We theoretically and experimentally evidence that fiber systems are convective systems since their nonlocal inherent properties, such as the dispersion and Raman effects, break the reflection symmetry. Theoretical analysis and numerical simulations carried out for a fiber ring cavity demonstrate that the third-order dispersion term leads to the appearance of convective and absolute instabilities. Their signature is an asymmetry in the output power spectrum. Using this criterion, experimental evidence of convective instabilities is given in a fiber cavity pumped by a pulsed laser.