# 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 .

Let us make use of the so-called 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 low-field 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 shows a numerical solution of the resonant layer equation for a high-field 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 low-field and high-field tokamak fusion reactors do not differ substantially from one another.