22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 8: Effect of a Vertical Field on Tokamak Equilibrium
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1 .65, MHD Thory of usion Systms Prof. ridrg Lctur 8: Effct of Vrticl ild on Tokmk Equilirium Toroidl orc lnc y Mns of Vrticl ild. Lt us riw why th rticl fild is imortnt. 3. or ry short tims, th cuum chmr cts lik rfctly conducting shll: t msc. 4. On longr tim scl, th filds diffus through th shll nd rticl fild is rquird for quilirium. 5. Anlytic rocdur: Assum n xtrnl rticl fild slowly ntrts highly conducting shll. Th shll thn coms surconducting. W tk th limit s th shll mos to infinity:. 6. Th limit is nontriil. To gin, w l th shll in lc. Influnc of th Vrticl ild. cuss shift of th lsm surfc with rsct to, th cntr of th shll.. Th lid rticl fild is gin y = V Z.65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg of 9
2 3. Assum V scls with th shift Δ. V is clrly comonnt of th oloidl fild. Considr, 4. Writ V in trms of ψ, th cuum rticl fild flux function. = z = V r sin + cos ψ ψ r r r. ψ = r cos 5. Th nw oundry condition including th rticl fild is gin y Sol th Grd Shfrno Eqution Using th Tokmk Exnsion to Account for. Zro ordr: sm s for: rdil rssur lnc. irst ordr: sm qution s for: ψ =... not cos DEPENDENCE ut with nw oundry condition: ( ) ( r, ) ( r ) ( r ) ( ) 3. Lt ψ = ψ + ψ cos ψ, = cos ψ =.65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg of 9
3 4. Th solution for ψ is found s follows:. d d ψ r = r r d dr dr Β dr. old hom ψ = ψ + ψ old ψ stisfis th qution with.c. ψ old ( ) = hom hom ψ stisfis th homognous qution with.c. ψ ( ) = 5. Homognous solution. ψ ' c = r. c c r ψ = + r dr c. Choos c = for rgulrity ψ = c r = r ( ) hom d. ( ) ( ) 6. Th full solution cn writtn s dx d ψ ( r) = y y dy + r dy ( ) x r x 7. Th nw Shfrno shift is gin y ( ).. ( ) ( ) ( ) ( ) old hom ' ' ' ψ ψ ψ Δ = = ψ ψ ψ ( ) ( ) Δ = Δold ( ) ( ) ( ) =Δold ( ).65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg 3 of 9
4 8. Thus Δ li ln V = β + + ( ) 9.. How much rticl fild do w nd to k th lsm cntrd?. St Δ =, ( ) = ( I π) c. I li = β + + ln 4π Th Limit. How much fild is rquird for Δ = if th shll is not rsnt?. Imgin th shll rcding furthr nd furthr wy so tht 3. Tk this limit in th xrssion for ln?.65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg 4 of 9
5 4. Wht is th difficulty?. Physiclly <. Also w ssumd. Hr is siml roximtion: ln( ) ln( ) Elctricl Enginring Drition of th ln / Limit. ln / rrsnts th forc du to th chng in mgntic nrgy twn th lsm nd th wll s th lsm shifts outwrd y n mount d =Δ. It is th nlog of th l i trm xct lid to th xtrnl fild 3. Extrnl fild chngs:.65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg 5 of 9
6 4. Th forc = (otntil nrgy) = (mgntic nrgy) s th lsm is dislcd y n mount d. Not tht sinc th lsm is rfct conductor, th flux linking th lsm rmins fixd during th dislcmnt. d r d = = d L I d d xtrnl inductnc. Constnt flux imlis LI = ψ = const. c. d I LI = = d L dl d 5. If th lsm is surroundd y shll L = ln I = ln 6. or lsm without shll (homwork rolm) L 8 = ln I 8 = ln 7. Thrfor, th ror limit is 8 ln ln = ln Sustitut into th formul I l 3 8 = + + ln 4π i β 9. This is widly usd formul in th dsign of circulr tokmks Summry of Ohmiclly Htd Tokmk Equiliri. low β : β, β t. q : rquird for stility.65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg 6 of 9
7 3. rdil rssur lnc: oloidl fild (Z inch) 4. toroidl forc lnc: rticl fild 5. toroidl fild: ndd only for stility to k q 6. Ordring β t β q δ Intuiti orm of Toroidl orc lnc. Multily th qution y π I. Thn I li 3 8 β n π I = l T = π I = IL forc du to th rticl fild cting on th currnt I 4. T 8 I I dl = ln = d forc du to th chng in th xtrnl fild 5. T 3 I li I Li 4π I L = = = i 4 π. ut π = r = LI i d rdr L i 4π = r dr Ι indndnt of.65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg 7 of 9
8 . Thus Li nd Li dli = so tht d i I I l dli = d c. T 3 = I dl i d 6. T = = r dr I I 6π 4 β I I = 8π r dr 4. cll th gnrl rssur lnc rltion I π rdr = + π 8π r dr lt ( r ) = + ( ψ ) + δ ( r) I 4 δ = 4π r dr+ 4π r dr. δ T4 = 8π rdr 4π rdr 4π rdr δ = 4π r dr 7. Summry I d δ π I = ( L + Li) + 4π rdr d rticl fild forc hoo forc tir tu forc / forc 8. Proof of th tir tu nd / forc. tt : = dr tt.65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg 8 of 9
9 = ( ) dr intgrts to zro = rd dr d = 4π rdr. J = = = ( ) = = + δ = δ c. = ( J ) dr = δ d r δ = δ dr 3 δ δ = dr 3 3 4π δ rdr rd d dr.65, MHD Thory of usion Systms Lctur 8 Prof. ridrg Pg 9 of 9
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