Colin P. McNally

GLOBAL SIMULATIONS OF PROTOPLANETARY DISKS WITH OHMIC RESISTIVITY AND AMBIPOLAR DIFFUSION

Protoplanetary disks are believed to accrete onto their central T Tauri star because of magnetic stresses. Recently published shearing box simulations indicate that Ohmic resistivity, ambipolar diffusion and the Hall effect all play important roles in disk evolution. In the presence of a vertical magnetic field, the disk remains laminar between 1-5au, and a magnetocentrifugal disk wind forms that provides an important mechanism for removing angular momentum. Questions remain, however, about the establishment of a true physical wind solution in the shearing box simulations because of the symmetries inherent in the local approximation. We present global MHD simulations of protoplanetary disks that include Ohmic resistivity and ambipolar diffusion, where the time-dependent gas-phase electron and ion fractions are computed under FUV and X-ray ionization with a simplified recombination chemistry. Our results show that the disk remains laminar, and that a physical wind solution arises naturally in global disk models. The wind is sufficiently efficient to explain the observed accretion rates. Furthermore, the ionization fraction at intermediate disk heights is large enough for magneto-rotational channel modes to grow and subsequently develop into belts of horizontal field. Depending on the ionization fraction, these can remain quasi-global, or break-up into discrete islands of coherent field polarity. The disk models we present here show a dramatic departure from our earlier models including Ohmic resistivity only. It will be important to examine how the Hall effect modifies the evolution, and to explore the influence this has on the observational appearance of such systems, and on planet formation and migration.

Rossby wave instability does not require sharp resistivitygradients

Rossby wave instability (RWI) at dead zone boundaries may play an important role in planet formation. Viscous hydrodynamics results suggest RWI is excited only when the viscosity changes over a radial distance less than two density scale heights. However in the disks around Solar-mass T Tauri stars, it is not viscosity but magnetic forces that provide the accretion stress beyond about 10 AU, where surface densities are low enough so stellar X-rays and interstellar cosmic rays can penetrate. Here we aim to explore the conditions for RWI in the smooth transition with increasing distance, from resistive and magnetically-dead to conducting and magnetically-active. We perform 3D unstratified MHD simulations with the Pencil Code, using static resistivity profiles. As a result, we find that in MHD, contrary to viscous models, the RWI is triggered even with a gradual change in resistivity extending from 10 to 40 AU (i.e., spanning 15 scale heights for aspect ratio 0.1). This is because magneto-rotational turbulence sets in abruptly when the resistivity reaches a threshold level. At higher resistivities the longest unstable wavelength is quenched, resulting in a sharp decline of the Maxwell stress towards the star. The sharp gradient in the magnetic forces leads to a localized density bump, that is in turn Rossby wave unstable. We conclude that even weak gradients in the resistivity can lead to sharp transitions in the Maxwell stress. The upshot is that the RWI is more easily activated in the outer disk than previously thought. Rossby vortices at the outer dead zone boundary thus could underlie the dust asymmetries seen in the outer reaches of transition disks.

TEMPERATURE FLUCTUATIONS DRIVEN BY MAGNETOROTATIONAL INSTABILITY IN PROTOPLANETARY DISKS

The magnetorotational instability (MRI) drives magnetized turbulence in sufficiently ionized regions of protoplanetary disks, leading to mass accretion. The dissipation of the potential energy associated with this accretion determines the thermal structure of accreting regions. Until recently, the heating from the turbulence has only been treated in an azimuthally averaged sense, neglecting local fluctuations. However, magnetized turbulence dissipates its energy intermittently in current sheet structures. We study this intermittent energy dissipation using high resolution numerical models including a treatment of radiative thermal diffusion in an optically thick regime. Our models predict that these turbulent current sheets drive order-unity temperature variations even where the MRI is damped strongly by Ohmic resistivity. This implies that the current sheet structures where energy dissipation occurs must be well-resolved to correctly capture the flow structure in numerical models. Higher resolutions are required to resolve energy dissipation than to resolve the magnetic field strength or accretion stresses. The temperature variations are large enough to have major consequences for mineral formation in disks, including melting chondrules, remelting calcium-aluminum-rich inclusions, and annealing silicates; and may drive hysteresis: current sheets in MRI active regions could be significantly more conductive than the remainder of the disk.

