The Effects of Radially Varying Diffusivities on Stellar Convection Zone Dynamics

TL;DR

This study probes how radially varying viscous and thermal diffusivities influence non-rotating, non-magnetic solar-like convection in a spherical shell using an anelastic, 3D framework. By spanning 33 models with $\nu,\kappa \propto \rho^{\alpha}$ and various boundary conditions, it demonstrates a diffusion-free-like bulk kinetic-energy regime with $Re \propto Ra_F^{0.337\pm0.009}$ that is largely independent of $\alpha$, while heat transport remains diffusion-influenced due to boundary-layer diffusion. Crucially, the diffusivity form redistributes turbulent power with depth and alters convective efficiency across the shell, implying that helioseismic inferences and dynamo-relevant quantities (e.g., kinetic helicity) may depend on the diffusivity prescription. These results highlight the need for careful diffusion modeling when comparing solar convection simulations to observations and when interpreting convection-driven magnetic field generation.

Abstract

Convection is ubiquitous in stellar and planetary interiors where it likely plays an integral role in the generation of magnetic fields. As the interiors of these objects remain hidden from direct observation, numerical models of convection are an important tool in the study of astrophysical dynamos. In such models, unrealistic large values of the viscous ($ν$) and thermal ($κ$) diffusivity are routinely used as an ad-hoc representation of the effects of subgrid scale turbulence which is otherwise too small-scale to resolve numerically. However, the functional forms of these diffusion coefficients can vary greatly between studies, complicating efforts to compare between results and against observations. We explore this issue by considering a series of non-rotating, non-magnetic, solar-like convection models with varying radial functions for the diffusivities and differing boundary conditions. We find that the bulk kinetic energy scales similarly regardless of the diffusivity parameterization. This scaling is consistent with a free-fall scaling, wherein viscosity plays a subdominant role in the force balance. We do not, however, observe such diffusion-free behavior in the convective heat transport. Our results also indicate that the functional form adopted for the diffusion coefficients can impact the distribution of turbulence within the convective shell. These results suggest that some care should be taken when comparing solar convection models directly against helioseismic observations.

Figures (8)

• Figure 1: Viscosity versus radius for $\mathrm{Ra_F = 10^4}$. Each curve denotes a different diffusivity parameter $\alpha$ employed in our models. Dashed grey lines indicate at what radial level upper and lower convection zone corresponds that is referenced in this study. For a given $Ra_F$, the volume-averaged viscosity is the same for all models.
• Figure 2: Snapshots of radial velocity for $\mathrm{Ra_F = 10^4}$ sampled at a radius of $\mathrm{r/R_o = 0.98}$ (top row) and $\mathrm{r/R_o} = 0.75$ (bottom row). As $\alpha$ is increased from negative to positive values, the region of strongest convection shifts from the lower convection zone to the upper convection zone.
• Figure 3: Time-averaged velocity power for $\mathrm{Ra_F =10^4}$ in the lower ($\mathrm{r/R_o = 0.75}$), middle ($\mathrm{r/R_o = 0.86}$), and upper convection zone ($\mathrm{r/R_o = 0.98})$. Each colored curve denotes a different value of $\alpha$. At a given $\mathrm{Ra_F}$, higher values of $\alpha$ result in more small-scale power in the upper convection zone, whereas lower, negative values of $\alpha$ tend to enhance small-scale power in the lower convection zone.
• Figure 4: Dependence of dimensional kinetic energy on $\mathrm{Ra_F}$ for internally-heated models. Each colored curve corresponds to a different value of the diffusivity exponent $\alpha$. The dashed line corresponds to the delineation between low-$\mathrm{Ra_F}$ and high-$\mathrm{Ra_F}$ behavior found in Featherstone2016a. At sufficiently high-$\mathrm{Ra_F}$, all models converge toward a similar value of KE, but those with negative values of $\alpha$ asymptote more rapidly than the $\alpha=0$ and $\alpha=0.5$ cases.
• Figure 5: Volume averaged $\mathrm{Re}$ versus $\mathrm{Ra_F}$. The orange line is fit to all internal heating data for $\mathrm{Ra_F \geq 10^4}$. Green dashed line is fit to same range but for fixed flux cases. FF refers to cases without internal heating and fixed flux and FE refers to cases with fixed entropy boundary conditions.