Nührenberg, J.

Zille, R.

English

MPG.PuRe

- 445 -Ballooning Modes in 3D Finite·p StcUarators J. NUHRENBERG, R. ZILLE Max-Planck-Institut fur Plasmaphysik, EURATOM Association, D-8046 Garching B P folo 030 The evaluation of Mercier1 and resistive interchange2 modes is followed up by generalizing the JMC code, which evaluates the current density from 3D equilibria as the basic quantity for the investigatioL of localized instabilities, in order to consider ballooning modesJ in general 3D equilibria. (Previous work 4 was restricted to low-p approximations. ) To ~his end, Boozer's coordinates~ are constructed from the cquilibria because they may conve-niently be used to analyze the ballooning -equation. A form of the Mercier criterion is used which is valid on field tines with rational twist. A typical field line of this type in a ste llarator with five periods (such as W VII-AS) would have, for example , tT = 2/5, where tT is the total twis~, so that localized m = 5, n = 2 modes are considered. The analysis of such modes should not be plagued by existence problems of the equilibrium associated with Lp (twist per period) being rational of much higher ord.er (2/25 in the above example). The ballooning equation is cast in a form which avoids numerically dangerous cancell::\.tiolls and shows, explici tly, the order with respect ~o /3 of the various terms. Thus, the prin-cipal stabilizing terms (magnetic well diminished by its diamagnetic part and shear) and destabilizing terms (square of the parallel current density and unfavourable cont ri butions from the field line curvature) are clearly borne out. Moreover, this form of the ballooning equation readily yields the asymptotic behaviour (for large arguments) which recovers the Mercier criterion in an analogous form. This feature not only facilitates testing of the code but may also be used to obtain the ballooning stability condition by asymptotic matching. Results are presented fo r tokamaks (as test cases) and 3D stei1arators such as ATF and · W VU-AS_ In Boozer's coordinates s, fJ, tP, where s is the flux label and fJ, tP are poloidal and toroidal angle· like variables, respectively, the local shear a can be written in the following form • = IVsl-'(Vs x B) -V x (Vs x B) = Ff!" /.,(9 + B--'1(1 gos / .,(91'1 si' - J g.s / .,(91'1 si') I Merdt-r, c., ill PI .... ' mll Physics and Contr. Nucl.Fusioll Ret. 1901. Nuri.Fu,.HI02. Supp\.2. IAEA. Vi~lIn,. ( !!:lu21 801; R. G roBnlllll n , F. Herrneg-gt-r, J . Nii hrl'nberg, Proe.She rwood Theory Coni .. Ariin gton ! !:lS:!, 1521 2J . Nilhrenb<'rg, R. Gruflm31l1l, Proc.S herwood Tht-ory Cou!., IlIciint- Villa.ge 1984. 2R6: 511, Int. Wo rkshop 011 Stellaraton, Sehl,,!! Rillgbt'rg 1984. COUJm.EC, EUR 96115 EN. I. 3:!!:l 3D. Correa-Restrl'po. Z.Nllturlorscb.3:l11 (1978) 789 ~H.L . Dl'rk. M.N. Rosellbluth , J .L. Shohct, Phys.Fluius 26 .( 1983) 2(jl(j ~ W.A. Cooper. T .O. Hellt l",. PI....,m/ll Ph)'u.nd Contr.Fu5. 26 (1984) 921 SA. Door.tr , Phy!.F luius 23 (1980) 004 - 446 -Here, all functions are normalized to one field period of the equilibrium; Fr , J toroidal Ilux and current, Fp, I poloidal Ilux and current, 'p = FplFt ,' = dlds , p'V' = I' Ft + J'Fj" ..;gB2 = - Ffl- FpJ . The ballooning equation can then be' reduced to the following equation for F (which indicates instability if F vanishes twice) Here, ~ is used as the variable along the field lines and the coefficients a- I, 8, i1, D, iJ have to be used as function of this variable , a =vgIV'I' it = - Ff.'IVsr~/B if =(/gOS - JgOS)/vgB D =p',' F~-l(l B-'J ,8 - J B -'J,.p) D =Ivgp" / B' - p'vg,s -p' B-'(IF, + J F~) + p'vgB · V(PB')IIF~ Here, iJ js the periodic part of the fundion f3 in the covariant representation of B (8 = Vx +PVs), P, D, and b can be reduced as follows p = -(F~g.s + Fpg,s) /Vi vgB Vp = p'(vg - V') x = fll/B Thus, it ,represents the stabilizing influence of the shear, b (for stellarators with vanish· ing net toroidal current) mainly the stabilizing influence of an outwardly increasing B"J diminished by the diamagnetic effect, D the tenus connected with ill and with the shear known from Mercier's criterion, In accordance with the t!xpl.icit resemblauce of this form of the ballooning equation with Mercier's criterion, the