Günter, S.

Lackner, K.

Pereverzev, G.

Strumberger, E.

English

MPG.PuRe

Analysis of ß-limits in tokamak reactor scenarios

Analysis of β-limits in tokamak reactor scenariosE. Strumberger, S. Gu¨nter, K. Lackner and G. PereverzevMax-Planck-Institut fu¨r Plasmaphysik, EURATOM Association,D-85748 GarchingIntroduction: In the framework of the European Power Plant Conceptual Studies (PPCS)environmental, safety and economic aspects of a future power plant, called DEMO, are stud-ied. Still many issues, such as geometrical size, pulsed or steady state operation need to beaddressed. In these proceedings we present results of ‘DEMO Physics Studies’ concerningthe analysis of β-limits in tokamaks. A high β-limit is a necessary condition for a workingtokamak reactor. However, instabilities associated with ideal internal and external modeslimit the plasma beta. Therefore, stability-optimized plasma profiles have to be found. At-tractive steady-state scenarios should also be characterized by a high bootstrap contributionto the plasma current. And, the stability-optimized plasma profiles have to be consistent withpossible heating and current drive scenarios in order to be relevant for tokamak reactor sce-narios. We present stability studies of basic plasma and conducting wall configurations fortokamak reactor scenarios, which have been performed with the equilibrium VMEC/NEMECcode [1] and the linear stability CASTOR FLOW code [2]. Ideal MHD stability limits are de-termined for pulsed and steady state scenarios with ad-hoc postulated pressure and q-profiles.Furthermore, the stabilization of n  1 ideal modes by infinitely-conducting walls is studiedusing a set of walls in a parameterized distance from the plasma surface. These investigationswill form a basis for later resistive-wall studies. Advanced tokamak scenarios with reversedq-profiles which are ideal stable may be unstable because of resistive coupled tearing modes.Therefore, the stability of these equilibria is also investigated with respect to resistive modes.Furthermore, the bootstrap contribution to the plasma current is computed for all consideredplasma configurations. In order to arrive at stable plasma scenarios with optimized transportproperties, the stability limit of a transport optimized equilibrium derived by the ASTRAcode [3] is also analyzed.Equilibria with monotonic q-profile. Although steady state advanced scenarios with re-versed q-profile are the preferred operation mode for a reactor, pulsed scenarios with con-ventional monotonic q-profile are admitted as a fall-back option. The ideal MHD stability oflow-n modes and the bootstrap currents of equilibria with monotonic q-profile are studied us-ing various pressure profiles. The geometrical and physical parameters are: major radius R0= 8.14 m, minor radius a0 = 2.80 m, aspect ratio A = 2.91, elongation E = 1.71, triangularity∆ = 0.35, toroidal vacuum magnetic field B0(R0) = 5.70 T, total plasma current Ip = 21.95MA, beta normalized βN = 3.59, safety factor at the magnetic axis qa = 1.36, safety factor atthe plasma boundary qb = 4.07. The pressure, total current and bootstrap current profiles areshown in Figs 1 and 2. The pressure profile A is a peaked ASDEX Upgrade-type profile with33rd EPS Conference on Plasma Phys. Rome, 19 - 23 June 2006 ECA Vol.30I, P-4.165  (2006)pedestal, while the profiles B, C and D are similar to the ones given in Ref. [4,5]. As Fig. 2illustrates, a peaked pressure profile (profile A) causes the boostrap current to peak near theplasma centre, whereas a broad pressure profile (profile D) causes the bootstrap current topeak near the plasma edge. Due to the pedestal and the steep pressure gradient at the plasmaedge of pressure profile A, the corresponding bootstrap current rises again at the plasma edge.In Fig. 3 the growth rates are plotted as function of the toroidal mode number n. For caseA the growth rate increases with rising n. This is due to the steep pressure gradient at theplasma boundary. No unstable solutions could be found for n3 for cases B and C. For caseD the growth rate is almost constant up to n=6 and then decreases. No unstable modes couldbe found for n7.0.0 0.2 0.4 0.6 0.8 1.0ρtor0.00.40.81.21.62.02.4p   [MPa]q=1.5 q=2.0ABCDFig. 