Inverting the Biot -Savart integral in the context of stellarator coil determination

by Schmitt, Ronald Frederick, Ph.D., COLUMBIA UNIVERSITY, 2008, 174 pages; 3292145


The design of power-efficient stellarator coils that are practical to engineer and allow access to the plasma is a difficult task that involves mathematical and physics issues in addition to the engineering concerns. The complexity of stellarator coils is a major factor in the cost and viability of stellarator fusion reactors. Coil determination is a complex inverse problem with many subtleties. The problem of coil determination is tantamount to inverting the Biot-Savart integral of magnetostatics. The inverse of the Biot-Savart operator tends to be ill-conditioned, dramatically emphasizing high-order Fourier modes. Using a simplistic method to invert this integral results in coils of impracticable shape and current magnitude. Two methods are introduced to optimize stellarator coil determination. The first technique involves the application of linear perturbation theory to the coil design problem. First, the flux modes that cause the N0 largest perturbations are determined, where N0 is an adjustable parameter that in practice is much smaller than the total number of Fourier modes. The next step is to project the coil equations into the smaller mathematical space of these dominant modes. This technique reduces the constraints on the coils such that the set of equations only includes those flux modes that are needed to approximate the shape of the plasma surface. This technique aims to produce smoother coils by avoiding the over-constrained nature of other methods. The second technique involves rank-ordering and Gram-Schmidt orthogonalization of the Fourier modes, Gmn (&thetas; p, ϕp), of the Biot-Savart integral between two surfaces. The rank-ordering of the modes in terms of magnitude, |Gmn|, gives a natural way to decompose the current in terms of the most influential modes. The solution for the coil current is then constrained to only use the most influential modes. The two methods are applied to the extant Helically Symmetric Experiment (HSX) stellarator and results are compared with the actual coil set used in HSX.

AdviserAllen H. Boozer
Source TypeDissertation
SubjectsPlasma physics
Publication Number3292145

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