We shall adopt model plasma equations of poloidal and toroidal angular motion that are analogous to those introduced in Section 3.14. Let us write
Following the analysis of Section 3.15, it is convenient to writewhere and
Now, the modified angular velocity profiles, and , are mostly localized in the vicinity of the th rational surface. Hence, it is a reasonable approximation to express Equations (14.162) and (14.163) in the simplified forms 
Let us write [4,17,16]where Here, is a Bessel function, and denotes its th zero . Note that Equations (14.168)–(14.171) automatically satisfy the boundary conditions (14.164) and (14.165).
It is easily demonstrated that 
Equations (14.69), (14.70), and (14.166)–(14.175) yieldHere, The values of , , , and at the rational surfaces in our example tokamak discharge are specified in Table 14.8. Incidentally, Equations (14.176) and (14.177) are the toroidal generalizations of the cylindrical equations (3.190) and (3.191), respectively.
Let us define the frequency shifts that develop at the various rational surfaces in the plasma in response to the electromagnetic torques:
Consider an tearing mode that reconnects magnetic flux principally at the th rational surface in our example tokamak discharge. Let be the normalized magnetic flux reconnected at the th surface, and let us suppose that is sufficiently large that the plasma response at the th surface lies in the nonlinear regime. Because of the assumed strong shielding at the other rational surfaces in the plasma, we expect the plasma responses at these surfaces to lie in the linear regime.
We can determine the normalized reconnected magnetic fluxes, , driven at the other rational surfaces from Equations (14.119), (14.123), (14.146), and (14.182). We find that
The normalized electromagnetic torque acting at the th (where ) rational surface iswhere use has been made of Equations (14.123), (14.146), (14.181), (14.182), and (14.187). The normalized electromagnetic torque acting at the th rational surface is obtained from angular momentum conservation (see Section 14.11):
According to Equation (14.151), the frequency shift at the th rational surface modifies the real frequency of the tearing mode. In fact,
The normalized linear layer response indicies, (where ), that appear in Equation (14.188), are functions of nine normalized layer parameters. (See Table 14.3.) Two of these parameters, and , are modified by the frequency shifts induced by the electromagnetic torques. In fact, it is clear from Equations (14.130), (14.131), (14.185), and (14.191) thatwhere the superscript indicates a quantity that is unaffected by the electromagnetic torques.
Equations (14.176), (14.177), (14.184), (14.188)–(14.190), (14.192), (14.193)—together with the numerical solution of the Riccati differential equation, (5.121), subject to the boundary conditions (5.122) and (5.123), which determines the — form a closed set of equations that allow us to determine the reconnected magnetic fluxes driven at the various rational surfaces in our example tokamak discharge by a tearing mode that reconnects magnetic flux primarily at the th rational surface. The only free parameter in the model is the normalized reconnected magnetic flux at the th rational surface, .
Consider an tearing mode in our example tokamak discharge that reconnects magnetic flux principally at the (i.e., ) rational surface. Such a mode could represent an neoclassical tearing mode. It is helpful to define the natural frequencies at the various rational surfaces in the plasma:
Figure 14.4 shows the natural frequencies in our example tokamak discharge calculated as functions of the normalized magnetic island width, , in a simulation in which the island width is slowly ramped up. This calculation is made using the previously mentioned closed set of equations, as well as the data given in Tables 14.1, 14.2, 14.3, 14.5, and 14.8. It can be seen that if lies close to zero then the natural frequencies all take the unperturbed values specified in Table 14.5. On the other hand, as increases, the electromagnetic torques that develop at the rational surfaces modify the natural frequencies. In particular, the torques cause the natural frequency, , and the natural frequency, , to approach one another. At a critical value of , which is approximately 0.41, the two natural frequencies suddenly snap together, indicating a sudden loss of shielding at the rational surface .
The aforementioned sudden loss of shielding is illustrated in Figure 14.5, which shows the driven normalized magnetic island widths at the , 4, and 5 rational surfaces in our example discharge as functions of the normalized island width. It can be seen that, prior to the loss of shielding, comparatively narrow magnetic island chains are driven at the , 4, and 5 rational surfaces. However, as soon as the shielding at the rational surface is lost, there is a very significant increase in the width of the magnetic island chain driven at the surface.
Finally, Figure 14.6 shows the normalized electromagnetic torques that develop at the rational surfaces in our example discharge close to the time at which shielding is lost at the rational surface. It can be seen that the sudden loss of shielding is due to comparatively large transient electromagnetic torques that develop at the and rational surfaces, and drive the corresponding natural frequencies together.
The calculation that we have just performed indicates that the shielding of rational surfaces from one another, as a consequence of sheared rotation, in our example tokamak discharge is a very robust effect. In fact, the shielding only breaks down when a tearing mode grows to a sufficient amplitude that the width of its magnetic island chain becomes a substantial fraction of the plasma minor radius. However, when shielding breaks down, it does so in a sudden and catastrophic manner . In fact, in our example calculation, the loss of shielding at the rational surface drives a magnetic island chain at that surface whose width is sufficient that the chain would overlap with the chain at the rational surface, leading to the sudden destruction of magnetic flux-surfaces , which could quite conceivably trigger a disruption .