Bonnin, X.

Mutzke, A.

Nührenberg, C.

Nührenberg, J.

Schneider, R.

English

MPG.PuRe

Calculation of magnetic coordinates for stellarator fields

Calculation of magnetic coordinates for stellarator fieldsX. Bonnin, A. Mutzke, C. Nührenberg, J. Nührenberg, R. SchneiderMax-Planck-Institut für Plasmaphysik, EURATOM-IPP Association,Wendelsteinstraße 1, D-17491 Greifswald, GermanyWe present an extension of the Nemov algorithm [1] to compute magneticcoordinates in stellarators and their associated metric coefficients using onlyfield line tracing, and covering not only the core of the plasma, but alsoisland chains and scrape-off layer regions in the plasma edge. The algorithmis applied to the full W7-X vacuum field geometry The method has beenoptimized for numerical accuracy by minimizing recourse to derivatives ofderived quantities. The only requirements are the foreknowledge of themagnetic field and its topological structure.1 IntroductionThe magnetic coordinate system for a given magnetic field plays a fundamental role inthe study of confined plasmas in stellarator devices. Very often, physics problems ofinterest are only tractable provided one works in a suitable magnetic coordinate system,for example one in which the field lines are straight [2, 3], or where some symmetryor quasi-symmetry of the system is made apparent [4]. Several methods have beendevelopped over the years to establish relevant coordinates and compute geometricalquantities such as metric coefficients for different magnetic field configurations [1, 3,5, 6, 7]. These are best adapted to cases with nested closed flux surfaces. We presenthere the extension of one such method [1] to the plasma edge of stellarators such asW7–X [8], generalized so as to be able to treat the presence of separatrices and ofdiverted islands. The method requires only a foreknowledge of the (total) magneticfield, and is optimized for numerical accuracy. It is however limited to non-ergodicfields. Ultimately, it delivers the full set of magnetic coordinates and metric coefficients.These can then be further used for plasma modelling, for example in the 3-D finite-volume plasma fluid code BoRiS [9].2 General algorithmOur goal is to obtain, starting from the magnetic field information only, and with goodnumerical accuracy, a set of Boozer-like coordinates, applicable throughout the edgeplasma of complicated devices like the W7–X stellarator, including the treatment of30th EPS Conference on Contr. Fusion and Plasma Phys., St. Petersburg, 7-11 July 2003 ECA Vol. 27A, P-4.10separatrices and island chains. The method described here involves only field line trac-ing, and minimizes the need for computing spatial derivatives of derived quantities. Werefer to our end coordinates as the BoRiS coordinate system, as the algorithm was ini-tially developped for use by the BoRiS code, but without loss of generality. We use botha Cartesian (x, y, z) and a cylindrical (R,ϕ, z) coordinate system for the spatial vari-ables. The Boozer coordinate system is noted (ψ, θ, ζ) and the BoRiS coordinate sys-tem (s, ϑ, φ). We assume the (non-ergodic) magnetic field is given on a cylindrical gridwithin a annular box domain [B] ranging over [Rmin . . . Rmax; 0 . . . 2π;Zmin . . . Zmax].In essence, we will be mapping the spatial box [B] to m + 1 boxes in Boozer space,with m domains ranging over[0 . . . FT (Ψislandsep ); 0 . . . 2π; 0 . . .2π]for the m islands ofthe plasma edge and one domain, covering [0 . . . FT (ΨSOL); 0 . . . 2π; 0 . . . 2π], which in-cludes both the core and SOL regions (beyond the island chain), where FT (Ψislandsep ) andFT (ΨSOL) represent the total toroidal flux enclosed within the island separatrix or thelast usable flux surface in the SOL, respectively. These domains will then get renormal-ized into m+ (1 + εSOL) unit cubes [0 . . . 1; 0 . . . 1; 0 . . . 