22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 9: The High Beta Tokamak
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1 .65, MHD Theory of Fusion Sysems Prof. Freidberg Lecure 9: The High e Tokmk Summry of he Properies of n Ohmic Tokmk. Advnges:. good euilibrium (smll shif) b. good sbiliy ( ) c. good confinemen ( τ nr ) d. good ohmic heing ( T T kev ). Disdvnges: e i. low β p β : β, β = = % b. exernl heing is reuired: joule heing is no deue. Exernl heing is expensive, bu lso rises β c. igh spec rio is reuired o rise β : his is echnologiclly difficul d. pulsed operion is reuired unless curren drive works efficienly 3. The high β okmk resolves he problem of low β. I llows okmk o opere higher β 5 %. This leds o more economic devices b. How do we chieve higher β? We pply ddiionl uxiliry heing, keeping fixed. This rises p relive o High β Tokmk Expnsion μ.. We gin ssume lrge spec rio: R R. Sbiliy is produced by lrge oroidl field: 3. Thus, s in he ohmic okmk r R, implying h.65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge of
2 4. Rdil pressure blnce, however is produced by he oroidl field = + δ dimgneic ' ' p + + ( r ) = μ μ r δ + p + μ μ r ' ' ( r ) = neglec, confines only β 5. To improve β over h chievble in he ohmic okmk we need p δ μ μ 6. Or in erms of β δ β 7. How lrge cn β nd δ ge? 8. The limiing condiion is deermined by oroidl force blnce which is sill ccomplished by combinion of I nd v 9. Incresing β increses he oroidl shif. The lrges possible β occurs when he shif becomes of order uniy Δ. Recll h in n ohmic okmk Δ R..65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge of
3 . Le us esime he shif using he smll shif relion ' ' b dr r r ' μ dp = y y dy r dy μp neglec since μ p. Therefore Δ R R μp μp β β ' p. For, hen Δ when β 3. This suggess he following ordering for he high β okmk δ β,, Δ Comprison of Expnsions Ohmic Tokmk High e Tokmk β μ p δ (pr) δ β μ p p β μ p (di) βp μ p Expnsion of Grd-Shfrnov Euion. Since Δ, oroidl force blnce nd rdil pressure blnce ener ogeher in zeroh order.. Good news: we need only he zeroh order euions. No firs order correcions re reuired. 3. d news: The zero oher euions re sill nonliner pril differenil euions..65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge 3 of
4 4. Expnsion:. = r, +..., rr b. p = p +..., μ p c. F R = F ( ) F = R p μ + new free funcion d. This uomiclly produces pinch pressure blnce p + μ cons 5. Subsiue he expnsion ino he Grd-Shfrnov euion dp d F = μ ( R + r cos) + cos sin = T R r r T cos = sin R r r R 3 neglec dp d F T = μ ( R + r cos ) d F dp + μrp μrr cos +... = d dp = R μp + + μ p μrr cos = R d ( ) R dp μrr cos μp.65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge 4 of
5 6. Therefore, o leding order he Grd-Shfrnov euion reduces o d = dp ( ) R μrr cos 7. Noe h μ RJ, so h on circulr plsm flux surfce μrj = R d verge over We see h d flux surfce. is proporionl o he verge oroidl curren wihin given 8. Even hough he euion is simpler, i is sill nonliner PDE. 9. In generl, i mus be solved numericlly.. The difficuly rises becuse he shifs re finie nd cnno be reed perurbively.. We shll deermine generl feures of high β okmk by exmining specil cse. Specil Cse. Choose dp μr = C C = cons d R = A A= cons. This implies p C (ssume (boundry)=) nd J A.65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge 5 of
6 3. Soluion: wih hese choices he Grd-Shfrnov euion becomes r + = A + Cr cos r r r r 4. oundry condiions: We ssume circulr plsm of rdius r = (, ) = (normlizion of flux funcion is rbirry) ( r, ) regulr for r The circulr ssumpion is mde for simpliciy nd cn be generlized o oher cross secions. 5. Soluion: (We need only cos n erms becuse of up-down symmery.) 3 r r pr = A + C cos 4 8 n n k4 cos hom 3 n n r n= = k + k ln r + k r cos + + r + b r cos n no regulr no regulr no regulr = krcos + k 3 only erms which re regulr nd reuired o blnce on he boundry r = pr For ll n, i follows h = n k ( ) 6. Choose k nd o mke 3 A C 3 ( r ) ( r ) ( r r), =, = + cos Then C = = Ar 3r cos R r R + 4 R R 3 C C C μ p = = A r + r r cos 8.65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge 6 of
7 Physicl Inerpreion Le us express A nd C in erms of more physicl uniies: β, I or euivlenly β, * I. * is prmeer reled o kink sbiliy nd he surfce MHD sfey fcor * A p μ RI π = μ RI. I = p dl = (, ) d μ ON A CIRCLE π C A πa = A d = R + cos π = R R 3. μr πa A = = * π R μ 4. μ μ β = p = pr dr d π C + μ C π 3 = r dr d A r r r cos 8 R π μ 4 π 4 5. Thus = AC = 8R 4R A = * C * C 4R = * β.65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge 7 of
8 6. Generl euilibrium relion for high β circulr okmk (minor degression). β p = pr dr d μ I 8 π 8π π = β μ I μ π = μi β * β, = R b. The generl euilibrium relion is given by β = βp * β* 7. Subsiue A nd C bck ino he soluions. Define ν = = βp, ρ = r * 3 = ρ + ν ρ ρ cos ν = + 3 cos ρ ρ * r μ p ( ρ ) ν = sin * = β ρ + νρ cos μ RJ = + ν cos * ( ρ ) cos - cos = ρ β ρ + νρ.65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge 8 of
9 Properies of High e Tokmk Euilibri. Skeched below re ypicl midplne profiles (Z=) showing rdil pressure blnce.. Shown here re p nd J profiles long he midplne for differen ν. Incresing ν implies higher β..65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge 9 of
10 . Observe he incresed shif of he mgneic xis s β increses. b. Observe he buildup of curren on he ouside of he orus o produce oroidl force blnce higher β. c. Observe J reversing on he inside of he orus when ν >. 3. The flux surfces re round, bu re no circles excep for he boundry 3 ρ + ν ρ ρ cos =cons 4. Le us clcule he mgneic xis shif Δ by finding he vlue of r where p r r. A = =. y symmery his occurs when =( orπ)..65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge of
11 3 ρ + ν ρ ρ b. Se Δ ρ =, = = ρ c. This yields Δ + ν 3 = Δ d. The vlue of Δ is given by Δ ν = ( ν ) + +3 Δ e. For he HT ν Δ ν Ohmic Tν 5. Find he shpe of he flux surfces ner he mgneic xis.. Le x = ρ cos y = ρ sin b. Then x + y + ν x Δ c. Expnd x = + δ x y = δy δx, δy nd define x =Δ d. Subsiue x + x δ x + δ x + δy + νx + νδ x = cons = x + ν x + δ x x + x + x ν ν ν or + δ y + νx + δ x + νx + ν x definiion of x.65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge of
12 ( νx )( x) ( νx )( y) + 3 δ + + δ = cons e. This is he euion of n ellipse elonged flux surfces, sushed ner he ouside f. The elongion κ is defined by b + 3ν x + ν x κ = = ( 3ν ) = + ( ν ) ν ( ν ) κ for ν nd κ = 3 for ν =..65, MHD Theory of Fusion Sysems Lecure 9 Prof. Freidberg Pge of
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