We describe in detail an annual seasonal process that occurs on the surface of the Russell Crater megadune on Mars. We give these features the name 'perennial rills', because their surface topographic expression persists from year-to-year and they form a distinctive, downstream branching network of small channels, or rills. We used time series images, elevation data from stereo photogrammetry and spectral data to characterise the evolution of these features over six Mars Years. Growth and modification of these networks occurs abruptly in spring (at approx. solar longitude 200°) after most of the seasonal CO2 ice has sublimated. We find that the peculiar morphology of perennial rills seems to be the only aspect that sets them apart from active linear dune gullies. By comparison to terrestrial analogues we identified two conditions favouring the production such a network: a) the presence of an impermeable layer and b) the repeated formation of obstacles in front of propagating channels. We find that the most plausible formation mechanisms that can explain the formation of both the perennial rills and active linear dune gullies are levitating CO2 blocks, or liquid debris flows of water/brine, but neither can completely satisfy all the observational evidence.