Linear Calculation

Consider an $n=1$ tearing mode that reconnects magnetic flux principally at the $k$th rational surface in our example tokamak plasma. Let us suppose that the amount of reconnected magnetic flux is sufficiently small that the plasma response at this surface lies in the linear regime, which is equivalent to supposing that the magnetic island width at the $k$th rational surface is much less than the linear layer width. (See Section 5.16.) Given that the mode is assumed not to interact strongly with the other rational surfaces in the plasma, due to the assumed strong shielding present at these surfaces, the analysis of Chapter 6 implies that the frequency of the mode can be written

$$\omega = {\rm i}\,\gamma _k+ \omega_{{\rm linear}\,k},$$ (14.141)

where $\vert\gamma_k\vert\ll \vert\omega_{{\rm linear}\,k}\vert$, and

$$\omega_{{\rm linear}\,k} = - n\left({\mit\Omega}_E+{\mit\Omega}_{\ast\,e}\right)_{r=r_k}.$$ (14.142)

The previous two equations, which are equivalent to Equation (6.1), imply that a linear tearing mode corotates with the electron fluid at the $k$th rational surface. It follows from Equation (14.130) that the normalized mode rotation frequency at the $k'$th rational surface, where $k'\neq k$, is

$$Q_{k'} =S_{k'}\,\omega_{{\rm linear}\,k}\,\tau_{H\,k'} =\frac{S_{... ...{\,1/3}}{S_k^{\,1/3}}\,(Q_{E\,k} + Q_{e\,k})\,\frac{\tau_{H\,k'}}{\tau_{H\,k}}.$$ (14.143)

Here, we have neglected the comparatively small growth-rate of the mode with respect to its comparatively large real frequency in the laboratory frame.

According to Equations (14.120) and (14.123), the linear dispersion relation of our tearing mode can be written

$${\mit\Delta\Psi}_k = E_{kk}\,{\mit\Psi}_k+ \sum^{k'\neq k}_{k'=1,K} E_{kk'}\,{\mit\Psi}_{k'},$$ (14.144)

with

$$({\mit\Delta}_{k'} - E_{k'k'})\,{\mit\Psi}_{k'} = E_{k'k}\,{\mit\Psi}_k + \sum^{k''\neq k,k'}_{k''=1,K} E_{k'k''}\,{\mit\Psi}_{k''},$$ (14.145)

for $k'\neq k$, where

$${\mit\Delta}_{k'} = S_{k'}^{\,1/3}\,\skew{5}\hat{\mit\Delta}_{k'}... ...E\,k'},Q_{e\,k'}, Q_{i\,k'}, \tau_{k'},D_{k'}, P_{\varphi\,k'}, P_{\perp\,k'}).$$ (14.146)

Table: 14.4 $n=1$ layer matching parameters in KSTAR discharge #18594 at time $t=6450$ ms.

$k$	$k'$	${\rm Re}({\mit\Delta}_{k'})$	${\rm Im}({\mit\Delta}_{k'})$	$k$	$k'$	${\rm Re}({\mit\Delta}_{k'})$	${\rm Im}({\mit\Delta}_{k'})$
1	2	$-47.5$	$+234$	3	1	$-21.2$	$-128$
1	3	$-713$	$+249$	3	2	$+18.2$	$-257$
1	4	$-419$	$+826$	3	4	$-229$	$+830$
2	1	$-57.1$	$-206$	4	1	$+0.06$	$-39.0$
2	3	$-728$	$+243$	4	2	$+0.41$	$-73.7$
2	4	$-403$	$+830$	4	3	$-0.07$	$-95.9$

