The closed-form expression for the mean squared beam width of annular beams propagating through atmospheric turbulence is derived by using the integral transform technique. The influence of turbulence on the spreading of annular beams is studied. On the other hand，the range of turbulence-independent propagation is obtained by examining the mean squared beam width，which indicates under what circumstances annular beams will be less affected by turbulence. It is shown that annular beams with larger obscure ratio ε，larger beam order M（N），larger wave length λ，and smaller outer radius w0 are less sensitive to the turbulence than those with smaller ε，M（N），λ，and larger w0. The main results obtained in this paper are explained physically.