  The interaction of accretion disks with the magnetospheres of young stars can
produce X-winds and funnel flows. With the assumption of axial symmetry and
steady state flow, the problem can be formulated in terms of quantities that
are conserved along streamlines, such as the Bernoulli integral (BI), plus a
partial differential equation (PDE), called the Grad-Shafranov equation (GSE),
that governs the distribution of streamlines in the meridional plane. The GSE
plus BI yields a PDE of mixed type, elliptic before critical surfaces where the
flow speed equals certain characteristic wave speeds are crossed and hyperbolic
afterward. The computational difficulties are exacerbated by the locations of
the critical surfaces not being known in advance. To overcome these obstacles,
we consider a variational principle by which the GSE can be attacked by
extremizing an action integral, with all other conserved quantities of the
problem explicitly included as part of the overall formulation. To simplify
actual applications we adopt the cold limit of a negligibly small ratio of the
sound speed to the speed of Keplerian rotation in the disk where the X-wind is
launched. We also ignore the obstructing effects of any magnetic fields that
might thread a disk approximated to be infinitesimally thin. We then introduce
trial functions with adjustable coefficients to minimize the variations that
give the GSE. We tabulate the resulting coefficients so that other workers can
have analytic forms to reconstruct X-wind solutions for various astronomical,
cosmochemical, and meteoritical applications.
