Rescaled Reduced Drift-MHD Model
Let
where
is the diamagnetic frequency at the rational surface [see Equation (5.47)], and
.
It follows that
in the immediate vicinity of the island chain.
It is helpful to define the
rescaled fields
,
,
,
, and
, where
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
Here,
,
 |
(8.21) |
and we have set
 |
(8.22) |
[see Equation (5.32)], where
is the normalized parallel plasma resistivity at the
rational surface [see Equation (5.14)].
Furthermore,
where
Here,
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.
Moreover,
![$\displaystyle L_p=\frac{5}{3}\,(1-c_\beta^2) \left[-\left(\frac{d\ln p}{dr}\right)_{r_s}\right]^{-1}$](img2700.png) |
(8.35) |
is the effective pressure gradient scale-length at the rational surface, and
.
Finally,
![$\displaystyle \tau_\parallel' = r_s^{\,2}\left/\left\{\frac{2}{3}\,(1-c_\beta^2...
...(\frac{\eta_i}{1+\eta_i}\right)\chi_{\parallel\,i}'(r_s)\right]\right\}\right.,$](img2702.png) |
(8.36) |
where
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
 |
(8.39) |
Equations (8.16)–(8.20) must be solved subject to the boundary conditions [see Equations (8.2)–(8.6) and (8.11)–(8.15)]
as
. Here,
where
is the E-cross-B frequency profile.
Note that
,
,
,
, 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.