1] in BoRiS coordinates, whereεSOL denotes the normalized thickness of the SOL in flux coordinates. This mappingand the calculation of the metric coefficients describing it are the goal of this work.We will thus compute:• The rotational transform: ι(ψ)• The flux surface function: FT ≡ ψ → s and its spatial radial derivatives∂FT∂R∣∣∣∣±z=0ϕ=0• The flux surface derivatives: F ′T ≡dFTdsand F ′P ≡dFPds• The toroidal and poloidal currents: J(ψ) and I(ψ), respectively• The Boozer and BoRiS Jacobians, J (ψ, θ, ζ) and J (s, ϑ, φ) ≡ √g respectively• The angle-like coordinates: θ = 2πϑ and χ ≡∫~B · d~̀→ ζ = 2πφ• The Clebsch vectors: ∇ψ and (∇θ − ι∇ζ)• Secondary metric coefficients: β̃, σ̃, gϑs, gss, gϑs and gφs.The algorithm proceeds as follows. We begin with a first pass at field-line tracing, inwhich we identify the location of the magnetic axes, as well as the islands and SOLstructure. A longer field line trace yields the field-line integral χ. We then re-arrangethe field-line data so as to define the flux surfaces. Next, we compute the rotationaltransform ι, the toroidal flux function FT and its radial derivatives. The toroidal (J)and poloidal (I) currents follow. The Jacobians for both the Boozer and the BoRiScoordinate systems can now be calculated. We continue with the angle-like coordinatesθ, ϑ, ζ and φ. We resort to the Nemov algorithm [1] in order to determine the Clebschcomponents of the field: ∇ψ and its conjugate (∇θ − ι∇ζ). This also yields ∇sand gss information. Lastly, for non-vacuum fields, one must determine the covariantcomponent of the magnetic field, β̃. Appropriate constitutive relations complete theset of needed metric coefficients and physical quantities.30th EPS Conference on Contr. Fusion and Plasma Phys., St. Petersburg, 7-11 July 2003 ECA Vol. 27A, P-4.10 2 of 43 The W7–X vacuum field caseWe consider the standard ι case, where ι tends to 1 at the separatrix, and the edge isinhabited by a 5/5 chain of large islands, which will partly intercept divertor structures.Beyond the chain of islands, there also exists a so-called “laminar” zone, where fluxsurfaces are still well-defined. We refer to this laminar zone as the far Scrape-Off Layer(SOL), in extension of the tokamak nomenclature.We show in Figures contour plots of gss for the two W7-X symmetry planes. It isclearly seen that gss goes to zero at all X- and O-points, but can exhibit rapid spatialvariations.References[1] V.V. Nemov, Calculations of the Magnetic Surface Function Gradient and Asso-ciated Quantities in a Torsatron, Nuclear Fusion 28 (1988) 1727–1736.[2] W.D. D’Haeseleer, W.N.G. Hitchon, J.D. Callen, J.L. Shohet, Flux coordinatesand Magnetic Field Structure, Springer-Verlag, Berlin, 1991.[3] A.H. Boozer, Establishment of Magnetic Coordinates for a given Magnetic Field,Phys. Fluids 25 (1982) 520–521.[4] J. Nührenberg and R. Zille, Quasi-Helically Symmetric Toroidal Stellarators, Phys.Lett. A 129 (1988) 113–117.[5] G. Kuo-Petravic and A.H. Boozer, Numerical Determination of the Magnetic FieldLine Hamiltonian, J. Comp. Phys. 73 (1987) 107–124.[6] A.H. Reiman and H.S. Greenside, Numerical Solution of Three-Dimensional Mag-netic Differential Equations, J. Comp. Phys. 75 (1988) 423–443.[7] J.A. Rome, An Improved Method of Calculating Flux Coordinates, J. Comp. Phys.82 (1989) 348–361.[8] W. Lotz, J. Nührenberg and C. Schwab, Plasma Physics and Controlled NuclearFusion Research, 1990, Nucl. Fusion Suppl. 1991, Vol. 2 (1991) 603.[9] M. Borchardt, J. Riemann, R. Schneider and X. Bonnin, W7–X modelling withthe 3D SOL fluid code BoRiS, J. Nucl. Mater. 290–293 (2001) 546–550.30th EPS Conference on Contr. Fusion and Plasma Phys., St. Petersburg, 7-11 July 2003 ECA Vol. 27A, P-4.10 3 of 44 FiguresFigure 1: The (s, ϑ) gridding of the(ϕ = 0) plane, according to aam10(in red), and VMEC (in dashed blue).Figure 2: The gss variation on the(ϕ = 0) symmetry plane, for the W7–X standard case vacuum field. The is-land values were scaled down by a fac-tor of 100.Figure 3: The rotational transform profilefor the W7–X vacuum field case, includingthe island and outer scrape-off regions.Figure 4: The variation of gss as a func-tion of angle and flux surface index, onthe (ϕ = 0) symmetry plane for theW7-X case. The results of a VMEC runare shown for comparison.30th EPS Conference on Contr. Fusion and Plasma Phys., St. Petersburg, 7-11 July 2003 ECA Vol. 27A, P-4.10 4 of 4

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Calculation of magnetic coordinates for stellarator fields

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