Let
be the perturbed current density associated with the tearing mode. We can write
Normally, according to our previous assumptions, all three contravariant components of
are zero. Consider, however, the
behavior in the vicinity of the th rational surface, , at which
. (See Section 3.7.) In general, ,
, and
are continuous across the surface, whereas is discontinuous [11,14].
Hence, we deduce that
where use has been made of Equations (14.45) and (14.54). Here, it is assumed that there are rational surfaces in the plasma,
numbered sequentially from 1 to , in the order of the innermost to the outermost. It is easily demonstrated from Equations (14.7)–(14.9), (14.19)–(14.22),
and (14.58)–(14.60) that
at a given rational surface. Thus, we conclude that a current sheet forms at
each rational surface in the plasma. Moreover, each sheet is made up of current filaments that flow *parallel* to the local equilibrium magnetic field.