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-$p$ limit, the solution to this equation takes the form

$$Y_e(p)\rightarrow Y_0\left[\frac{\skew{6}\hat{\mit\Delta}}{\pi\,p} + 1+ {\cal O}(p)\right].$$ (5.115)

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

$$\frac{d^2 Y_e}{dp^2} - \frac{P_\perp}{(1+1/\tau)\,D^2}\,p^2\,Y_e \simeq 0.$$ (5.116)

This is a parabolic cylinder equation [1] whose most general large-$p$ solution is

$$Y_e(p) = \frac{A\,{\rm e}^{-\alpha\,p^2/2} + B\,{\rm e}^{+\alpha\,p^2/2}}{p^{1/2}}\left[1+{\cal O}\left(\frac{1}{p^2}\right)\right],$$ (5.117)

where $A$ and $B$ are arbitrary constants, and

$$\alpha =\left[\frac{P_\perp}{(1+1/\tau)\,D^2}\right]^{1/2}.$$ (5.118)

Obviously, the physical solution of Equation (5.116) does not blow up at large $p$. Hence, we must select $B=0$ in Equation (5.117), which implies that

$$Y_e(p) = \frac{A\,{\rm e}^{-\alpha\,p^2/2} }{p^{1/2}}\left[1+{\cal O}\left(\frac{1}{p^2}\right)\right],$$ (5.119)

at large $p$.

Figure: 5.5 Numerical solution of the resonant layer equations for a low-field tokamak fusion reactor. The vertical dashed lines correspond to $Q-Q_E=Q_i$, 0, and $Q_e$, respectively, in order from the left to the right.

Let us make use of the so-called Riccati transformation [5,18],

$$W(p) = \frac{p}{Y_e}\,\frac{dY_e}{dp}.$$ (5.120)

Equation (5.78) yields

$$\frac{dW}{dp} = \left[\frac{2\,p}{-{\rm i}\,(Q-Q_E-Q_e)+ p^2}-\fr... ...- \frac{W^{\,2}}{p} + p\left[-{\rm i}\,(Q-Q_E-Q_e)+p^2\right]\frac{B(p)}{C(p)}.$$ (5.121)

According to Equation (5.115), the small-$p$ behavior of the solution to the previous equation is

$$W(p)= - 1 + \frac{\pi\,p}{\skew{6}\hat{\mit\Delta}} + {\cal O}(p^2).$$ (5.122)

Likewise, according to Equation (5.119), the large-$p$ behavior of the solution is

$$W(p) =-\alpha\,p^2 -\frac{1}{2} + {\cal O}\left(\frac{1}{p^2}\right).$$ (5.123)

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

$$\skew{6}\hat{\mit\Delta}= \lim_{p\rightarrow 0}\left(\frac{\pi}{dW/dp}\right).$$ (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 $Q_\ast=2.002$, $D=3.021$, $P_\varphi=874$, $P_\perp=287$, and $\tau =1$, assuming that $Q$ is real. (See Table 5.1.) Note that $\skew{6}\hat{\mit\Delta}$ 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 $Q=Q_E+Q_e$, which is equivalent to $\omega=\omega_E+\omega_{\ast\,e}$. 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 $Q-Q_E-Q_e$ when $\vert Q-Q_E-Q_e\vert\ll 1$, but saturates in magnitude as $\vert Q-Q_E-Q_e\vert\rightarrow{\cal O}(1)$.

Figure: 5.6 Numerical solution of the resonant layer equations for a high-field tokamak fusion reactor. The vertical dashed lines correspond to $Q-Q_E=Q_i$, 0, and $Q_e$, respectively, in order from the left to the right.

Figure 5.6 shows a numerical solution of the resonant layer equation for a high-field tokamak fusion reactor. This calculation is made with $Q_\ast=1.496$, $D=2.257$, $P_\varphi=874$, $P_\perp=287$, and $\tau =1$, assuming that $Q$ 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.