ON VERTICALLY GLOBAL, HORIZONTALLY LOCAL MODELS FOR ASTROPHYSICAL DISKS

Disks with a barotropic equilibrium structure, for which the pressure is only a function of the density, rotate on cylinders in the presence of a gravitational potential, so that the angular frequency of such a disk is independent of height. Such disks with barotropic equilibria can be approximately modeled using the shearing box framework, representing a small disk volume with height-independent angular frequency. If the disk is in baroclinic equilibrium, the angular frequency does generally depend on height, and it is thus necessary to go beyond the standard shearing box approach. In this paper, we show that given a global disk model, it is possible to develop approximate models that are local in horizontal planes without an expansion in height with shearing-periodic boundary conditions. We refer to the resulting framework as the vertically global shearing box (VGSB). These models can be non-axisymmetric for globally barotropic equilibria but should be axisymmetric for globally baroclinic equilibria. We provide explicit equations for this VGSB which can be implemented in standard magnetohydrodynamic codes by generalizing the shearing-periodic boundary conditions to allow for a height-dependent angular frequency and shear rate. We also discuss the limitations that result from the radial approximations that are needed in order to impose height-dependent shearing periodic boundary conditions. We illustrate the potential of this framework by studying a vertical shear instability and examining the modes associated with the magnetorotational instability.

Low mass planet migration in magnetically torqued dead zones – I. Static migration torque

Motivated by models suggesting that the inner planet forming regions of protoplanetary discs are predominantly lacking in viscosity-inducing turbulence, and are possibly threaded by Hall-effect generated large-scale horizontal magnetic fields, we examine the dynamics of the corotation region of a low-mass planet in such an environment. The corotation torque in an inviscid, isothermal, dead zone ought to saturate, with the libration region becoming both symmetrical and of a uniform vortensity, leading to fast inward migration driven by the Lindblad torques alone. However, in such a low viscosity situation, the material on librating streamlines essentially preserves its vortensity. If there is relative radial motion between the disc gas and the planet, the librating streamlines will no longer be symmetrical. Hence, if the gas is torqued by a large scale magnetic field so that it undergoes a net inflow or outflow past the planet, driving evolution of the vortensity and inducing asymmetry of the corotation region, the corotation torque can grow, leading to a positive torque. In this paper we treat this effect by applying a symmetry argument to the previously studied case of a migrating planet in an inviscid disc. Our results show that the corotation torque due to a laminar Hall-induced magnetic field in a dead zone behaves quite differently from that studied previously for a viscous disc. Furthermore, the magnetic field induced corotation torque and the dynamical corotation torque in a low viscosity disc can be regarded as one unified effect.

PHOTOPHORESIS IN A DILUTE, OPTICALLY THICK MEDIUM AND DUST MOTION IN PROTOPLANETARY DISKS

We derive expressions for the photophoretic force on opaque spherical particles in a dilute gas in the optically thick regime where the radiation field is in local thermal equilibrium. Under those conditions, the radiation field has a simple form, leading to well defined analytical approximations for the photophoretic force that also consider both the internal thermal conduction within the particle, and the effects of heat conduction and radiation to the surrounding gas. We derive these results for homogeneous spherical particles; and for the double layered spheres appropriate for modeling solid grains with porous aggregate mantles. Then, as a specific astrophysical application of these general physical results, we explore the parameter space relevant to the photophoresis driven drift of dust in protoplanetary disks. We show that highly porous silicate grains have sufficiently low thermal conductivities that photophoretic effects, such as significant relative velocities between particles with differing porosity or levitation above the midplane, are expected to occur.

