Magnetic Field and Current Density Perturbations
Consider a tearing mode perturbation that has periods in the poloidal direction,
and periods in the toroidal direction, where , , and
. We shall assume that
all perturbed scalar and vector quantities vary as
respectively,
where
is a simulated toroidal angle.
Given that tearing modes in tokamak plasmas are relatively lowamplitude (i.e.,
) [15], global
(i.e.,
) [see Equation (2.352)], relatively slowlygrowing (i.e.,
) [see Equation (2.362)] instabilities, it follows from the analysis of Section 2.25 that they are governed by the linearized forms of the
equations of marginallystable idealMHD, (2.375)–(2.380). In particular, the
linearized form of the curl of the force balance criterion, (2.377), combined with the linearized forms of
Maxwell's equations, (2.349)–(2.351), give
where
and
are the perturbed magnetic field and current density, respectively.
Equations (3.1), (3.3), and (3.9)–(3.12) yield
and

(3.15) 
with
where
If we write

(3.20) 
then, after some algebra, Equations (3.13)–(3.17) reduce to
and [6,10]

(3.23) 
where
Now, a global tearing instability in a low, large aspectratio, tokamak plasma is characterized by [6]
It follows from Equations (3.4), (3.5), (3.7), and (3.19) that
Thus, in the low, large aspectratio limit, Equations (3.20)–(3.25) simplify considerably to give
and

(3.35) 
It is also easily demonstrated that
Hence, we conclude that the magnetic field and current density perturbations associated with a
tearing mode in a low, large aspectratio, tokamak plasma are specified by Equations (3.32)–(3.38).
From now on, we shall treat as approximately independent of , in accordance with Equation (3.29).