22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 19

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1 . tability of the straight tokamak.65, MHD Theory of Fusion ystems Prof. Freidberg Lecture 9. ressure driven modes (uydams Criterion). internal modes 3. external modes. Tokamak Ordering Bθ ar B μ q or B z z 3. uydams Criterion a q Bz z rb q a μ μ Bz z 8, ιb ( q q). Over most of the lasma the destabilizing term in uydams criterion is much smaller than the stabilizing contribution. uydams criterion satisfies over most of the lasma. Excetion: near r= " r r q " r ( r) q + q q " q ( r ) q r.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page of

2 q z rb 8 q + μ > " Bq z 3 " q r + 8μ r > dominates near r= 3. Resolutions: straight case 4. Resolution: Toroidal Case a. In toroidal case there are imortant modifications to uydams criterion: Mercier criterion. These corrections can eliminate the need for flattening the rofile b. imle, low β circular limit of Mercier criterion q z + rb 8μ q > q toroidal correction c. For q >, ressure term is stabilizing: average curvature is formable. 5. Conclusion: Localized interchange modes are not very imortant in a straight tokamak because β is very small. Near r=, we need either flattening (straight) or q > (toroidal).65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page of

3 Internal Modes in a traight Tokamak Lz = πr m oloidal wave number ar π πr n λ = = k= k n R B B n toroidal wavenumber θ z. Use this ordering to simlify f and g a. rf f = k k k m n m m = + = + r R r r mbθ nbz mbθ mbz n BθB F = kbz + = + = + r R r R m rbz mbz n mb n = R q m R q m 3 3 mb rb q m q m ( a ) r n n f = = m R R B b. k μ n r n r βb = = = k mr R m a g (small) g (small) 3 4 k mbθ nbr n B 3 = k B 4 z = 4 rk a r Rm m q 4 kr rb n B = ( ) kr q m a R g rf m.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page 3 of

4 . Therefore δw F B n rdr r ( m ξ + ) ξ m q π R μ R tability of Internal Modes. m stable, both terms ositive.. m = nq r > (n= worst) q I ξ > q.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page 4 of

5 3. m= q(r) < somewhere use the following trial function a. ξ= < r < rs δ s ( s)( s) q r r r = qr qr q r s x = rs δ < r < rs + δ δ = q r r s = r > r s + δ.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page 5 of

6 b. δwf B δ = rdr + qx r π R μ R ξ δ B 3 = rq ( x) dx R δ rs B = rq δ 6R rs 6 δ c. δwf as δ d. with an m= resonant surface in the lasma, the system is marginally q < stable in leading order; i.e. if e. to test stability for this case we must calculate δ W to next order for the m= mode. Calculate Next Order δw for m=, n= Mode.. 3 z rmb f = n R m + n r R q + 4 kr k rk m k r k μ k mbθ g = rf + + kbz r F nr rb μ + n n n R q q R q nr = μ + R rb 4 n 3n q q R 3. W π R μ δ F = dr f g ξ + ξ.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page 6 of

7 Use same trial function as before ummary of Internal Modes in a traight Tokamak. m stable. m=, n= worst case for n=, requires q > for stability 3. internal modes do not limit β, or I ( q( a )), but clam q by sawtooth oscillations 4. To show stability we needed to calculate 4 δ W = δ W + δw 4 Consider now External Modes const.. Vacuum is force free fields. m= Kruskal hafranov limit 3. High m external kinks ubtle Issues For External Modes Vacuum as force free Plasma.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page 7 of

8 . cold, but highly conducting lasma surrounds core - more realistic than vacuum. Is there any difference in stability in these cases. I σ= II σ= might anticiate big difference 3. But! Vac. δ W v = B dr FFP δ W FFP = dr B + γ ξ + ξ J B + ξ ξ * * in FFP J = = in equilibrium δ WFFP = dr B Thus, FFP same as Vac. might anticiate no difference in stability since δw s are the same for each. 4. How do we calculate δw, δw. Minimizing condition is v FFP B = B = vacuum fields Vac: BC. n B = n B = n ξ B w FFP n B = n B = n ξ B and B ( = ξ B ) w 5. In the FFP we must check that a well behaved B always gives rise to well behaved ξ. This is an additional constraint that can make the FFP more stable 6. Examle: cylindrical screw inch B r ιb +ιfξ ξ= F r a. if k, m are such that F in FFP region then ξ is well behaved and δ W v = δw FFP.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page 8 of

9 b. Usually, however F = in FFP for external modes. Then, ξ is unbounded dislacement leads to infinite energy. This is not an allowable c. Calculation must be redone with new boundary condition B r =. Thus is an additional constraint, which is equivalent to lacing a conducting wall at r = r s r s external internal mode with wall at singular surface. d. FFP is more stable than Vac if F r = in FFP region. 7. But!! most realistic case is neither vacuum nor FFP, but a lasma with a small resistivity In that case δ W B η = dr B and ιη = v v B ηj B = ( ξ B) B t ω s Careful analysis choose that ξ is bounded at the resonant surface. tability boundary is the same as Vacuum case, but growth rate is smaller, deending uon resistivity.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page 9 of

10 ummary Vacuum: certain stability boundary, growth rate ν T R Ideal FFP: same stability boundary, growth rate if k B much more stable ( γ= ) if k B = Resistive FFP: same boundary as vacuum but τmhd γ γ MHD < ν < τre ν.65, MHD Theory of Fusion ystems Lecture 9 Prof. Freidberg Page of