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Recent measurements of Jupiter's gravitational field (by Juno) and seismologyof Saturn's rings (by Cassini) strongly suggest that both planets have astably-stratified core that still possesses a primordial gradient in theconcentration of heavy elements. The existence of such a "diffusely" stratifiedcore has been a surprise as it was long expected that the Jovian planets shouldbe fully convective and hence fully mixed. A vigorous zone of convection,driven by surface cooling, forms at the surface and deepens through entrainmentof fluid from underneath. In fact, it was believed that this convection zoneshould grow so rapidly that the entire planet would be consumed in less than amillion years. Here we suggest that two processes, acting in concert, present asolution to this puzzle. All of the giant planets are rapidly rotating and havea cooling rate that declines with time. Both of these effects reduce the rateof fluid entrainment into the convection zone. Through the use of an analyticprescription of entrainment in giant planets, we demonstrate that these twoeffects, rotation and dwindling surface cooling, result in a convection zonewhich initially grows but eventually stalls. The depth to which the convectiveinterface asymptotes depends on the rotation rate and on the stratification ofthe stable interior. Conversely, in a nonrotating planet, or in a planet thatmaintains a higher level of cooling than current models suggest, the convectionzone deepens forever, eventually spanning the entire planet.

Dwindling Surface Cooling of a Rotating Jovian Planet Leads to a Convection Zone that Grows to a Finite Depth