As before (see Section 8.2), it is assumed that , where is the linear layer width. (See Chapter 6.) In other words, the width of the island chain is assumed to be much greater than the linear layer width, but much less than the minor radius of the rational magnetic flux-surface. Let . Reusing the analysis of Sections 5.3 and 8.2, we find thatin the limit (i.e., many island widths from the rational surface). Here, , , , , and , where is the E-cross-B velocity profile in the outer region [see Equation (5.21)], the diamagnetic velocity profile [see Equation (5.29)], and the magnetic shear profile [see Equation (5.28)]. Moreover, , and . The parameter is introduced into the analysis in order to take into account the fact that the E-cross-B velocity profile in the outer region (i.e., everywhere in the plasma apart from the immediate vicinity of the magnetic island chain) develops a gradient discontinuity at the rational surface in response to the localized electromagnetic torque that emerges at the surface. (See Section 8.2.) Note, finally, that in neglecting any dependance of on in Equation (11.58) we are making use of the so-called constant- approximation [9,19], which is valid as long as [4,6]. Here, is defined in Equations (3.73) and (3.183).
Equations (11.58)–(11.62) are analogous to the boundary conditions in our previous reduced non-neoclassical drift-MHD model, (8.2)–(8.6), apart from Equation (11.61). The latter equation is derived on the assumption that ion neoclassical poloidal flow damping relaxes the ion poloidal velocity profile to its neoclassical value (see Section 2.18) many island widths away from the rational surface.
Let and , where is the diamagnetic frequency at the rational surface [see Equation (5.47)], and . It follows that in the immediate vicinity of the island chain. It is helpful to define the rescaled fields , , , , and , whereThese fields are (essentially) the same as those used in our previous rescaled reduced non-neoclassical drift-MHD model. (See Section 8.2.)
Equations (11.50)–(11.56) rescale to giveHere, , , and we have set
Equations (11.68)–(11.72) must be solved subject to the boundary conditions [see Equations (11.58)–(11.62) and (11.63)–(11.67)]as . Here, where is the E-cross-B frequency at the rational surface. Note that , , , , and are all quantities in the inner region. Note, further, that the boundary conditions (11.83)–(11.87), as well as the symmetry of the rescaled reduced neoclassical drift-MHD equations, (11.68)–(11.72), ensure that , , and are even functions of , whereas and are odd functions.
Finally, asymptotic matching between the inner region and the surrounding plasma yields (see Section 8.10)