a positive $\Delta'$ corresponds to an unstable mode [@FKR]. Similarly, also in non-linear theory, $\Delta'$ can be considered a measure of the stability of the mode [@Rutherford; @MP].

We focus now on the modifications to the stability parameter due to local changes of the profile of $J_{eq}'/\psi_{eq}'$. We first assume a given equilibrium magnetic flux $\psi_{eq0}$, associated with current density $J_{eq0}$ (and therefore to a safety factor $q_0$). We then superimpose a “zonal” field, $\delta\psi_{eq}$, which can be thought as the $m=0$, $n=0$ component of the magnetic flux perturbation (and is accompanied by a “zonal” current $\delta
J_{eq}$ and a $\delta q$). The physical origin of the corrugations, as discussed in the introduction, is suggested to arise due to a combination of short scale turbulence and zonal flows. We again emphasize that, in this calculation, we assume that they are sufficiently consistent in time and space such that a new “equilibrium” is temporarily created (hence the subscript *eq*). Obviously, there needs to be a constant source of corrugations or otherwise they would dissipate on a time scale proportional to $a^2/\eta$, where $a$ (of the order of the collisionless skin depth) is the characteristic length scale of the zonal field. In 