22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 20

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1 .615, MHD Theoy of Fusion Systes Pof. Feideg Lectue Resistive Wll Mode 1. We hve seen tht pefectly conducting wll, plced in close poxiity to the pls cn hve stong stilizing effect on extenl kink odes.. In ctul expeients, the etllic vcuu che suounding the pls is good ppoxition to pefectly conducting wll. 3. Howeve, its conductivity is not infinite ut is finite. 4. In fct we do not wnt the conductivity too high nd/o, too thick ecuse it would tke too long extenlly pplied feedck fields to penette the shell nd intect with the pls. 5. Also, highe esistivity, slle cuents e induced in the che duing tnsients, lleviting powe supply equieents. 6. The question ised hee concens the effect of finite esistivity of the wll on extenl kink stility. 7. Thee e thee possile situtions nd only one is elly inteesting. 8. In the fist cse the pls is stle to extenl kinks with the wll t Hee, since the pls is ledy stle, wll, eithe idel o esistive does not ffect stility. This cse is uninteesting. 9. In the second cse, the pls is unstle with the wll t nd with the wll t its ctul position, ssuing the wll is pefectly conducting. Since the pls is unstle with pefectly conducting wll s =, king the wll esistive does not help. This cse is lso uninteesting. 1. The inteesting cse is when the pls is unstle with the wll t, ut stle with pefectly conducting wll t =. Does the esistivity of the wll destoy wll stiliztion? 11. To ddess this issue we investigte the pole in stight cylindicl geoety. Howeve, the esults e vlid fo genel tooidl geoety s well..615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 1 of 1

2 Pln of ttck 1. The nlysis of the esistive wll ode is cied out in fou steps.. Fist, efeence vlues of δ W e clculted fo n idel wll locted s ( δ W ) nd t = ( δw ) 3. The full eigenvlue pole is solved egion y egion ssuing slow gowing odes - on the scle of the wll diffusion tie. 4. Thid, the fields within the esistive wll e clculted using the then wll ppoxition. This gives ise to set of jup conditions coss the wll. 5. The esulting set of coupled eqution nd oundy conditions e solved yielding the dispesion eltion. The Refeence Cses 1. Recll tht δw fo genel scew pinch suounded y pefectly conducting wll is given y δw FF ΛF = f g d ξ + ξ + + ξ π R μ k whee B B F = kb z +, F = kbz 1+ Λ k = k + 1. The exct iniizing ξ stisfies ( F ) g ξ ξ = ξ = ξ ξ egul F 3. Recll tht F = nd fo n extenl ode with esonnt sufce outside k the pls this iplies tht F in the pls. 4. Thus the vitionl eqution fo ξ is non-singul. Its solution is ipotnt, ut oing. 5. Assue the solution fo ξ is known, eithe nlyticlly o coputtionlly. 6. If we ultiply the eqution fo ξ y ξ d we find tht.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge of 1

3 F g ξ + ξ d = Fξξ 7. This llows us to wite δw FF ΛF F ξ = + + π R μ k k ξ ξ 8. Note tht ( ξ ξ) is known quntity fo the solution fo ξ. 9. The fist efeence cse coesponds to the wll t : Λ=Λ = 1 Fo this cse δw FF F Λ F ξ ξ = + + ξ π R μ k k 1. The second efeence cse coesponds to the wll t Λ=Λ = Keep in ind tht Λ > Λ (well stiliztion) 1. Fo oth efeence cses 13. These eltions llow us to wite δw δ W F = + Λ Λ π R μ π R μ ξ ξ is the se. It is unffected y the wll. 14. The inteesting cse unde considetion coesponds to δ < unstle with the wll t W δ W > stle with pefect wll t = The eigenvlue pole with esistive wll 1. We solve the full eigenvlue pole with the esistive wll. Howeve, we cn ke use of uch of wht we hve ledy done y ssuing slow gowing odes - esistive wll diffusion te. 3. Exple: =.3, R = 1,Tc = Tc kev, ξ 4. Then 6 τ = R ν =.3 1 sec. MHD T c.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 3 of 1

4 5. Conside stinless steel vcuu che of thickness d = 1. Then, with 8 = 11 1 Ω τ D = μd = sec. 6. Fo thick coppe wll d=1 c, τ D = μ d =. sec. 8 = Ω. 7. Clely τ τ fo eithe cse. D MHD 8. The ipliction is tht in the pls eigenfunction eqution, ω k ν, ω kν nd k fo extenl ode. Theefoe we cn ignoe ω in the pls egion. τ ι 9. The esulting eqution fo ξ thus coesponds to the idel ginl stility eqution which is ou old fiend. ( f ) ξ gξ = 1. The ωs will ppe whee we discuss the wll. 11. The egion etween the pls nd the wll stisfies B = φ, φ = φ k + φ = I I I I I 1. The solution, neglecting k fo siplicity (to hve polynoils the thn Bessel functions) is given y φ I = c1 + c 13. We will find c 1 nd c shotly y tching jup conditions 14. A siil nlysis holds fo the oute vcuu egion whee B = φ φ = II II II 15. The solution hee hs only decying solution since the fields ust e egul s. Thus φ II = c3 The wll solution 1. Now lets look within the wll.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 4 of 1

