Resonant Layer Equations
In a conventional tokamak plasma, the Lundquist number, , which is the nominal ratio of the plasma inertia term to the resistive diffusion term in the plasma Ohm's law [14], is very much greater than unity. In fact, according to Table 1.5, typically exceeds in a tokamak fusion reactor.
However, a resonant layer is
characterized by a balance between plasma inertia and resistive diffusion [17]. Such a balance is
only possible if the layer is very narrow in the radial direction (because a narrow layer enhances radial derivatives, and, thereby, enhances
resistive diffusion). Let us define the stretched radial variable [2]

(5.56) 
Assuming that
in the layer (i.e., assuming that the layer thickness is roughly of order
),
and making use of the fact that , we deduce that
.
Hence, the linear equations (5.39)–(5.42) reduce to the following set of resonant layer equations [7,15]:
Here,
Table: 5.1
Dimensionless parameters appearing in resonant layer equations for a lowfield and a highfield
tokamak reactor. See Equations (4.5), (5.65), (5.66), (5.53), (5.54), and (4.65).

LowField 
HighField 

5.0 
12.0 

1.0 
1.0 




3.02 
2.26 

874 
874 

287 
287 

0.635 
0.635 

0.128 
0.128 

Table 5.1 gives estimates for the values of the dimensionless parameters that characterize the resonant layer equations, (5.57)–(5.60), in a lowfield
and a highfield tokamak fusion reactor. (See Chapter 1.) These estimates are made using the following assumptions:
(lowfield) or
(highfield),
,
,
(where and are the deuteron and triton masses, respectively),
, , , (where is the minor radius of the plasma), , ,
, and
. The parallel energy diffusivities,
and
, are estimated from Equations (2.319) and (2.320), respectively,
using
, which is the typical parallel wavenumber of the tearing perturbation at the edge of a resistive layer
whose characteristic thickness is
. Note that
, which allows us to neglect the
parallel transport terms (i.e., the terms involving
) in Equation (5.58). The neglect of these terms is justified because the linear layer width is much less than the critical width,

(5.68) 
below which parallel transport is unable to constrain the perturbed electron temperature to be constant along magnetic fieldlines [12].