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.

arXiv comment17 pages, 11 figures, 3 tables

References in corpus (28)

• Large scale dynamics in turbulent Rayleigh-Benard convection
• Scaling in thermal convection: A unifying theory
• Radial boundary layer structure and Nusselt number in Rayleigh-Benard convection
• Flow reversals in thermally driven turbulence
• Turbulent superstructures in Rayleigh-Bénard convection
• Anomalously Weak Solar Convection
• Flow reversals in turbulent convection via vortex reconnections
• Transition to the ultimate regime in two-dimensional Rayleigh-Bénard convection
• Phenomenology of buoyancy-driven turbulence: recent results
• Flow organization in non-Oberbeck-Boussinesq Rayleigh-Benard convection in water
• Resolving the fine-scale structure in turbulent Rayleigh-Benard convection
• Seismic Sounding of Convection in the Sun
• 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
• The Spectral Amplitude of Stellar Convection and its Scaling in the High-Rayleigh-Number Regime
• Scaling of heat flux and energy spectrum for "very large" Prandtl number convection
• Large-scale mean patterns in turbulent convection
• Horizontal Structures of Velocity and Temperature Boundary Layers in 2D Numerical Turbulent Rayleigh-Bénard Convection
• Logarithmic mean temperature profiles and their connection to plume emissions in turbulent Rayleigh-Bénard convection
• Consequences of high effective Prandtl number on solar differential rotation and convective velocity
• 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
• Similarities between 2D and 3D convection for large Prandtl number
• Convective heat transport in slender cells is close to that in wider cells at high Rayleigh and Prandtl numbers
• Effects of variable thermal diffusivity on the structure of convection
• Reversals in infinite-Prandtl-number Rayleigh-Bénard convection

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• 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 $\mathbf{Pr}=10^{-3}$