Turbulent exchange processes in the atmospheric boundary layer (ABL) over the polar ocean regions are strongly influenced by the characteristics of the surface. While thick sea ice minimizes turbulent energy transport by its insulating effect, a large part of it can occur through leads, which are elongated open-water channels in sea ice. Especially between late fall and spring, temperature differences between the near-surface ABL and the surface of leads can amount up to 40K causing strong convective plumes over the leads. Our aim is to improve the understanding of the main processes that drive turbulent transport in the convective plumes by means of developing turbulence parametrizations for such a strongly heterogeneous ABL. We model three cases of the turbulent flow over leads observed by aircraft and use a plume- but non-eddy-resolving model either with a local or a non-local closure for turbulence. The latter uses a lead-width dependent approach for an idealized, lead-perpendicular, and near-neutral inflow in an ABL of 300m height. The main goal of our work is to elaborate the quality of both parametrizations when applied to the non-idealized cases including an evaluation of potential advantages of the non-local closure as compared to the local one. While the main observed ABL structures over the leads are represented with both closures, the non-local approach shows a physically more realistic representation of regions where the observations hint at gradient-independent transport or with observed vertical entrainment. Moreover, by introducing two modifications for the non-local closure, an improved representation is obtained for the observed effects by more stable inflow conditions and by allowing for increased interaction between the plume and the inversion layer as compared to the idealized cases. We consider our results as another important step towards a physically reasonable representation of heterogeneous ABLs by small-scale, non-eddy-resolving modeling.