Understanding how upper-ocean heat content evolves and affects sea ice in the polar regions is necessary to predict past, present, and future weather and climate. Sea ice, a composite of individual floes, varies significantly on scales as small as meters. Lateral gradients in surface forcing across sea-ice concentration gradients can energize subgrid-scale ocean eddies that mix heat in the surface layer and control sea-ice melting. Here the development of baroclinic instability near floe edges is investigated using a high-resolution ocean circulation model, an idealization of a single grid cell of a climate model partially covered in thin, nearly static sea ice. From the resulting ocean circulation we characterize the strength of eddy-induced lateral mixing and heat transport, and the effects on sea-ice melting, as a function of state variables resolved in global climate models. Plain Language Summary Sea ice is intrinsically tied to the ocean it forms out of, but the evolution of the Arctic ocean remains poorly understood thanks in part to the sea ice itself, which makes both travel and remote sensing extremely difficult. The increasing computing power available to climate modelers may be a poisoned chalice: Sea-ice models are built on a continuum framework and cannot therefore realize the sharp and heterogeneous concentration differences that may energize ocean circulation and thereby control sea-ice melting. We begin the process of incorporating these kind of effects in sea-ice models by describing and parameterizing the summertime response of the ocean to an idealized sharp sea-ice edge, providing guidance on how this methodology can be simplified and further implemented in continuum sea-ice models while maintaining the impacts of these coupled effects.