Glacier dirt cones are meter-scale cones of ice covered with sediment and rock. The cones develop through a process known as differential melt, whereby ice underlying thick debris melts more slowly than bare ice. We report observations of dirt cones on the Kuskulana Glacier, Alaska and develop a model that simulates the growth of dirt cones from debris-filled pits in the ice. With this model, we vary ice melt rates, hillslope debris diffusion rates and pit geometry. Cone heights scale with the square root of debris volume and growth occurs in three distinct stages: emergence, flux-controlled growth and melt-controlled growth. Using dimensional analysis, we derive a characteristic length composed of the ratio of the debris diffusion rate (D) and the bare ice melt rate (b0). Shorter characteristic lengths produce taller, steeper cones. The characteristic length ($\ell = D/b_0$) determines, in part, the relative duration of each growth stage because it controls debris flux as relief increases. These experiments suggest increasing melt rates and low-mobility debris increase relief on hummocky debris-covered glaciers. Furthermore, the modeling approach demonstrates a method for handling debris transport over an irregular ice surface and could serve as a component in more comprehensive debris-covered glacier models.