Magnetohydrodynamics of Turbulent Accretion Disks and Planet Formation

Abstract: Planets have long been believed to form in disks of gas and dust around youngstars, interacting with their surroundings via a setof complex and highly nonlinear processes. In the core accretion scenario forgiant planet formation dust coagulates first into km-sized icyand rocky planetesimals that further collide, formingprogressively larger solid bodies that eventually give rise to cores of severalEarth masses. If a critical mass is attained, these cores become gas giantplanets by undergoing runaway accretion of gas. Otherwise, just a small amount ofnebular gas is retained by the core, which ends up as an ice giant.Although explaining the overall architecture of the solar system, this picture was shaken by the discovery of the extra-solar planets, which present a diversity far greater than the standard scenario above could predict. Planet-disk interaction seems to be one of the obvious candidates to accountfor this diversity, since planets exchange angular momentum with the disk,leading to either inward or outward migration. An understanding of thephysical state of accretion disks is therefore essential to provide a detailed picture ofthe effect of migration on planetary orbits. Such understanding can be achieved through hydrodynamical modelling of the environment where planets are formed.On the same token, the study of accretion disks is in itself an interesting field of research, since ourcurrent understanding of the physics of accretion is only marginal: the accretion rates observed in circumstellar environments are far too great to be explained by molecular viscosity alone. A milestone was achieved 15 years ago with the realization that the magneto-rotational instability (MRI) generates a vigorous turbulence that appears to be present under a wide range of conditions in accretion disks. Much work was done since in an effort to understand the properties of turbulent magnetized accretion disks and its implications to stellar accretion and planet formation.The equations of hydrodynamics are nonlinear and we must therefore resort to numericalsimulations. For all simulations we employ the \pencilcodec, a 3D high-order finite-difference MHD codeusing Cartesian coordinates. A cylindrical formulation of {\sc Pencil} is presented, but was not yet used in a publication. Planets and stars are treated as particlesevolved with an $N$-body scheme, which is also introduced. In paper I we test the reliability of the numerical setup by contributingto a code comparison project that consisted of performing numerical simulations of a disk-planet systemusing various grid-based andsmoothed particle hydrodynamics codes. Surface density contours, potential vorticityand smoothed radial profiles were compared at several times, as well asthe disk mass and gravitational torque time evolution. Our results were consistent between the manycodes and outperformed the only other Cartesian code used in the comparison.In paper II we develop our models into three-dimensional magnetohydrodynamical simulationsof disks of gas and solids, studying the turbulence generated by the magneto-rotational instabilitythat is believed to be the source of the aforementioned anomalous viscosity. We find that Cartesian gridsare well-suited for accretion disk problems. Thedisk-in-a-box models based on Cartesian grids presented in paper II develop andsustain MHD turbulence, in good agreement with published results achieved withcylindrical codes. We investigate the dependence of the magneto-rotationalinstability on disk scale height, finding evidence that the turbulencegenerated by the magneto-rotational instability grows with thermal pressure. Theturbulent stresses depend on the thermal pressure obeying a power law with index $0.24\pm0.03$, compatible with the value of $0.25$ found in shearing boxcalculations. The ratio of stresses decreased with increasing temperature. Wealso study the dynamics of boulders in the hydromagnetic turbulence. Thevertical turbulent diffusion of the embedded boulders is comparable to theturbulent viscosity of the flow. Significant overdensities arise in the solidcomponent as boulders concentrate in high pressure regions, with interesting implications for planet formation.We went further, on paper III, to study whether the predictions based on local shearing box simulationsare also valid for global reference frames. By doing so, we help on the disentangling of physics andnumerics of simulated hydromagnetic turbulence, since it has been shown that the saturated state of the MRI depends not only on physical parameters, but also on numerical choices such as box length and resolution. We find that the saturation predictor derived from shearing box data also applies for the case of global disks in the minimum saturation regime where the fastest growing wavelength of the MRI is not resolved. A simulation where this wavelength is resolved shows that the scaling law is also accurate for global disks in this regime. Analytical considerations of the ratio of stresses, however, do not agree with the values found in the non-linear saturated state of the turbulence in the global simulations reported here. A brief summary of ongoing projects is presented in the last chapter.