We present Direct Numerical Simulations of the transport of heat and heavy elements across a double-diffusive interface or a double-diffusive staircase, in conditions that are close to those one may expect to find near the boundary between the heavy-element rich core and the hydrogen-helium envelope of giant planets such as Jupiter. We find that the non-dimensional ratio of the buoyancy flux associated with heavy element transport to the buoyancy flux associated with heat transport lies roughly between 0.5 and 1, which is much larger than previous estimates derived by analogy with geophysical double-diffusive convection. Using these results in combination with a core-erosion model proposed by Guillot et al. (2004), we find that the entire core of Jupiter would be eroded within less than 1Myr assuming that the core-envelope boundary is composed of a single interface. We also propose an alternative model that is more appropriate in the presence of a well-established double-diffusive staircase, and find that in this limit a large fraction of the core could be preserved. These findings are interesting in the context of Juno's recent results, but call for further modeling efforts to better understand the process of core erosion from first principles.

Oscillatory double-diffusive convection (ODDC; also known as semi-convection) refers to a type of doublediffusive instability that occurs in regions of planetary and stellar interiors that have a destabilizing thermal stratification and a stabilizing mean molecular weight stratification. In this series of papers, we use an extensive suite of three-dimensional (3D) numerical simulations to quantify the transport of heat and chemical species by ODDC. Rosenblum et al. first showed that ODDC can either spontaneously form layers that significantly enhance the transport of heat and chemical species compared to microscopic transport or remain in a state dominated by large-scale gravity waves, in which there is a more modest enhancement of the turbulent transport rates. Subsequent studies in this series focused on identifying under what conditions layers form and quantifying transport through layered systems. Here we proceed to characterize transport through systems that are unstable to ODDC, but do not undergo spontaneous layer formation. We measure the thermal and compositional fluxes in nonlayered ODDC from both two-dimensional (2D) and 3D numerical simulations, and show that 3D simulations are well approximated by similar simulations in a 2D domain. We find that the turbulent mixing rate in this regime is weak and can, to a first-level approximation, be neglected. We conclude by summarizing the findings of papers I through III into a single prescription for transport systems unstable to ODDC.