5 . Assue the wll is then d. The wll looks ectngul 3. Let = + x, = y 4. The eqution fo B in the wll is otined s follows B t = E = J = B = B μ μ 5. Focus on the (i.e. x coponent), nd ssue t t B e ιω ωι e ω =gowth te. ι B x μω k + Bx = x t B x 6. B y nd B z e found fo B = nd the ssuption Jx = (then wll ppox - ll cuent flows pllel to the sufce): 7. We do not need B y nd ex B = B z so we will not clculte the. 8. Then wll odeing: Assue ω ι μ d J τ D 9. Then ι μω μ 1 d d x 1 1 μ d 1 μ d 1. Also B μωb ( ) = ι x d μω ι d 11. This iplies tht the k + B x whee B = const, B B d 1 X x1 x 1. The eqution nd solution fo B e given y x1 cn e neglected nd tht B = B +B ( x) x x x1 B x x1 = μω ι B x μω B B ι B x = x + x x 13. Fo thin wll d, this solution tnsltes into the following two jup conditions.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 5 of 1

6 + μω ι d x x x x B = B + B B + μω ιbxd μω ιb Bx = d 14. O μωdb B = B = ι The jup condition nd dispesion eltion 1. Thee e fou unknowns in the pole c 1, c, c 3, ω. Thee e fou jup conditions. t the wll given ove, nd on the pls we ust now deteine 3. The fist is the usul n B condition n B = n ξ B B =ιfξ 1 4. The second is the pessue lnce jup condition (lots of wok) B B μ p1 + B B1 + ξ μ p + = B B1 + ξ 5. Fo no sufce cuents nd p, p s = vnishing this educes to 1 B ξ B = B B 6. Vcuu pt B B 1 = B φ I = ιfφ I 7. Pls pt B ( B ) ξ = ξ B B ξ B B = t the edge B B B = ξ = ξ ξ z 8. Now B = B + B. Ne the edge Bz = const nd B k. Theefoe B B : e nd B ξ ξ B =.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 6 of 1

7 9. The lst te is B ξ whee 1 1 Be Be ξ = ( ξ ) + = ( ξ ) + B B z z ( kb ) 1 ιbz 1 ιg ι = ( ξ ) + ι kb = ( ξ ) + G( ξ) ξ B B k B Bz ι G = kb G( ) = ξ + kb ξ k B 1. Note ( kb G) 11. Coine te 1 F = kb kb ( ξ) F kgb kb ξ = ξ kb 1. Collect te ( ξ) kgb k B F B ( ξ B) = ξ + ξ k F B F kgb = ξ + + ξ k k k F FF = ξ ξ k k 13. Pessue lnce oundy condition o F FF ιfφ I = ξ ξ k k ι ξf φ I = F + ξ k ξ.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 7 of 1

8 Suy of whee we e φ I = c1 + c ϕ II = c3 As = B =, B = μ ω ι db As ι ξf = B = ιf ξ, φ = F + ξ k ξ Apply B.C (note: B =ϕ) 1 I As = B = ( c c ) = ( c ) As μωdb + 1 ι = B ( 1) c1 ( 1) c = c 3 = + As c1 + 1 = B1 = ιfξ W c 1 F W = ι ξ = W ι As ξf c1 ι ξf = φ = F + ξ + c W = F + k W k ξ ξ ( c c ) 1 μω ι d Solve fo c,c,c 1 3 fo 3 equtions c1 c + c3 = 1 c W c = ιfw ξ ιw ξf c1 + W c = F + ξ k ξ.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 8 of 1

9 Solution c c ιw F F ξ F = + + ξ k ξ k 1 ι F F ξ F = + + ξ W k ξ k ι F F F c3 = c c1 = ( W + 1) + ( W 1) + ( W 1 ) W k k Dispesion Reltion (lst eqution) ( 1) c ( 1) c ( 1) c ( c c ) = ξ ξ Define τ D = μ d n, set c3 = c c 1 μω ι d Then c1 ωτ ι D = c 3 Siplify F F ξ F W + + k ξ k ωτ ι D = 1 F F ξ W + 1 F ( W 1) + + W k k ξ W 1 Recll δw F F F ξ = F + + π R μ k k ξ δw F F W + 1 F ξ = F + + R k W 1 π μ k ξ.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 9 of 1

10 Theefoe W δw ωτ ι D = W 1 δw Resistive wll ode is unstle!! Gowth te 1 τd δ W < δ W > h.615, MHD Theoy of Fusion Systes Lectue Pof. Feideg Pge 1 of 1

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