Critical balance and scaling of strongly stratified turbulence at low Prandtl number

Abstract

We extend the scaling relations of strongly (stably) stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $Pr\,Rb<1$ , which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of the Boussinesq equations in the $Pr\,Rb\ll 1$ regime. We find that all the standard scaling relations from the unity $Pr$ limit of strongly stratified turbulence simply carry over if the Froude number, $Fr$ , is replaced by a modified Froude number, $Fr_M\equiv Fr/(Pr\,Rb)^{1/4}$ . The geophysical and astrophysical regimes are thus smoothly connected across the $Pr\,Rb=1$ transition. Applications to vertical transport in stellar radiative zones and modification to the instability criterion for the small-scale dynamo are discussed.

The Combined Effects of Vertical and Horizontal Shear Instabilities in Stellar Radiative Zones

“…This regime is valid as long as the vertical scale of the turbulent eddies is much smaller than the domain size, yet large enough to not be influenced by viscosity. The scaling law in Equation (3) is consistent 4 with the model predictions from Zahn (1992), Lignières (2020), Skoutnev (2023), andShah et al (2023) in their respective thermally diffusive regimes, which is rather surprising, as the scalings for w rms and l z individually differ in each model. However, the constraint Pe < 1 is rarely satisfied in stars.…”

“…Unfortunately, as Pe t is an emergent quantity (rather than an input parameter), a model is needed to predict when the transition from nondiffusive to thermally diffusive turbulence occurs as a function of the input parameters. Here, the disagreement between the models is significant; for instance, Skoutnev (2023) finds that the solar tachocline would lie close to the regime transition, while Shah et al (2023) argue instead that it lies squarely in the nondiffusive regime. Finally, we note that running DNS in the regime Pe ?…”

“…By contrast with the case of vertical shear instabilities, where models generally agree with each other, and with the data from numerical simulations, quantifying mixing by horizontal shear instabilities is a topic of ongoing investigation, where no consensus has yet been reached. Several theoretical models have been put forward, starting with Zahn (1992), followed by the more recent works of Garaud (2020a), Cope et al (2020), Lignières (2020), Chini et al (2022), Skoutnev (2023), andShah et al (2023). These models vary significantly in their detailed predictions, because of differences in their respective founding assumptions and in the mathematical approach selected and because their domains of validity are sometimes limited.…”