Consider a magnetic island chain of width that reconnects magnetic flux at the th rational surface in the plasma. The toroidal equivalent of the cylindrical island width evolution equation, (12.15), is
where , , is specified in Equation (14.125), , where is defined in Equation (A.98), and Here, denotes the noninductive “bootstrap” component (i.e., the component that is driven by pressure gradients, rather than the parallel electric field) of the equilibrium plasma current (see Section 2.20), is a fluxsurface average operator [see Equation (A.82)], and . In writing Equation (14.196), we have neglected the relatively unimportant (for neoclassical tearing modes) island saturation term in the Rutherford equation. (See Section 12.2.) We have also neglected the ion polarization term (because we have previously shown that this term is negligible). (See Section 12.4.) Finally, in accordance with the previous analysis in this chapter, we have assumed that there is perfect shielding at all of the other rational surfaces in the plasma (i.e., for ).Equation (14.196) states that the width of the magnetic island chain evolves on the local (to the th rational surface) resistive timescale, , in response to the effective tearing stability index, [20], the destabilizing (because is usually positive) effect of the loss of the bootstrap current inside the chain's magnetic separatrix [3], and the stabilizing (because is usually negative) effect of magnetic fieldline curvature [28].
Making use of Equation (A.99), it can be seen that the curvature term (i.e., the third term on the righthand side) in Equation (14.196) is entirely consistent with the corresponding term in Equation (12.15). However, our new curvature term is more general than our previous cylindrical version because the calculation of the dimensionless curvature parameter outlined in Section A.8 makes no assumptions about the geometry of the equilibrium magnetic fluxsurfaces (other than that they are axisymmetric) [21].
Making use of Equation (14.197), the bootstrap term (i.e., the second term on the righthand side) in Equation (14.196) is equivalent to the corresponding term in Equation (12.15) provided that the parallel bootstrap current takes the form [see Equation (2.265)]
(14.198) 
(14.200) 
Equations (14.197) and (14.199) yield
(14.201) 
(14.202) 
As was mentioned in Section 12.4, the bootstrap and curvature terms in the generalized Rutherford equation, (14.196), depend crucially on an assumed flattening of the plasma pressure profile inside the magnetic separatrix of the island chain. However, if the island width falls below a certain threshold value then the transport of heat and particles along magnetic fieldlines cannot compete with the anomalous transport of heat and particles across magnetic fluxsurfaces, and the pressure flattening within the magnetic separatrix is lost [9]. Under these circumstances, we would expect a modification of the bootstrap and curvature terms in the generalized Rutherford equation [9,30].
According to the analysis of Reference [9], the critical value of below which the electron temperature fails to flatten inside the magnetic separatrix of our magnetic island chain can be written
where , and Here, is the electronion collision time [see Equation (2.20)], the electron thermal speed [see Equation (2.17)], the (anomalous) electron perpendicular energy diffusivity, and the electron parallel energy diffusivity. Equation (14.205) specifies the parallel diffusivity predicted by the short meanfreepath theory of Braginskii [2]. [See Equation (2.54).] Note that this expression has been corrected into order to take into account the presence of impurity ions in the plasma. Equation (14.206) specifies the parallel diffusivity predicted by the long meanfreepath theory of Section 2.23 [24,25] on the assumption that(14.207) 
According to the analysis of Reference [9], the critical value of below which the ion temperature fails to flatten inside the magnetic separatrix of our magnetic island chain can be written
(14.208) 
Finally, according to the analysis of Reference [9], the critical value of below which the electron number density fails to flatten inside the magnetic separatrix of our magnetic island chain can be written
(14.212) 
Table 14.9 specifies the critical island widths below which the local electron temperature, the ion temperature, and the electron number density profiles fail to flatten in our example tokamak discharge. These critical widths are calculated from the equilibrium and profile data shown in Figures 14.1–14.3. It can be seen that the critical island width below which the electron temperature profile fails to flatten, , is of order 2% of the plasma minor radius. On the other hand, the critical island width below which the ion temperature profile fails to flatten, , is significantly larger than (because ions stream along magnetic fieldlines at a considerably slower rate than electrons) [9]. Finally, the critical island width below which the electron number density fails to flatten, , lies between and .
Let us generalize the righthand side of the generalized Rutherford equation, (14.196), to take into account the incomplete flattening of the electron temperature, ion temperature, and electron number density profiles within the magnetic separatrix of the island chain. In order to achieve this goal, we need to identify which components of the bootstrap and curvature terms are associated with the flattening of the electron temperature, ion temperature, and electron number density profiles, and then modify these components in the appropriate manner. We shall assume that the majority ion and impurity ion temperature profiles both fail to flatten below the same critical island width. Likewise, we shall assume that the electron, majority ion, and impurity ion number density profiles all fail to flatten below the same critical island width. Taking these considerations into account, and making use of the analysis of References [9] and [30], we arrive at
where(14.217)  
(14.218)  
(14.219)  
(14.220)  
(14.221)  
(14.222)  
(14.223)  
(14.224)  
(14.225)  
(14.226)  
(14.227)  
(14.228)  
(14.229)  
(14.230)  
(14.231)  
(14.232)  
(14.233) 

Table 14.10 specifies the bootstrap and curvature parameters that appear on the righthand sides of the generalized Rutherford equations, (14.216), of the tearing modes in our example tokamak discharge. These parameters are calculated from the equilibrium and profile data shown in Figures 14.1–14.3, as well as the neoclassical theory set out in Appendix A. It can be seen that the , , and parameters are all positive, indicating that the bootstrap terms in the Rutherford equations are destabilizing. On the other hand, the , , and parameters are all negative, indicating that the curvature terms in the Rutherford equations are stabilizing. It it also clear that the magnitudes of the , , and parameters generally exceed those of the , , and parameters, which implies that the destabilizing effect of the perturbed bootstrap current is larger than the stabilizing effect of magnetic fieldline curvature.
Finally, the righthand sides of the generalized Rutherford equations of the tearing modes in our example tokamak discharge are plotted as functions of the normalized island widths in Figure 14.7. These righthand sides are calculated from Equation (14.216) using the data set out in Tables 14.2, 14.9, and 14.10. It can be seen that only the (i.e., ) mode has a righthand side that rises above zero. It follows that only the neoclassical tearing mode is is potentially unstable. (See Section 12.4.) The critical island width above which the mode is triggered (i.e., the smaller zerocrossing of the righthand side) is about 1% of the plasma minor radius. The saturated island width (i.e., the larger zerocrossing of the righthand side) is about 17% of the plasma minor radius.