Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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.
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