Non-Boussinesq convection at low Prandtl numbers relevant to the Sun

Convection in the Sun occurs at Rayleigh numbers, Ra, as high as 1022 and molecular Prandtl numbers, Pr, as low as 10-6, under conditions that are far from satisfying the Oberbeck-Boussinesq (OB) idealization. The effects of these extreme circumstances on turbulent heat transport are unknown, and no comparable conditions exist on Earth. Our goal is to understand how these effects scale (since we cannot yet replicate the Sun's conditions faithfully). We study thermal convection by using direct numerical simulations, and determine the variation with respect to Pr, to values as low as 10-4, of the turbulent Prandtl number, Prt, which is the ratio of turbulent viscosity to thermal diffusivity. The simulations are primarily two-dimensional but we draw upon some three-dimensional results as well. We focus on non-Oberbeck-Boussinesq (NOB) conditions of a certain type, but also study OB convection for comparison. The OB simulations are performed in a rectangular box of aspect ratio 2 by varying Pr from O(10) to 10-4 at fixed Grashof number Gr equivalent to Ra/Pr = 109. The NOB simulations are done in the same box by letting only the thermal diffusivity depend on the temperature. Here, the Rayleigh number is fixed at the top boundary while the mean Pr varies in the bulk from 0.07 to 5 x 10-4. The three-dimensional simulations are performed in a box of aspect ratio 25 at a fixed Rayleigh number of 105, and 0.005 Pr 7. The principal finding is that Prt increases with decreasing Pr in both OB and NOB convection: Prt similar to Pr-0.3 for OB convection and Prt similar to Pr-1 for the NOB case. The Prt dependence for the NOB case especially suggests that convective flows in the astrophysical settings behave effectively as in high-Prandtl-number turbulence.