Local barotropic instabilities developing in weakly non-parallel large-scale flows are investigated for weak supercriticality building-up on the WKB analysis of Bar-Sever and Merkine (1988). It is found that when the supercriticality is weak, in the parallel flow sense, its associated growth rate can be off-set by the decay rate induced by non-parallelism to generate a marginally unstable local instability. The marginal stability analysis pivots about the turning point region of the WKB solution which interacts with the critical layer of the local parallel problem where the gradient of the potential vorticity of the basic state vanishes. It is also found that for a finite range of supercriticality another critical layer must be encountered by the local eigenfunction of the upstream branch of the WKB solution. A criterion for this additional critical layer encounter is determined. The singularity of this upstream critical layer is removed by local non-parallel effects without affecting the downstream structure of the unstable wavepacket and its eigenfrequency. The instability seems to terminate when it encounters the critical layer. The analysis is supported by numerical simulations of the corresponding initial value problem.