[Paper Review] Global magnetohydrodynamical models of turbulence in protoplanetary disks I. A cylindrical potential on a Cartesian grid and transport of solids

Wladimir Lyra, Anders Johansen+2 more|ArXiv.org|May 28, 2007

Astrophysics and Star Formation Studies52 references61 citations

TL;DR

This paper presents global 3D magnetohydrodynamical simulations of protoplanetary disks using a Cartesian grid with a cylindrical gravitational potential, demonstrating that such setups can sustain magnetorotational instability (MRI)-driven turbulence. The study finds that turbulent stresses scale with thermal pressure as a power law of 0.24±0.03, and that solids experience vertical turbulent diffusion with Schmidt numbers of 1.0±0.2 and 0.78±0.06, indicating strong coupling between gas and solid phases.

ABSTRACT

We present global 3D MHD simulations of disks of gas and solids, aiming at developing models that can be used to study various scenarios of planet formation and planet-disk interaction in turbulent accretion disks. A second goal is to show that Cartesian codes are comparable to cylindrical and spherical ones in handling the magnetohydrodynamics of the disk simulations, as the disk-in-a-box models presented here develop and sustain MHD turbulence. We investigate the dependence of the magnetorotational instability on disk scale height, finding evidence that the turbulence generated by the magnetorotational instability grows with thermal pressure. The turbulent stresses depend on the thermal pressure obeying a power law of 0.24+/-0.03, compatible with the value of 0.25 found in shearing box calculations. The ratio of stresses decreased with increasing temperature. We also study the dynamics of boulders in the hydromagnetic turbulence. The vertical turbulent diffusion of the embedded boulders is comparable to the turbulent viscosity of the flow. Significant overdensities arise in the solid component as boulders concentrate in high pressure regions.

Motivation & Objective

• To develop global 3D MHD simulations of accretion disks using Cartesian grids with a cylindrical potential to study turbulence and planet formation.
• To investigate the dependence of MRI turbulence on disk thermal pressure and magnetic field strength.
• To model the dynamics of solid boulders in turbulent gas using Lagrangian superparticles and quantify vertical diffusion.
• To assess the role of turbulent stresses in preventing radial and vertical settling of solids in protoplanetary disks.
• To establish the suitability of Cartesian grids for global disk simulations with embedded solids and magnetic fields.

Proposed method

• The Pencil Code, a high-order finite-difference MHD code in Cartesian coordinates, is used to solve ideal MHD equations with an isothermal equation of state.
• A cylindrical gravitational potential is applied on a Cartesian grid to simulate a differentially rotating disk, enabling global simulations without coordinate singularities.
• Solid particles are modeled as superparticles that experience a linear drag force proportional to their relative velocity with respect to the gas.
• Turbulent stresses are computed from the Reynolds and Maxwell stress tensors, and the ratio of Maxwell to Reynolds stress is analyzed as a function of temperature.
• Vertical diffusion of solids is quantified via the scale height of the solid layer, from which turbulent Schmidt numbers are derived.
• Shock viscosity, hyperdiffusion, and anisotropic dissipation are applied to stabilize numerical solutions and suppress high-wavenumber noise.

Experimental results

Research questions

• RQ1 Can Cartesian grid simulations with a cylindrical potential accurately reproduce MRI-driven turbulence in protoplanetary disks?
• RQ2 How does the turbulent stress depend on thermal pressure and sound speed in MRI-active disks?
• RQ3 To what extent does turbulent diffusion prevent vertical settling of solid particles in a 3D turbulent disk?
• RQ4 What is the effective Schmidt number for vertical turbulent diffusion of solids, and how does it compare to that of the gas?
• RQ5 How do overdensities in solid-to-gas ratios form in turbulent regions, and what are their maximum values?

Key findings

• The Cartesian grid with a cylindrical potential successfully sustains MRI-driven turbulence, matching results from cylindrical codes.
• Turbulent stress scales with thermal pressure as a power law of 0.24±0.03, consistent with the expected 0.25 from shearing box simulations.
• The ratio of Maxwell to Reynolds stress decreases from 5 to 1 when sound speed increases by a factor of 4, while magnetic field strength is held constant.
• Solids form a finite vertical layer due to turbulent diffusion, with scale heights implying globally averaged Schmidt numbers of 1.0±0.2 for α≈10⁻³ and 0.78±0.06 for α≈10⁻¹.
• Despite low average solid density (ρₚ = 6.0×10⁻¹¹ kg m⁻³), local overdensities reach solid-to-gas ratios of up to 85, far exceeding the initial value of 0.01.
• The turbulent diffusion of solids is comparable in strength to the turbulent viscosity acting on the gas, indicating strong coupling between phases.