To summarize, the modification of the stability parameter of a constant-$\psi$ tearing mode depends linearly on the amplitude of the “zonal” magnetic field and has a strong cubic dependence on its wavelength : $$\label{9} \delta \Delta' \approx C\cos(\alpha) A K^3,$$ where $C$ is a constant depending on the equilibrium and the system configuration (in the simple case treated here $C=\pi$). When considering the features of the current corrugation, the scaling is linear both in its wavelength ($K$) and amplitude ($AK^2$). Although this scaling was obtained with a basic model, it applies also to more complicated geometries, as will be shown in Section IV.

As an application, which is also useful to verify the validity of Eq.\[9\], we study numerically the dispersion relation of a tearing mode generated by a corrugated equilibrium. The calculation is performed in the same 2D slab double periodic box, equilibrium and corrugation described above. The equation that we solve are the plasma vorticity equation and Ohm’s law (resistive reduced MHD model): $$\begin{aligned}
\label{10} \frac{d U}{dt}&=&\nabla_{\|}J,\\
\label{11} \frac{d \psi}{dt}&=&S^{-1}(J_{eq}-J),\end{aligned}$$ where all the quantities are normalized: the lengths with respect to typical macroscopic scale, the time to the relevant Alfven time, $S$ is the Lundquist number, and all 