The combined effects of vertical and horizontal shear instabilities

Shear instabilities can be the source of significant amounts of turbulent mixing in stellar radiative zones. Past attempts at modeling their effects (either theoretically or using numerical simulations) have focused on idealized geometries where the shear is either purely vertical or purely horizontal. In stars, however, the shear can have arbitrary directions with respect to gravity. In this work, we use direct numerical simulations to investigate the nonlinear saturation of shear instabilities in a stably stratified fluid, where the shear is sinusoidal in the horizontal direction, and either constant or sinusoidal in the vertical direction. We find that, in the parameter regime studied here (non-diffusive, fully turbulent flow), the mean vertical shear does not play any role in controlling the dynamics of the resulting turbulence unless its Richardson number is smaller than one (approximately). As most stellar radiative regions have a Richardson number much greater than one, our result implies that the vertical shear can essentially be ignored in the computation of the vertical mixing coefficient associated with shear instabilities for the purpose of stellar evolution calculations, even when it is much larger than the horizontal shear (as in the solar tachocline, for instance).Comment: 26 pages, 8 figures, resubmitted to Ap

A new model for mixing by double-diffusive convection (semi-convection): I. The conditions for layer formation

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 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. (2011) 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 numerically accessible "planetary" parameter regime, and can easily 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.Comment: Submitted to Ap

Particle-Gas Dynamics with Athena: Method and Convergence

The Athena MHD code has been extended to integrates the motion of particles coupled with the gas via aerodynamic drag, in order to study the dynamics of gas and solids in protoplanetary disks and the formation of planetesimals. Our particle-gas hybrid scheme is based on a second order predictor-corrector method. Careful treatment of the momentum feedback on the gas guarantees exact conservation. The hybrid scheme is stable and convergent in most regimes relevant to protoplanetary disks. We describe a semi-implicit integrator generalized from the leap-frog approach. In the absence of drag force, it preserves the geometric properties of a particle orbit. We also present a fully-implicit integrator that is unconditionally stable for all regimes of particle-gas coupling. Using our hybrid code, we study the numerical convergence of the non-linear saturated state of the streaming instability. We find that gas flow properties are well converged with modest grid resolution (128 cells per pressure length \eta r for dimensionless stopping time tau_s=0.1), and equal number of particles and grid cells. On the other hand, particle clumping properties converge only at higher resolutions, and finer resolution leads to stronger clumping before convergence is reached. Finally, we find that measurement of particle transport properties resulted from the streaming instability may be subject to error of about 20%.Comment: 33 pages, accepted to ApJ

Growth and migration of solids in evolving protostellar disks I: Methods and Analytical tests

This series of papers investigates the early stages of planet formation by modeling the evolution of the gas and solid content of protostellar disks from the early T Tauri phase until complete dispersal of the gas. In this first paper, I present a new set of simplified equations modeling the growth and migration of various species of grains in a gaseous protostellar disk evolving as a result of the combined effects of viscous accretion and photo-evaporation from the central star. Using the assumption that the grain size distribution function always maintains a power-law structure approximating the average outcome of the exact coagulation/shattering equation, the model focuses on the calculation of the growth rate of the largest grains only. The coupled evolution equations for the maximum grain size, the surface density of the gas and the surface density of solids are then presented and solved self-consistently using a standard 1+1 dimensional formalism. I show that the global evolution of solids is controlled by a leaky reservoir of small grains at large radii, and propose an empirically derived evolution equation for the total mass of solids, which can be used to estimate the total heavy element retention efficiency in the planet formation paradigm. Consistency with observation of the total mass of solids in the Minimum Solar Nebula augmented with the mass of the Oort cloud sets strong upper limit on the initial grain size distribution, as well as on the turbulent parameter \alphat. Detailed comparisons with SED observations are presented in a following paper.Comment: Submitted to ApJ. 23 pages and 13 figure

The Structure of the DoAr 25 Circumstellar Disk

We present high spatial resolution (< 0.3" = 40$AU) Submillimeter Array observations of the 865 micron continuum emission from the circumstellar disk around the young star DoAr 25. Despite its bright millimeter emission, this source exhibits only a comparatively small infrared excess and low accretion rate, suggesting that the material and structural properties of the inner disk may be in an advanced state of evolution. A simple model of the physical conditions in the disk is derived from the submillimeter visibilities and the complete spectral energy distribution using a Monte Carlo radiative transfer code. For the standard assumption of a homogeneous grain size distribution at all disk radii, the results indicate a shallow surface density profile,$\Sigma \propto r^{-p}$ with p = 0.34, significantly less steep than a steady-state accretion disk (p = 1) or the often adopted minimum mass solar nebula (p = 1.5). Even though the total mass of material is large (M_d = 0.10 M_sun), the densities inferred in the inner disk for such a model may be too low to facilitate any mode of planet formation. However, alternative models with steeper density gradients (p = 1) can explain the observations equally well if substantial grain growth in the planet formation region (r < 40 AU) has occurred. We discuss these data in the context of such models with dust properties that vary with radius and highlight their implications for understanding disk evolution and the early stages of planet formation.Comment: ApJL in pres

The dynamics of the radiative envelope of rapidly rotating stars. I. A spherical Boussinesq model

Context: The observations of rapidly rotating stars are increasingly detailed and precise thanks to interferometry and asteroseismology; two-dimensional models taking into account the hydrodynamics of these stars are very much needed. Aims: A model for studying the dynamics of baroclinic stellar envelope is presented. Methods: This models treats the stellar fluid at the Boussinesq approximation and assumes that it is contained in a rigid spherical domain. The temperature field along with the rotation of the system generate the baroclinic flow. Results: We manage to give an analytical solution to the asymptotic problem at small Ekman and Prandtl numbers. We show that, provided the Brunt-Vaisala frequency profile is smooth enough, differential rotation of a stably stratified envelope takes the form a fast rotating pole and a slow equator while it is the opposite in a convective envelope. We also show that at low Prandtl numbers and without $\mu$-barriers, the jump in viscosity at the core-envelope boundary generates a shear layer staying along the tangential cylinder of the core. Its role in mixing processes is discussed. Conclusions: Such a model provides an interesting tool for investigating the fluid dynamics of rotating stars in particular for the study of the various instabilities affecting baroclinic flows or, even more, of a dynamo effect.Comment: 17 pages, accepted in Astronomy and Astrophysic

Type-1.5 superconductivity in multicomponent systems

In general a superconducting state breaks multiple symmetries and, therefore, is characterized by several different coherence lengths $\xi_i$, $i=1,...,N$. Moreover in multiband material even superconducting states that break only a single symmetry are nonetheless described, under certain conditions by multi-component theories with multiple coherence lengths. As a result of that there can appear a state where some coherence lengths are larger and some are smaller than the magnetic field penetration length $\lambda$: $\xi_1\leq \xi_2... < \sqrt{2}\lambda<\xi_M\leq...\xi_N$. That state was recently termed "type-1.5" superconductivity. This breakdown of type-1/type-2 dichotomy is rather generic near a phase transition between superconducting states with different symmetries. The examples include the transitions between $U(1)$ and $U(1)\times U(1)$ states or between $U(1)$ and $U(1)\times Z_2$ states. The later example is realized in systems that feature transition between s-wave and $s+is$ states. The extra fundamental length scales have many physical consequences. In particular in these regimes vortices can attract one another at long range but repel at shorter ranges. Such a system can form vortex clusters in low magnetic fields. The vortex clustering in the type-1.5 regime gives rise to many physical effects, ranging from macroscopic phase separation in domains of different broken symmetries, to unusual transport properties