HigherOrder Solution
In order to include the perpendicular transport term, we need to evaluate Equation (8.16) to third order in . Doing so,
we obtain

(8.77) 
where

(8.78) 
and use has been made of Equations (8.49)–(8.53), (8.57), (8.58), (8.61), (8.62), and (8.75).
The fluxsurface average of Equation (8.77) yields

(8.79) 
where use has been made of Equations (8.68), (8.70), and (8.71). Hence, we
can write

(8.80) 
where use has been made of Equation (8.76).
The first term on the righthand side of the previous equation represents the parallel current driven
inductively when the reconnected flux at the rational surface varies in time.
The second term on the righthand side of the previous equation represents the parallel return current driven by the perpendicular polarization current associated with the
acceleration of the ion fluid around the magnetic separatrix of the island chain [10,11].
In fact, it is easily seen that if the ion fluid could pass freely through the separatrix (i.e.,
and
) then the polarization term would be zero.
In order to include the perpendicular transport term, we need to evaluate Equation (8.17) to first order in . Doing so,
we obtain

(8.81) 
where use has been made of Equations (8.49)–(8.53), (8.55), (8.57), and (8.58). The fluxsurface average of the
previous equation yields

(8.82) 
where use has been made of Equation (8.68). Here, we have made use of the easily proved result that
, where
.
We can solve the previous equation, subject to the boundary condition (8.59), to give

(8.83) 
Here, we have taken into account the previously mentioned fact that within the island separatrix.
Note that
is discontinuous across the island separatrix, which implies that the pressure gradient—and, hence,
the diamagnetic velocity—is also discontinuous across the separatrix. Of course, there is not a real
discontinuity. In fact, we would expect the discontinuity to be resolved in a layer on the separatrix whose width is obtained by balancing the
parallel and perpendicular energy transport terms in Equation (8.17) [3].
In other words, we expect the layer thickness to correspond to the island width at which
.
Hence, according to Equations (8.28) and (8.29), the characteristic layer thickness is , where

(8.84) 
and
and are specified in Equations (8.33) and (8.34), respectively. (The factor of 4 ensures that
has the same value as the standard critical island width, , defined
in Reference [3].)
As is clear from Table 8.1, is of order half a centimeter in a tokamak fusion reactor.
In order to include the perpendicular transport term, we need to evaluate Equation (8.18) to first order in . Doing so,
we obtain

(8.85) 
where use has been made of Equations (8.49)–(8.53), (8.55), (8.57), (8.58), (8.61), (8.62), and (8.76). The fluxsurface average of the previous equation yields

(8.86) 
We can solve the previous equation, subject to the boundary conditions (8.59) and (8.63), to give

(8.87) 
[5].
Here, we have rejected, as unphysical, the solution that blows up as
as
.
We have also made use of the fact that within the magnetic separatrix of the island chain. Finally, we have demanded that the ion
fluid velocity—and, hence, the function
—be continuous across the separatrix, because the ion fluid possesses finite perpendicular viscosity.
Note, however, that the discontinuity in the function
across the island separatrix [see Equation (8.83)] implies that the
electron and MHD fluid velocities are discontinuous across the separatrix. (See Section 8.7.) As previously
mentioned, these discontinuities are resolved in a layer of characteristic thickness on the separatrix.
In order to include the perpendicular transport term, we need to evaluate Equation (8.19) to second order in . Doing so,
we obtain

(8.88) 
where use has been made of Equations (8.49)–(8.53) (8.55), (8.57), (8.58), (8.61), (8.62), and (8.65). The fluxsurface average of the
previous equation yields

(8.89) 
The previous equation can be solved, subject to the boundary condition (8.43), to
give

(8.90) 
Hence, we conclude that the lowestorder ion parallel flow is unaffected by the presence of the island chain.