It is convenient to work in a frame of reference that co-rotates with the island chain. [This goal can be achieved by making the transformation . See the following paragraph for the definitions of these quantities.] In the co-rotating reference frame, the normalized reconnected flux at the rational surface, [see Equation (3.184)], is assumed to be a positive real quantity. It is helpful to define the reduced (by a factor four) radial width of the magnetic island chain that develops in the inner region:
[see Equation (5.129)]. Here, is the magnetic shear-length at the rational surface [see Equation (5.27)].In the following, 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 Section 5.3, we find that
in 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)], the ion parallel velocity profile [see Equation (5.19)], the magnetic shear profile [see Equation (5.28)], and the collisionless ion skin-depth at the rational surface [see Equation (4.24)]. Moreover, is the rotation frequency of the tearing mode in the laboratory frame,(8.7) |
(8.8) |