Saturn hexagon as a form of internal Stokes waves

Observations show a hexagon of the very regular shapes with equal corners of practically 120 degrees on the Saturn surface. There are already several theories aimed to explain appearance of these corners. On the other hand, such corners are inherent to Stokes waves and these waves can have the various nature. In this study a simplified 2D problem on inner Stokes waves within a circle with two incompressible fluids of slightly diverse densities and a vortex with the center coinciding with the center of this circle originates is considered; the steady flows exist inside and outside the inner wave surface separating two fluids. Its shape is determined via solving the corresponding nonlinear free-surface problems. Numerical solutions for various ratios of fluid densities and circle radius to hexagon side size are compared with the observed hexagon.