Why Double Diffusion Doesn’t Mix the Cores of Jupiter and Saturn
Double-diffusive convection in the "fuzzy" cores of Jupiter and Saturn is investigated as a mechanism for heavy element transport. Transport via convective staircases was added to the APPLE evolutionary code; calculations were performed for models with composition gradients set by accretion. Evolution was followed for 4.56 Gyr, accounting for convection, diffusion, and double diffusion. Result: less than ~1 M⊕ of heavy material is mixed over the entire cooling time; initial composition gradients persist. The cause: buoyancy work for material transport is limited by the thermal energy of the interior, whereas idealized Boussinesq simulations encourage layer merging. Thus, double-diffusive convection alone does not erase gradients; observed heavy element distributions point to additional mechanisms—such as giant collisions—that provide broader initial mixing than in standard accretion models.
Context
Understanding the interior structure of gas giants, from our own Jupiter to distant exoplanets, is key to unraveling their formation and evolution. Data from the Juno and Cassini missions point to fuzzy cores, where heavy elements (water, carbon) are spread across an extended region rather than confined to a sharp boundary. Double diffusive convection—a process that arises when a destabilizing temperature gradient coexists with a stabilizing composition gradient—has long been considered a likely mixing mechanism. Yet its real efficiency under planetary interior conditions, where heat transport dominates over substance diffusion, has remained unknown.
Methods
To assess the role of double diffusion, the authors have, for the first time, self-consistently implemented Spruit's (2013) recipe for convective 'staircases' into the APPLE planetary evolution code. Initial conditions were taken from formation models of Jupiter and Saturn, which already featured fuzzy cores due to partial vaporization of accreting planetesimals—essentially comets. The code accounted for ordinary convection (according to the Schwarzschild criterion), molecular diffusion, and double diffusive transport. The key parameter is the stability ratio 𝑅𝜌, which depends on the superadiabatic gradient and composition gradient. Effective transport coefficients for composition and heat were set as functions of 𝑅𝜌 and the microscopic properties of hydrogen and helium. The evolution was tracked on a Lagrangian mesh over 4.56 billion years, solving the structure and transport equations jointly using the Henyey method.
Results
The main result: double diffusive convection is catastrophically inefficient. In the baseline simulation, Jupiter redistributed only ≈0.11 Earth masses of heavy elements from the core to the envelope, and Saturn ≈0.07 M⊕. Artificially boosting the coefficients by a factor of 100 only raised these values to ≈0.55 and 0.61 M⊕, respectively, while a thousandfold boost yielded ≈2 and 1.3 M⊕. Strikingly, even this extreme exaggeration of transport does not lead to significant smearing of the profiles. The reason is self-regulation: enhanced heat transport lowers the superadiabatic gradient, reducing 𝑅𝜌, on which the effective coefficients depend nonlinearly. A negative feedback loop emerges, preventing runaway mixing. Thus, the primordial composition gradients imprinted during formation survive for billions of years.
Implications
The results imply that double diffusion cannot be the main architect of fuzzy cores. The observed distributions of heavy elements must be explained either by the accretion process itself (which may produce more extended gradients than previously thought) or by rare events—for instance, head-on collisions with bodies of ~10 M⊕. This calls into question numerous evolutionary models that relied on efficient post-formation mixing, and opens the door to revising transport mechanisms in planetary interiors. As Chandrasekhar already emphasized in the context of stars, mixing in compressible environments is energetically expensive, and planetary conditions only amplify this effect.
Future development
The immediate next step is to carry out hydrodynamic simulations in a fully compressible setting, without the Boussinesq approximation that artificially eases mixing. Calculations with a larger number of scale heights are already being developed, which will allow quantitative linking of the energy budget and transport efficiency. Moreover, detailed formation models (accounting for, say, migration of bodies and gases) must be coupled with long-term evolution to trace the fate of heavy elements from planetesimals to the structures measured by Juno and Cassini.
Impact
The work directly impacts the physics of gas giants and exoplanets, astrophysical fluid dynamics, and the interpretation of observational data, including future spectroscopy of distant world atmospheres. New insights into the energetics of convective mixing could reshape approaches to modeling brown dwarfs and even massive exoplanets, where similar processes may influence observed heat flows and chemical composition.
Next steps
Incorporating realistic impact physics into evolutionary codes and performing joint “formation + evolution” simulations capable of reproducing both gravitational moments and elemental abundances. Also needed are laboratory experiments on multilayered convection under planetary-like conditions, and further development of theoretical models of turbulent transport in stably stratified environments.
Key open problems
The study exposes a fundamental problem: the applicability of mixing theories developed for laboratory scales to planetary interiors, where compressibility alters the energy balance. This is part of a broader question of how thermal evolution distributes elements—directly relevant to the puzzle of fuzzy cores and the production of entropy in planetary interiors. The results also point to limitations of standard accretion models: perhaps early mixing of material was far more vigorous than commonly assumed.
🎯 The idea of double diffusive 'staircases' originated in oceanography: there, water layers with different temperatures and salinities form stepped profiles. In Jupiter's interior, such layers could stretch for hundreds of kilometers, if they existed. And the concept of a fuzzy core became popular after the Juno probe revealed that Jupiter's gravitational field is incompatible with a sharp boundary.
Key numbers
- Transported mass (Jupiter, baseline): 0.11 M⊕
- Transported mass (Saturn, baseline): 0.07 M⊕
- Boost ×1000 (Jupiter): ~2 M⊕
- Boost ×1000 (Saturn): ~1.3 M⊕
- Simulation age: 4.56 Gyr