When a strong, cold wind blows over open polar leads, ice grows in the form of small crystals which are herded downwind by both wind waves and a wind-driven surface current to pile up at the downwind edge of the lead. As this process continues, the piled-up ice, called ‘grease ice,’ advances upwind until the entire lead is ice covered. In a numerical model of ice growth in a horizontally one-dimensional lead with a wind blowing along its length we use the heat flux taken from existing formulations to determine the ice production rate. Similarly, we use the wave field and surface current, which are functions of wind speed and fetch, to calculate the ice pileup depth and the ice cover advance rate. Because this ice growth mechanism allows water at its freezing point to remain in contact with the cold air, high heat and salt fluxes are maintained. The ice growth predicted from this model is an order of magnitude greater than one-dimensional vertical growth.