Consider a wave mode propagating through a plasma—for example, the ionosphere— in which the direction of the background magnetic field is slowly changing. We have just demonstrated that so long as B is not almost perpendicular to k, we can use the quasi-longitudinal approximation, the difference in phase velocity between the two eigenmodes is ∝ B · k, and the integral for the magnitude of the rotation of the plane of polarization is ∝∫n_e B · dx.

Now, suppose that the parallel component of the magnetic field changes sign. It has been implicitly assumed that the faster eigenmode, which is circularly polarized in the quasi-longitudinal approximation, becomes the slower eigenmode (and vice versa) when the field is reversed, and the Faraday rotation is undone. However, if we track the modes using the full dispersion relation, we find that the faster quasi-longitudinal eigenmode remains the faster eigenmode in the quasi-perpendicular regime, and it becomes the faster eigenmode with opposite sense of circular polarization in the field-reversed quasi-longitudinal regime. Now, let there be a second field reversal where an analogous transition occurs. Following this logic, the net rotation should be ∝ ∫n_e|B · dx|. What is going on?