Turbine with pressurised cavities

A turbine engine comprising a turbine comprising a first rotor and a second rotor configured to pivot about a longitudinal axis (X) according to two opposite directions of rotation, the first rotor including a radially outer drum from which blades extend radially inwards, the first rotor and the second rotor being surrounded by a stator annular part, wherein said stator annular part delimits with the radially outer drum at least one upstream annular cavity and one downstream annular cavity separated from each other by sealing means, the upstream annular cavity being connected to a first cooling air supply at a first pressure and the downstream annular cavity being connected to a second cooling air supply at a second pressure lower than the first pressure, the first cooling air supply being connected, at an upstream end, to a compressor and configured for drawing air from the compressor and/or wherein the second cooling air supply is connected, at an upstream end, to an annular secondary air flow path which extends radially outwards of said stator annular part, the second cooling air supply being configured for drawing air from the annular secondary air flow path.