is the diamagnetic frequency at the rational surface [see Equation (5.47)], and
It follows that
in the immediate vicinity of the island chain.
Rescaled Reduced Drift-MHD Model
It is helpful to define the
Here, a dimensionless measure of the plasma pressure at the rational surface [see Equation (4.65)].
The reduced drift-MHD model specified in Section 5.2 rescales to give
and we have set
[see Equation (5.32)], where
is the normalized parallel plasma resistivity at the
rational surface [see Equation (5.14)].
is the ion sound radius, the resistive diffusion time [see Equation (5.49)],
the toroidal momentum confinement time [see Equation (5.50)], and
the energy confinement time [see Equation (5.52)]. All of these quantities are evaluated at the rational surface.
is the effective pressure gradient scale-length at the rational surface, and
Here, and are the electron and ion thermal velocities, respectively [see Equation (2.17)].
Note that the parallel electron and ion energy diffusivities
have been estimated from Equations (2.319) and (2.320), respectively,
on the assumption that
, which is the typical parallel wavenumber of the tearing perturbation at the edge of a magnetic island chain of reduced width . Note that in writing Equation (8.18)
we have made use of the easily proved identity
Equations (8.16)–(8.20) must be solved subject to the boundary conditions [see Equations (8.2)–(8.6) and (8.11)–(8.15)]
is the E-cross-B frequency profile.
, , and are all
quantities in the inner region.
Note, further, that the boundary conditions (8.40)–(8.44),
as well as the symmetry of the rescaled, reduced, drift-MHD equations, (8.16)–(8.20), ensure that
, , and
are even functions of , whereas and
are odd functions.