Cooper, W.

Graves, J.

Hirshman, S.

Merkel, P.

English

MPG.PuRe

Free Boundary Three-Dimensional Anisotropic Pressure EquilibriaW.A. Cooper1, J.P. Graves1, S.P. Hirshman2, P. Merkel31 Ecole Polytechnique Fédérale de Lausanne (EPFL),Centre de Recherches en Physique des Plasmas (CRPP), Lausanne, Switzerland2 Fusion Energy Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA3 Max Planck Institut für Plasmaphysik, Garching, GermanyThe VMEC2000 code has been extended to model anisotropic pressure three-dimensionalequilibria. A variant of a bi-Maxwellian distribution function is applied to describe the ener-getic particle species such that it satisfies the constraint imposed by the lowest order solution ofthe Fokker-Planck equation that the hot particles are uniformly distributed along the magneticfield lines. This particular anisotropic pressure model has been previously adapted to a fixedboundary version of the VMEC code [1]. The implementation in VMEC2000, which we havelabelled as the ANIMEC (Anisotropic Neumann Inverse Moments Equilibrium Code) also treatsfree boundary conditions previously considered in an earlier version of the VMEC code [2].We have tested the energetic particle driven anisotropic pressure model that has been developedon a 2-field period quasiaxisymmetric stellarator (QAS) [3] with off-axis fast particle depo-sition which provides very stringent conditions for convergence. Choosing〈β〉 ≃ 4.5%, with〈β〉 ≡∫d3xµ0(p|| + p⊥)/∫d3xB2 wherep||(p⊥)corresponds to the parallel (perpendicular)pressureof the entire plasma, we specifically consider the following four cases:1. High field (HF) side deposition of energetic particles about a layer around a critical fieldBC = 4.9T with p⊥ > p||(p⊥/p|| ∼ 3.4),2. Low field (LF) side deposition of fast particles about a layer around a critical fieldBC =4.2T with p⊥ > p||(p⊥/p|| ∼ 3.4),3. High field (HF) side deposition of energetic particles about a layer around a critical fieldBC = 4.9T with p|| > p⊥(p||/p⊥ ∼ 3.4),4. Low field (LF) side deposition of fast particles about a layer around a critical fieldBC =4.2T with p|| > p⊥(p||/p⊥ ∼ 3.4).Under vacuum conditions, the equilibrium calculations closely recover the vacuum magneticflux surfaces obtained from Poincaré plots resulting from magnetic field line tracing. At finiteβ (〈β〉 ≃ 4.5%), the entire plasma column shifts significantly away from the major axis. The35th EPS Conference on Plasma Phys. Hersonissos, 9 - 13 June 2008 ECA Vol.32D, P-4.061  (2008)flux surface shapes change with pressure, but are not significantly affected byp||/neqp⊥. In factthe cases we have examined are indistinguishable except for the LFp⊥ > p|| example whichballoons slightly further away from the major axis. This is particularly noticeable in the bean-shaped up-down symmetric cross-section.The convergence towards an equilibrium state is measured by following the residual horizontalforce. Typically when this force reaches a level of 1× 0−9, the matric preconditioner is acti-vated. Under vacuum conditions, this is achieved in under 1000 iterations and the preconditionerreduces the residual force to below 1× 0−17 within 20 iterations. At finite〈β〉 ≃ 4.5%, it takesabout 3400 iterations before reaching the preconditioner triggering level of a residual force of1×10−9 after which the force level decreases below 1× 0−16 in less than 10 iterations. Theexception is the LFp⊥ > p|| case which converges much more slowly requiring 6440 iterationsto trigger the preconditioner. Nevertheless, we still reach after that a comparably small residualhorizontal force under action of the preconditioner. This compares favourably with a previouslydeveloped poorly conditioned anisotropic pressure version of the VMEC code where it is diffi-cult to obtain residual horizontal force levels of 1×10−9.To verify the quality of the equilibrium state achieved, we calculate aposteriori the flux sur-face averaged radial force balance (appropriately normalised). For axisymmetric systems, thiscorresponds to the Grad-Shafranov equation. At〈β〉 ≃ 4.5%, the poorly conditioned solutionobtained with the earlier anisotropic pressure VMEC code version only reaches levels of 10−3-10−5 for the four cases investigated. With the current fully preconditioned ANIMEC code, theradial