  The global ideal kink equation, for cylindrical geometry and zero beta, is
simplified in the high poloidal mode number limit and used to determine the
tearing stability parameter, $\Delta^\prime$. In the presence of a steep
monotonic current gradient, $\Delta^\prime$ becomes a function of a parameter,
$\sigma_0$, characterising the ratio of the maximum current gradient to
magnetic shear, and $x_s$, characterising the separation of the resonant
surface from the maximum of the current gradient. In equilibria containing a
current "spike", so that there is a non-monotonic current profile,
$\Delta^\prime$ also depends on two parameters: $\kappa$, related to the ratio
of the curvature of the current density at its maximum to the magnetic shear,
and $x_s$, which now represents the separation of the resonance from the point
of maximum current density. The relation of our results to earlier studies of
tearing modes and to recent gyro-kinetic calculations of current driven
instabilities, is discussed, together with potential implications for the
stability of the tokamak pedestal.
