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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 only needs to 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.

We study the properties of oscillatory double-diffusive convection (ODDC) in the presence of a uniform vertical background magnetic field. ODDC takes place in stellar regions that are unstable according to the Schwarzschild criterion and stable according to the Ledoux criterion (sometimes called semiconvective regions), which are often predicted to reside just outside the core of intermediate-mass main-sequence stars. Previous hydrodynamic studies of ODDC have shown that the basic instability saturates into a state of weak wave-like convection, but that a secondary instability can sometimes transform it into a state of layered convection, where layers then rapidly merge and grow until the entire region is fully convective. We find that magnetized ODDC has very similar properties overall, with some important quantitative differences. A linear stability analysis reveals that the fastest-growing modes are unaffected by the field, but that other modes are. Numerically, the magnetic field is seen to influence the saturation of the basic instability, overall reducing the turbulent fluxes of temperature and composition. This in turn affects layer formation, usually delaying it, and occasionally suppressing it entirely for sufficiently strong fields. Further work will be needed, however, to determine the field strength above which layer formation is actually suppressed in stars. Potential observational implications are briefly discussed. [ABSTRACT FROM AUTHOR]

Turbulent mixing in the radiative regions of stars is usually either ignored or crudely accounted for in most stellar evolution models. However, there is growing evidence that such mixing is present and can affect various aspects of a star's life. Here, we present a first attempt at quantifying mixing by horizontal shear instabilities in stars using direct numerical simulations. The shear is driven by a body force, and rapidly becomes unstable. At saturation, we find that several distinct dynamical regimes exist, depending on the relative importance of stratification and thermal diffusion. In each of the regimes identified, we propose a certain number of theoretically motivated scaling laws for the turbulent vertical eddy scale, the turbulent diffusion coefficient, and the amplitude of temperature fluctuations (among other quantities). Based on our findings, we predict that the majority of stars should fall into one of two categories: high Péclet number stratified turbulence, and low Péclet number stratified turbulence. The latter is presented in a related paper by Cope et al., while the former is discussed here. Applying our results to the solar tachocline, we find that it should lie in the high Péclet number stratified turbulence regime, and predict a substantial amount of vertical mixing for temperature, momentum, and composition. Taken as is, the new turbulence model predictions are incompatible with the Spiegel & Zahn model of the solar tachocline. However, rotation and magnetic fields are likely to affect the turbulence, and need to be taken into account in future studies. [ABSTRACT FROM AUTHOR]

It has long been known that solar-type stars undergo significant spin-down, via magnetic braking, during their main-sequence lifetimes. However, magnetic braking only operates on the surface layers; it is not yet completely understood how angular momentum is transported within the star and how rapidly the spin-down information is communicated to the deep interior. In this work, we use insight from recent progress in understanding internal solar dynamics to model the interior of other solar-type stars. We assume, following Gough & McIntyre, that the bulk of the radiation zone of these stars is held in uniform rotation by the presence of an embedded large-scale primordial field, confined below a stably stratified, magnetic-free tachocline by large-scale meridional flows downwelling from the convection zone. We derive simple equations to describe the response of this model interior to spin-down of the surface layers, which are identical to the two-zone model of MacGregor & Brenner, with a coupling timescale proportional to the local Eddington-Sweet timescale across the tachocline. This timescale depends both on the rotation rate of the star and on the thickness of the tachocline, and it can vary from a few hundred thousand years to a few Gyr, depending on stellar properties. Qualitative predictions of the model appear to be consistent with observations, although they depend sensitively on the assumed functional dependence of the tachocline thickness on the stellar rotation rate. [ABSTRACT FROM AUTHOR]

Regions of stellar and planetary interiors that are unstable according to the Schwarzschild criterion, but stable according to the Ledoux criterion, are subject to a form of oscillatory double-diffusive (ODD) convection often called "semi-convection." In this series of papers, we use an extensive suite of three-dimensional (3D) numerical simulations to quantify the transport of heat and composition by ODD convection, and ultimately propose a new 1D prescription that can be used in stellar and planetary structure and evolution models. The first paper in this series demonstrated that under certain conditions ODD convection spontaneously transitions from an initial homogeneous state of weak wave-breaking turbulence into a staircase of fully convective layers, which results in a substantial increase in the transport of heat and composition. Here, we present simulations of ODD convection in this layered regime, we describe the dynamical behavior of the layers, and we derive empirical scaling laws for the transport through layered convection. [ABSTRACT FROM AUTHOR]

The process referred to as "semi-convection" in astrophysics and "double-diffusive convection in the diffusive regime" in Earth and planetary sciences occurs in stellar and planetary interiors in regions which are stable according to the Ledoux criterion but unstable according to the Schwarzschild criterion. In this series of papers, we analyze the results of an extensive suite of three-dimensional (3D) numerical simulations of the process, and ultimately propose a new 1D prescription for heat and compositional transport in this regime which can be used in stellar or planetary structure and evolution models. In a preliminary study of the phenomenon, Rosenblum et al. showed that, after saturation of the primary instability, a system can evolve in one of two possible ways: the induced turbulence either remains homogeneous, with very weak transport properties, or transitions into a thermo-compositional staircase where the transport rate is much larger (albeit still smaller than in standard convection). In this paper, we show that this dichotomous behavior is a robust property of semi-convection across a wide region of parameter space. We propose a simple semi-analytical criterion to determine whether layer formation is expected or not, and at what rate it proceeds, as a function of the background stratification and of the diffusion parameters (viscosity, thermal diffusivity, and compositional diffusivity) only. The theoretical criterion matches the outcome of our numerical simulations very adequately in the computationally accessible "planetary" parameter regime and can be extrapolated to the stellar parameter regime. Subsequent papers will address more specifically the question of quantifying transport in the layered case and in the non-layered case. [ABSTRACT FROM AUTHOR]

To date, no self-consistent numerical simulation of the solar interior has succeeded in reproducing the observed thinness of the solar tachocline and the persistence of uniform rotation beneath it. Although it is known that the uniform rotation can be explained by the presence of a global-scale confined magnetic field, numerical simulations have thus far failed to produce any solution where such a field remains confined against outward diffusion. We argue that the problem lies in the choice of parameters for which these numerical simulations have been performed. We construct a simple analytical magnetohydrodynamic model of the solar interior and identify several distinct parameter regimes. For realistic solar parameter values, our results are in broad agreement with the tachocline model of Gough & McIntyre. In this regime, meridional flows driven at the base of the convection zone are of sufficient amplitude to hold back the interior magnetic field against diffusion. For the parameter values used in existing numerical simulations, on the other hand, we find that meridional flows are significantly weaker and, we argue, unable to confine the interior field. We propose a method for selecting parameter values in future numerical models. [ABSTRACT FROM AUTHOR]