Reusing the analysis of Sections 5.7–5.9, let us again suppose that there are two layers in space (i.e., Fourier space). The
small- layer turns out to be of width
Here, is the hydromagnetic timescale defined in Equation (5.43). Given that we are effectively assuming that
, the condition for the separation of the layer solution into two layers (i.e., that the width of the small- layer is less than that of the large- layer) is always satisfied. The large- layer is governed by the equation
Here, is the normalized diamagnetic frequency [see Equation (5.65)], and
are magnetic Prandtl numbers [see Equations (5.53) and (5.54)], and is the ratio of the electron
to the ion pressure gradient at the rational surface [see Equation (4.5)].
The boundary conditions on Equation (6.5) are
that is bounded as
. Here, is an arbitrary constant.
In the various constant- linear growth-rate regimes considered in the next section, Equation (6.5) reduces to an equation of the form
where is real and non-negative, and is a complex constant. As described in Section 5.8, the solution of this
equation that is bounded as
can be matched to the small- asymptotic form (6.8) to give
. The width of the large- layer in space is