Convection in the interiors of large planets

Modelling convection has been an early challenge of fluid mechanics and continues to be a source of questions. Oberbeck and Boussinesq came up with a simplification that is so successful that their model is often considered to be the only option for convection. In planetary sciences, the concept of adiabatic gradient is widely used but the associated energy dissipation (by viscosity or Joule heating) and its role in compressible convection is less understood. Where does it take place in the convection domain and what are the consequences on the structure of the flow? What model of compressible convection can we and should we use? What is the ultimate regime of compressible convection? In Lyon, we have recently obtained an upper bound of heat transfer in the simplest model of compressible convection (anelastic liquid approximation) but there is much more to do! We have also attempted to predict what convection in the large mantles of super-Earths would look like. The choice of an equation of state (EoS) is important for this exercise and it is even more important to use consistent parameters, such as volume expansion coefficient or incompressibility. It appears that, for a wide range of EoS, the enhanced adiabatic gradient near the surface of the planet is in favour of a stagnant-lid regime for super-Earths. Finally, after a first attempt ten years ago, we will start a new program of experiments specifically devoted to compressible convection and I will discuss what we expect to investigate.