Kißlinger, J.

Nührenberg, C.

Strumberger, E.

Wobig, H.

German

MPG.PuRe

26th EPS Conf. on Contr. Fusion and Plasma Physics, Maastricht, 14 - 18 June 1999      ECA Vol.23J (1999) 1485 - 14881485Equilibrium and Stability Properties of a Helias ReactorE. Strumberger1, H. Wobig1, J. Kilinger1,C. Nuhrenberg21Max-Planck-Institut fur Plasmaphysik, IPP-Euratom Association,85748 Garching, Germany2Max-Planck-Institut fur Plasmaphysik, Teilinstitut Greifswald,IPP-Euratom Association, 17489 Greifswald, GermanyHelias (Helical Advanced Stellarator) congurations are studied at the IPP Garchingwith respect to their usefulness for a Helias reactor. In this paper a conguration is inves-tigated with respect to its equilibrium and MHD stability properties up to an averagedvolume beta of hi = 5%. It is characterized by a major radius of R0 = 22 m, a plasmaradius of a0 = 2 m and low shear, that is, the rotational transform value of the vacuummagnetic eld amounts to  = 0:875 on the magnetic axis and  = 5=5 in the edge region.There, ve macroscopic islands surround the last closed magnetic surface (LCMS). Thecoil system consists of ve eld periods with ten coils per period which produce a vac-uum magnetic eld of 5 T on the magnetic axis. In comparison to a previous design [1]the geometry of the coils has been slightly changed in order to reduce the shear and toincrease the magnetic mirror. Here the mirror eld amounts to approximately 10%.For the MHD equilibrium and stability computations a system of numerical codes isused. Three-dimensional free-boundary equilibria and the corresponding magnetic eldsare computed self-consistently for various pressure proles and -values up to hi = 5% bymeans of the NEMEC, MFBE and GOURDON codes. The NEMEC code [2] computesa free-boundary Helias equilibrium for a given toroidal ux and the MFBE code [3]calculates the magnetic eld of this equilibrium on a grid inside and outside the plasmaboundary. This magnetic eld serves as input to the GOURDON code, which is usedto determine the LCMS of this eld. If the LCMS does not coincide with the plasmaboundary obtained by the NEMEC code, the toroidal ux, which is a free parameter inthe NEMEC code, is modied, i.e. the toroidal ux is determined iteratively. Figure 1shows the magnetic elds for hi = 0 and 4%. The edge region ergodizes for hi  2%.The width of the 5/5 islands increases with , while the positions of the X- and O-pointsof these macroscopic islands are almost unchanged. This is a favourable behaviour withrespect to divertor operation.Comparisons of the resulting nite- magnetic elds with the corresponding vac-uum eld produced by the external coils yield informations about the variations of theShafranov shift, the aspect ratio, the magnetic well and -prole with increasing plasmabeta. Table I contains the values of the rotational transform at the magnetic axis, the26th EPS CCFPP 1999 ; E.Strumberger et al.: Equilibrium and Stability Properties of a Helias Reactor1486Shafranov shift, the plasma volume, the aspect ratio and the magnetic well in dependenceon hi.FIG. 1: Upper halves of the Poincare plots at the symmetric bean-shaped cross-sectionfor the vacuum magnetic eld (left gure) and the nite- magnetic eld with hi = 4%(right gure).TABLE I: Rotational transform 0 on the magnetic axis, Shafranov shift R=a0 (R =mean shift of the magnetic axis, a0 = plasma radius), volume V enclosed by the LCMS,aspect ratio A and magnetic well V 00 for various hi.hi [%] 0 R=a0 [%] V [m3] A V 00 [%]0 0.875 - 1828.4 10.60 0.681 0.867 6.3 1744.3 10.98 2.592 0.860 12.3 1681.6 11.20 4.373 0.849 18.1 1623.5 11.45 6.004 0.837 23.4 1570.5 11.68 7.615 0.820 28.9 1521.3 11.91 9.14One of the main results is the small Shafranov shift. It is in the expected range ofa Helias conguration and fullls the requirement of a fusion reactor. From this point ofview a further optimization of Helias congurations is not necessary. Further, equilibriawith a sucient plasma volume exist up to hi = 5%. Figure 2 illustrates the behaviourof the -prole. The rotational transform decreases with rising plasma pressure leadingto the existence of low order rational surfaces in the plasma centre where the onset ofMHD instabilities is to be expected. The rst surface of this kind is the surface with = 5=6, which appears around hi = 4:2% and a peak value of (0) = 10:3%.26th EPS CCFPP 1999 ; E.Strumberger et al.: Equilibrium and Stability Properties of a Helias Reactor1487Investigations of Mercier and resistive interchange criteria and local ballooningmodes, made with the JMC code [4], show stability up to hi = 4%. For hi = 5%Mercier and resistive interchange criteria show instability around the 5/6 resonance (seeFig. 