Criteria for the stability of an axisymmetric, compressible, differentially rotating, non self-gravitating fluid with a purely poloidal magnetic field are derived from a global analysis using an effective variational principle. These are applied to the example of geometrically thin accretion discs in which it has been recently suggested that unstable axisymmetric modes could be important in leading to angular momentum transport thus providing an effective viscosity. In the case of an externally imposed magnetic field it is possible to estimate the strength of the field above which the disc will become stable to this sort of mode. To give such an estimate, a global analysis of the type given here is necessary. The estimated field strength was found to be such that the typical Alfven speed is proportional to the product of the orbital velocity and the square root of the ratio of the disc thickness to radius.