1: Various pressure pro-files named A, B , C , D .0.0 0.2 0.4 0.6 0.8 1.0ρtor0.00.51.0<j. B>/B 0   [MA/m2]bootstrap currenttotal currentADCBFig. 2: Total currents andbootstrap currents.0 2 4 6 8 10n0.0e+005.0e+041.0e+051.5e+052.0e+05γ   [1/s]ADBCR−Rb=5mFig. 3: Growth rates as func-tion of n.0 1 2 3 4 5R−Rb   [m]01000020000300004000050000γ   [1/s]planned wall positionn=1n=2n=3n=4case A0 1 2 3 4 5R−Rb   [m]0100002000030000γ   [1/s]planned wall positionn=1n=2case B0 1 2 3 4 5R−Rb   [m]0100002000030000γ   [1/s]planned wall positionn=1n=2n=3n=4case DFig. 4: Growth rates as function of the distance of the ideal wall (R Rb, with Rb being theoutermost radial coordinate of the plasma boundary) for cases A,B,D and n=1-4. The greyshaded area marks possible positions of an external wall in a distance between 60 and 70 cm.In Fig. 4. the growth rates of cases A, B and D and toroidal mode numbers n=1-4 are plottedas function of the wall distance (case C is similiar to case B). While modes with n3 are sta-bilized within this distance, the n=4 mode of case A stabilizes only for smaller wall distances.Furthermore, it is expected from these results that modes with n>5 can only be stabilized byan ideal wall located very closely to the plasma boundary. The high-n modes of case A arelocalized at the plasma edge. These modes are so-called edge localized modes (ELMs). Incontrast to case A, the high-n modes of case D are mainly localized inside the plasma. Whilethe considered βN = 3:59 is already the limit for cases A and D, cases B and C would allowslightly higher values if no higher n-modes appear.Transport consistent scenario. This equilibrium was derived by 1.5 transport modelingtaking into account tokamak heating and current drive systems, as well as bootstrap current.The geometrical and physical parameters of this equilibrium are: R0 = 8.10 m, a0 = 2.80 m,A = 2.89, E = 1.71, ∆ = 0.42, B0 = 5.68 T, Ip = 20.08 MA, βN = 1.55.33rd EPS 2006; E.Strumberger et al. : Analysis of beta-limits in tokamak reactor scenarios 2 of 40.0 0.2 0.4 0.6 0.8 1.0ρtor12345qFig. 5: Reversed q-profile.0.0 0.2 0.4 0.6 0.8 1.0ρtor0.00.20.40.60.8p   [MPa]Fig. 6: Pressure profile.0 1 2 3 4R−Rb   [m]100100010000100000γ   [1/s]planned wall positionn=1 external mode(3,1) coupled tearing modeFig. 7: Growth rate.Already the n=1 mode can not be stabilized by an external wall within a reasonable distance(Fig. 7), because of the steep pressure gradient at the plasma edge (Fig. 6). Furthermore,when the ideal mode becomes stabilized a resistive coupled tearing mode apears. Due to theshape of the q-profile (Fig. 5) the major poloidal harmonic of the resisitive mode is m=3.This mode can not be stabilized by an external wall, but also appears in case of an ideal walllocated at the plasma boundary. Not shown here is the bootstrap current. Its shape fits verywell the profile of the total current, but its magnitude is too small.Stability and bootstrap current optimized scenario. Here we use optimized profiles ofsafety factor (Fig. 9), pressure and density (Fig. 8) developed for the advanced tokamakpower plant ARIES AT by C.E. Kessel et al. [5].0.0 0.2 0.4 0.6 0.8 1.0ρtor0.00.20.40.60.81.01.2densitytemperaturepressureFig. 8: p;T;n-profiles.0.0 0.2 0.4 0.6 0.8 1.0ρtor2.02.53.03.54.0qFig. 9: q-profile.0.0 0.2 0.4 0.6 0.8 1.0ρtor0.00.51.01.5<j. B>/B 0   [MA/m2]bootstrap currenttotal currentβN = 3.9βN = 5.0Fig. 10: Current profiles.0 1 2 3 4R−Rb   [m]020000400006000080000100000γ   [1/s]planned wall position            n=1βN = 3.9βN = 5.0Fig. 11: Growth rates asfunction of the wall distance.0 2 4 6 8 10n0e+002e+044e+046e+048e+041e+05γ   [1/s]βN=5.0βN=3.9R−Rb=5mFig. 12: Growth rates asfunction of n.0 1 2 3 4R−Rb   [m]0200004000060000γ   [1/s]planned wall position   βN=3.9n=1n=3n=2n=4Fig. 13: Growth rates asfunction of the wall distance.Using these profiles, equilibria for two βN-values are investigated. The geometrical and phys-ical plasma parameters are: R0 = 8.10 m, a0 = 2.80 m, A = 2.89, E = 1.70, ∆ = 0.48, B0 =5.64, Ip = 24.