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Global instabilities in swirling flows can significantly alter the flame and flow dynamics of swirl-stabilized flames, such as those in modern power generation gas turbine engines. In this study, we characterize the interaction between the precessing vortex core (PVC), which is the consequence of a global hydrodynamic instability, and thermoacoustic instabilities, which are the result of a resonant coupling between combustor acoustics and the unsteady heat release rate of combustion. This characterization is performed using experimental data obtained from a model gas turbine combustor system employing two concentric swirling nozzles of air, separated by a ring of fuel injectors, operating at 5 bar pressure. The flow split between the two streams is systematically varied to observe the impact of flow structure variation on the flow and flame dynamics. High-speed sPIV, OH-PLIF, and acetone PLIF are used to obtain information about the velocity fields, flame, and fuel flow behavior, respectively. Spectral POD and spatial frequency analysis are used to identify and characterize the dominant oscillation mechanisms driving the system. Three dominant modes are seen: two thermoacoustic modes and the precessing vortex core. Our results show that in the cases where the frequency of the PVC overlaps with either of the thermoacoustic modes, the thermoacoustic modes are suppressed. A weakly nonlinear asymptotic analysis shows that the suppression of the axisymmetric shear layer shedding, and hence thermoacoustic mode, is the result of a nonlinear coupling between the PVC and the axisymmetric mode of the swirling jet. Evolution equations for both the symmetric and PVC modes are derived to show the controlling parameters that drive this suppression. We conclude by discussing ways in which thermoacoustic instability suppression can be achieved through combustor flow field design.