Geiger, J.

Kasilov, S.

Kernbichler, W.

Nemov, V.

Seiwald, B.

Suzuki, Y.

English

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Calculations of on effective ripple for a stellarator magnetic field computed by the HINT2 code

Nemov, V. V.

Kasilov, S. V.

Max Planck Society eDoc Server

Calculations of an effective ripple for a stellarator magnetic fieldcomputed by the HINT2 code∗V. V. Nemov1,2, S. V. Kasilov1,2, W.Kernbichler2, B. Seiwald2, Y. Suzuki3, J. Geiger41Institute of Plasma Physics, National Science Center “Kharkov Instituteof Physics and Technology”, Ukraine2Association EURATOM-ÖAW, Institut für Theoretische Physik-Computational Physics,TU Graz, Austria3National Institute for Fusion Science, Japan4Association EURATOM-IPP Greifswald, GermanyIntroductionIn numerical studies of plasma confinement in stellarators it is desirable to use the mostrealistic magnetic field of the device. Especially, this is necessary for a final stage where itis desirable to use real-space configurations and to take finite plasma pressure and externalcurrents into account. As of today, there exist two three-dimensional MHD finite beta equilib-rium codes, PIES [1] and HINT [2], which are used for stellarator plasma equilibrium stud-ies in real-space coordinates. These codes were developed without the a priori assumption ofnested magnetic surfaces. However, the requirement of large computer resources (large amountof memory, high speed supercomputers) is characteristic for these codes. Recently, the codeHINT has been strongly improved in computational speed, convergence and memory usage,resulting in code HINT2 [3]. The magnetic field,B in HINT2 can be presented in real-spacecoordinates and it can be conveniently used for an investigation of stellarator confinementproperties. In this work, the effective ripple,εeff, as measure of 1/ν-transport is computed forW7-AS and W7-X equilibria with various averageβ -values. Those equilibrium data are pro-vided as output from the HINT2 code. Theεeff computations are performed using the codeNEO [4].Magnetic field data and magnetic surfacesHere, a number of sets of HINT2 data forB are used, which had been obtained as results ofan equilibrium study for W7-AS withβ values of 0.5%, 1.0% and 1.5%, as well as for W-7Xwith β≈2% andβ≈4%. The HINT2 output forB is represented in cylindrical coordinates,ρ,ϕ,z. The dependencies ofBρ ,Bϕ ,Bz on ρ,ϕ andz are given on a three-dimensional grid ofdiscrete mesh points. Between these mesh points theB components can be computed using aninterpolation procedure. For this procedure either three-dimensional cubic splines or Lagrangepolynomial interpolation are used.The magnetic surfaces obtained as results of a field line following code are presented in Figs. 1and 2 for W7-AS (β = 0.5% and 1.5%) and W7-X (forβ≈ 2% and 4%), respectively. Itfollows from Fig. 1 that for W7-AS with increasingβ inner islands corresponding toι = 5/9(ι is the rotational transform) and corrugation of the outer magnetic surfaces are increasing. Inaddition, the magnetic axis is slightly shifted towards the outer side of the torus. The magneticconfiguration forβ = 1% is not shown in Fig. 1 since it is of intermediate character with respect∗This work, supported by the European Communities under the contract of Association between EURATOMand the Austrian Academy of Sciences, was carried out within the framework of the European Fusion Devel-opment Agreement. The views and opinions expressed herein do not necessarily reflect those of the EuropeanCommission. Additional funding is provided by the Austrian Science Foundation, FWF, under contract numberP16797-N08.34th EPS Conference on Plasma Phys. Warsaw, 2 - 6 July 2007 ECA Vol.31F, P-4.063  (2007)to the presented configurations. One can note that the shapes of the presented magnetic surfacecross-sections are qualitatively in good agreement with results for configurations having aboundaryι-value of 5/9 in Refs. [5] and [6].Fig. 1. Magnetic surfaces of W7-AS in theϕ = 0 plane and after a half of the field period forβ = 0.5%(left, "w7as-d-p05") andβ = 1.5% (right, "w7as-d-p15"). Sizes are given in meters.From Fig. 2 it can be seen that in W7-X the confinement region (shown in black) is surroundedby a stochastic region (shown in red) which contain rather big magnetic islands withι = 5/5.Close to the boundary of the confinement region, small magnetic islands are seen which aresmaller forβ≈ 2%. Forβ≈ 4% the results in Fig. 2 are in good agreement with the results forthe finiteβ equilibrium of the high mirror configuration (β ≈ 4.36%) in Ref. [7].Effective ripple computationsTo study the 1/ν transport for these configurations the numerical field line following codeNEO [4] is used. For an arbitrary stellarator type magnetic field, this code computes the param-eterε3/2eff with εeff being the so called equivalent helical or effective ripple. The 1/ν transportcoefficients are proportional toε3/2eff and it is the part which contains all the influence of themagnetic field geometry.The computational results forε3/2eff are presented in Fig. 3 (left) as functions of the distance,ξ , of starting points of integration from the magnetic axis in theϕ = 0 plane. Starting pointsfor every magnetic surface are chosen in theϕ = 0 plane at the outer side of mid plane withrespect to the magnetic axis. The results are obtained for integration intervals correspondingto 250 field periods for intact magnetic surfaces and to 3000 field periods for island and nearisland magnetic surfaces. It follows from the results thatε3/2eff values for W7-AS are