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

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 Pr = ν/κ, where ν is the kinematic viscosity and κ is the thermal diffusion; in stars, Pr is extremely low, Pr ≈ 10 −7 . The influence of Pr on the convective motions at the heart of the dynamo is not well understood since most numerical studies are limited to using Pr ≈ 1. We systematically vary 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 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 Pr. Importantly, we find that Pr plays a key role in determining when the free-fall regime is reached by controlling the thickness of the thermal boundary layer.

3D Simulations and MLT. I. Renzini’s Critique

“…The formal Rayleigh number may be infinity (no explicit radiative diffusion 5 ). The 1D model used for guidance (Cristini, et al 2017) had a very high Péclet number (Pe ∼ 10 8 ), but the turbulent cascade implies that the results are insensitive to Pe (see also Orvendahl, et al (2018)). The convective Mach numbers are small (< 0.02).…”

On the inverse cascade and flow speed scaling behaviour in rapidly rotating Rayleigh–Bénard convection

“…A well-known example is the so-called 'free-fall' scaling of the form Re ∼ (Ra/Pr) 1/2 observed in non-rotating convection (e.g. Ahlers, Grossmann & Lohse 2009;Orvedahl et al 2018). The free-fall scaling arises from a balance between nonlinear advection and the buoyancy force in the momentum equation, and represents a 'diffusion-free' scaling in the sense that the flow speeds are independent of both the thermal and viscous diffusion coefficients.…”

Prandtl number dependence of stellar convection: Flow statistics and convective energy transport

“…On the other hand, Brandenburg et al (2005) found that the correlation coefficients of velocity and temperature fluctuations with the enthalpy flux remained Prdependent at least to Reynolds numbers of the order of 10 3 . Finally, Orvedahl et al (2018) studied non-rotating anelastic convection in spherical coordinates and found that the overall velocity amplitude increases as the Prandtl number decreases while the spectral distribution of velocity is insensitive to Pr.…”