Self-Consistent Evolution Models Show Weak Double-Diffusive Mixing in Jupiter and Saturn
Double-diffusive convection in the ``fuzzy'' cores of giant planets has been widely discussed as a mechanism for redistributing heavy elements, but its efficiency in evolutionary models remains uncertain. Previous estimates rely on idealized compositional structures and have not treated double-diffusive transport self-consistently in planetary evolution calculations. Here we implement a prescription for transport across convective staircases in the planetary evolution code \texttt{APPLE} and apply it to post-formation interior models of Jupiter and Saturn containing compositional gradients produced during formation. These models are evolved for 4.56 Gyr including convection, diffusion, and double-diffusive transport. We find that double-diffusive convection produces limited mixing between the deep interior and the envelope. In both Jupiter and Saturn, less than $\sim 1\,M_\oplus$ of heavy material is redistributed over the full cooling history, leaving the primordial compositional gradients largely intact. This inefficiency arises because the buoyancy work available to drive compositional transport is constrained by the thermal energy budget of the deep interior, in contrast to idealized Boussinesq simulations that operate in regimes more favorable to layer merging and efficient mixing. As a result, double-diffusive convection alone cannot significantly erode the compositional gradients generated during formation. The observed heavy-element distributions in Jupiter and Saturn therefore likely require additional transport mechanisms or formation pathways, including large collisional events, that produce broader initial mixing than standard accretion models predict.