THREEDIMENSIONAL^{ }MAGNETOHYDRODYNAMIC^{ }SIMULATIONS^{ }OF^{ }ACCRETION^{ }TO^{ }AN^{ }INCLINED^{ }ROTATOR:^{ }THE^{ }"CUBED^{ }SPHERE"^{ }METHOD Authors: Koldoba, A.V., Romanova, M.M., Ustyugova, G.V., Lovelace R.V.E. We
describe^{ }a threedimensional, Godunovtype 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^{ }zaxis 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 m
at an angle^{ }
J
= 30^{°} to^{ }the
star's rotation axis^{ }W,
which is aligned^{ }with the normal to^{ }the disk. Results are^{
}given for different grids^{ }(N_{R} × N^{2})
from^{ }26 × 15^{2} (coarsest)^{ }to 50 × 29^{2
}(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 threedimensional results^{ }for J
= 0^{ }with the
axisymmetric (twodimensional),^{ }spherical coordinate system
simulations^{ }of funnel flows of^{ }Romanova et al. Two^{ }important
new threedimensional 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^{ }xz cross section of^{ }the flow containing
mand W,
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 threedimensional funnel^{ }flows
at different inclination^{ }angles J.

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