well-known asymptotic analysis, F ex ~v + FI ~v- I +"" readily recovers that criterion in the form of the indicial equation v = -.5 ± 1(.5 - (aD/u'))' - (a/o')«aD' /0') + (D) - (D))I' where [) = J Dd!fo = ,'X, This equation may also be used to obtain a partial test of four of the five coefficients of the ballooning equation which is solved in tbe following way, Relatively low order rational toroidal twist numbers are considered and tbe contracted variable ~ = N -I m;-I!fo is used where LT = ~I N number of field periods, The coefficients of the balloon ing equation are periodic in ;P,~~ that t: . above 'averages () are well defined , Stellarator equilibria of tbe usua.l symmetry are considered, so that it is possible to consider ballooning mo~es symmetric with respect to~. Thus, F(O) = 1, (d FldI,6 HO) = 0 i3 used and a ballooning instability is found if F = 0 for ;p > 0, - 447 -The 3D equilibrium code used so far is BETAI; the JMC evaluation code only uses the flux invariants and the geometry of the equilibrium obtained, reconstructs t he remainder of the equilibrium information, constructs Boozer's coordinates and obtains the coefficients of the ballooning equation. By way of example, Fig.l shows a, jj for W VII· AS for 'T = i, note the three different scales: the equilibrium period, the poloidal period, and the period of the closed field line. All results shown are obtained with fil> ~ te bu t reasonably fine meshes - the cboices being guided by the experience in extrapolatiIo,:t Mercier's criterion - and are being complemented by extrapolation studies; Fourier convergence problems typical of magnetic coordinates also need further study. o Fig.l. The coefficlen.ts a (+) and jj (x) for a W VU·AS equilibrium with IT (P) = 0.028. i and Test results have been obtained for axisymmetric equilibria and a simple unstable l = 2 stellarator. Figure 2a 'shows F for ' T = ~ in a standard circular tokamak at (p) = 0.02; 0.03, below and above the ballooning stability boundary. Figure 2b shows F for 'T = ~ in an l = 2 stellarator with five periods. Since for this twist value and a purely elli ptic boundary no vacuum magnetic well is present, instability is seen at ' all f3 values and the ballooning character (F = 0 at ~ < n~l) prevails for (P) > 0.002. Results for W VII-AS and ATF are shown in Fig.3. The common feature is that tbe insta-bilities found are of the Mercier type, i.e. driven by the parallel current density ({a0 2/ q2}) and occurring in the asymptotic range. Apparently, in these cases ballooning instability proper occurs at larger p·values than Mercier instability (in contrast to the tokamak sit-uation). It will be in teresting to see whether this behaviour persists in configurations in which Mercier instability occurs at higher p·values. In this connection we mention an l "", 1, 2, 3 equili brium encountered in the search for stable medium-p stellarators. The ' equilibrium shown in FigA is Mercier stable at (p) = 0.05. Ballooning stability results for this configuration are being obtained. . IF. Dau .. r. O. Bet/l.l\court. ",nd P. Gn.rn.b~di"'n M",gnetohydrodynmnic EquilibriullI /l.nd St",bility of St~l· lto.r&to~ (Springe r, New Yo rk. 198 .. - 448 -o -----o - - - - -=;--:=-=- -=::-- -=--=;--O~.O"----O:5 \.0 0 .0 0.5 " 1.0 1.5 2.0 " Fig.2. a. The solutions F of 'he ballooning equation for a circular tokamak with aspect ratio 3, Lr{O) = l,tr(1} = ~ fOrtT = ~ and {.o} = 0.02 (stable) and {P} = 0.03 (unstable). b. F for an l = 2 stellarator with baU-axes ratio 2, aspect ratio 7.5, 5 periods, LT = ~ and (P) = 0.0012;0.01. 20 o ~--:....:;---=--:--=;-::=:- :::-:;:::::-:::-~- -==-~-. • 2 O~-----~------o'O " Fig.3. a. F for W VII-AS equiHbria with (.0) = 0.014 (stable) a.nd 0.028 (unstable) for LT = ~. b. F for an ATF equil..i brium (standard flux-conserving case) with (,8) = 0.033 for LT = ~. This case is weakly Merder unstable; F = 0 at ~ ~ 100. FigA. Flux surfaces of an l = I, 2, 3 equilibrium which is stable witb respect to Mcrcier and resist ive interchange modes at {P} = 0.05. . 

Ballooning Modes in 3D Finite-ß Stellarators

http://hdl.handle.net/21.11116/0000-0008-FB7E-1

Ballooning Modes in 3D Finite-ß Stellarators

Abstract

Similar works

Full text

Available Versions

MPG.PuRe