The observation of a Hopf bifurcation in fluid dynamics is commonly associated with the occurrence of a global hydrodynamic instability. The increase of a control parameter above a critical value causes a change from a stable to an unstable flow condition. This behaviour is well understood for the occurrence of instabilities in laminar flows. In turbulent flows, however, the occurrence of hydrodynamic instabilities is similarly observed but less clear. The current work examines the use of stochastic models to describe the supercritical Hopf bifurcation of the global mode in a turbulent swirling jet. The consideration of the interaction between the global mode and the stochastic turbulent perturbations allows a consistent description of the experimental observations. This opens up extensive possibilities for describing and interpreting the occurrence of hydrodynamic instabilities in turbulent flows.