The flow around the rotor of an axial turbine, operating in the wake of an upstream one, is resolved using large-eddy simulation on a cylindrical grid consisting of 3.8 billion points. Three distances from the upstream rotor are considered, ranging from 6 to 10 diameters. The inflow boundary conditions for the simulation of the downstream rotor were generated by a precursor simulation of the upstream one. The impact on the dynamics of the tip vortices and the statistics at their core is compared across cases. Results demonstrate the strong sensitivity of the tip vortices shed by the downstream rotor to the disturbance produced by the wake of the upstream one. The onset of their instability moves very close to the rotor plane and is almost independent of the distance between the upstream and downstream rotors, at least in the range of simulated distances. This makes the development of the wake of the downstream rotors, driven by the instability of the tip vortices, much faster and very similar across distances from the upstream one. The results explain the phenomena of performance stabilization of downstream turbines in linear arrays, recently reported in the literature.