  We study the azimuthal modulational instability of vortices with different
topological charges, in the focusing two-dimensional nonlinear Schr{\"o}dinger
(NLS) equation. The method of studying the stability relies on freezing the
radial direction in the Lagrangian functional of the NLS in order to form a
quasi-one-dimensional azimuthal equation of motion, and then applying a
stability analysis in Fourier space of the azimuthal modes. We formulate
predictions of growth rates of individual modes and find that vortices are
unstable below a critical azimuthal wave number. Steady state vortex solutions
are found by first using a variational approach to obtain an asymptotic
analytical ansatz, and then using it as an initial condition to a numerical
optimization routine. The stability analysis predictions are corroborated by
direct numerical simulations of the NLS. We briefly show how to extend the
method to encompass nonlocal nonlinearities that tend to stabilize solutions.
