Island Rotation Frequency
Equations (11.144) and (12.3) yield

(12.4) 
It follows from Equation (11.136) that
. Hence, Equations (11.113) and (11.130) imply that the
MHD streamfunction of an isolated magnetic island chain takes the form

(12.5) 
Here,
is a magnetic fluxsurface label that takes the value zero at the
island Opoints, unity on the magnetic separatrix, and is much larger that unity far from the island chain [see Equation (11.101)]. Moreover, is a dimensionless measure of the ion temperature gradient at the rational surface [see Equation (4.4)], and the ratio of the electron and ion pressure gradients at the rational surface
[see Equation (4.5)]. The quantity

(12.6) 
where is the fraction of trapped particles at the rational surface [see Equation (2.202)], parameterizes the influence of ion neoclassical
poloidal flowdamping [see Equations (2.217), (2.218), and (11.5)].
Finally, the function
is specified in Section 8.11. Note that
as
. It follows from Equation (12.5) that

(12.7) 
However, Equations (11.109) and (12.4) yield

(12.8) 
where [see Equation (11.88)]

(12.9) 
Here, is the rotation
frequency of magnetic island chain in the laboratory frame (see Section 5.4),
the EcrossB frequency at the rational surface [see Equation (5.44)], and
the diamagnetic frequency at the rational surface [see Equation (5.47)].
The previous three equations give

(12.10) 
which implies that

(12.11) 
where

(12.12) 
is the ion diamagnetic frequency at the rational surface [see Equation (5.46)].
We conclude that, as a result of ion neoclassical poloidal flowdamping (i.e.,
), an isolated magnetic island chain does not exactly
corotate with the unperturbed ion fluid at the rational surface (i.e.,
). (Recall that nonneoclassical theory leads to the
prediction that a wide magnetic island chain does exactly corotate with the unperturbed ion fluid at the rational surface. See Section 9.5.)
Instead, assuming that
, and given that
under most circumstances [see Equation (12.6)], we deduce that a wide magnetic island chain propagates in the electron diamagnetic direction with respect to the unperturbed ion fluid at the rational surface, but in
the ion diamagnetic direction with respect to the unperturbed MHD fluid [8,9,33]. This behavior should be contrasted with that of
a linear tearing mode, which propagates in the electron diamagnetic direction with respect to the unperturbed MHD fluid. (See Chapter 6.)
Figure: 12.1
Normalized perpendicular velocity profiles across the Opoints (lefthand panel) and the Xpoints (righthand panel)
of an isolated magnetic island chain in the island rest frame. The solid, dashed, and dotted
lines correspond to the electron, MHD, and ion fluids, respectively. Here, , , and
. (See Table 11.1.)

According to Equations (8.75), (8.76), (11.130), and (12.5), the electron and ion fluid streamfunctions take the respective forms
The poloidal components of the perpendicular velocities of the
electron, MHD, and ion fluids in the island rest frame are related to their respective streamfunctions, , , and , according to Equations (8.72)–(8.74).
Figure 12.1 shows typical normalized perpendicular velocity profiles of the electron, MHD, and ion fluids across the Opoints and
the Xpoints of an isolated magnetic island
chain in the island rest frame, as determined from Equations (8.72)–(8.74),
(12.5), (12.13), and (12.14). Here, is the magnetic fluxsurface minor radius,
, and is the poloidal mode number of the tearing mode. It can be seen that all three fluids corotate with the island
chain inside the magnetic separatrix. However, unlike the nonneoclassical case (see Figure 9.3), ion neoclassical poloidal flowdamping (i.e.,
) causes the ion fluid velocity profile
to be modified by the presence of the island chain.
In fact, all three velocity 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. It is clear from the figure that the island chain rotates in the ion diamagnetic (i.e., negative) direction with respect to
the unperturbed EcrossB frame (i.e., the unperturbed MHD fluid) at the rational surface, but in the electron diamagnetic (i.e. positive) direction with respect to the unperturbed ion fluid.