Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton The general results show that, given only poloidal field magnetic measurements (i.e. no diamagnetic signal), then β I and l i can only be separated for significantly non circular equilibria. From our discussions of β I +l i /2, we know we can measure β I +L i /2 exactly anyway. If we have reconstructed the plasma surface then we can evaluate the integrals s 1 and s 2 on that plasma surface, and then L i = l i . For circular equilibria it is found that only the sum β I + l i /2 can be found. Of course, adding a diamagnetic measurement allows the separation for any shape, as we have seen. Also adding a pressure profile form, or the pressure on axis, is enough to allow separation even for circular discharges. It is generally found that a distortion of the shape to b/a ~ 1.3 is needed to make the separation. A significant problem is that, having made specific assumptions about the form of the current density profile, we do not know how general our results are. 114
Magnetic fields and tokamak plasmas Alan Wootton 16. FAST SURFACE RECONSTRUCTION Here we want to investigate 'fast' methods of determining the plasma boundary, and fields on that boundary. If we can do this, then we can exactly derive parameters such as β I + l i /2, and q, on the boundary, as well as parameterizing the boundary shape itself. Here we specifically do not want to obtain any information on fields and fluxes inside the plasma boundary. In the vacuum region S φvacuum bounded by the contour l plasma and l, the flux function satisfies the homogeneous equation L*ψ = 0. A solution for this equation which agrees with boundary conditions on l, and that is valid in a region containing S φvacuum , must then be valid on the plasma boundary itself. In general we choose to approximate ψ by a series solution N ψ ≈ ψ est = ψ 0 + ∑ c j χ j 16.1 j =1 With ψ 0 any known terms, and the basis functions all satisfy the homogeneous equilibrium equation on some region S φ0 which includes the vacuum region S φvacuum . Following the discussion in section 8 we can write ψ = ψ ext + ψ int from Equations 8.12 through 8.15. Usually we take an interior solution ψ int associated with current distributions inside S φ , and an exterior part ψ ext associated with currents in the exterior region S φext. For example ψ ext may be known from measured currents in external conductors, or calculated directly from the measurements using Green's functions. Various representations for the plasma current have been used; an expansion in toroidal eigenfunctions, discrete current filaments, single layer potential on a control surface, To date no purely analytic answers are available; generally the coefficients in an expansion, of position or currents in filaments, are altered numerically to provide a minimum "chi squared" between some measured and computed fields or fluxes. For example, suppose the actual measurements are represented by q i , with i running from 1 to M. There is a response matrix Q such that 0 q est ,i = q est,i N + ∑ c j Q ij 16.2 j =1 115
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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