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

Abstract

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, $P{r}_t$, 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 $0.005\le Pr\le 7$. The principal finding is that $P{r}_t$ increases with decreasing $Pr$ in both OB and NOB convection: $P{r}_t\sim P{r}^{-0.3}$ for OB convection and $P{r}_t\sim P{r}^{-1}$ for the NOB case. The $P{r}_t$-dependence for the NOB case especially suggests that convective flows in the astrophysical settings behave effectively as in high-Prandtl-number turbulence.

17 pages, 11 figures, 3 tables

References in corpus (7)

• Thermal Boundary Layer Equation for Turbulent Rayleigh-Benard Convection
• Scaling of large-scale quantities in Rayleigh-Bénard convection
• Global and local statistics in turbulent convection at low Prandtl numbers
• Thermal boundary layer structure in low-Prandtl-number turbulent convection
• Mean Temperature Profiles in Turbulent Thermal Convection
• Non-Boussinesq low-Prandtl number convection with a temperature-dependent thermal diffusivity
• Convective heat transport in slender cells is close to that in wider cells at high Rayleigh and Prandtl numbers

Cited by in corpus (7)

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• Strongly superadiabatic and stratified limits of compressible convection
• Turbulent mesoscale convection in the Boussinesq limit and beyond
• Turbulent Prandtl number from isotropically forced turbulence
• Compressible turbulent convection: The role of temperature-dependent thermal conductivity and dynamic viscosity
• Scale-resolved turbulent Prandtl number for Rayleigh-Bénard convection at $\mathbf{Pr}=10^{-3}$