Table 14.4 shows the dimensionless layer matching parameters, ${\mit\Delta}_{k'}$, where $k'\neq k$, calculated from the layer parameters given in Table 14.3, with the $Q_{k'}$ specified by Equation (14.143). It can be seen that all of the ${\mit\Delta}_{k'}$ have magnitudes that are much greater than unity, indicating a strong shielding response at the rational surfaces other than the $k$th surface. This strong shielding is a consequence of sheared rotation in the plasma [14]. Roughly speaking, a given rational surface can only reconnect magnetic flux that corotates with the local electron fluid. However, as a consequence of sheared rotation, if a tearing perturbation corotates with the local electron fluid at a given rational surface in the plasma then it does not corotate with the local electron fluids at any of the other surfaces. This is illustrated in Table 14.5, which specifies the linear natural frequencies, $\omega_{{\rm linear}\,k}$, associated with the various $n=1$ rational surfaces present in the plasma. It is clear that the natural frequencies all differ substantially from one another. Consequently, a linear $n=1$ tearing mode can only reconnect magnetic flux at one rational surface at a time in the plasma.

Table: 14.5 Natural frequencies of $n=1$ tearing modes in KSTAR discharge #18594 at time $t=6450$ ms.

$k$	$m_k$	$w_{{\rm linear}\,k} ({\rm krad/s})$	$w_{{\rm nonlinear}\,k} ({\rm krad/s})$
1	2	$-45.9$	$-72.4$
2	3	$-36.2$	$-55.9$
3	4	$+27.6$	$-56.6$
4	5	$+263$	$-114$

Treating $\vert{\mit\Delta}_{k'}-E_{k'k'}\vert^{-1}$ as a small parameter, Equation (14.145), yields

$$\frac{{\mit\Psi}_{k'}}{{\mit\Psi}_k} \simeq \frac{E_{k'k}}{{\mit\Delta}_{k'}-E_{k'k'}},$$ (14.147)

for $k'\neq k$, which implies that $\vert{\mit\Psi}_{k'}\vert/\vert{\mit\Psi}_k\vert\ll 1$. In other words, the strong rotational shielding present at the $k'$th rational surface does indeed ensure that very little magnetic flux is reconnected at that surface compared to that reconnected at the $k$th surface (i.e., the rational surface at which the mode corotates with the local electron fluid). The previous equation can be substituted back into Equation (14.144) to give

$$\frac{{\mit\Delta\Psi}_k}{{\mit\Psi}_k} \simeq E_{kk} + \delta E_{kk},$$ (14.148)

where

$$\delta E_{kk}= \sum^{k'\neq k}_{k'=1,K} \frac{\vert E_{kk'}\vert^2}{{\mit\Delta}_{k'} - E_{k'k'}}.$$ (14.149)

Here, $E_{kk}$ is the tearing stability index of the mode that reconnects magnetic flux at the $k$th rational surface in the limit of full shielding (i.e., ${\mit\Delta\Psi}_{k'}=0$) at the other rational surfaces, whereas $\delta E_{kk}$ is the correction to this index due to the fact that the shielding is not actually perfect. Table 14.6 gives the $E_{kk}$ and $\delta E_{kk}$ values calculated from the data in Tables 14.2 and 14.4 for the four possible linear $n=1$ tearing modes in our example tokamak discharge. It can be seen that $\vert\delta E_{kk}\vert\ll \vert E_{kk}\vert$ for all modes. In other words, in all cases, the correction due to residual reconnection at the other rational surfaces is essentially negligible.

Table: 14.6 Linear $n=1$ tearing stability indices in KSTAR discharge #18594 at time $t=6450$ ms.

$k$	$E_{kk}$	${\rm Re}(\delta E_{kk})$	${\rm Im}(\delta E_{kk})$
1	$-4.81$	$-4.43\times 10^{-3}$	$-1.05\times 10^{-2}$
2	$-7.92$	$-2.07\times 10^{-2}$	$+2.26\times 10^{-3}$
3	$-15.2$	$-1.75\times 10^{-2}$	$-1.45\times 10^{-2}$
4	$-17.4$	$+1.24\times 10^{-1}$	$+8.01\times 10^{-1}$