Prandtl-Number Effects in High-Rayleigh-Number Spherical Convection

TL;DR

This study investigates how the Prandtl number influences convective motions in stellar environments, revealing that lower Prandtl numbers lead to faster flows and affect the boundary layer, with implications for solar dynamo modeling.

Contribution

It systematically explores the effects of varying Prandtl and Rayleigh numbers on high-Rayleigh-number spherical convection, highlighting the role of Prandtl number in reaching the free-fall regime.

Findings

Lower Prandtl numbers produce faster convective flows.

Spectral velocity distribution is largely insensitive to Prandtl variations.

Prandtl number controls the thermal boundary layer thickness and the onset of free-fall regime.

Abstract

Convection is the predominant mechanism by which energy and angular momentum are transported in the outer portion of the Sun. The resulting overturning motions are also the primary energy source for the solar magnetic field. An accurate solar dynamo model therefore requires a complete description of the convective motions, but these motions remain poorly understood. Studying stellar convection numerically remains challenging; it occurs within a parameter regime that is extreme by computational standards. The fluid properties of the convection zone are characterized in part by the Prandtl number $\mathrm{Pr}=\nu/\kappa$, where $\nu$ is the kinematic viscosity and $\kappa$ is the thermal diffusion; in stars, $\mathrm{Pr}$ is extremely low, $\mathrm{Pr}\approx 10^{-7}$. The influence of $\mathrm{Pr}$ on the convective motions at the heart of the dynamo is not well understood since most numerical studies are limited to using $\mathrm{Pr}\approx1$. We systematically vary $\mathrm{Pr}$ and the degree of thermal forcing, characterized through a Rayleigh number, to explore its influence on the convective dynamics. For sufficiently large thermal driving, the simulations reach a so-called convective free-fall state where diffusion no longer plays an important role in the interior dynamics. Simulations with a lower $\mathrm{Pr}$ generate faster convective flows and broader ranges of scales for equivalent levels of thermal forcing. Characteristics of the spectral distribution of the velocity remain largely insensitive to changes in $\mathrm{Pr}$. Importantly, we find that $\mathrm{Pr}$ plays a key role in determining when the free-fall regime is reached by controlling the thickness of the thermal boundary layer.