrbwSڰ\aaS(υxTly통Y'1=a}O a ͊tNlڵ̇%cCeak1!ܷ!A)v\c2V%ſr4Al/̷9ĴEnM'\<1-m$f1g_mؠS&߳vƧbCE 5] }zj>X+0EN(gO蜐7带#5+ YkSn)6})c]%"=!C|9+bZ㒘Ѯ͇vmj1-ZK7_Ūq cOfLL ý)k)6ڹ6]2@ekGcjSlƵ)1i +Ǵ񜽷rN=}мl׋S}ofC{t|ؑ}zRﲯ#XXn>Ğmc1"ߗBx3̛IC(SiإP/ 0¸!!|ӛ[S632r&NC_:&C8p}pY6K}Ӱvnh jl#},%1Bxn~p5~jiuIO1k_V*/df\m*#mK/=mzN䈭ވňiqbEY!Ŵ!-ĹgǮu#oۭأhG[%ω|9>$5oľ8ճ8쯸ɲ5)l7촡s!ߚ믳)7/C<Rl{L9׶>>!;-9G B8OLskj 8A7Ɓ6eqcZU.ߔ:8V+O=l8onHY;csnk!oy5]kS5n6 Xj] zƹ&;hYzA8?C/؎üv͡a/p}^V@λaY2\ï: cʼn]ׅPO!B!B!ۭek;#24J |M_x /2'CľBxuo؞-#?K9ֵ?۞(5Z(υ}b%{HeLCzv6ǎ5= \ Frb֮sc#@3f_h_׵ž>aj לHl(̹v_>c!o35ZiJzS"Je Z q>g-bρ-@[ c@k%gtk+_][ǿ5li9K8-5µ3^-_:Bx{#l~+OZB?/_~IIL le 6Z}/iYRK1Em#/#l$E az==tL\3/vnl=G|@L `C{tc_Tc;O|Λ+!1-b|;tyC)ߩCAZ: C7ULb'k(`{aCiĴp }E9~iŁrLLk{Nµ9vb:}IĴL"a>\dgo[*kP+! d:Ko_V'^cjӀo dbmmD9Ĩ2'uwٖ3OX0&apk6}^!ً 6ٖB!B!XVOkN@@B!.6mu|t=w;_67߼j|nj~]>snD ?=?`@@c@c19>[~D`~B`aM.//+R\Nc0;lat  nnn1X*g 1X*:΍N^F}qqW,s?v)xp su=Y\5ӱ3[m?79c'gTn4M BܭtAdVon$^fM!" 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The Ohmic heating power then scales as A reactor can not be heated to the temperatures needed for fusion reactions to occur using only the Ohmic heating since it s efficiency decreases with increasing temperature PUjSF?Additional forms of heating Several types of waves can be absorbed by the plasma (will not be discussed further) One can inject neutral particles with high energies into the plasma This is known as  Neutral beam heating This is the most efficient form of heating applied today CA real discharge again E Current profile Temperature / conductivity / current profile is peaked Safety factor is smaller at smaller radius A critical value of 3 at the edge does lead to the stability of the plasma as a whole but not locally A&Saw-tooth instability / Internal kink  G!I"*Transport in a homogeneous magnetic field ++ Particles undergo scattering. 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L l  6A'PPjJ?"  kQ  6RPPjJ" e j IThe vector potential  2PPXB @ 0DPPjJ@ ^ ,l  6A)PPjJ?" @  l   6A&PPPPjJ?"` -l   6A*PPjJ?" 4 N  XB   0DPPjJ \ XB  @ 0DPPjJ f   6PPjJ"  ]  U!Assuming the velocity is constant" 2"H  0޽h ? )))̙̙380___PPT10.c % 0 (  X  C       S Ā  0    H  0޽h ? 3380___PPT10.crO w%P1Z( / 0DArialngsRomanTTdKܖ 0ܖDTimes New RomanTTdKܖ 0ܖ DWingdings@ .  @n?" dd@  @@`` D<pm    RomanTTdKܖ 0ܖ@ .  @n?" dd@  @@`` D<pm    12 CD L MNOPQR SVW[ \b$YV7DuٽoE/b$MBMbPVN b$i.&:a2$s.;("H 2$Ɩ2,ahvzGk{Yb$2]83fǍ./B2$!@tqvӖH2$r}zFPq2$bw$ǖubKd2$/o`$wU`9`2$+wk2$0BY]ve$,P'2$lX*ژHkQ2$Kr~ADYZQow2$ݩ$#Ny.J2$sdUyGphZ 2-T3b$t@ydb%e2$.Qx茸Hb#UTs#2$aU}m-AYt`2$\M%,ty1 ,8XR$DžSIA,b$MP%#;bVVb$a&ozA s APPAjJ" PP@8wʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0T@K <4BdBd x 0T80___PPT10 pp?  %Physics of fusion power "Lecture 12: Diagnostics / heating +Contents<Heating Ohmic Neutral beam Transport Classical transport 4  = /6  Collisions Previously derived Velocity scaling surprising related to For larger velocity, also a larger change in velocity is necessary to deflect the particle over a given angle The contact time scales as 1/vZ Mean velocity The mean electron velocity can be modelled by the equation This has the solution Such that the velocity decays on a typical time given by 1 / collision frequency Z Ohm s law Adding the electric field acceleration The current follows Ohm s law The conductivity scales as  Heating power The change in energy (of one particle) follows from The total energy per unit of volume is Therefore the energy put in the plasma due to the electric field is ;Ohmic heating  1This energy change Is the Ohmic heating power =Key things to remember Current density follows Ohm s law with the conductivity increasing with temperature Ohmic heating power is the product of current and electric field Since it is the current that is limited in a tokamak reactor (kink) at higher temperatures one must use a smaller electric field. 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