Combustion instability is a serious problem limiting the operating envelope of present day gas turbine systems using a lean premixed combustion strategy. Gas turbine combustors employ swirl as a means for achieving fuel-air mixing as well as flame stabilization. However swirl flows are complex flows comprised of multiple shear layers as well as recirculation zones which makes them particularly susceptible to hydrodynamic instability. We perform a local stability analysis on a family of base flow model profiles characteristic of swirling flow that has undergone vortex breakdown as would be the case in a gas turbine combustor. A temporal analysis at azimuthal wavenumbers m = 0 and m = 1 reveals the presence of two unstable modes. A companion spatio-temporal analysis shows that the region in base flow parameter space for constant density density flow, over which m = 1 mode with the lower oscillation frequency is absolutely unstable, is much larger that that for the corresponding m = 0 mode. This suggests that the dominant self-excited unstable behavior in a constant density flow is an asymmetric, m=1 mode. The presence of a density gradient within the inner shear layer of the flow profile causes the absolutely unstable region for the m = 1 to shrink which suggests a possible explanation for the suppression of the precessing vortex core in the presence of a flame.