For values of r greater than the coherence length \ensuremath{\xi}, the axially symmetric Ginzburg-Landau equations are solved for a flux vortex carrying a longitudinal current. The field is not force free, and it is shown that there are no regular solutions to the force-free field equations that decay exponentially with increasing penetration into a superconductor. It is also shown, in this approximation, that in the case of a vortex carrying a nonzero longitudinal current, the Ginzburg-Landau equations are equivalent to the radial pressure-balance equilibrium relation in ideal magnetohydrodynamics. The techniques developed in this field to address stability issues can then be used to answer questions related to vortex stability.