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
is a simulated toroidal angle.
Magnetic Field and Current Density Perturbations
Given that tearing modes in tokamak plasmas are relatively low-amplitude (i.e.,
) , global
) [see Equation (2.352)], relatively slowly-growing (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 marginally-stable ideal-MHD, (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
are the perturbed magnetic field and current density, respectively.
Equations (3.1), (3.3), and (3.9)–(3.12) yield
If we write
then, after some algebra, Equations (3.13)–(3.17) reduce to
Now, a global tearing instability in a low-, large aspect-ratio, tokamak plasma is characterized by 
It follows from Equations (3.4), (3.5), (3.7), and (3.19) that
Thus, in the low-, large aspect-ratio limit, Equations (3.20)–(3.25) simplify considerably to give
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 aspect-ratio, 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).