3), and local ballooning modes are even unstable for higher mode numbers, e.g. = 5=6; 6=7; 7=8 and 8/9.FIG. 2: Rotational transform proles for the vacuum eld (upper curve) andthe nite- magnetic elds up to hi =5% (lower curve) versusps with s thenormalized ux label.FIG. 3: Mercier (+) and resistive inter-change () criteria versusps for hi =5%. Instability occurs atps  0:5 for = 5=6.0.0 0.2 0.4 0.6 0.8 1.0sequence parameter-20-15-10-50102  eigenvaluestableunstableFIG. 4: Result of a global ideal MHD stability study of the hi = 4% equilibrium:CAS3D2 eigenvalues versus an articially introduced factor which was applied to themain stabilizing term in the MHD energy functional, so that for sequence parameterunity the full MHD energy functional is obtained. The stability limit is reached atapproximately 0.7 in the sequence parameter.In addition, the equilibrium with hi = 4% has been investigated with respectto the global ideal MHD stability by using the CAS3D code [5] in its xed-boundaryversion CAS3D2 which uses the incompressibility constraint on the MHD displacement26th EPS CCFPP 1999 ; E.Strumberger et al.: Equilibrium and Stability Properties of a Helias Reactor1488vector. In order to assess the stability of a single equilibrium here a 1-parametric sequenceof stability calculations has been generated by articially varying the amount of themain stabilizing term, such that for vanishing sequence parameter the stabilizing termsare considerably reduced and that for sequence parameter unity the full MHD energyfunctional is obtained. Various perturbation Fourier tables have been used, especiallyincluding both mode families present in a 5-periodic device. The computation parametershave been chosen to be comparable to those used for the investigation of the W7-Xstability [6] (64 radial grid point, 115 perturbation Fourier harmonics, N=2 mode family).As an example Fig. 4 shows CAS3D2 eigenvalues as obtained in such series of calculationsfor N=2 mode family perturbations. The results show that in this sequence a stabilitylimit is obtained for a value of the sequence parameter of approximately 0.7. Similarresults have been obtained also for the N=1 mode family, so that it may concluded thatthis equilibrium is stable with a good safety margin.Equilibrium and stability properties of the Helias reactor solely depend on the betaprole while connement and fusion power output depend on the details of the temper-ature and density proles. A parameter set which is compatible with the limits listedabove is the following (Table II):TABLE II: Design values of the reactor plasma.Magnetic eld 5 TPeak temperature T (0) 14 keVPeak density n(0) 2:4  1020 m 3Average beta hi 4.25 %Peak beta (0) 10.5 %Fusion power 2.9 GWConnement time 2.1 sIn summary, the present analysis of MHD equilibrium and stability shows that thereduction of the Shafranov shift and the stability limits are suciently optimized for thepurpose of a Helias reactor.References[1] Beidler, C.D., et al., Proc. 16th In. Conf. on Fusion Energy, Montreal 1996, IAEA,Vienna 1997, Vol.3, 407.[2] Hirshman, S.P., et al., Comput. Phys. Comm. 43 (1986) 143.[3] Strumberger, E., Nuclear Fusion 37 (1997) 19.[4] Nuhrenberg, J., et al., in Proc. Workshop on Theory of Fusion Plasma, Varenna1987, Eds. A. Bondesin, E. Sindoni, F. Troyon, 1988, EUR 11336 EN, 3.[5] Nuhrenberg, C., Phys. Fluids B 5 (1993) 3195.[6] Nuhrenberg, C., Phys. Plasmas 3 (1996) 2401.

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