New bounds for heat transport in internally heated convection at infinite Prandtl number

We prove new bounds on the heat flux out of the bottom boundary, FB, for a fluid at infinite Prandtl number, heated internally between isothermal parallel plates, under two kinematic boundary conditions. In uniform internally heated convection, the supply of heat equally leaves the domain entirely by conduction in the absence of a flow. When the heating, quantified by the Rayleigh number, R, becomes sufficiently large, turbulent convection ensues, reducing the heat leaving the domain through the bottom boundary. In the case of no-slip boundary conditions, with the background field method, we prove that FB≳R−2/3−R−1/2⁡log(1−R−2/3), up to a positive constant independent of the Rayleigh and Prandtl numbers. Whereas between stress-free boundaries we show that, FB≳R−40/29−R−35/29⁡log(1−R−40/29). We perform a numerical study of the system in two dimensions up to a Rayleigh number of 5 × 109 using the spectral solver Dedalus. The numerical investigations indicate that FB∼R−0.092 and FB∼R−0.12 for the two kinematic boundary conditions respectively. The gap between the bounds and simulations, along with the constructions used in the proofs highlight the potential for further optimisation of bounds for FB in internally heated convection.