Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Figure 20.3. The geometry of an iron core and
a linked plasma of current I. The contour l
considered in the text is shown.
Now consider the case shown in Figure 20.3, an iron core with a central limb as on TEXT. If the
total number of turns linking the center core is 0, and there is no current in the iron (Equation
20.4) then ∫H . dl = 0 for any contour l inside the iron. Then H derives still from a single
potential, and all our previous steps are valid. However, if a finite number of amp turns I links
the iron, then ∫H . dl = I. This means that we can still write the field in the iron as ∇Φ, but now Φ
is multi valued, increasing by ±I once around the contour l. That is, in this case, Equations 20.8
and 20.9 apply in the iron, but with the additional constraint
∫ ∇Φ • dl = I
20.10
l
In this case, had we assumed perpendicularity, we would have obtained ∫H . dl = 0, while in reality
∫H . dl = I. Thus the field lines only enter the iron core perpendicularly if no current flows in the
iron, and the net ampere turns is zero.
The only place that the iron really affects magnetic diagnostics is in the equilibrium
reconstruction. No longer can we use the free space expressions for a circular current filament,
but they must be modified to satisfy the boundary condition. In toroidal plasma devices we
almost always satisfy the conditions necessary for the boundary condition Equation 20.7. One
way to model the effect of iron is by placing additional circular filaments inside the iron itself,
with currents chosen to satisfy Equation 20.7 at a given number of locations. This is illustrated
in Figure 20.4: the 'image" filaments must have a current flowing in the same direction as the
filament in air, so that there is an attraction between the filament in air and the iron itself. As the
filament in air gets closer to the iron, the "image" current must increase, and so the attraction
must increase. This means that an iron core can lead to an axisymmetric, or n = 0, instability.
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