rathersmall near the magnetic axis. With increasingξ these values are increasing and come close tothe level for stellarators which are not-optimized with respect to neoclassical transport (see,e.g., [4]). In this outer region they are much bigger than those for W7-X (plots 4 and 5 inFig.3). With increasingβ ε3/2eff for W7-AS also increases. For W7-X,ε3/2eff is somewhat biggerfor β ≈ 2% than forβ ≈ 4% in the central part of the confinement regions. The boundaries ofthe confinement region areξ ≈ 0.2 (β ≈ 4%) andξ ≈ 0.24 (β ≈ 2%). In the stochastic regionsat higherξ much largerε3/2eff results are obtained which are shown conditionally with rhombi.However, when the starting points of integration fall into regions of islands atι = 5/5, bigbut finite values ofε3/2eff are obtained. Within the confinement regions, for W7-AS as well as34th EPS 2007; B.Seiwald et al. : Calculations of an effective ripple for a stellarator magnetic field compute... 2 of 4for W7-X irregularities ofε3/2eff are seen for island magnetic surfaces and in their vicinity. ForW7-X this effect is more pronounced in theβ ≈ 4% case.Fig.2. Magnetic surfaces of W-7X in theϕ=0 plane (left, right) and after a half of the field period(middle) forβ≈4% (left, middle) andβ≈2% (right). Sizes are given in meters.Fig.3. Parametersε3/2eff (left) andreff (right) as functions ofξ=Rst−Rax; 1: W7-AS,β = 0.5%, Rax ≈1.994 (black); 2: W7-AS,β = 1.0%, Rax ≈ 2.003 (blue); 3: W7-AS,β = 1.5%, Rax ≈ 2.013 (red);4: W7-X, β ≈ 4%, Rax ≈ 5.98 (black); 5: W7-X,β ≈ 2%, Rax ≈ 5.94 (blue); rhombi mark stochasticregions in W7-X.Rst andRax are starting point of integration and magnetic axis position, resp., in theϕ = 0 plane.Effective radiusTogether withε3/2eff , an auxiliary parameterreff is also calculated. This parameter is definedasreff =2VS, (1)withVS=13limLs→∞Ls∫0dsBr · ∇ ψLs∫0dsB|∇ ψ|−1. (2)Here,S is the magnetic surface area andV is the volume enclosed by this surface. The ratioV/S is computed using integration over the magnetic field line length,s, as it is seen from (2),34th EPS 2007; B.Seiwald et al. : Calculations of an effective ripple for a stellarator magnetic field compute... 3 of 4whereψ is the magnetic surface label andr is the radius vector. The integration interval is thesame as forε3/2eff computations. The computational results forreff are presented in Fig. 3 (right).For magnetic surfaces with nearly circular cross-sections,reff is close to the mean radius of themagnetic surface cross-section. In general,reff is a convenient approximation of this quantity.However, such an estimate can not be valid for complicated shapes of magnetic surfaces. It isseen from Fig. 3 that for W7-AS when approaching the plasma edgereff decreases whereasthe mean radius of the magnetic surface cross-section should increase there. The reason forthis is the fact that the areaS increases faster thanV because of increasing corrugation of themagnetic surface shape (see Fig. 1). In addition, rapid drops ofreff in Fig. 3 are clear signs ofmagnetic islands.Note that with increasingV the energy content of the plasma is increasing. At the same time theenergy losses are increasing with increasingS. Therefore, theV/Sratio can be considered as anadditional parameter for stellarator optimization. An increase ofV/S corresponds to a betteroptimization and a decrease ofreff in the vicinity of the plasma boundary is an undesirableeffect from this viewpoint.SummaryFor the analysis five sets of HINT2 data forB are used, which had been obtained as resultsof an equilibrium study for W7-AS withβ values of 0.5%, 1.0% and 1.5% (boundaryι of 5/9[5,6]), as well as for W7-X withβ ≈ 2% andβ ≈ 4% (high mirror configuration [7]). TheHINT2 output forB is given in cylindrical coordinates,ρ,ϕ,z. The dependencies ofBρ ,Bϕ ,Bzon ρ,ϕ andz are given on a three-dimensional grid of discrete mesh points. Between thesemesh points theB components were computed using interpolation either by three-dimensionalcubic splines or Lagrange polynomial interpolation. From the computational results it followsthat for W7-AS configurationsε3/2eff increases with increasingβ . For W-7X,ε3/2eff is somewhatbigger forβ ≈ 2% than forβ ≈ 4%. An increase ofε3/2eff is found for magnetic islands and fornon-island magnetic surfaces which are close to islands. One has also to note that an undesir-able effect of a decrease of theV/S-ratio takes place for W7-AS in the vicinity of the plasmaboundary.The obtained results demonstrate the possibility of using HINT2 data for study of neoclassicaltransport in stellarators.References[1] A. H. Reiman and H. Greenside, J. Comput. Phys.75, 423 (1988)[2] K. Harafuji, et al., J. Comput. Phys.81, 169 (1989)[3] Y. Suzuki, et al., Nucl. Fusion46, L19 (2006)[4] V. V. Nemov, S. V. Kasilov, W. Kernbichler and M. F. Heyn, Phys. Plasmas,6, 4622(1999).[5] J. Geiger, T. Hayashi, Proc. 29th EPS Conf. on Plasma Phys. and Contr. Fusion (Mon-treux), 17-21 June 2002 ECA Vol.26B, P-5.035 (2002).[6] J. Geiger, Y. Suzuki, Proc. 33rd EPS Conf. on Plasma Phys. Rome, 19-23 June 2006 ECAVol.30I, P-2.115 (2006).[7] Y. Suzuki, 2nd Joint meeting of US-JAPAN Workshop and Kyoto University 21st Cen-tury COE symposium on "New approaches in plasma confinement experiments in helicalsystems", Nov. 13-15, 2006.34th EPS 2007; B.Seiwald et al. : Calculations of an effective ripple for a stellarator magnetic field compute... 4 of 4

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Calculations of on effective ripple for a stellarator magnetic field computed by the HINT2 code

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