Gyroscopic pumping of large-scale flows in stellar interiors, and application to Lithium Dip stars

The maintenance of large-scale differential rotation in stellar convective regions by rotationally influenced convective stresses also drives large-scale meridional flows by angular--momentum conservation. This process is an example of ``gyroscopic pumping'', and has recently been studied in detail in the solar context. An important question concerns the extent to which these gyroscopically pumped meridional flows penetrate into nearby stably stratified (radiative) regions, since they could potentially be an important source of non-local mixing. Here we present an extensive study of the gyroscopic pumping mechanism, using a combination of analytical calculations and numerical simulations both in Cartesian geometry and in spherical geometry. The various methods, when compared with one another, provide physical insight into the process itself, as well as increasingly sophisticated means of estimating the gyroscopic pumping rate. As an example of application, we investigate the effects of this large-scale mixing process on the surface abundances of the light elements Li and Be for stars in the mass range 1.3-1.5 solar masses (so-called ``Li-dip stars''). We find that gyroscopic pumping is a very efficient mechanism for circulating material between the surface and the deep interior, so much in fact that it over-estimates Li and Be depletion by orders of magnitude for stars on the hot side of the dip.However, when the diffusion of chemical species back into the surface convection zone is taken into account, a good fit with observed surface abundances of Li and Be as a function of stellar mass in the Hyades cluster can be found for reasonable choices of model parameters.Comment: Submitted to Ap

On the Penetration of Meridional Circulation below the Solar Convection Zone II: Models with Convection Zone, the Taylor-Proudman constraint and Applications to Other Stars

The solar convection zone exhibits a strong level of differential rotation, whereby the rotation period of the polar regions is about 25-30% longer than the equatorial regions. The Coriolis force associated with these zonal flows perpetually "pumps" the convection zone fluid, and maintains a quasi-steady circulation, poleward near the surface. What is the influence of this meridional circulation on the underlying radiative zone, and in particular, does it provide a significant source of mixing between the two regions? In Paper I, we began to study this question by assuming a fixed meridional flow pattern in the convection zone and calculating its penetration depth into the radiative zone. We found that the amount of mixing caused depends very sensitively on the assumed flow structure near the radiative--convective interface. We continue this study here by including a simple model for the convection zone "pump", and calculating in a self-consistent manner the meridional flows generated in the whole Sun. We find that the global circulation timescale depends in a crucial way on two factors: the overall stratification of the radiative zone as measured by the Rossby number times the square root of the Prandtl number, and, for weakly stratified systems, the presence or absence of stresses within the radiative zone capable of breaking the Taylor-Proudman constraint. We conclude by discussing the consequences of our findings for the solar interior and argue that a potentially important mechanism for mixing in Main Sequence stars has so far been neglected.Comment: 42 pages, 13 figures. Submitted to Ap

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

On rotationally driven meridional flows in stars

A quasi-steady state model of the consequences of rotation on the hydrodynamical structure of a stellar radiative zone is derived, by studying in particular the role of centrifugal and baroclinic driving of meridional motions in angular-momentum transport. This nonlinear problem is solved numerically assuming axisymmetry of the system, and within some limits, it is shown that there exist simple analytical solutions. The limit of slow rotation recovers Eddington-Sweet theory, whereas it is shown that in the limit of rapid rotation, the system settles into a geostrophic equilibrium. The behaviour of the system is found to be controlled by one parameter only, linked to the Prantl number, the stratification and the rotation rate of the star.Comment: 5 pages, submitted to MNRAS Letter

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 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

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