22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 18

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1 .65, MHD Theoy of Fuion Sytem Pof. Feidbeg Lectue 8. Deive δw fo geneal cew pinch. Deive Suydam citeion Scew Pinch Equilibia μ p + + ( ) = μ J = μ J= Stability ( ) m k ξ=ξ e ι +ι ξ=ξ e +ξ e +ξ e =ξ +ξ b Note: b = e + e eη = b e = e e e,e,b othogonal unit vecto η. Cay out calculation in tem of ξ, ηξ, ξ, ξ, ξ ξ = ξ + ξ ξ=ξ +ξ b η=ξ ξ ξ = ξ e + η e η ξ=ξ.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page of

2 . Check Incompeibility ξ ξ a. ξ= ξ + = ξ + b. cala = + cala ιm = + ιk cala Define m m F = + k = k, k = e +ke cala = ι F cala c. To make ξ= to minimie δ W, we mut chooe ξ o that ξ ξ +ι F = o ξ = ι ξ F d. If k and m ae uch that F fo < < a, then ξ i bounded and we can chooe ξ=. Thi i the uual ituation fo extenal mode e. Suppoe k and m ae choen o that F= at iolated intenal point < < a. Uual cae fo intenal mode. ξ ha the fom At, ξ i not bounded (not allowable), but only at one point.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page of

3 ιf f. Reolution: Chooe ξ = ξ F +σ ξ i now bounded, but ξ i no longe eo. g. Calculate contibution to δ WF ιfξ ιf ιf σ ξ = ξ + = ξ + ξ = ξ F +σ F +σ Aume now that σ i vey mall, but finite Main contibution to γp ξ come fom aound = whee F = h. Expand about : F = F + F F x, x = σ δ W = p d p dd γ ξ = γ ξ d 4 ( F +σ ) σ =πl p γ ξ dx σ + F x 4 p γ ξ =πl σ F i. Fo mall but finite σδ, W Concluion: Even fo iolated ingula uface, the plama compeibility tem make no contibution to δw.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page 3 of

4 Minimie Remainde of δw. Sepaate tem Q = ξ = Q e + Q η e Q =ιfξ η Qη =ιfη+ξ. κ= b b = e 3. ξ ιgη m ξ + ξ κ= ξ+ G = k = e k 4. ( p)( ) * p ξ ξ κ = ξ J = J = ξ * ( * Q = Qη Qηξ * ) Subtitute { line bending δ WF = d F ξ + ι Fη + ξ ξ ιgη + ξ + mag. comp. + p ξ peue diven * * F J ι ξη ξ η ξ kink.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page 4 of

5 Simplify. Note that η appea only algebaically. Complete the quae and minimie with epect to η i η= G ( ξ ) + k ξ k k m = + k. Remaining δ W a WF ( R) d δ = π π a b c( ξ + ξξ + ) ξ () a. integate () by pat b. lot of algeba, uing equilibium elation 3. Reult: d F g a π R μ k δw a F kz m = ξ + ξ + ξ a F f = k k g = p k + F + k m μ k 4 k k k F Complete Calculation by Computing δw, δ Wv. Aume no uface cuent: δ W =. Vacuum Enegy: δ W v = d = = μ 3. Wite = φ with φ =.C. a. Wall a b φ = n = = () b b.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page 5 of

6 b. n = n = n ( ξ ) = ι F ξ( a ) a+ξ a a φ a () Solution: m k φ = c Im k + ckm k e ι +ι φ = kc I + kc K e ιm+ιk m m Chooe c and c o that () and () ae atified Then * * Wv d d d Subtitute * δ = = φ φ = φ φ φ φ μ μ μ * π Ra * φ = ds φ n φ = φ μ μ a δw v ΛF = ξ π R μ m a ( a) ( b a) ( b a) mk K I I K Λ= kak K I I K a a b a b a + a b ab m m kb m ka ka kb Summay kb ka δ W fo geneal cew pinch a δw k m ΛF = f g d F ξ + ξ + + ξ m a π R μ k intenal mode: ξ ( a) = extenal mode: ξ( a) ( a ).65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page 6 of

7 Suydam Citeion a. Neceay condition fo tability b. Tet againt localied intechange (extenal mode) c. Only neceay, becaue a pecial localied tial function i ued Mathematical Motivation a. Chooe k uch that F( ) = fo ame in < < a b. Aound thi point k F,g p < detabiliing k c. A localied mode can till give a finite contibution if ξ i lage. fξ lage mall Phyical Motivation a. intechange plama and field: plama want to expand, field line want to contact b. intechange i moe difficult with hea. A intechange take place, field line ae bent fom one uface to anothe. Deivation. look a δw F in the vicinity of x =. aume intenal mode o that ξ ( a) = 3. aume localied intenal mode + = F F F x F x.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page 7 of

8 Then f 3 F k + m x k p g k μ + m and = ξ π R μ k + m 3 δwf F dξ dx x D dx D k p μ = F 4. Simplify D a 5. Wite q = m =, k + = definition R Then F( ) m m = k + = k + k kr q but, at kr R = = = eonant condition m q o that F( ) ( ) q = k q ( ) q F = k + k q q q q q = k q.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page 8 of

9 D μpq q = only a function of equilibium quantitie (no m and k ) 6. δw dx x D ξ ξ a. if p >, D < tability < b. aume p inteeting cae, D > tability? ξ ξ+δξ ξ 7. Vay to detemine minimiing d F g ξ + ξ Fξ gξ = dx x D ξ ξ x ξ + Dξ = 8. We can olve Eule Lagange equation: aume pp ( + ) + D = p = + 4D, 9. Need to ditinguih two cae: D > 4, D < 4 P ξ= x. Note: x D dx x px + p ξ + ξ = ξξ = p > bounded altenate function p < unbounded not allowable p = logaithmic divegence not bounded. Conide 4 D < ξ= c in k ln x + x k = 4D c co ( k ln x ).65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page 9 of

10 . Show ocillatoy oot alway lead to intability by chooing a modified tial function a. In I and V, ξ=ξ = δ W =δ W = I V ξ b. In II and IV atifie x ξ + D ξ = = x D dx dx x D ξ + ξ ξ = ξ + ξ + x ξξ δ W.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page of

11 II δ W = x ξξ = IV 4 δ W = x ξξ = 3 c. Region III ξ=ξ = cont, ξ = δ W = x D ξ ξ d = D ξ Δ Δ = III 3 d. by aumption D > 4 δ W = D ξ Δ < intability e. when D < 4 no ocillatoy olution exit. One oot i not allowable, the othe i allowable Concluion: when D < 4 not localied, ocillatoy tial function can be choen. Sytem i table to localied intechange when D > 4 localied teat function exit which make δ W < D < Suydam citeion 4 q + 8μ p > q fo tability Detabiliing tem: 8μ peue gadient, bad cuvatue p.65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page of

12 Stabiliing tem: q q hea, line bending magnetic enegy Ocillation theoem If uydam citeion i violated, thee i alway a eo mode, macocopic mode with maximum gowth ate. Thi i ignificance of Suydam citeion..65, MHD Theoy of Fuion Sytem Lectue 8 Pof. Feidbeg Page of