The Spectral Amplitude of Stellar Convection and Its Scaling in the High-Rayleigh-Number Regime

Abstract

Convection plays a central role in the dynamics of any stellar interior, and yet its operation remains largely hidden from direct observation. As a result, much of our understanding concerning stellar convection necessarily derives from theoretical and computational models. The Sun is, however, exceptional in that regard. The wealth of observational data afforded by its proximity provides a unique test bed for comparing convection models against observations. When such comparisons are carried out, surprising inconsistencies between those models and observations become apparent. Both photospheric and helioseismic measurements suggest that convection simulations may overestimate convective flow speeds on large spatial scales. Moreover, many solar convection simulations have difficulty reproducing the observed solar differential rotation owingto this apparent overestimation. We present a series of three-dimensional stellar convection simulations designed to examine how the amplitude and spectral distribution of convective flows are established within a star's interior. While these simulations are nonmagnetic and nonrotating in nature, they demonstrate two robust phenomena. When run with sufficiently high Rayleigh number, the integrated kinetic energy of the convection becomes effectively independent of thermal diffusion, but the spectral distribution of that kinetic energy remains sensitive to both of these quantities. A simulation that has converged to a diffusion-independent value of kinetic energy will divide that energy between spatial scales such that low-wavenumber power is overestimatedand high-wavenumber power is underestimated relative to a comparable system possessing higher Rayleigh number. We discuss the implications of these results in light of the current inconsistencies between models and observations.

Convective Velocity Suppression via the Enhancement of the Subadiabatic Layer: Role of the Effective Prandtl Number

“…As a result, the entropy difference between the top boundary and the bulk of the domain ∆ s ≡ |max s − min s | increases as Pr increases. However, Pr-dependence of the upper boundary layer is small compared with the previous studies (O 'Mara et al 2016), having no scaling relations between the thermal diffusivity κ, the thickness of the thermal boundary layer d t , and the entropy difference ∆ s , such as d t ∝ κ 1/2 or ∆ s ∝ κ −1/2 which can only hold for simulations imposing a diffusion-type upper boundary condition where the energy flux is released through the thermal conduction term (Featherstone & Hindman 2016).…”

“…For example, since we impose large viscous and thermal SGS diffusivities with the typical Reynolds number Re eff calculated as 30−40, the parameter regime studied in our simulations is laminar. It is expected that, if SGS ν is decreased and the Reynolds number Re (and thus Rayleigh number Ra) increases, the thermal effect would become ineffective and v rms would cease to diminish being independent of the values of diffusivities (Featherstone & Hindman 2016).…”

“…For specifying the radiative heating term, the functional form presented in Featherstone & Hindman (2016) is adopted which can mimic the normalized radiative energy flux tabulated in a standard solar model. The radiative heating Q heat is assumed to be proportional to the background pressure,…”

Stellar Mixing Length Theory With Entropy Rain

“…This has been partially confirmed by Hotta et al (2015), who have discussed the possible role of small-scale magnetic fields in suppressing the formation of large-scale flows. Furthermore, Featherstone & Hindman (2016) have found that with increasing Rayleigh numbers, there is more kinetic energy at small scales and less at large scales such that the total kinetic energy is unchanged by this rearrangement of energy.…”

“…Greer et al 2015) or smaller (e.g. Featherstone & Hindman 2016) than those at the surface. If most of the kinetic energy were to reside on small scales throughout the convection zone even at a depth of several tens of megameters and beneath, E(k) is expected to decrease toward smaller wavenumbers k either like white noise ∝ k 2 or maybe even with a Batchelor spectrum ∝ k 4 (Davidson 2004).…”