0.045$.

The presence of a corrugation with wavelength $K$ can significantly change the growth rate of the mode, as shown in Fig.\[fig3\].

In the figure the crosses, circles and squares represent the numerical solutions obtained with the linear code for corrugations of amplitude $5\times 10^{-5}$, $1\times 10^{-4}$, $2\times
10^{-4}$, respectively. When $K<40$ (i.e. $K\lambda<1.82$), these values lead to current corrugations of maximum amplitude smaller than 7%, 15% and 30% of the equilibrium current density. We remark that even small amplitude corrugations can strongly modify the growth rate of the mode. For example, in the case shown here a corrugation that amounts to 4% of the amplitude of the equilibrium current density (e.g. $A=1\times 10^{-4}$ and $K\lambda=0.91$) can lead to a growth rate three times as large than in the “smooth” case. For values of $K\lambda$ smaller than 1 ($K<20$ in our example), the theoretical predictions obtained from the dispersion relation Eq.\[12\] together with the scaling Eq.\[9\], are in excellent agreement with the numerical data. The disagreement for larger values of the corrugation’s wavelength is due to the limitations of the validity of the scaling. Furthermore, also the dispersion relation Eq.\[12\] is not correct when the constant-$\widetilde{\psi}$ approximation does not hold 