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

120 6. Toroidal Flux Coordinates 6.1 Straight Field-Line Coordinates 121
V
8,=8 and [,=[-2nT. (6.1.22b)
Yiol
Note that these transformations are consistent with (4.6.2). The function v in
terms of 8, and [, becomes:
Again, v = constant is the equation of a straight line
)jtoref - eel[, = constant . (6.1.24)
The contravariant components of B are still defined by (6.1.6) but with 8
replaced by 8, and [ by [,:
From these equations for Be' and &r, we observe that
Be,
-- - !!iol - flux function ,
fi ytor
thus proving a result that we anticipated before, in (4.8.10), during the "derivation"
of the expression for the rotational transform. The fact that Bef/Bcr is a flux
function is crucial for the straightness of B. From the equation for a field line,
we see that the slope of the line is indeed constant on the flux surface, and equal to
the rotational transform t.
In summary, a transformation of one of the coordinates is sufficient to make
the field lines straight. In tokamaks, it is customary to retain the symmetry angle
(which measures rotation about the major axis) ( = [,. Consequently, 8 then must
be changed.
We should remark that the angles 8, and [, are not the only ones in which
the magnetic field appears straight. Indeed, for example, if we further transform
the 8, and [, coordinates into new coordinates 8, and (, as follows:
where G is an arbitrary periodic function, we see that upon substitution of these
equations into (6.1.23) for v, we obtain
showing that 8, and [, are flux coordinates as well. One can use this degree of
freedom to deform the coordinates further to make other quantities look simpler.
In Hamada coordinates, the system is deformed until the current-density lines
become straight in addition to the magnetic-field lines. Boozer uses this freedom
to free the magnetic scalar potential (that would exist in vacuum) from a periodic
part. In axisymmetric devices, it is customary to choose 8, such that V4 1 V(,,
where 5, is the ignorable coordinate. These three choices have been discussed
qualitatively above, but will be considered in detail below.
Using the flux coordinates 8, and [,, the magnetic field in Clebsch form is
equal to:
In terms of the poloidal flux through a disk, !Piol, the expression for B is
Here we have used the fact that d qO1 = - d Y,"Ol (see (4.7.4)).
If the flux surface volume V is chosen as a flux-surface label, (6.1.30) becomes
In tokamaks, it is customary to use the poloidal flux as a flux label, and to
use the safety factor q. If one takes Q = YioI/2n = Y, one obtains from (6.1.30)
-
- For the
-
poloidal disk flux as a flux label, the expression is exactly the same as in
(6.1.33), but with Q = Y YkI/2n.
In stellarators, the choice of flux label Q is commonly Ytor, such that with
Q = $ YtOr/2n, we obtain