Let
be the perturbed magnetic field associated with a tearing mode to which the plasma is subject.
Now,
,
so we obtain

(14.34) 
where use has been made of Equation (14.13).
It is easily demonstrated from Equations (14.1)–(14.5) that
Suppose, for the moment, that the tearing perturbation has periods in the poloidal direction and periods in the
toroidal direction. Let us adopt the simplifying approximation that the perturbed current density,
, is negligible in the regions lying between the
various rational surfaces in the plasma [16]. Given that
, it follows from Equations (14.13)–(14.15) that
Assuming that
, the previous three equations imply that

(14.40) 
Hence, we deduce that

(14.41) 
Consequently, the final term on the lefthand side of Equation (14.34) is of order
smaller than the other two terms. Let us assume that this final term
is negligible, as would be the case in a large aspectratio (i.e.,
) torus. It follows that

(14.42) 
where use has been made of Equation (14.3), (14.35), and (14.36). Finally, Equations (14.35) and (14.39) yield

(14.43) 
Let
where the sum is over all relevant poloidal harmonics of the perturbed magnetic field. Here, we are now taking account of the fact that a tearing mode in a toroidal tokamak plasma possesses a unique toroidal mode number, but consists of many coupled poloidal harmonics with different poloidal mode numbers [5,6,11,14,22,31].
Operating on Equations (14.42) and (14.43) with
, we
obtain
where
Incidentally, we can recover the approximate relations (14.46) and (14.47) from the
completely general analysis of Reference [11] by neglecting the equilibrium plasma current, as well as by taking the limit that
. This procedure is roughly
equivalent to neglecting the term involving in the cylindrical tearing mode equation, (3.60). Hence, by analogy
with this equation, we would expect our toroidal tearing mode to be classically stable (given that the
drive for the classical tearing instability in the cylindrical tearing mode equation derives from the term involving ). However, this does not preclude the possibility that our
toroidal tearing mode could be unstable as a neoclassical tearing mode. (See Chapter 12.)
Finally, it is readily demonstrated that