Shallow water magnetohydrodynamic theory with application to Planetary ring

Introduction: Planetary rings, such as those of Saturn, are thin, rotating systems composed of numerous small particles. Due to their small aspect ratio, they can be approximated as shallow fluid layers in the continuum limit.

Scientific context: We apply rotating shallow water theory to model these rings, adapting it to account for the parent planet’s gravitational potential and magnetic field. The dynamics are further influenced by particle charging via solar wind, satellite interactions, photoionization, and triboelectric effects. These introduce electromagnetic forces, motivating the use of shallow water magnetohydrodynamics (SWMHD).

Method and Results: We derive the SWMHD equations for a rotating annular channel subject to radial gravity and an external magnetic field, representing the planetary ring system. Analytical exploration of the resulting wave modes reveals the dependence of wave frequencies on geometric and physical parameters like aspect ratio, channel width, fluid column height, and gravity harmonics. We identify critical parameter thresholds where certain wave modes cease to exist. In the asymptotic limit of a large-radius channel, the system behavior converges to that of classical rotating channel flows, validating the model. 

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