-25 MA, βN = 3.9 -5.0. The bootstrap current profiles aligne very well withthe total current profiles (Fig. 10). In case of βN = 5:0 the bootstrap current fraction exceeds50%. The growth rates as function of the ideal wall distance and the toroidal mode numberare plotted for two βN-values in Figs 11 and 12. As expected, the growth rates increase with33rd EPS 2006; E.Strumberger et al. : Analysis of beta-limits in tokamak reactor scenarios 3 of 4increasing plasma beta, whereas the stabilizing distance of the wall is reduced. For n=1-4the growth rates as function of the wall distance are shown for βN=3.9 in Fig. 13. Both, thegrowth rate and the stabilizing wall distance decrease with rising n.Conclusions. Within the framework of linear MHD theory it is possible to design high-βtokamak equilibria with appropriate profile and magnitude of the bootstrap current, and de-sirable stability properties. The discussed optimized equilibrium is at least stable up to βN=5,and the bootstrap current fraction exceeds 50%. The shape of the bootstrap current is wellaligned with the total current profile. Nevertheless, none of the investigated equilibria is sta-ble without external wall. This result underlines the need of stabilization structures, that is,resistive wall plus feedback system, in order to reach stable high-β plasma equilibria. Thesudies of various types of equilibria further show that also modes with n>2 may play an im-portant role. Usually, the stabilizing distance of the external wall decreases with increasingtoroidal mode number. Some of the equilibria become more and more unstable with increas-ing toroidal mode number. This is due to their steep pressure gradient at the plasma boundary.The transport consistent equilibrium demonstrates that if an equilibrium turns out to be idealstable, its stability behaviour with respect to resistive modes should also be investigated indetail. In linear ideal MHD theory only equilibria with rational surfaces outside the plasmaboundary (m/n> qb) can be unstable with respect to external kink modes. That is, an equi-librium limited by a separatrix (q!∞) would be stable with respect to these modes. For thepresented ideal MHD stability studies we used hypothetical plasma configurations with finiteq-value at the plasma boundary, namely qb = 3:8  4:2, and plasma shapes (no separatrix)with elongation E = 1:70  1:96 and triangularity ∆ = 0:35  0:57. But, whether a plasmais stable with respect to an external kink mode, or whether this mode can be stabilized byan external wall located in a technically feasible distance, depends sensitively on the choiceof these parameters. Stability computations for the same core plasma, but slightly differentplasma boundaries yield different results. Therefore, more realistic computations should beperformed. As a first step, free-boundary equilibria should be calculated in order to obtainprofound information on the overall equilibrium. Further, in contrast to the assumptions ofthe used ideal MHD model, there is a smooth transition from an almost ideal core plasma tothe surrounding non-conducting vacuum. In the boundary region of a real plasma the resis-tivity increases continously due to the decreasing temperature. And, the external wall is alsoresistive. For future computations we therefore suggest to take these resistivities into accountand to perform the stability studies for plasma boundaries sufficiently close to the separatrix.References[1] Hirshman, S.P. et al., Comput. Phys. Commun. 43 (1986) 143.[2] Strumberger, E. et al., Nuclear Fusion 45 (2005) 1156.[3] Pereverzev, G., Yushmanov, P.N., Laboratory Report IPP 5/98, Garching 2002.[4] Kessel, C.E., Nuclear Fusion 34 (1994) 1221.[5] Kessel, C.E. et al., Fusion Engineering and Design 80 (2006) 63.33rd EPS 2006; E.Strumberger et al. : Analysis of beta-limits in tokamak reactor scenarios 4 of 4

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Analysis of ß-limits in tokamak reactor scenarios

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