J. Geophys. Res. 117, A04215 (2012)
Electromagnetic activity in Saturn's magnetosphere varies over a period that is almost, but not exactly, the same as the planet's 10.7-hour rotational period. The two may be linked, but correlation is not the same as causation. Moreover, Saturn's planetary magnetic field is well-aligned with its axis of rotation, so it isn't clear how its field and its rotation could be linked, even in principle.
Xianzhe Jia and colleagues suggest that they might be connected by long-lived disturbances in Saturn's upper atmosphere. They postulate the existence of cyclone-like features that drive perturbations of its ionosphere, which in turn interact with its magnetosphere and cause its electromagnetic activity to vary. Such features should rotate almost in time with the planet's rotation, but lag slightly behind.
Magnetohydrodynamic simulations by Jia et al. reproduce much of the periodic behaviour observed in Saturn's magnetosphere, and the authors argue that this is the most comprehensive explanation so far. However, it remains to be seen whether the ionospheric cyclones they postulate actually exist.
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Gerstner, E. Cyclones on Saturn. Nature Phys 8, 442 (2012). https://doi.org/10.1038/nphys2343