(7.43) 
(7.44) 
Figure 7.5 shows errorfield penetration calculations made for a lowfield tokamak fusion reactor. The calculation parameters are , , , , and , as well as , , and , and, finally, , , , , m, , and . (See Tables 2.1, 5.1, and 6.1.) The topleft 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 errorfield 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., —see Section 5.15) are sufficient to keep well above , unless the unperturbed electron fluid rotation frequency at the rational surface, , falls significantly below the electron diamagnetic frequency, . The bottomleft 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 topright 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., ). Finally, the bottomright 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 errorfield 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 errorfield penetration calculations made for a highfield tokamak fusion reactor. The calculation parameters are , and , as well as , , and , and, finally, , , , , m, , and . (See Tables 2.1, 5.1, and 6.1.) It can be seen, by comparison with Figure 5.5, that errorfield penetration in a highfield tokamak fusion reactor is fairly similar to that in a lowfield reactor. The main differences are that the shielding factors and driven island widths are somewhat smaller in the highfield case compared to the lowfield case.
