Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton From these two signals we can calculate δΦ. The subtractions are performed electronically (i.e. analog); the constant k is determined experimentally so that δΦ is zero without plasma (toroidal field only). In reality there are problems. One in particular is caused by the discreteness of the toroidal field system, and the current redistribution in the toroidal field coils during a shot. When the toroidal field current is initiated, the current flows at the inner edge of the conductors to minimize the linked flux. As the pulse proceeds, the current redistributes and approximates a uniform distribution (not exactly because of repulsion of current channels). The time for this redistribution to occur is approximately the radial penetration time of the poloidal current into the conductor of radial extent w: τ ≈ πµ 0 σw 2 /16, typically 200 ms. If the toroidal field system was a perfect toroidal solenoid this redistribution would leave the fields unaffected. However, because of the discrete number of toroidal fields, there is now a time varying toroidal field ripple. The size of the changing field ripple depends on where a pickup coil is placed: therefore two coils linking the same steady state flux can link different transient fluxes. It is best to place the coils between the toroidal field coils, where the redistribution effect is smallest. Another problem is due to poloidal eddy currents in any conducting vacuum vessel. This produces a non zero change in the toroidally averaged B φ , not just a local ripple. Therefore it couples strongly to the pickup loops. Compensation for both the effects discussed has been performed successfully using software, by simulating redistribution and eddy currents as simple circuits, coupled to both the primary B φ coil current and a secondary pickup coil. Further problems occur if the loops are not exactly positioned, so that they couple to the poloidal fields produced by the primary, vertical field and shaping windings. That any such effects exist can be checked for by firing discharges with positive (+) and negative (-) B φ . Let ∆Φ p be the signal caused by poloidal field coupling, ∆Φ(+) the signal obtained with positive B φ , and ∆Φ(-) the signal obtained with negative B φ . While the toroidal field coupling effects will change sign with reversing B φ , any poloidal field effects will not. Therefore ∆Φ ( + ) − ∆Φ ( −) = 2 δΦ + ∆Φ p ( ) 14.27 ∆Φ ( + ) + ∆Φ ( −) = 2∆Φ p 14.28 Any finite ∆Φ p can be correct for using a circuit model again, with the coupling between any winding (including vacuum vessel) and the pickup coil written in terms of mutual inductances. 112
Magnetic fields and tokamak plasmas Alan Wootton 15. FULL EQUILIBRIUM RECONSTRUCTION The problem under discussion is how to reconstruct as much as is possible about the equilibrium from external measurements. In particular, if we knew everything about the fields outside the plasma, what could we uniquely determine Could we separate β I and l i (In principle yes for toroidal systems). Could we go further and actually uniquely determine the plasma current distribution of a toroidal plasma (I don't know) If we want to allow measurements of dp(ψ)/dψ and FdF(ψ)/dψ we must have redundancy beyond that required to solve the equilibrium equation with a known current density profile. We see immediately a problem in straight geometry, because with straight circular cross sections the magnetic measurements must be consistent with a solution that has concentric circular surfaces. In this case the measurements are consistent with any profile function which gives the correct total plasma current, so there is an infinite degeneracy. In practice the mathematical subtleties of what can and cannot in principle be determined are not discussed. Instead the usual technique for equilibrium reconstruction is to choose a parameterization for j φ , with a restricted number of free parameters. These free parameters are chosen to minimize the chi squared error or cost function between some measured and computed parameter, for example the poloidal field component on some contour. Obviously we need some boundary conditions, as discussed in section 8. Information available might consist of the flux ψ on a contour, the fields B n and B τ on a contour, and currents in conductors, the total current, and if lucky the diamagnetism. Typical parameterizations are ⎡ j φ = aβ R + ( 1− β) R ⎤ 0 ⎣ ⎢ R 0 R ⎦ ⎥ g δψ ( ) 15.1 j φ = ( α 1 δψ + α 2 δψ 2 +α 3 δψ 3 )R + ( b 1 δψ)R −1 15.2 with δψ = (ψ-ψ boundary )(ψ mag axis -ψ boundary ). In each case the term proportional to R represent the part of j φ proportional to dp/dψ, and the part of j φ proportional to R -1 is proportional to FdF/dψ. R o is some characteristic radius. The assumed function g might be of the form g(δψ,γ) = exp(-γ 2 (1-δψ) 2 , dψ γ or δψ + γδψ 2 . If in equation 15.1 the function g(δψ) is 1, then the description of j φ is called quasi uniform: for a circular outer boundary there is an exact analytic solution of the equilibrium equation. 113
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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