Near-magnetic-axis geometry of a closely quasi-isodynamic stellarator
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NearMagneticAxis Geometry of a Closely QuasiIsodynamic Stellarator1 M. I. Mikhailova, J. Nührenbergb, and R. Zilleb a
National Research Centre Kurchatov Institute, pl. Akademika Kurchatova 1, Moscow, 123182 Russia b MaxPlanckInstitut für Plasmaphysik, EURATOM Assoziation, Teilinstitut Greifswald, Wendelsteinstr. 1, Greifswald, 17491 Germany email: [email protected] Received June 4, 2013
Abstract—It is shown that the condition of stationary second adiabatic invariant at the magnetic axis of a stel larator configuration can be satisfied to good accuracy for all reflected particles. DOI: 10.1134/S1063780X14010073 1
1. INTRODUCTION
magnetic coordinates, the relevant equation [1, 5] reads2
In the course of the introduction of quasiisodyna micity (QI) [1], deeply to moderately deeply reflected particles trapped in one period of a stellarator were considered. Inclusion of all reflected particles led to configurations with poloidally closed contours of the magnetic field strength on magnetic surfaces [2]. An integrated physics optimization of such a configura tion led to promising neoclassical and magnetohydro dynamic properties [3]. An analysis of the QI quality of this configuration near its magnetic axis was per formed in [4] and showed that QI is qualitatively but not closely realized. Here, it is investigated whether the QI property to the lowest order in the distance from the magnetic axis, i.e., the stationarity of the sec ond adiabatic invariants for all reflected particles, can be achieved more closely. A twostage procedure is adopted; first, this prob lem is investigated in terms of the structure of the mag netic field strength in magnetic coordinates near the magnetic axis; then the solution is substantiated by finding the associated lowest order fluxsurface geom etry.
cos ιφ
∑ b cos ( nφ ) + sin ιφ ∑ b sin ( nφ ) = 0 . n
n
(1)
Obviously the equation does not have a smooth solu tion. With the convention that φ = 0 correspond to the minimum of B0(φ) and φ = π to its maximum, the area around φ = 0 determines the behavior of the deeply trapped particles, the area around φ = π determines the behavior of the barely trapped ones. If the plasma β is not very large, deeply trapped particles are most endangered to be lost by radial drift. Barely trapped/transitional particles can get lost through col lisionless diffusion. So, it might be useful to select approximate solutions satisfying Eq. (1) near the extrema of B0. This is done here and shown that the QI condition can be closely approximated with few first order Fourier components of the field strength. The three equations for firstorder accuracy (in the dis tances from the endpoints 0 and π) are ∑ bn = 0 ,
∑(−1)
n
∑
n
bn = 0 , (−1) nbn = 0 . In addition, the derivative at φ = 0 of the odd part of B along a field line is required to vanish, ∑ nbn = 0 , to avoid an extra local minimum of B near φ = 0.
2. PROCEDURE For the type of configurations discussed here, with stellarator symmetry and poloidally
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