Modelling planetary rings through shallow water theory

Planetary rings comprise of innumerable particles with the largest solid bodies being kilometer sized, but
the abundant population is in the range of 1 − 10 cm. Such a particulate system may be modeled as a
continuum if the associated particle mean-free path λ is much smaller than the characteristic length scale
L of the problem, i.e. the Knudsen number, Kn = λ /L ≪ 1. Mean free path for rings is of the order of
particle dimensions, so Knudsen number is indeed small. Moreover, the aspect ratio of such ring systems
is generally very small. For example, Saturn’s ring system is 282, 000 km across and thickness is typically
about 10 m. These considerations motivate the present attempt to model planetary rings through shallow
water theory, with the added challenge that a radial gravity field due to the central planet is present. To
begin with, we consider an incompressible and inviscid rotating shallow fluid in an annular domain with a
radial gravity field. Modified shallow water equations are derived. The dependence of different
wave mode frequencies on the width of the annular domain and its location is investigated. Critical values
of these parameters are found for which some of the conventional shallow water wave modes cease to exist.
We also find that a radially varying central gravity field plays the same role as that of a topography with
cross-channel pressure gradient that are introduced in the usual analysis of topographic Rossby waves.