Asymptotic Matching
Consider a tearingstable, cylindrical, tokamak plasma of minor radius , and simulated major radius . In accordance with the analysis of Chapter 3, suppose that
the plasma is surrounded by a concentric, radiallythin, rigid, resistive wall of minor radius . Likewise,
suppose that the wall is surrounded by a concentric, radiallythin, magnetic fieldcoil, of minor radius , that carries a nonrotating (in the laboratory frame) helical
current of amplitude . Let the current possess periods in the poloidal direction, and periods in the toroidal direction. The static magnetic
field generated by the fieldcoil constitutes our errorfield. The errorfield resonates with the
plasma at the rational magnetic fluxsurface, minor radius , at which the safetyfactor
(see Section 3.2) takes the value .
Setting (because the errorfield is static) in Equations (3.187) and (3.188),
we obtain

(7.1) 
Here,
where , , are conventional cylindrical coordinates,
the perturbed magnetic field, and the equilibrium toroidal magnetic fieldstrength.
(See Sections 3.3, 3.8, and 3.15.)
Clearly,
is a measure of the reconnected magnetic flux driven at the rational surface by the errorfield, whereas
is a measure of the helical current sheet that is induced at the surface. Moreover,

(7.4) 
is the (real dimensionless) tearing stability index (of a tearing mode with poloidal mode number , and
toroidal mode number ) in the absence of the wall,

(7.5) 
the
(real dimensionless) tearing stability index in the presence of a perfectly conducting wall at (see Section 3.8), and

(7.6) 
Here, the (real dimensionless) plasmawall coupling parameter, , and the (real dimensionless) wall stability index,
, are defined in Sections 3.9 and 3.17, respectively. We expect
, because the plasma is assumed to be tearing stable (see Chapter 6), and
.
In fact, if we make the approximation that the equilibrium plasma current external to the rational magnetic fluxsurface is
negligible [i.e., in Equation (3.77)] then it is easily demonstrated that
In the radiallythin, resonant layer that surrounds the rational magnetic fluxsurface, Equation (5.70) yields

(7.9) 
where is the the Lundquist number at the rational surface, and the complex layer matching parameter,
, is defined in
Equation (5.69).
Hence, asymptotic matching between the inner and the outer regions (see Section 4.1) yields

(7.10) 
where use has been made of Equations (7.1) and (7.9).
The previous equation specifies the (normalized) reconnected magnetic flux,
, driven at the rational surface by the (normalized)
errorfield coil current,
. The complex layer parameter,
, specifies the strength of a shielding current
that is induced in the resonant layer, and acts to prevent driven magnetic reconnection. Note that the
resistive wall has no influence on
because no eddy currents are induced in the wall by a static
errorfield.