22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 22

Transcription

1 allooning mode equaion π δ W = ψ ψ μ d W.615, MHD Theory o Fusion Sysems Pro. Freidberg Lecure μ R W Jd k k k k k J χ dψ 1 X d ( ψ ) = χ ( n + ) ( κn nκ ) Mercier Crierion X 1. In using he quasimode reresenaion we have had o assume he soluion X converges suicienly raidly as χ +. ϕ. Wheher or no convergence is acceable deends uon equilibrium roiles and arameers. 3. Analysis o Euler-Lagrange equaion or X indicaes ha here are wo classes o soluions or large χ deending on roiles Oscillaing soluions give rise o unbounded energy: W. Srong convergence gives rise o bounded energy 6. Oscillaory case imlies ha ballooning mode ormaion is no valid as χ. However, or his case a rial uncion o he orm.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 1 o 11

2 leads o δ W < (insabiliy) 7. For exonenial soluions, one sars wih he srongly converging soluion as χ= and inegraes o he righ. 8. The condiion o oscillaory soluions is known as he Mercier crierion. When he Mercier crierion is violaed, he soluions oscillae or large χ. The ballooning mode equaion is no valid bu his does no maer as he sysem is already unsable o inerchanges. 9. When he soluion s do no oscillae he Mercier crierion is saisied and he sysem is sable o inerchanges, and he ballooning mode ormalism is solid. In his case one inegraes he equaion and looks o see i here is a zerocrossing. I here is one, he sysem is unsable o ballooning modes 1. Relaion o Mercier-Suydam Suydam: local behavior as x near regular surace in sace Mercier: behavior as χ in seudo angle Mercier is acually ourier ransorm o Suydam like analysis χ ιs J e : Sα( ψ ψ ) dχ χ ψ R.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page o 11

3 ψ ψ ( ψ) dq χ dψ seudo angle radial localizaion xk ransorm variable Forms o he Mercier Crierion 1. Exac orm DM < 1 4 or sabiliy μ Rκn Λ 1 Λ R M = + 4 D q J R Λ= F μf J q ψ Q π π Q = Jdχ Jdχ R RP. For okamaks, ressure is low:,. As wih Suydam crierion, Mercier crierion is saisied over mos o he discharge because o low. Only roblem is near he origin. a. For a 1 okamak wih circular cross secion, Mercier becomes rq q + 4r 1 q > Near r= q is very small and we require q > 1 b. Near he origin or non-circular okamaks, he crierion becomes 4 3 κ 1 δ 1 < q 1 κ + + κ κ + 1 κ κ + 1 ( κ 1).615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 3 o 11

4 or κ= 1, riangulariy and have no eec. or κ> 1, is desabilizing, +δ sabilizing good Mercier: elongaion and ouward riangulariy, moderae q > 1 and c. Why do oroidal eecs inroduce such big changes since hey are o order. Comue κ n = n ( b b ) Circle: Torus: κ n r n r er κ R b ez + e b e + e 1 r 1 cos +... er cos e sin r R R κ q r 1 r.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 4 o 11

5 allooning Modes 1. Simle limi ballooning mode equaion or lasma wih gradien in. r 1 1, circular cross secion. r cos χ J κn R R R r d 1 d d ψ r d r χ χ J 1 μ rq and χ= dχ = q ( ) + ( sin sin ) ψ R R R d μ F dψ μr d = sin χ R dr 3. This gives or he Euler-Lagrange equaion X 1+Λ +α Λ sin+ cos X = ( in ) Λ= S α sin s rq μ r S = α= = q q R R 4. Solved numerically gives S vs α diagram.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 5 o 11

6 I irs region o sabiliy (goes unsable a high α ) II second region o sabiliy (evenually becomes sable a high α ) 5. Region I maximum. Se α 6S o deermine criical roile or maximum. For q 1,q = 1 a 3 (circle) q a 6. Numerical sudies by Sykes, Yamazaki κ A < = μ R I q * q* cross secional area I =.44 a For oimized roiles wih elongaion and ouer riangulariy 7. For κ=,q = 1.5, = 1 3 1% Second Sabiliy 1. Why does such a region exis. Examine local shear 1 J q ψ = q ψ, χ dχ q = π π R local shear r q = S αcos q r shear ressure driven modulaion.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 6 o 11

7 3. Noe: bad curvaure occurs as = due o oroidal ield. Sabilizing erm due o shear α rq q Exernal Kinks 1. Consider surace curren model. =cons, circular cross secion δ W = δ W + δ Ws + δw v 1 δ W = dr > μ v 1 δ Wv = dr > μ δ Ws = ds n ξ n + μ 1 s.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 7 o 11

8 1 + = ξ κ + ( κ κ ) Q Q ds n n n n = μ curvaure erm kink erm oroidal curvaure δw s 1 π = dξ n + cos π R μ π. Modes have he srucure o ressure driven kinks high ballooning eec kink erm 3. Sabiliy diagram (shar boundary model) 4. Full numerical sudies using oimized roiles and cross-secions.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 8 o 11

9 and q κ I = 14 = 8 q a > q * * *min Noe, no dried q *min or ollowing modes. 5. For κ= 1, q* = 1.7 =.8 For oimized Troyon q = 1,5, κ= 1.6 Se = 13 = 5% *min =.15 Near he regime o reacor ineres!! Requiremens on 1. There are wo basic usion energy requiremens where eners. Igniion and Wall Loading 3. Igniion ( T = T = T) e ι a. n σv n T σv P = Q = Q 4 4 T = 4μ nt b. P 4 Q σv = 4 T 16μ as 1 3 kev, σ v = 1 m sec c P =.5 1 ω m d. P = Pe e. 3nT 3 P e = = =.6 ω m τ τ 4μ τ =.6 τ.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 9 o 11

10 1. τ = 4. Wall Loading.4 a. 6 4 E =η =η =η π π P P P V.5 1 R a b. 3 P 4 1 R a 4 1 R a a 1. 1 a E = E P = 1. 1 a c. P = P A W W Pπ Ra = P 4π ar P P = a W P.5 1 = a 6 4 W d. Eliminae a rom se b. 13 PE 1 3 PE a = = ( 1. 1 ) 3 = PE a e. Subsiue a ino (C) P W P E.5 1 = 34 3 PW 14 E = 3 1 P. For W 6 P = 4 1 ω m and P = 1 was E g. For =5T a R R hen =.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 1 o 11

11 <6% 5. The Troyon limi I <.3 a IMD (%) < N N = 3 a or κ < = 6% q *.615, MHD Theory o Fusion Sysems Lecure Pro. Freidberg Page 11 o 11