  Convection in the Sun occurs at Rayleigh numbers, $Ra$, as high as $10^{22}$,
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, $Pr_t$, 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 $Gr \equiv Ra/Pr = 10^9$. 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 $5 \times 10^{-4}$. The three-dimensional
simulations are performed in a box of aspect ratio 25 at a fixed Rayleigh
number of $10^5$, and $0.005 \leq Pr \leq 7$. The principal finding is that
$Pr_t$ increases with decreasing $Pr$ in both OB and NOB convection: $Pr_t \sim
Pr^{-0.3}$ for OB convection and $Pr_t \sim Pr^{-1}$ for the NOB case. The
$Pr_t$-dependence for the NOB case especially suggests that convective flows in
the astrophysical settings behave effectively as in high-Prandtl-number
turbulence.
