A condition for small bootstrap current in three-dimensional toroidal configurations
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A Condition for Small Bootstrap Current in Three-Dimensional Toroidal Configurations1 M. I. Mikhailova, *, J. Nührenbergb, and R. Zillec a National
Russian Research Center Kurchatov Institute, Moscow, 123182 Russia für Plasmaphysik, Greifswald, 17491 Germany c Max-Planck-Institut für Plasmaphysik, Garching, 85748 Germany *e-mail: [email protected]
b Max-Planck-Institut
Received December 24, 2015; in final form, April 21, 2016
Abstract—It is shown that, if the maximum of the magnetic field strength on a magnetic surface in a threedimensional magnetic confinement configuration with stellarator symmetry constitutes a line that is orthogonal to the field lines and crosses the symmetry line, then the bootstrap current density is smaller compared to that in quasi-axisymmetric (qa) [J. Nührenberg et al., in Proc. of Joint Varenna−Lausanne Int. Workshop on Theory of Fusion Plasmas, Varenna, 1994, p. 3] and quasi-helically (qh) symmetric [J. Nührenberg and R. Zille, Phys. Lett. A 129, 113 (1988)] configurations. DOI: 10.1134/S1063780X16110052
Three-dimensional configurations offer the possibility to implement the island divertor concept [1]. Configurations with small bootstrap current facilitate the operation of an island divertor. In Wendelstein 7-X [2], the small bootstrap current is provided by a suitable adjustment of the toroidal and helical plasma curvatures [3]. Here, we consider a different condition for reducing the bootstrap current. It is shown that, if the maximum of the magnetic field strength on a magnetic surface in a configuration with stellarator symmetry2 constitutes a line that is orthogonal to the field lines and crosses the symmetry line, then the bootstrap current density is smaller compared to that in quasi-axisymmetric (qa) [4, 5] and quasi-helically (qh) symmetric [6, 7] configurations, which could be exploited in a future stellarator design. A standard method for assessing the long-meanfree-path limit of the bootstrap current density on a magnetic surface in a three-dimensional magnetic confinement configuration is the evaluation of the socalled bootstrap current coefficient G b , conveniently summarized, e.g., in [8] and more directly derived in [9]. A key ingredient of this procedure is the consideration of the field line passing through the maximum of the field strength, Bmax , on the considered magnetic surface and evaluation of the particle motion along this field line. Accordingly, the mechanism for boot1 The article was translated by the authors. 2 In the case of stellarator symmetry, the points
with cylindrical coordinates r , ϕ, z and r , −ϕ, −z are equivalent; the line z = 0, ϕ = 0 (or π) is then a symmetry line.
strap current generation can be visualized by considering particle orbits barely passing the point of the maximum field strength. Figure 1 shows such orbits for a qa [4] and a qh [6] configuration and illustrates why the bootstrap current density is directed opposite in the qa and qh configurations. In nearly quasi-isodynamic (qi) [10, 11] configuratio
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