vanish far from the region where the current corrugation is located (i.e. $d\delta \widetilde{\psi}/dr= 0$ both for $r = 0$ and for $r = \infty$). From Eqs.\[5\], \[6a\] and \[6b\] we obtain: $$\label{7} \delta \Delta'=\frac{1}{r_s}\lim_{\epsilon\rightarrow
0}\left[\int_{0}^{r_s-\epsilon} rdr \left( \frac{\delta
J_{eq}'}{\psi_{eq}'}\frac{\widetilde{\psi}_0}{\widetilde{\psi}_{r_s}}\right)+\int_{r_s+\epsilon}^{\infty}
rdr \left( \frac{\delta
J_{eq}'}{\psi_{eq}'}\frac{\widetilde{\psi}_0}{\widetilde{\psi}_{r_s}}\right)\right]=
\frac{1}{r_s}P\int_{0}^{\infty} rdr \left( \frac{\delta
J_{eq}'}{\psi_{eq}'}\frac{\widetilde{\psi}_0}{\widetilde{\psi}_{r_s}}\right),$$ where $P$ stands for the principal value of the integral (necessary because the integrand is singular around the resonant surface).

In the next Section we explicitly calculate $\delta\Delta'$ for a given current density corrugation in slab geometry. This allows us to obtain a semi-analytic scaling of $\delta \Delta'$ with respect to the features of $\delta \psi_{eq}$.

Slab case and $\delta \Delta'$ scaling
======================================

In order to understand how a current corrugation affects the stability parameter, we first investigate its effect in a simple slab configuration. The problem is analyzed in a double periodic rectangular box, where the normalized “radial” variable is $-\pi
\leq x \leq \pi$, and the normalized “helicoidal” variable is $-\pi/k_y \leq y \leq \pi/k_y$, where $k_y$ is the aspect ratio of the box (and also the wave number of the mode). We assume a given normalized equilibrium current density, $J_{eq0}=\cos(x)$ \[consequently, $\psi_{eq0}=\cos(x)$\]. With this choice, resonant surfaces form 