Regimes of stratified stellar turbulence

Quantifying transport by strongly stratified turbulence in stellar interiors is important for the development of accurate stellar evolution models. Stratified turbulence in stars and planets differs fundamentally from geophysical turbulence because the Prandtl number, Pr, is very small there. In geophysical flows, when Pr = O(1), turbulent flows (Re » 1) are only weakly thermally diffusive, as reflected by the largeness of the Péclet number (Pe = PrRe) which is O(Re) » 1 when Pr = O(1). With Pr « 1, by contrast, Pe « Re, so it is possible to have Pe « 1 « Re, i.e. regimes of thermally diffusive stratified turbulence, which are not possible in geophysical fluids. Motivated by numerical simulations showing the emergence of anisotropic structures and scale-separated features, we perform a multiscale analysis of the governing equations, demonstrating the existence of several distinct regimes depending on the emergent buoyancy Péclet number Pe_b = Pr Re_b, where Re_b = α² Re, α is the aspect ratio of the turbulence and Re is the input Reynolds number. Scaling relationships linking the aspect ratio to the strength of the stratification naturally emerge from our analysis. For Pe_b « 1 flows, our results recover scaling laws that have been empirically obtained from direct numerical simulations, namely α ∝ (Fr² / Pe)^1/3, where Fr is the Froude number. For Pe_b > 1, the aspect ratio scales as Fr, consistent with published work on strongly stratified geophysical turbulence at Pr = O(1). Finally, we have identified a new regime at intermediate values of Pe_b, in which slow, large-scales are thermally diffusive while fast, small-scales are not. Using these results, we construct regime diagrams that identify transitions between non-diffusive stratified turbulence, diffusive stratified turbulence, and viscously-controlled regimes at Pr « 1 and Pr = O(1).