Numerical Solution of Resonant Layer Equations

Let

$$q = p\,D.$$ (6.27)

Equations (6.5)–(6.7) yield

$$\frac{d^{2} Y_e}{dq^2} - \frac{\hat{E}(q)}{\hat{F}(q)} \,Y_e =0,$$ (6.28)

where

$$\hat{E}(q) = -(1+1/\tau)^{-1}\,\hat{Q}_\ast^{\,2} -{\rm i}\,\hat{Q}_\ast\,(\hat{P}_\varphi+\hat{P}_\perp)\,q^2+ \hat{P}_\varphi\,\hat{P}_\perp\,q^4,$$ (6.29)

$$\hat{F}(q) \simeq \hat{P}_\perp - {\rm i}\,\hat{Q}_\ast +(1+1/\tau)\,\hat{P}_\varphi\,q^2,$$ (6.30)

and

$$\hat{Q}_\ast = \frac{Q_\ast}{D^4},$$ (6.31)

$$\hat{P}_\varphi = \frac{P_\varphi}{D^6},$$ (6.32)

$$\hat{P}_\perp = \frac{P_\perp}{D^6}.$$ (6.33)

Equation (6.28) must be solved subject to the constraint that $Y_e(q)$ is bounded as $q\rightarrow \infty$, and

$$Y_e(q) = Y_0\left[1- \frac{(1+1/\tau)^{1/2}\,\delta_s\,q}{\pi\,d_\beta} + {\cal O}(q^2)\right]$$ (6.34)

as $q\rightarrow 0$. Here, use has been made of Equations (6.8), (6.26), and (6.27). As is easily demonstrated (see Section 5.14), the solution of Equation (6.28) that is bounded as $q\rightarrow \infty$ is

$$Y_e(q) = \frac{A\,{\rm e}^{-\alpha\,q^2/2}}{q^{1/2}}\left[1+{\cal O}\left(\frac{1}{q^2}\right)\right],$$ (6.35)

where

$$\alpha = \left(\frac{\hat{P}_\perp}{1+1/\tau}\right)^{1/2},$$ (6.36)

and $A$ is an arbitrary constant.

Figure: 6.2 Real part of $\delta _s/\delta _\beta$ calculated as a function of $\hat{Q}_\ast\equiv Q_\ast/D^4$ and $\hat{P}\equiv P/D^6$.

Let us again (see Section 5.14) make the Riccati transformation [2,5]

$$W(q) = \frac{q}{Y_e}\,\frac{dY_e}{dq}.$$ (6.37)

Equation (6.28) yields

$$\frac{dW}{dq} = \frac{W}{q}-\frac{W^2}{q}+ \frac{q\,\hat{E}(q)}{\hat{F}(q)}.$$ (6.38)

According to Equation (6.34), the small-$q$ behavior of the solution to the previous equation is

$$W(q)= - \frac{(1+1/\tau)^{1/2}\,\delta_s\,q}{\pi\,d_\beta} + {\cal O}(q^2).$$ (6.39)

Likewise, according to Equation (6.35), the large-$q$ behavior of the solution is

$$W(q) =-\alpha\,q^2 -\frac{1}{2} + {\cal O}\left(\frac{1}{q^2}\right).$$ (6.40)

Equation (6.38) is conveniently solved numerically by launching a solution of the form (6.40) at large $q$, and then integrating backward to small $q$ [5]. It follows from Equation (6.39) that

$$\frac{\delta_s}{d_\beta}= -\lim_{q\rightarrow 0}\left[\frac{\pi}{(1+1/\tau)^{1/2}}\,\frac{dW}{dq}\right].$$ (6.41)

Figure: 6.3 Imaginary part of $\delta _s/\delta _\beta$ calculated as a function of $\hat{Q}_\ast\equiv Q_\ast/D^4$ and $\hat{P}\equiv P/D^6$.

Table 6.2 gives estimates for the normalized resonant layer parameters, $\hat{Q}_\ast$, $\hat{P}_\varphi$, and $\hat{P}_\perp$, that appear in Equations (6.28)–(6.30), in a low-field and a high-field tokamak fusion reactor. These estimates are made using the data shown in Table 6.1. Table 6.2 also gives estimates for the linear layer thicknesses and tearing mode growth-rates in such reactors. These estimates are obtained via numerical solution of the resonant layer equation. It can be seen that the typical radial thickness of a linear tearing layer in a tokamak fusion reactor is only a few millimeters. Furthermore, linear tearing modes in tokamak fusion reactors grow on timescales that typically lie between a tenth of a second and a second [assuming that $E_{ss}\sim {\cal O}(1)$.] Finally, the real frequencies of such modes, in a frame of reference that co-rotates with the electron fluid at the resonant surface, (i.e., the imaginary components of $\gamma$) are very much smaller (by a factor of order $10^6$) than a typical diamagnetic frequency. (See Table 6.1.) In other words, linear tearing modes do indeed co-rotate with the electron fluid at the resonant surface to a very high degree of fidelity.

Figures 6.2 and 6.3 show values of ${\rm Re}(\delta_s/d_\beta)$ and ${\rm Im}(\delta_s/d_\beta)$, respectively, evaluated numerically as functions of $\hat{Q}_\ast$ and $\hat{P}$. In producing these figures, it is assumed that $\hat{P}_\varphi=\hat{P}_\perp=\hat{P}$. The fact that ${\rm Re}(\delta_s)>0$ for all values of $\hat{Q}_\ast$ and $\hat{P}$ confirms that tearing modes are linearly unstable for $E_{ss}<0$, and stable otherwise [4]. [See Equation (6.25).] It is clear that the layer width increases with increasing normalized diamagnetic frequency, $\hat{Q}_\ast$, and also with increasing normalized magnetic Prandtl number, $\hat{P}$.