Evolution of Semi-convective Staircases in Rotating Flows: Consequences for Fuzzy Cores in Giant Planets

Recent observational constraints on the internal structure of Jupiter and Saturn suggest that these planets have ``fuzzy" cores, i.e., radial gradients of the concentration of heavy elements that might span $50\%$ to $70\%$ of each planet's radius. These cores could be composed of a semi-convective staircase, i.e., multiple convective layers separated by diffusive interfaces arising from double-diffusive instabilities. However, to date, no study has demonstrated how such staircases can avoid layer mergers and persist over evolutionary time scales. In fact, previous work has found that these mergers occur rapidly, quickly leading to only a single convective layer. Using 3D simulations of convective staircases in non-rotating and rotating flows, we demonstrate that rotation prolongs the lifetime of a convective staircase by increasing the timescale for both layer merger and erosion of the interface between the final two layers. We present an analytic model for the erosion phase, predicting that rotation increases the erosion time by a factor of approximately $\mathrm{Ro}^{-1/2}$, where $\mathrm{Ro}$ is the Rossby number of the convective flows (the ratio of the rotation period to the convective turnover time). For Jovian conditions at early times after formation (when convection is vigorous enough to mix a large fraction of the planet), we find the erosion time to be roughly $10^{9}~\mathrm{yrs}$ in the non-rotating case and $10^{11}~\mathrm{yrs}$ in the rotating case. Thus, the current existence of convective staircases within the deep interiors of giant planets is a strong possibility, and rotation could be an important factor in the preservation of their fuzzy cores.