Numerical Solution of Resonant Layer Equations
It is possible to solve the resonant layer equation (5.78) numerically. We already know that, in the small
limit, the solution to this equation takes the form

(5.115) 
[See Equation (5.83).] In the large limit, Equations (5.78)–(5.81) reduce to

(5.116) 
This is a parabolic cylinder equation [1] whose most general large solution is

(5.117) 
where and are arbitrary constants,
and

(5.118) 
Obviously, the physical solution of Equation (5.116) does not blow up at large . Hence, we
must select in Equation (5.117), which implies that

(5.119) 
at large .
Figure: 5.5
Numerical solution of the resonant layer equations for a lowfield tokamak fusion reactor.
The vertical dashed lines correspond to , 0, and , respectively, in order from the
left to the right.

Let us make use of the socalled Riccati transformation [5,18],

(5.120) 
Equation (5.78) yields

(5.121) 
According to Equation (5.115), the small behavior of the solution to the previous equation is

(5.122) 
Likewise, according to Equation (5.119), the large behavior of the solution is

(5.123) 
Equation (5.121) is conveniently solved numerically by launching a solution of the form (5.123) at large , and then
integrating backward to small [18]. Equation (5.122) yields

(5.124) 
Figure 5.5 shows a numerical solution of the resonant layer equation for a lowfield tokamak
fusion reactor. This calculation is made with
, ,
,
, and , assuming that is real. (See Table 5.1.) Note that
parameterizes the amplitude and phase of
a shielding current that is driven inductively at the rational surface, in response to a rotating tearing perturbation in the
outer region, and acts to suppress magnetic reconnection at the surface [11]. It can be seen that
the shielding current is zero when , which is equivalent to
. In other words, the shielding current is zero when the tearing perturbation in the outer region
rotates at the frequency of a naturally unstable tearing mode at the rational surface [2,11]. (See Chapter 6.) The shielding
current clearly increases linearly with when
, but saturates in magnitude as
.
Figure: 5.6
Numerical solution of the resonant layer equations for a highfield tokamak fusion reactor.
The vertical dashed lines correspond to , 0, and , respectively, in order from the
left to the right.

Figure 5.6 shows a numerical solution of the resonant layer equation for a highfield tokamak
fusion reactor. This calculation is made with
, ,
,
, and , assuming that is real. (See Table 5.1.) Note that the figure is very similar to Figure 5.5, indicating that the resonant layer responses in lowfield and highfield tokamak fusion reactors do not differ substantially from one another.