Bounds on heat transfer and dissipation in the core

An upper bound of heat transfer has been published recently [1] for the first time in the case of compressible convection. This concerns only the anelastic liquid approximation, but the best place where such an approximation is valid might well be the core of terrestrial planets. In this work, I will apply this result to the specific case and geometry of the outer core. A big difference is also the fact that not only viscous dissipation but also Joule heating are sources of entropy production.

For a given forcing (Rayleigh number), we will see that there is a trade-off between a maximal heat flux and a maximal Joule dissipation. An upper bound can be obtained for both quantities, but they cannot both reach that bound.

We shall also consider the case of terrestrial planets of larger radii than the Earth. A number of exoplanets are suspected to be in that case. We will investigate the consequences of larger compressibility on their internal structure [2] and obtain upper bounds of heat flux and Joule dissipation.

[1] T Alboussière, Y Ricard, S Labrosse, "Upper bound of heat flux in an anelastic model for Rayleigh–Bénard convection", JFM 999, 2024

[2] Y Ricard, T Alboussière, "Compressible convection in super-Earths", PEPI 341, 2023