Benchmark of the Bootstrap Current Simulation in Helical Plasmas

Abstract

The importance of the parallel momentum balance on the bootstrap current evaluation in non-axisymmetric systems is demonstrated by the benchmarks among the local drift-kinetic equation solvers, i.e., the Zero-Orbitwidth (ZOW) model, DKES, and PENTA. The ZOW model is extended to include the ion parallel mean flow effect on the electron-ion parallel friction. Compared to DKES code in which only the pitch-angle-scattering term is included in the collision operator, PENTA code employs the Sugama-Nishimura method to correct the momentum balance. The ZOW model and PENTA codes, both of which conserve the parallel momentum in like-species collisoins and include the electron-ion parallel frictions, agree each other well on the calculations of the bootstrap current. The DKES results without the parallel momentum conservation deviates significantly from those from the ZOW model and PENTA. This work verifies the reliability of the bootstrap current calculation with the ZOW model and PENTA for the helical plasmas. The study of the bootstrap current is necessary to reproduce accurately the MHD equilibrium for high-beta plasmas. For the axisymmetric magnetic geometry, reliable analytic formulas of bootstrap current is available [1]. For the non-axisymmetric system, one needs to rely on numerical methods to evaluate the bootstrap current, which is complicatedly dependent on the magnetic geometry, the collision frequency, and the radial electric field. The past studies [2] presented the benchmark between the Monte-Carlo global model VENUS+δ f and the local semi-analytical solution SPBSC [3] in LHD. The bootstrap current between the VENUS+δ f and the SPBSC codes shows a systematic difference. Although the difference may be caused in part by the finite-orbit-width effect, a missing discussion in that paper is about the treatment of collision term. The VENUS+δ f code did not treat the friction force between electrons and ions, while SPBSC solved the balance between parallel viscosity and friction force as shown in later in Eq. (3) by analytic formula. In order to carry out a more direct investigation on the impact of the parallel friction on the bootstrap current calculations, this paper performs the benchmark among the ZOW model [4], DKES [5], and PENTA [6], which are all based on local neoclassical models.The ZOW model [7] solves the radially-local driftkinetic equation by the δ f Monte-Carlo method, and the parallel friction F is treated as follows. For the likeauthor's e-mail: huang.botsz@nifs.ac.jp species collisions, the linearized collision operators are employed and this satisfies the parallel momentum balance, i.e., F ,ee = F ,ii = 0. For ion, the ion-electron friction F ,ie is neglected because of the large mass ratio, m e /m i 1. For electron, in the previous work, the electron-ion collision was only approximated as the pitchangle scattering operator with the stationary background Maxwellian ion distribution, i.e., C ei L ei . In the present work, not only the pitch-angle scattering but also the ion parallel mean flow U ...

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Measurements of the parameter dependencies of the bootstrap current in the W7-X stellarator

“…Various stellarators report on measuring or neoclassically calculating their bootstrap current, and also on benchmarking between codes. At the Wisconsin Levitated Octupole [20], Proto-Cleo [21], Heliotron-E [22], Uragan-3 [23], Wendelstein 7-A [24], LHD [25,26] ATF [27], Wendelstein 7-AS [28], CHS [29], Heliotron-J [30], HSX [31], Uragan-3M [32] and TJ-II [33,34] measurements and neoclassical calculations generally showed good agreement, with only one of the above [29] as an exception.…”

“…Various stellarators report on measuring or neoclassically calculating their bootstrap current, also benchmarking between codes. W.L.OCTUPOLE [Zarnstorff 1984], Proto-Cleo [Treffert 1984], Heliotron-E [Besshou 1984], Uragan-3 [Gutarev 1987], Wendelstein 7-A [Gasparino 1988], LHD [Okamoto 1990, Huang 2017, ATF [Murakami 1991], Wendelstein 7-AS [Erckmann 1992], CHS [Isobe 2002], Heliotron-J [Nagasaki 2006], HSX [Schmitt 2007], Uragan-3M [Pashnev 2008], TJ-II [Velasco 2011, Camacho 2017. Measurements and neoclassical calculations generally showed good agreement, with only one of the above [Isobe 2002] as an exception.…”