Abstract Motivated by Saturn’s circumpolar hexagonal wave, we derive, at leading order in the thin-shell parameter, a consistent set of nonlinear governing equations for the dynamics of flows confined to a thin layer in a narrow, zonal, cloud band on Saturn and driven by internal heat forcing. Some exact solutions of this nonlinear system are derived in the Lagrangian setting. These solutions describe oscillations superimposed on a sheared high-speed zonal jet, with hypotrochoidal particle trajectories in the shape of a regular polygon with rounded corners. We show that the specific number of corners is determined by the heat forcing and by the location of the hexagonal pattern. The speeds and the temperature associated with this model for Saturn’s hexagon are in good agreement with observations. Significance Statement Saturn’s long but narrow circumpolar hexagonal wave in its Northern Hemisphere has fascinated scientists since it was first glimpsed in the 1980s. Although several phenomenological models have been able to reproduce such flow patterns, a self-consistent model for how large-scale and high-speed polygonal jets might form in the upper troposphere is lacking. Observations show that an assortment of smaller vortices that are caught up in the hexagon’s jet stream rotate clockwise. On the other hand, the circles of latitude that delimit the zonal band to which the hexagon is confined rotate counterclockwise due to the eastward jets along them. We derive a consistent set of nonlinear governing equations for the dynamics of flows confined to a thin layer in a narrow, zonal, cloud band that admit solutions with particles moving on paths shaped like regular polygons with rounded corners, which arise as trajectories of the movement of two circular motions in opposite directions. In Saturn’s case, the specific number of corners is linked to the internal heat forcing and to the location of the pattern.
Constantin et al. (Tue,) studied this question.