Bounding convection heat flux in planets and stars by Thierry Alboussiere - IP4RG

Bounding convection heat flux in planets and stars

The intensity of convective flows in planets and stars is extreme in terms of the dimensionless Rayleigh number, so that it makes sense to look for an asymptotic upper bound of the heat flux for very large Rayleigh numbers. We consider a Rayleigh-Benard configuration, where a fluid between two horizontal planes is heated from below: the lower boundary is maintained at a higher temperature than that of the upper boundary. Howard [1] determined rigorously an upper bound for the convective heat flux in 1963. In dimensionless numbers, the result is Nu < c Ra^{1/2}. His proof is not easy to follow. In 1996, Doering and Constantin [2] found another method, leading to the same result, based on a temperature decomposition into an arbitrary vertical profile satisfying the temperature conditions at the top and bottom and homogeneous temperature fluctuations. This is slightly easier to follow. In 2015, Seis [3] came up with a third method, where the heat flux is analyzed from the bottom to the top, in particular when the convective part of the flux must take over the conduction part. This method is significantly easier to understand and I will present it. All methods have been put in a common frame by Chernyshenko [4] under the general form of auxiliary functionals. Recently, we have extended bounding methods to a non-Boussinesq model of convection [5], the anelastic liquid model, and I will also present it. This is a first step in obtaining rigorous bounds in compressible convection.

[1] L.N. Howard. Heat transport by turbulent convection. Journal of Fluid Mechanics, 17(3): 405–432, 1963
[2] C.R. Doering and P. Constantin. Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E, 53(6):5957–5981, 1996
[3] C. Seis. Scaling bounds on dissipation in turbulent flows. Journal of Fluid Mechanics, 777: 591–603, 2015
[4] S. Chernyshenko, Relationship between the methods of bounding time averages. Phil. Trans.
R. Soc. A, 380:20210044, 2022
[5] T. Alboussiere, Y. Ricard and S. Labrosse, Upper bound of heat flux in an anelastic model for Rayleigh-Benard convection, arXiv:2403.04358, 2024