other words, $\delta\psi_{eq}$ directly modifies $\psi_{eq0}$, thus generating the new “equilibrium”: $\psi_{eq}=\psi_{eq0}+\delta\psi_{eq}$ (together with $J_{eq}=J_{eq0}+\delta J_{eq}$ and $q=q_{0}+\delta q$). This new configuration affects the perturbed magnetic flux through Eq.\[4\] and therefore the stability parameter through Eq.\[5\]. In order to explicitly separate the effect of the “zonal” field, we write the solution of Eq.\[4\] as $\widetilde{\psi} = \widetilde{\psi}_0 + \delta \widetilde{\psi}$, where $\widetilde{\psi}_0$ is the eigenfunction associated with the old equilibrium, while $\delta\widetilde{\psi}$ is the net effect of the corrugations on the mode. Similarly, also the stability parameter becomes: $\Delta' = \Delta'_0 + \delta
\Delta'$.

Following Refs.[@Zakharov1989; @Westerhof1990], we assume that the localized current density corrugation is such that $\delta
\widetilde{\psi} \ll \widetilde{\psi}_0$, while $\delta
\widetilde{\psi}' \sim  \widetilde{\psi}'_0$. Furthermore, we assume that $\delta q \ll q_0$, which is valid when $\int dr (r
\delta J_{eq}) \ll \int dr (r J_{eq0})$ (as in the case of a localized zero-average oscillating $\delta J_{eq}$). With these approximations \[i.e. neglecting terms $O(\delta
\widetilde{\psi}/\widetilde{\psi}_0,\delta q/q_0)$\] we obtain from Eq.\[4\]: $$\label{6}
\frac{d}{dr}\left(r\frac{d}{dr}\delta\widetilde{\psi}\right)= -r
\frac{\delta J_{eq}'}{\psi_{eq}'}\widetilde{\psi}_0,$$ and consequently, after integration: $$\begin{aligned}
\label{6a} \left.\frac{d}{dr}\delta\widetilde{\psi}\right|_{r_s-\epsilon}&=& -\int^{r_s-\epsilon}_{0} dr\frac{r}{r_s-\epsilon} \frac{\delta J_{eq}'}{\psi_{eq}'}\widetilde{\psi}_0,\\
\label{6b}
\left.\frac{d}{dr}\delta\widetilde{\psi}\right|_{r_s+\epsilon}&=&\int^{\infty}_{r_s+\epsilon}
dr \frac{r}{r_s+\epsilon} \frac{\delta
J_{eq}'}{\psi_{eq}'}\widetilde{\psi}_0,\end{aligned}$$ where we have employed the fact that any change in the perturbed flux function must 