Flux Coordinates and Magnetic Field Structure - University of ...

192 11. The Dynamic Equilibrium of an Ideal Tokamak Plasma
In this derivation we have used the fact that
since a/aco = 0 and Vc; VY = 0 = VC; V8, all because of axisymmetry. We
employed E = ~,e, = Eco VC, where Vc, = e,/~. Faraday's law is then, from
(1 1.4.8) and (1 1.4.9),
This means that
ay
- = RE, + flux function .
at
With Y = - Y,d,1/2n, and with ET = EkA), the net electric field, we see that the
flux function can be chosen to be zero. (This follows from the integral form of
Amp6re's law where a C, loop is taken through point A of Fig. 11.3.) The time
rate of change of the poloidal flux through a disk bounded by a fixed C, loop is
thus
1 % fix, = - 2nREkA) .
at disk
After substitution of this result in (1 1.4.6), the latter can be solved for the radial
flux-surface "velocity":
(In u,d, s stands for surface and d for disk.) Taking into account that Y = - Y$,/2n,
we observe from (1 1.4.4) and (1 1.4.13) that the plasma E x B inward pinch
velocity is exactly equal to the radial velocity of the flux surfaces. This is a
statement of the frozen flux theorem: plasma and flux surfaces are tied together.
An observer sitting on (and moving with) the plasma will find that the
fluid-pinch motion generates a motional EMF. The electric field connected with
this motional EMF is exactly the negative of the electric field that caused the
E x B inward motion in the first place. This is as it should be: an observer moving
with the plasma, or thus the plasma itself, does not see any electric field. The
electric field is only present in the lab frame.
11.4.3 Motion of Flux Surfaces Defined by the Poloidal Ribbon Flux Ppl
Suppose now that we wish to label a flux surface by the poloidal ribbon flux (see
Fig. 11.3). To actually show that the flux surfaces labeled by 'Y",, = constant
move with the same radially inward velocity as the plasma, we return to (1 1.4.10)
or (1 1.4.1 1). When we now identify Y = Vpo1/2n, we must recognize that the
11.4 Motion of the Plasma and the Flux Surfaces 193
ribbon flux does not know about the transformer; all it contains is the "reaction
flux" due to an increase of B,, and (Jll),. The integration constant that we called
the "flux function" in (1 1.4.1 1) is the ubiquitous space independent transformer
loop voltage (devided by 2n) e(t)/2n = = $E-dl/2n = Id !P/dtl/2n that
we encountered in Fig. 11.1 (e(t) is spatially uniform, but E(A),tr a 1/R). Instead
of (1 1.4.12), we obtain for the case of the ribbon flux:
The appropriate signs can be understood from (1 1.4.12): Y = !PL,/2n and E$A'-react
- ~kA),reaCt 6, A = - I ET (A),react 16,; A thus EiA).reac' is a negative quantity (see Fig. 11.3).
With the flux surfaces, qo, = constant defined in a similar way to (1 1.4.5),
we find for their radial "velocity"
From Fig. 11.3, it is clear that EkA)." + E(A),reac'
T = EkA)vtr - 1 Er).reactl = Er), where
Er) is the magnitude of the net toroidal electric field (EkA) > 0). Thus
A comparison with (1 1.4.4) and (1 1.4.13) shows that, indeed,
Thus plasma and flux surfaces labeled by a constant poloidal (ribbon or disk)
flux move together.
11.4.4 Motion of Flux Surfaces Defined by the Toroidal Flux Yt0,
Finally, we show that the toroidal-flux label Ytor leads to a similar conclusion.
We could invoke the general applicability of the frozen-flux theorem: the plasma
and the flux surfaces (regardless of their label) move together. Simple manipula-
tions with formulae, however, provide some physical insight: the toroidal-flux
change should be connected with the poloidal electric field.
To find the inward E x B plasma velocity, we first derive a geometric
relationship for the component perpendicular to B in terms of poloidal and
toroidal components. From (1 1.2.2) we obtain:
The only induced E-field component present is in the perpendicular direction