Figure 10.9 shows the critical island width required to trigger a rotation collapse calculated as a function of the wall thickness, for various different unperturbed island rotation frequencies, using the improved torque balance model. Separate calculations are made for a low-field and a high-field tokamak fusion reactor. (See Chapter 1.) The calculation parameters are determined using the following assumptions: (low-field) or (high-field), , , (where and are the deuteron and triton masses, respectively), , , , and . The plasma equilibrium is assumed to be of the Wesson type (see Section 9.4), with and . The poloidal and toroidal mode numbers of the tearing mode are and , respectively. It follows that . The perfect-wall saturated island width is . The wall radius and resistivity are assumed to be and (which is the electrical resistivity of stainless steel), respectively.
As is clear from Figure 10.9, the critical island width required to trigger rotation collapse increases with increasing wall thickness (because the wall becomes less electrically resistive) until a critical thickness is reached above which the critical island width becomes independent of the wall thickness. Of course, the critical wall thickness is that above which the thin-wall approximation breaks down. The eddy current induced by the rotating island chain in a wall whose thickness is greater than the critical thickness (which corresponds to a skin-depth in the wall material) only penetrates a skin-depth into the wall from its inner boundary, which implies that the effective thickness of the wall becomes the skin-depth, rather than its actual thickness. According to the figure, for a plasma with diamagnetic levels of ion fluid rotation (i.e., ), the critical island width is below 10% of the plasma minor radius for thin (i.e., ) resistive walls. On the other hand, the critical island width is about twice this value for thick (i.e., ) conducting walls. As before, it is apparent 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, and consequent lower ion fluid rotation.