  (abridged) Accretion from disks onto young stars is thought to follow
magnetic field lines from the inner disk edge to the stellar surface. The
accretion flow thus depends on the geometry of the magnetic field. This paper
extends previous work by constructing a collection of orthogonal coordinate
systems, including the corresponding differential operators, where one
coordinate traces the magnetic field lines. This formalism allows for an
(essentially) analytic description of the geometry and the conditions required
for the flow to pass through sonic points. Using this approach, we revisit the
problem of magnetically controlled accretion flow in a dipole geometry, and
then generalize the treatment to consider magnetic fields with multiple
components, including dipole, octupole, and split monopole contributions. This
approach can be generalized further to consider more complex magnetic field
configurations. Observations indicate that accreting young stars have
substantial dipole and octupole components, and that accretion flow is
transonic. If the effective equation of state for the fluid is too stiff, the
flow cannot pass smoothly through the sonic points in steady state. For a
multipole field of order \ell, we derive a constraint on the polytropic index,
n>\ell+3/2, required for steady transonic flow to reach free-fall velocities.
For octupole fields, inferred on surfaces of T Tauri stars, n>9/2, so that the
flow must be close to isothermal. The inclusion of octupole field components
produces higher densities at the stellar surface and smaller hot spots, which
occur at higher latitudes; the magnetic truncation radius is also modified.
This contribution thus increases our understanding of magnetically controlled
accretion for young stellar objects and can be applied to a variety of
additional astrophysical problems.
