A numerical study on a single wind turbine wake has been carried out focusing on the instability properties of the trailing tip vortices shed from the turbine blades. The numerical model is based on large-eddy simulations (LES) of the Navier-Stokes equations together with the actuator line method to simulate the wake behind the Tjaereborg wind turbine. The wake is perturbed by low amplitude stochastic excitations located in the neighborhood of the tip spiral, giving rise to spatially developing instabilities. Dynamic mode decomposition (DMD) is then utilized for identification of the coherent flow structures. The DMD results indicate that the amplification of specific waves along the spiral is responsible for triggering the instability leading to wake breakdown. Two types of dynamic structures dominates the flow; low and high frequency groups. Examination of these structures reveals that the dominant modes have the largest spatial growth.