Assuming that the chain possesses the rotation frequency in the laboratory frame, so that , and that (i.e., there is no error-field), Equation (3.188) yieldsHere, is the normalized reconnected magnetic flux at the rational surface [see Equations (3.72) and (3.184)], is the normalized helical magnetic flux that penetrates the wall [see Equations (3.82) and (3.192)], the real dimensionless parameters and are defined in Equations (3.87) and (3.195), respectively,
Equations (3.187) and (10.1) yield
It follows from the previous equation that
Equations (8.108), (10.7), and (10.8) yield the following modified Rutherford island width evolution equation :Here, is the minor radius of the rational surface, the resistive evolution time [see Equation (5.49)], the hydromagnetic time [see Equation (5.43)], the toroidal momentum confinement time [see Equation (5.50)], a dimensionless measure of the plasma pressure at the rational surface [see Equation (4.65)], the ratio of the electron and ion pressure gradients at the rational surface [see Equation (4.5)], the radial width of the magnetic island chain, the ion sound radius, the collisionless ion skin-depth at the rational surface [see Equation (4.24)], the magnetic shear-length at the rational surface [see Equation (5.27)], and the effective pressure gradient scale-length at the rational surface [see Equation (8.35)]. Furthermore, , , and . (See Section 8.11.)
The first term on the right-hand side of the previous equation governs the growth and saturation of the magnetic island chain when its rotation frequency is sufficiently large that the wall acts as a perfect conductor. The second term describes the loss of wall stabilization when the chain's rotation frequency is sufficiently small that perturbed magnetic field can penetrate through the wall [3,10]. The third and fourth terms represent the destabilizing effect of the ion polarization current induced in the vicinity of the rational surface when the ion fluid is diverted around the island chain's magnetic separatrix [7,13]. Unlike the case of an isolated island chain (see Chapter 9), the polarization effect is non-zero because, according to Equation (8.74), (8.87), (8.101), and (10.8), the electromagnetic braking torque exerted on the plasma in the immediate vicinity of the rational surface, due to interaction with the conducting wall, generates finite ion flow in the island rest frame.
Assuming that the island chain co-rotates with the ion fluid at the rational surface (see Section 9.5), Equations (3.189) and (9.31) imply thatwhere , which is defined in Equation (9.32), is the unperturbed (by any electromagnetic torques that develop at the rational surface) rotation frequency. Here, we are treating and as constants because the electromagnetic torque that develops at the rational surface has no explicit time dependance (assuming that the width of the island chain grows on a timescale that is much greater than and ). Equations (3.180), (3.190), (3.191), (4.23), (5.27), (5.43), (5.50), (7.28), and (7.34)–(7.35) can be combined to give where Here, is the poloidal flow-damping time [see Equation (7.28)], , the simulated major radius of the plasma, the minor radius of the plasma, , the poloidal mode number of the island chain, and the toroidal mode number of the island chain. Equations (10.8), (10.10), and (10.11) can be combined to give the torque balance equation  The left-hand side of the previous equation represents the viscous restoring torque that acts to prevent changes in the plasma rotation at the rational surface, whereas the right-hand side represents the electromagnetic braking torque acting on the plasma in the vicinity of the rational surface due to the eddy current induced in the conducting wall.
The steady-state solutions of the torque balance equation, (10.18), correspond to the roots of the cubic polynomial
In the limit , we can find approximate solutions of the torque balance equation, (10.18). The high-rotation solution branch is characterized by , and is such that
It is clear from Equation (10.17) that the slowing down of the island chain's rotation due to the eddy current induced in the conducting wall has a destabilizing effect on the chain. In fact, if there is no substantial slowing down then the normalized modified Rutherford equation (10.17) reduces to
Figure 10.2 shows a numerical solution of Equations (10.17) and (10.18) made for a low-field and a high-field tokamak fusion reactor. (See Chapter 1.) The simulation parameters are determined using the following assumptions: (low-field) or (high-field), , , (where and are the deuteron and triton masses, respectively), , , , , . Here, is the electron diamagnetic frequency [see Equation (5.45)]. The poloidal and toroidal mode numbers of the magnetic island chain are are , respectively. The wall parameters are and , which correspond to a moderately conducting, close-fitting wall. The plasma equilibrium is assumed to be of the Wesson type (see Section 9.4), with and . It follows that . The perfect-wall saturated island width is . Finally, the various dimensionless parameters appearing in Equations (10.17) and (10.18) take the values (low-field) or (high-field), , , , , and .
It can be seen from Figure 10.2 that as the normalized width, , of the island chain grows in time, the chain's normalized rotation frequency, , is gradually reduced, until it has been reduced to about half of its original value, at which point there is a sudden collapse in the rotation frequency to a very low value. The collapse in the rotation frequency causes the chain to be further destabilized due to the loss of wall stabilization, and ion polarization effects. Consequently, the final saturated width of the island chain is greater than the perfect-wall saturated island width (i.e., ). Note that a low-field tokamak fusion reactor is more susceptible to rotation braking than a high-field fusion reactor because of its lower diamagnetic frequency (see Table 6.1), and consequent lower ion fluid rotation. The slowing-down curves shown in the figure are similar in form to those observed experimentally when a wide magnetic island chain interacts electromagnetically with a resistive wall in a toroidal confinement device .
Figure 10.3 displays the results of a series of simulations of the type shown in Figure 10.2 in which the radius of the rational surface is scanned over a range of values [by changing while keeping fixed]. The wall parameters are and . It can be seen that if the rational surface lies well inside the plasma boundary then the island chain attains its final, perfect-wall saturated width without a collapse in its rotation frequency. On the other hand, if the rational surface lies closer to the plasma boundary then a collapse in the rotation frequency is triggered before the chain attains its final saturated width. The critical island width at which the rotation collapse is triggered lies between 20% and 10% of the plasma minor radius. Moreover, the final saturated width of the island chain exceeds the perfect-wall value due to the loss of wall stabilization, and ion polarization effects. Finally, it is again clear that a low-field tokamak fusion reactor is more susceptible to rotation braking than a high-field fusion reactor.
Figure 10.4 shows the results of a series of simulations of the type shown in Figure 10.2 in which the unperturbed island chain rotation frequency is scanned over a range of values for various different wall time-constants. The plasma parameters are and , which corresponds to , whereas the wall radius is . It is clear that if the wall is very resistive (i.e., s) then the critical island width above which the chain's rotation frequency collapses is only weakly dependent on the unperturbed rotation frequency, and is about 10% of the plasma minor radius. On the other hand, for the case of a highly conducting wall (i.e., s), diamagnetic levels of plasma rotation are sufficient to completely suppress the collapse in the island rotation frequency, unless the unperturbed rotation frequency lies close to zero. As before, it is clear that a low-field tokamak fusion reactor is more susceptible to rotation braking than a high-field fusion reactor.