  In an electrically conducting fluid, vortices stretch out a weak, large-scale
magnetic field to form strong current sheets on their edges. Associated with
these current sheets are magnetic stresses, which are subsequently released
through reconnection, leading to vortex disruption, and possibly even
destruction. This disruption phenomenon is investigated here in the context of
two-dimensional, homogeneous, incompressible magnetohydrodynamics. We derive a
simple order of magnitude estimate for the magnetic stresses --- and thus the
degree of disruption --- that depends on the strength of the background
magnetic field (measured by the parameter $M$, a ratio between the Alfv\'en
speed and a typical flow speed) and on the magnetic diffusivity (measured by
the magnetic Reynolds number $\mbox{Rm}$). The resulting estimate suggests that
significant disruption occurs when $M^{2}\mbox{Rm} = O(1)$. To test our
prediction, we analyse direct numerical simulations of vortices generated by
the breakup of unstable shear flows with an initially weak background magnetic
field. Using the Okubo--Weiss vortex coherence criterion, we introduce a vortex
disruption measure, and show that it is consistent with our predicted scaling,
for vortices generated by instabilities of both a shear layer and a jet.
