Bridging the Prandtl number gap: 3D simulations of thermohaline convection in astrophysical regimes

Abstract

Thermohaline convection (also known as fingering convection or thermohaline mixing) occurs in stellar radiation zones where a sufficient inversion of the mean molecular weight is present. This process mixes chemicals radially and occurs in a variety of stars, including near the luminosity bump on the red giant branch and potentially in polluted white dwarfs. Previous efforts to characterize this process using 3D simulations have been restricted to regimes far from actual stars: The Prandtl number $\Pr$--the ratio of the kinematic viscosity to thermal diffusivity--assumes values as low as $10^{-6}$ in stars, but 3D simulations have been restricted to $\Pr \gtrsim 10^{-2}$. For this reason, disagreements between observations and simulations are routinely dismissed as stemming from this $\Pr$ gap. This letter bridges this gap and demonstrates that 3D simulations of thermohaline convection can be performed in stellar parameter regimes. Using a suite of simulations spanning previously studied regimes with $\Pr \gtrsim 10^{-2}$ down to $\Pr = 10^{-6}$, we demonstrate that the chemical mixing model of Brown, Garaud, & Stellmach (2013) remains consistent with 3D simulations across both regimes. Therefore, tensions between this model and observations cannot be dismissed as resulting from a $\Pr$ gap, and must be resolved by considering additional physics.

Figures (3)

• Figure 1: Different regimes in which thermohaline convection occurs (shaded regions) are compared against simulations presented in this work (black points) in terms of the diffusivity ratio $\tau$ (characterizing the thermal diffusivity) and the density ratio $R_0$ (characterizing the stratification). Filled contours show the estimated Reynolds number of the flows (detailed in main text). Thermohaline convection in oceans typically occurs with $\tau \sim 10^{-2}$; in stars, the possible values of $\tau$ vary dramatically, but are generally much smaller. Previous 3D simulations have explored $\tau \gtrsim 10^{-2}$ extensively, while the present work considers simulations with much smaller $\tau$.
• Figure 2: Snapshots of the vertical velocity $\tilde{u}_z$ at $\tilde{y} = 0$ in the saturated state for simulations with $\Pr = 10^{-6}$, $\tau = \Pr/2$, and (a) $\mathcal{R} = 1.1$, (b) $\mathcal{R} = 11$, and (c) $\mathcal{R} = 301$ (or $\varepsilon = 0.1$, $10$, and $300$). As $\mathcal{R}$ increases, velocity fluctuations increase and develop finer structures.
• Figure 3: Turbulent compositional flux $|\tilde{F}_C|$ is shown across a range of parameters against (a) the reduced density ratio [see Eq. \ref{['eq:r-def']}] and (b) supercriticality $\varepsilon = \mathcal{R} - 1$. Points represent results from 3D simulations while curves show predictions from the BGS13 model, with color corresponding to the value of $\Pr$; throughout, we fix $\tau = \Pr/2$ ($\mathrm{Sc} = 2$). Vertical lines in panel (b) indicate the Ledoux threshold for convective instability for $\tau = 0.05$ (blue) and $\tau = 0.005$ (purple). Across all values of $\Pr$ considered, the BGS13 model remains in good agreement with simulations. Additionally, the data collapses more uniformly when plotted against $\varepsilon$ than it does when plotted against $r$, indicating the former may be a more practical control parameter when studying thermohaline convection at small $\Pr$ and $\tau$.