Error-Field Penetration

If $\bar{I}_{c\,pen}$ is the critical value of the normalized error-field coil current needed to trigger error-field penetration then the critical radial magnetic field (in the absence of a shielding current) that must be generated at the rational surface in order to trigger penetration is

$$\left(\frac{b_r}{B_z}\right)_{pen} = \frac{2\,m\,s_s}{(-{\mit\Del... ...}\right)\right]^{-1/2}\left(\frac{P_\varphi}{S}\right)^{1/2}\tilde{I}_{c\,pen}.$$ (7.43)

Here, use has been made of Equations (7.2), (7.10), and (7.40). Moreover, shielding factor (i.e., the ratio of the magnetic flux that would be driven at the rational surface in the absence of a shielding current to that which is actually driven) takes the form

$${\mit\Sigma}(Q_E) = \frac{\left([\zeta+\skew{6}\hat{\mit\Delta}_r(Q_E)]^2 + [\skew{6}\hat{\mit\Delta}_i(Q_E)]\right)^{1/2}}{\zeta}.$$ (7.44)

Finally, the reduced [by a factor four—see Equation (8.1)] width of the locked magnetic island chain driven at the rational surface is

$$w=\frac{W}{4}=\left(\frac{1}{n\,s_s\,\epsilon_s}\,\frac{b_r}{B_z}\,\frac{1}{{\mit\Sigma}}\right)^{1/2}\,r_s,$$ (7.45)

where use has been made of Equations (5.27), (5.129), and (7.2).

Figure 7.5 shows error-field penetration calculations made for a low-field tokamak fusion reactor. The calculation parameters are $\tau =1$, $Q_{e}=1.00$, $D=3.02$, $P_\varphi=874$, and $P_\perp=287$, as well as $S=1.35\times 10^9$, $m=2$, and ${\mit\Delta}_{nw}=-2\,m$, and, finally, $n=1$, $s_s=1$, $\tau_i=1.36\times 10^{-2}\,{\rm s}$, $\tau_\varphi=1.60\,{\rm s}$, $R_0=7.59$ m, $a=R_0/3$, and $r_s=a/2$. (See Tables 2.1, 5.1, and 6.1.) The top-left panel shows shows the critical radial magnetic field that must be induced at the rational surface (in the absence of shielding currents) in order to trigger error-field penetration as a function of the unperturbed electron fluid rotation frequency at the rational surface. It can be seen that diamagnetic levels of rotation (i.e., $\omega_{\perp\,e}\sim\omega_{\ast\,e}$—see Section 5.15) are sufficient to keep $(b_r/B_z)_{\rm pen}$ well above $10^{-4}$, unless the unperturbed electron fluid rotation frequency at the rational surface, $\omega_{\perp\,e\,0}= \omega_{E\,0}+\omega_{\ast\,e}$, falls significantly below the electron diamagnetic frequency, $\omega_{\ast\,e}$. The bottom-left panel shows the electron fluid rotation frequency at the rational surface as a fraction of the unperturbed rotation frequency just before (solid curve) and just after (dashed curve) penetration. It can be seen that the electron fluid rotation frequency needs to be reduced by a substantial factor (i.e., at least, a factor of two) before penetration occurs. However, after penetration, the electron fluid rotation frequency is effectively reduced to zero. The top-right panel shows the shielding factors just before (solid curve) and just after (dashed curve) penetration. It can be seen that, prior to penetration, diamagnetic levels of rotation are sufficient to reduce the amount of driven magnetic reconnection at the rational surface by factors that exceed 200 (unless the unperturbed electron fluid rotation frequency at the resonant surface falls significantly below the electron diamagnetic frequency). However, after penetration, there is no shielding at all at the rational surface (i.e., ${\mit\Sigma}\simeq 1$). Finally, the bottom-right panel shows the reduced width of the locked magnetic island chain driven at the rational surface just before (solid curve) and just after (dashed curve) penetration. It can be seen that, prior to penetration, the driven island width is of order 0.5 cm, which is about the same as the linear layer width estimated in Table 6.2. It follows that using linear theory to determine the error-field penetration threshold is a reasonable approximation (recall that linear theory would be completely invalidated were the island width to become much greater than the linear layer width—see Section 5.16.) On the other hand, after penetration, the driven island width typically exceeds 10 cm. Such a substantial island chain would significantly degrade the energy confinement properties of the plasma [1,5], and might even trigger a disruption. It is also clear that, after penetration, the driven tearing perturbation is governed by nonlinear, rather than linear, physics.

Figure 7.6 shows error-field penetration calculations made for a high-field tokamak fusion reactor. The calculation parameters are $Q_{e}=0.75$, and $P_\varphi=874$, as well as $S= 5.63\times 10^8$, $m=2$, and ${\mit\Delta}_{nw}=-2\,m$, and, finally, $n=1$, $s_s=1$, $\tau_i=2.36\times 10^{-3}\,{\rm s}$, $\tau_\varphi=2.78\times 10^{-1}\,{\rm s}$, $R_0=3.16$ m, $a=R_0/3$, and $r_s=a/2$. (See Tables 2.1, 5.1, and 6.1.) It can be seen, by comparison with Figure 5.5, that error-field penetration in a high-field tokamak fusion reactor is fairly similar to that in a low-field reactor. The main differences are that the shielding factors and driven island widths are somewhat smaller in the high-field case compared to the low-field case.

Figure 7.6: Analytically determined error-field penetration threshold in a high-field tokamak fusion reactor. The panels are the same as those described in Figure 7.5.