This paper is concerned with the transformation of an upper level trough into a shear line and a tropopause cyclone or cut-off low. During this event, the isentropic potential vorticity of air columns between fixed isentropes approaching the trough axis at its level of maximum intensity from the west is well conserved over a period of about 12 to 24 hours, but there is a significant repartitioning of the constituent parts, from a relatively large static stability to a relatively large absolute vorticity. This indicates stretching of vertical columns of air. The effect that is inducing the vortex stretching is identified within the context of quasi-geostrophic theory as ‘opposite vector-frontogenesis’. This implies that Q-vectors point in opposite directions, respectively, above and below the level of maximum trough-intensity. Isentropic analysis indicates that the criterion for unstable isentropic downgliding is fulfilled in the upper troposphere in a restricted area, covering several hundred thousand square kilometres, near the base of the trough when and where the cut-off low is formed. This area of unstable isentropic downgliding, to which we assign the term ‘baroclinic downburst’, and which maintains its identity during a period of time in the order of 24 hours, coincides approximately with the dry intrusion as identified on a ‘water vapour’ image. It is argued that unstable isentropic downgliding is in fact necessary in order to realise a cyclone-core characterised by relatively high static stability, relatively high absolute vorticity and relatively high potential vorticity, as is demanded by the invertibility principle for potential vorticity. The events described above are triggered by the interaction of two potential vorticity anomalies.