Island Rotation Frequency

Figure 9.3 shows typical normalized perpendicular velocity profiles of the electron, MHD, and ion fluids across the O-points and the X-points of an isolated magnetic island chain in the island rest frame, as determined from Equations (8.72)–(8.74) and (9.28)–(9.30) [7]. Here, , where is the poloidal mode number of the tearing mode, and is the diamagnetic frequency at the rational surface [see Equation (5.47)]. It can be seen that all three fluids co-rotate with the island chain inside the magnetic separatrix. The ion fluid velocity profile is completely unaffected by the presence of the chain. On the other hand, the electron and the MHD fluid velocity profiles are modified by the island chain. In particular, both profiles are discontinuous across the magnetic separatrix. As was explained in Section 8.9, these discontinuities are resolved in a thin layer of thickness on the separatrix [see Equation (8.84) and Table 8.1]. It is clear from the figure that the island chain rotates in the ion diamagnetic (i.e., negative) direction with respect to the unperturbed E-cross-B frame at the rational surface [7].

Equations (8.45), (8.63), (9.27), and (9.29) yield , or

where and(9.33) |

Finally, it is reasonable to suppose that the critical island width that must be exceeded before a magnetic island chain switches from rotating in the electron direction, with respect to the unperturbed local E-cross-B frame, to rotating in the ion direction is that above which the pressure profile flattens within the separatrix [14].