Three Dimensional MHD Simulations of Accretion to an Inclined Rotator: The "Cubed Sphere" Method

Abstract

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 numerical method uses a "cubed sphere" coordinate system which has advantages of Cartesian and spherical coordinate systems but does not have the singular axis ofthe spherical system. The grid is formed by a sequence of concentric spheres of radii \(R_j \propto q^j\) with \(j=1..N_R\) and \(q={\rm const}>1\). 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. Simulation results are discussed for the funnel flows (FF) to a star with dipole moment \(\rvecmu\) at an angle \(\Theta=30^\circ\) to the star's rotation axis \({\bf \Omega}\) which is aligned with the normal to the disk. Two important new 3D features are found in these simulations: (1) The funnel flow to the stellar surface is mainly in two streams which approach the star from opposite directions. (2) In the \(x-z\) cross section of the flow containing \(\rvecmu\) and \({\bf \Omega}\), 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 3D funnel flows at different inclination angles \(\Theta\).