Double-Diffusive Convection

Much progress has recently been made in understanding and quantifying vertical mixing induced by double-diffusive instabilities such as fingering convection (usually called thermohaline convection) and oscillatory double-diffusive convection (a process closely related to semiconvection). This was prompted in parts by advances in supercomputing, which allow us to run Direct Numerical Simulations of these processes at parameter values approaching those relevant in stellar interiors, and in parts by recent theoretical developments in oceanography where such instabilities also occur. In this paper I summarize these recent findings, and propose new mixing parametrizations for both processes that can easily be implemented in stellar evolution codes.Comment: To be published in the proceedings of the conference "New Advances in Stellar Physics: from microscopic to macroscopic processes", Roscoff, 27-31st May 201

Turbulent transport by diffusive stratified shear flows: from local to global models. III. A closure model

Being able to account for the missing mixing in stellar radiative zones is a key step toward a better understanding of stellar evolution. Zahn (1974) argued that thermally diffusive shear-induced turbulence might be responsible for some of this mixing. In Part I and Part II of this series of papers we showed that Zahn's (1974, 1992) mixing model applies when the properties of the turbulence are local. But we also discovered limitations of the model when this locality condition fails, in particular near the edge of a turbulent region. In this paper, we propose a second-order closure model for the transport of momentum and chemical species by shear-induced turbulence in strongly stratified, thermally diffusive environments (the so-called low P\'eclet number limit), which builds upon the work of Garaud \& Ogilvie (2005). Comparison against direct numerical simulations (DNSs) shows that the model is able to predict the vertical profiles of the mean flow and of the stress tensor (including the momentum transport) in diffusive shear flows, often with a reasonably good precision, and at least within a factor of order unity in the worst case scenario. The model is sufficiently simple to be implemented in stellar evolution codes, and all the model constants have been calibrated against DNSs. While significant limitations to its use remain (e.g. it can only be used in the low P\'eclet number, slowly rotating limit), we argue that it is more reliable than most of the astrophysical prescriptions that are used in stellar evolution models today

2D or not 2D: the effect of dimensionality on the dynamics of fingering convection at low Prandtl number

Fingering convection (otherwise known as thermohaline convection) is an instability that occurs in stellar radiative interiors in the presence of unstable compositional gradients. Numerical simulations have been used in order to estimate the efficiency of mixing induced by this instability. However, fully three-dimensional (3D) computations in the parameter regime appropriate for stellar astrophysics (i.e. low Prandtl number) are prohibitively expensive. This raises the question of whether two-dimensional (2D) simulations could be used instead to achieve the same goals. In this work, we address this issue by comparing the outcome of 2D and 3D simulations of fingering convection at low Prandtl number. We find that 2D simulations are never appropriate. However, we also find that the required 3D computational domain does not have to be very wide: the third dimension need only contain a minimum of two wavelengths of the fastest-growing linearly unstable mode to capture the essentially 3D dynamics of small-scale fingering. Narrow domains, however, should still be used with caution since they could limit the subsequent development of any large-scale dynamics typically associated with fingering convection.Comment: Submitted to Ap

Turbulent transport in a strongly stratified forced shear layer with thermal diffusion

This work presents numerical results on the transport of heat and chemical species by shear-induced turbulence in strongly stratified but thermally diffusive environments. The shear instabilities driven in this regime are sometimes called "secular" shear instabilities, and can take place even when the gradient Richardson number of the flow (the square of the ratio of the buoyancy frequency to the shearing rate) is large, provided the P\'eclet number (the ratio of the thermal diffusion timescale to the turnover timescale of the turbulent eddies) is small. We have identified a set of simple criteria to determine whether these instabilities can take place or not. Generally speaking, we find that they may be relevant whenever the thermal diffusivity of the fluid is very large (typically larger than $10^{14}$cm$^2$/s), which is the case in the outer layers of high-mass stars ($M\ge 10 M_\odot$) for instance. Using a simple model setup in which the shear is forced by a spatially sinusoidal, constant-amplitude body-force, we have identified several regimes ranging from effectively unstratified to very strongly stratified, each with its own set of dynamical properties. Unless the system is in one of the two extreme regimes (effectively unstratified or completely stable), however, we find that (1) only about 10% of the input power is used towards heat transport, while the remaining 90% is viscously dissipated; (2) that the effective compositional mixing coefficient is well-approximated by the model of Zahn (1992), with $D \simeq 0.02 \kappa_T /J$ where $\kappa_T$ is the thermal diffusivity and $J$ is the gradient Richardson number. These results need to be confirmed, however, with simulations in different model setups and at higher effective Reynolds number.Comment: Submitted to Ap

Weakly non-Boussinesq convection in a gaseous spherical shell

We examine the dynamics associated with weakly compressible convection in a spherical shell by running 3D direct numerical simulations using the Boussinesq formalism [1]. Motivated by problems in astrophysics, we assume the existence of a finite adiabatic temperature gradient $\nabla T_{\rm{ad}}$ and use mixed boundary conditions for the temperature with fixed flux at the inner boundary and fixed temperature at the outer boundary. This setup is intrinsically more asymmetric than the more standard case of Rayleigh-B\'{e}nard convection in liquids between parallel plates with fixed temperature boundary conditions. Conditions where there is substantial asymmetry can cause a dramatic change in the nature of convection and we demonstrate that this is the case here. The flows can become pressure- rather than buoyancy- dominated leading to anomalous heat transport by upflows. Counter-intuitively, the background temperature gradient $\nabla\bar{T}$ can develop a subadiabatic layer (where $\boldsymbol{g}\cdot\nabla\bar{T}<\boldsymbol{g}\cdot\nabla T_{\rm{ad}}$, where $\boldsymbol{g}$ is gravity) although convection remains vigorous at every point across the shell. This indicates a high degree of non-locality.Comment: 19 figure