To zeroth order in , Equation (8.20) yields

(8.54) 
Solving this equation subject to the boundary condition (8.40), we obtain

(8.55) 
Thus, we conclude that, to lowest order in our expansion, the magnetic fluxsurfaces in the island region have the constant
structure pictured in Figure 5.7. The island Opoints correspond to
and
(where is an integer), the
Xpoints correspond to
and
, and the magnetic separatrix corresponds to
.
To zeroth order in , Equation (8.19) yields

(8.56) 
where use has been made of Equation (8.55). Given that is an odd function of , it
follows that

(8.57) 
We conclude that parallel
ion acoustic waves, whose dynamics are described by Equation (8.19), smooth out
any variations in the lowestorder normalized plasma pressure,
, around magnetic fluxsurfaces [9].
By symmetry,
inside the magnetic separatrix of the island chain. In other words, the plasma pressure profile (and, by implication, the electron number density, electron temperature, and ion temperature profile—see Section 4.1) is completely flattened inside
the separatrix.
It is helpful to define

(8.58) 
It follows that
. Furthermore, Equations (8.41) and (8.55) imply that

(8.59) 
To zeroth order in , Equation (8.16) yields

(8.60) 
where use has been made of Equations (8.55) and (8.57). Given that
is an odd function of , it follows that

(8.61) 
We conclude that the lowestorder normalized
MHDfluid streamfunction,
, is constant on magnetic fluxsurfaces.
By symmetry,
inside the magnetic separatrix of the island chain. In other words, the streamfunction profile is completely flattened inside the separatrix.
It is helpful to define

(8.62) 
It follows that
. Furthermore, Equations (8.42) and (8.55) imply that

(8.63) 
To zeroth order in , Equation (8.17) yields

(8.64) 
where use has been made of Equations (8.55), (8.57), and (8.61). Given that
is an even function of , we can write

(8.65) 
In other words, the lowestorder normalized parallel ion velocity,
, is also constant on
magnetic fluxsurfaces.
Finally, to zeroth order in , Equation (8.18) yields

(8.66) 
where use has been made of Equations (8.55), (8.57), (8.58), (8.61), and (8.62). Moreover,
.