force balance error ranges around 10−8-10−11. The absolute values of the flux surfaceaveraged radial force balance error profiles for the non-preconditioned and the preconditionedfinite β equilibria we have computed are shown in Fig. 2). Specifically depicted are the radialforce balance error profiles with hot particle deposition on the HF side with(p⊥/p||∼3.4)(bluecurves), on the LF side with(p⊥/p|| ∼ 3.4)(red curves), on the HF side with(p||/p⊥ ∼ 3.4)(magenta curves) and on the LF side with(p||/p⊥ ∼ 3.4)(green curves). The solid (dotted)lines correspond to the fully preconditioned (non-preconditioned) results. That is, the precon-ditioned result achieves error levels that are 5 orders of magnitude than the poorly conditionedresult of the previously developed code. The improvements realised enhances the confidence ofthe reliability and accuracy of computed ANIMEC equilibria for further stability and transportanalyses.Under current-free conditions, the rotational transform decreases significantly at finiteβ (〈β〉 ≃4.5%). This decrease is more important for the cases withp|| > p⊥ at least in the outer 3/4 ofthe plasma volume. Forp|| > p⊥, the rotational transform profile with HF fast particle deposi-35th EPS 2008; W.A.Cooper et al. : Free Boundary Three-Dimensional Anisotropic Equilibria 2 of 4tion almost identically matches that for LF hot particle depositi n. However, forp⊥ > p||, theLF energetic particle deposition case has a transform profile larger (smaller) in magnitude thanthat of HF fast particle deposition in the inner (outer) third of the plasma volume.The magnetic well at〈β〉 ≃ 4.5% becomes stronger in the outer 80% of the plasma volume,but develops a magnetic hill in the inner 20%. Forp|| > p⊥, the differential volume is indis-tinguishable for LF versus HF energetic particle desposition. The differential volume profile ofthe p⊥ > p|| cases is larger than those forp|| > p⊥ (except very close to the magnetic axis).Furthermore, the magnetic well (hill) for the LFp⊥ > p|| case is stronger than that of the HFp⊥ > p|| case in the outer 80% (inner 20%) of the plasma volume.For p|| > p⊥, the hot particle pressure contours do not differ significantly whether high fieldor low field fast particle deposition is applied and the pressures remainmore or less uniformon a flux surface. On the other hand, forp⊥ > p||, the energetic particle perpendicular pressurecontours concentrate in the region of deposition, for example on the high field side for HF depo-sition (see Fig. 2). This can be understood by the fact that the trapped hot particles spend mostof their orbit time around their deposition region locally enhancing the perpendicular pressure.For p⊥ > p||, the energetic particle parallel pressure contours localise in the low field region.AcknowledgmentsThis research was partially sponsored by the Fonds National Suisse de la Recherche Scientifiqueand Euratom. We thank J. Kisslinger and H. Wobig for providing the filament coils for thesystem investigated.References[1] W.A. Cooper, J.P. Graves, S.P. Hirshman et al., Nucl. Fusion46, 683 (2006)[2] S.P. Hirshman, W.I. van Rij and P. Merkel, Comput. Phys. Commun.43 143 (1986)[3] S. Okamura, K. Matsuoka, M. Fujiwara, M. Drevlak, P. Merkel and J. Nührenberg,J. Plasma Fusion Res.1, 164 (1998)35th EPS 2008; W.A.Cooper et al. : Free Boundary Three-Dimensional Anisotropic Equilibria 3 of 40 0.2 0.4 0.6 0.8 110−1510−1010−5100flux surface averaged radial force balances<β>∼  4.5%HF p⊥>p||LF p⊥>p||HF p||>p⊥LF p||>p⊥non−preconditionedpreconditionedFigure 1: The absolute values of the flux surface averaged radial force balance errorprofiles8 10 12 14 16−6−4−20246v=π/2RZph⊥ contours in 2−period QAS system0.10.20.30.40.50.68 10 12 14 16−6−4−20246v=π/2RZph⊥ contours in 2−period QAS system0.20.40.6Figure 2: The hot particle perpendicular pressure contours for HF (left) and LF (right)deposition35th EPS 2008; W.A.Cooper et al. : Free Boundary Three-Dimensional Anisotropic Equilibria 4 of 4

Free Boundary Three-Dimensional Anisotropic Pressure Equilibria

Cooper, W. A.

Graves, J. P.

Hirshman, S. P.

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Free Boundary Three-Dimensional Anisotropic Pressure Equilibria

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