Resonant surface of a perturbation

Equation (260) indicates that, for the differential equation (256), there is a resonant response to a perturbation $e^{i (m \theta - n \zeta)}$ on a magnetic surface with $m - n q = 0$. Therefore, the magnetic surface with $q = m / n$ is called the ``resonant surface'' for the perturbation $e^{i (m \theta - n \zeta)}$. The phase change of the perturbation along a magnetic field is given by $m \Delta \theta - n \Delta \zeta$, which can be written as $\Delta \theta (m - n q)$. Since $m - n q = 0$ on a resonant surface, the above formula indicates that there is no phase change along a magnetic field line on a resonant surface, i.e., the parallel wavenumber $k_{\parallel}$ is zero on a resonant surface.