PHOTOPHORETIC LEVITATION AND TRAPPING OF DUST IN THE INNER REGIONS OF PROTOPLANETARY DISKS

In protoplanetary disks, the differential gravity-driven settling of dust grains with respect to gas and with respect to grains of varying sizes determines the observability of grains, and sets the conditions for grain growth and eventually planet formation. In this work we explore the effect of photophoresis on the settling of large dust grains in the inner regions of actively accreting protoplanetary disks. Photophoretic forces on dust grains result from the collision of gas molecules with differentially heated grains. We undertake one dimensional dust settling calculations to determine the equilibrium vertical distribution of dust grains in each column of the disk. In the process we introduce a new treatment of the photophoresis force which is consistent at all optical depths with the representation of the radiative intensity field in a two-stream radiative transfer approximation. The levitation of large dust grains creates a photophoretic dust trap several scale heights above the mid-plane in the inner regions of the disk where the dissipation of accretion energy is significant. We find that differential settling of dust grains is radically altered in these regions of the disk, with large dust grains trapped in a layer below the stellar irradiation surface in where the dust to gas mass ratio can be enhanced by a factor of a hundred for the relevant particles. The photophoretic trapping effect has a strong dependence on particle size and porosity.

Low-mass planet migration in magnetically torqued dead zones – II. Flow-locked and runaway migration, and a torque prescription

We examine the migration of low mass planets in laminar protoplanetary discs, threaded by large scale magnetic fields in the dead zone that drive radial gas flows. As shown in Paper I, a dynamical corotation torque arises due to the flow-induced asymmetric distortion of the corotation region and the evolving vortensity contrast between the librating horseshoe material and background disc flow. Using simulations of laminar torqued discs containing migrating planets, we demonstrate the existence of the four distinct migration regimes predicted in Paper I. In two regimes, the migration is approximately locked to the inward or outward radial gas flow, and in the other regimes the planet undergoes outward runaway migration that eventually settles to fast steady migration. In addition, we demonstrate torque and migration reversals induced by midplane magnetic stresses, with a bifurcation dependent on the disc surface density. We develop a model for fast migration, and show why the outward runaway saturates to a steady speed, and examine phenomenologically its termination due to changing local disc conditions. We also develop an analytical model for the corotation torque at late times that includes viscosity, for application to discs that sustain modest turbulence. Finally, we use the simulation results to develop torque prescriptions for inclusion in population synthesis models of planet formation.

Low mass planet migration in Hall-affected disks

Recent developments in non-ideal magnetohydrodynamic simulations of protoplanetary disks suggest that instead of being traditional turbulent (viscous) accretion disks, they have a largely laminar flow with accretion driven by large-scale wind torques. These disks are possibly threaded by Hall-effect generated large-scale horizontal magnetic fields. We have examined the dynamics of the corotation region of a low mass planet embedded in such a disk and the evolution of the associated migration torque. These disks lack strong turbulence and associated turbulent diffusion, and the presence of a magnetic field and radial gas flow presents a situation outside the applicability of previous corotation torque theory. We summarize the analytical analysis of the corotation torque, give details on the numerical methods used, and in particular the relative merits of different numerical schemes for the inviscid problem.

Orbital Advection with Magnetohydrodynamics and Vector Potential

Orbital advection is a significant bottleneck in disk simulations, and a particularly tricky one when used in connection with magnetohydrodynamics. We have developed an orbital advection algorithm suitable for the induction equation with magnetic potential. The electromotive force is split into advection and shear term, and we find that we do not need an advective gauge as solving the orbital advection implicitly precludes the shear term from canceling the advection term. We prove and demonstrate the third order in time accuracy of the scheme. The algorithm is also suited for non-magnetic problems. Benchmarked results of (hydrodynamical) planet-disk interaction and the of the magnetorotational instability are reproduced. We include detailed descriptions of the construction and selection of stabilizing dissipations (or high frequency filters) needed to generate practical results. The scheme is self-consistent, accurate, and elegant in its simplicity, making it particularly efficiently in straightforward finite-difference methods. As a result of the work, the algorithm is incorporated in the public version of the Pencil Code, where it can be used by the community.