Three-dimensional Magnetohydrodynamic Simulations of Accretion to an Inclined Rotator: The "Cubed Sphere" Method

We describe a three-dimensional, Godunov-type numerical magnetohydrodynamics (MHD) method designed for studying disk accretion to a rotating magnetized star in the general case where the star's rotation axis, its magnetic moment, and the normal to the disk all have different directions. The equations of ideal MHD are written in a reference frame rotating with the star, with the z-axis aligned with the star's rotation axis. The numerical method uses a "cubed sphere" coordinate system that has advantages of Cartesian and spherical coordinate systems but does not have the singular axis of the spherical system. The grid is formed by a sequence of concentric spheres of radii R_j ∝ q^j, with j = 1, ... , N_R and q = constant > 1. The grid on the surface of the sphere consists of six sectors, with the grid on each sector topologically equivalent to the equidistant grid on the face of a cube. The magnetic field is written as a dipole component plus deviations, and only the deviations are calculated. Simulation results are discussed for the funnel flows to a star with dipole moment μ at an angle Θ = 30° to the star's rotation axis Ω, which is aligned with the normal to the disk. Results are given for different grids (N_R × N²) from 26 × 15² (coarsest) to 50 × 29² (finest). We observe that the qualitative features of the accretion flows are very similar for the different grids, but the coarser grid is affected by numerical viscosity. We compare our three-dimensional results for Θ = 0 with the axisymmetric (two-dimensional), spherical coordinate system simulations of funnel flows of Romanova et al. Two important new three-dimensional features are found in these simulations: (1) The funnel flow to the stellar surface is mainly in two streams that approach the star from opposite directions. (2) In the x-z cross section of the flow containing μ and Ω, the funnel flow often takes the longer of the two possible paths along magnetic field lines to the surface of the star. A subsequent paper will give a detailed description of the method and results on three-dimensional funnel flows at different inclination angles Θ.