Even in the central polar ocean regions, the sea ice cover cannot be regarded as a homogeneous and permanently closed surface. Due to divergent sea ice drift, even in winter elongated open-water channels, so-called leads, develop. Leads play an important role for surface-atmosphere interactions since they enable a direct contact between ocean and atmosphere. Especially in winter, when temperature differences amount up to 40 K, strong convective plumes are generated over leads with upward heat fluxes in the order of 10^2 Wm^(-2) . This has considerable effects on the structure of the atmosphere. Those effects mainly depend on the meteorological conditions close to the lead, but also on the lead geometry as, for example, the lead width. To understand the determining processes which lead to the formation and the decay of the plumes, we use numerical model simulations of a lead-perpendicular flow. We use a small-scale atmosphere model, which resolves the entire plume, but not the transport due to sub-grid scale turbulence. Hence, that transport has to be parametrized to close the system of the model equations. Thus, we developed an improved parametrization of the convection over leads, mainly for the turbulent heat flux. A previous parametrization, where one particular lead width (1 km) was considered, acts as a starting point and we derived a more general approach by including the lead width as a parameter. To validate our results, we use time-averaged results of a large eddy simulation (LES) model. We show that also for different lead widths small-scale model results obtained with our modified parametrization agree well with LES. In addition, our approach is still robust against variations of wind speed and surface temperature differences. Thus, our parametrization represents a clear improvement since now variable lead widths can be considered for a detailed investigation of the effects of lead-generated convection on the polar atmosphere.